+++ /dev/null
-Sequence-manipulation functions that complement basic functions
-provided by subr.el.
-
-All functions are prefixed with "seq-".
-
-All provided functions work on lists, strings and vectors.
-
-Functions taking a predicate or iterating over a sequence using a
-function as argument take the function as their first argument and
-the sequence as their second argument. All other functions take
-the sequence as their first argument.
-
-All functions are tested in test/automated/seq-tests.el
+++ /dev/null
-This is the file .../info/dir, which contains the
-topmost node of the Info hierarchy, called (dir)Top.
-The first time you invoke Info you start off looking at this node.
-\1f
-File: dir, Node: Top This is the top of the INFO tree
-
- This (the Directory node) gives a menu of major topics.
- Typing "q" exits, "?" lists all Info commands, "d" returns here,
- "h" gives a primer for first-timers,
- "mEmacs<Return>" visits the Emacs manual, etc.
-
- In Emacs, you can click mouse button 2 on a menu item or cross reference
- to select it.
-
-* Menu:
-
-The Algorithmic Language Scheme
-* SICP: (sicp). Structure and Interpretation of Computer
- Programs
+++ /dev/null
-;;; sicp-autoloads.el --- automatically extracted autoloads -*- lexical-binding: t -*-
-;;
-;;; Code:
-
-(add-to-list 'load-path (directory-file-name
- (or (file-name-directory #$) (car load-path))))
-
-\f
-;; Local Variables:
-;; version-control: never
-;; no-byte-compile: t
-;; no-update-autoloads: t
-;; coding: utf-8
-;; End:
-;;; sicp-autoloads.el ends here
+++ /dev/null
-(define-package "sicp" "20200512.1137" "Structure and Interpretation of Computer Programs in info format" 'nil :commit "4002d83083d520c6b5ede2df36cc2cee885d450a" :authors
- '(("Hal Abelson")
- ("Jerry Sussman")
- ("Julie Sussman"))
- :maintainers
- '(("Hal Abelson"))
- :maintainer
- '("Hal Abelson")
- :url "https://mitpress.mit.edu/sicp")
-;; Local Variables:
-;; no-byte-compile: t
-;; End:
+++ /dev/null
-;;; sicp.el --- Structure and Interpretation of Computer Programs in info format
-
-;;; Copyright (C) 1993 Hal Abelson, Jerry Sussman, and Julie Sussman
-
-;; Author: Hal Abelson
-;; Jerry Sussman
-;; Julie Sussman
-;; Homepage: https://mitpress.mit.edu/sicp
-
-;; This file is not part of GNU Emacs.
-
-;; This program is free software; you can redistribute it and/or modify
-;; it under the terms of the GNU General Public License as published by
-;; the Free Software Foundation, either version 3 of the License, or
-;; (at your option) any later version.
-
-;; This program is distributed in the hope that it will be useful,
-;; but WITHOUT ANY WARRANTY; without even the implied warranty of
-;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-;; GNU General Public License for more details.
-
-;; You should have received a copy of the GNU General Public License
-;; along with this program. If not, see <http://www.gnu.org/licenses/>.
-
-;; The included sicp.info and sicp.texi files reatain their
-;; origin Creative Commons Attribution-ShareAlike 4.0
-;; International Public License
-;; See http://creativecommons.org/licenses/by-sa/4.0/
-;; and http://creativecommons.org/licenses/by-sa/4.0/legalcode.
-
-
-;;; Commentary:
-
-;; The second edition of the book Structure and Interpretation of
-;; Computer Programs (SICP) in info format.
-
-;; This library provides the feature `sicp' and does nothing else.
-;; This allows making the info file available on Melpa. The texi
-;; file was taken from http://www.neilvandyke.org/sicp-texi. The
-;; html version can be found at https://mitpress.mit.edu/sicp.
-
-;; If you want to recreate the info file, you can do so using
-;;
-;; makeinfo -–no-split sicp.texi -o sicp.info
-
-;;; Code:
-
-(provide 'sicp)
-
-;;; sicp.el ends here
+++ /dev/null
-This is sicp.info, produced by makeinfo version 4.8 from sicp.texi.
-
-INFO-DIR-SECTION The Algorithmic Language Scheme
-START-INFO-DIR-ENTRY
-* SICP: (sicp). Structure and Interpretation of Computer Programs
-END-INFO-DIR-ENTRY
-
-\1f
-File: sicp.info, Node: Top, Next: UTF, Prev: (dir), Up: (dir)
-
-Structure and Interpretation of Computer Programs
-=================================================
-
-Second Edition
-by Harold Abelson and Gerald Jay Sussman, with Julie Sussman
-foreword by Alan J. Perlis
-(C) 1996 Massachusetts Institute of Technology
-
-Unofficial Texinfo Format version 2.neilvandyke4 (January 10, 2007)
-
-* Menu:
-
-* UTF:: Unofficial Texinfo Format
-* Dedication:: Dedication
-* Foreword:: Foreword
-* Preface:: Preface to the Second Edition
-* Preface 1e:: Preface to the First Edition
-* Acknowledgements:: Acknowledgements
-* Chapter 1:: Building Abstractions with Procedures
-* Chapter 2:: Building Abstractions with Data
-* Chapter 3:: Modularity, Objects, and State
-* Chapter 4:: Metalinguistic Abstraction
-* Chapter 5:: Computing with Register Machines
-* References:: References
-* Index:: Index
-
- --- The Detailed Node Listing ---
-
-Programming in Lisp
-
-* 1-1:: The Elements of Programming
-* 1-2:: Procedures and the Processes They Generate
-* 1-3:: Formulating Abstractions with Higher-Order Procedures
-
-The Elements of Programming
-
-* 1-1-1:: Expressions
-* 1-1-2:: Naming and the Environment
-* 1-1-3:: Evaluating Combinations
-* 1-1-4:: Compound Procedures
-* 1-1-5:: The Substitution Model for Procedure Application
-* 1-1-6:: Conditional Expressions and Predicates
-* 1-1-7:: Example: Square Roots by Newton's Method
-* 1-1-8:: Procedures as Black-Box Abstractions
-
-Procedures and the Processes They Generate
-
-* 1-2-1:: Linear Recursion and Iteration
-* 1-2-2:: Tree Recursion
-* 1-2-3:: Orders of Growth
-* 1-2-4:: Exponentiation
-* 1-2-5:: Greatest Common Divisors
-* 1-2-6:: Example: Testing for Primality
-
-Formulating Abstractions with Higher-Order Procedures
-
-* 1-3-1:: Procedures as Arguments
-* 1-3-2:: Constructing Procedures Using `Lambda'
-* 1-3-3:: Procedures as General Methods
-* 1-3-4:: Procedures as Returned Values
-
-Building Abstractions with Data
-
-* 2-1:: Introduction to Data Abstraction
-* 2-2:: Hierarchical Data and the Closure Property
-* 2-3:: Symbolic Data
-* 2-4:: Multiple Representations for Abstract Data
-* 2-5:: Systems with Generic Operations
-
-Introduction to Data Abstraction
-
-* 2-1-1:: Example: Arithmetic Operations for Rational Numbers
-* 2-1-2:: Abstraction Barriers
-* 2-1-3:: What Is Meant by Data?
-* 2-1-4:: Extended Exercise: Interval Arithmetic
-
-Hierarchical Data and the Closure Property
-
-* 2-2-1:: Representing Sequences
-* 2-2-2:: Hierarchical Structures
-* 2-2-3:: Sequences as Conventional Interfaces
-* 2-2-4:: Example: A Picture Language
-
-Symbolic Data
-
-* 2-3-1:: Quotation
-* 2-3-2:: Example: Symbolic Differentiation
-* 2-3-3:: Example: Representing Sets
-* 2-3-4:: Example: Huffman Encoding Trees
-
-Multiple Representations for Abstract Data
-
-* 2-4-1:: Representations for Complex Numbers
-* 2-4-2:: Tagged data
-* 2-4-3:: Data-Directed Programming and Additivity
-
-Systems with Generic Operations
-
-* 2-5-1:: Generic Arithmetic Operations
-* 2-5-2:: Combining Data of Different Types
-* 2-5-3:: Example: Symbolic Algebra
-
-Modularity, Objects, and State
-
-* 3-1:: Assignment and Local State
-* 3-2:: The Environment Model of Evaluation
-* 3-3:: Modeling with Mutable Data
-* 3-4:: Concurrency: Time Is of the Essence
-* 3-5:: Streams
-
-Assignment and Local State
-
-* 3-1-1:: Local State Variables
-* 3-1-2:: The Benefits of Introducing Assignment
-* 3-1-3:: The Costs of Introducing Assignment
-
-The Environment Model of Evaluation
-
-* 3-2-1:: The Rules for Evaluation
-* 3-2-2:: Applying Simple Procedures
-* 3-2-3:: Frames as the Repository of Local State
-* 3-2-4:: Internal Definitions
-
-Modeling with Mutable Data
-
-* 3-3-1:: Mutable List Structure
-* 3-3-2:: Representing Queues
-* 3-3-3:: Representing Tables
-* 3-3-4:: A Simulator for Digital Circuits
-* 3-3-5:: Propagation of Constraints
-
-Concurrency: Time Is of the Essence
-
-* 3-4-1:: The Nature of Time in Concurrent Systems
-* 3-4-2:: Mechanisms for Controlling Concurrency
-
-Streams
-
-* 3-5-1:: Streams Are Delayed Lists
-* 3-5-2:: Infinite Streams
-* 3-5-3:: Exploiting the Stream Paradigm
-* 3-5-4:: Streams and Delayed Evaluation
-* 3-5-5:: Modularity of Functional Programs and Modularity of
- Objects
-
-Metalinguistic Abstraction
-
-* 4-1:: The Metacircular Evaluator
-* 4-2:: Variations on a Scheme -- Lazy Evaluation
-* 4-3:: Variations on a Scheme -- Nondeterministic Computing
-* 4-4:: Logic Programming
-
-The Metacircular Evaluator
-
-* 4-1-1:: The Core of the Evaluator
-* 4-1-2:: Representing Expressions
-* 4-1-3:: Evaluator Data Structures
-* 4-1-4:: Running the Evaluator as a Program
-* 4-1-5:: Data as Programs
-* 4-1-6:: Internal Definitions
-* 4-1-7:: Separating Syntactic Analysis from Execution
-
-Variations on a Scheme -- Lazy Evaluation
-
-* 4-2-1:: Normal Order and Applicative Order
-* 4-2-2:: An Interpreter with Lazy Evaluation
-* 4-2-3:: Streams as Lazy Lists
-
-Variations on a Scheme -- Nondeterministic Computing
-
-* 4-3-1:: Amb and Search
-* 4-3-2:: Examples of Nondeterministic Programs
-* 4-3-3:: Implementing the `Amb' Evaluator
-
-Logic Programming
-
-* 4-4-1:: Deductive Information Retrieval
-* 4-4-2:: How the Query System Works
-* 4-4-3:: Is Logic Programming Mathematical Logic?
-* 4-4-4:: Implementing the Query System
-
-Implementing the Query System
-
-* 4-4-4-1:: The Driver Loop and Instantiation
-* 4-4-4-2:: The Evaluator
-* 4-4-4-3:: Finding Assertions by Pattern Matching
-* 4-4-4-4:: Rules and Unification
-* 4-4-4-5:: Maintaining the Data Base
-* 4-4-4-6:: Stream Operations
-* 4-4-4-7:: Query Syntax Procedures
-* 4-4-4-8:: Frames and Bindings
-
-Computing with Register Machines
-
-* 5-1:: Designing Register Machines
-* 5-2:: A Register-Machine Simulator
-* 5-3:: Storage Allocation and Garbage Collection
-* 5-4:: The Explicit-Control Evaluator
-* 5-5:: Compilation
-
-Designing Register Machines
-
-* 5-1-1:: A Language for Describing Register Machines
-* 5-1-2:: Abstraction in Machine Design
-* 5-1-3:: Subroutines
-* 5-1-4:: Using a Stack to Implement Recursion
-* 5-1-5:: Instruction Summary
-
-A Register-Machine Simulator
-
-* 5-2-1:: The Machine Model
-* 5-2-2:: The Assembler
-* 5-2-3:: Generating Execution Procedures for Instructions
-* 5-2-4:: Monitoring Machine Performance
-
-Storage Allocation and Garbage Collection
-
-* 5-3-1:: Memory as Vectors
-* 5-3-2:: Maintaining the Illusion of Infinite Memory
-
-Registers and operations
-
-* 5-4-1:: The Core of the Explicit-Control Evaluator
-* 5-4-2:: Sequence Evaluation and Tail Recursion
-* 5-4-3:: Conditionals, Assignments, and Definitions
-* 5-4-4:: Running the Evaluator
-
-An overview of the compiler
-
-* 5-5-1:: Structure of the Compiler
-* 5-5-2:: Compiling Expressions
-* 5-5-3:: Compiling Combinations
-* 5-5-4:: Combining Instruction Sequences
-* 5-5-5:: An Example of Compiled Code
-* 5-5-6:: Lexical Addressing
-* 5-5-7:: Interfacing Compiled Code to the Evaluator
-
-\1f
-File: sicp.info, Node: UTF, Next: Dedication, Prev: Top, Up: Top
-
-Unofficial Texinfo Format
-*************************
-
-This is the second edition SICP book, from Unofficial Texinfo Format.
-
- You are probably reading it in an Info hypertext browser, such as
-the Info mode of Emacs. You might alternatively be reading it
-TeX-formatted on your screen or printer, though that would be silly.
-And, if printed, expensive.
-
- The freely-distributed official HTML-and-GIF format was first
-converted personually to Unofficial Texinfo Format (UTF) version 1 by
-Lyssa Ayth during a long Emacs lovefest weekend in April, 2001.
-
- The UTF is easier to search than the HTML format. It is also much
-more accessible to people running on modest computers, such as donated
-'386-based PCs. A 386 can, in theory, run Linux, Emacs, and a Scheme
-interpreter simultaneously, but most 386s probably can't also run both
-Netscape and the necessary X Window System without prematurely
-introducing budding young underfunded hackers to the concept of "thrashing".
-UTF can also fit uncompressed on a 1.44MB floppy diskette, which may
-come in handy for installing UTF on PCs that do not have Internet or
-LAN access.
-
- The Texinfo conversion has been a straight transliteration, to the
-extent possible. Like the TeX-to-HTML conversion, this was not without
-some introduction of breakage. In the case of Unofficial Texinfo
-Format, figures have suffered an amateurish resurrection of the lost
-art of ASCII art. Also, it's quite possible that some errors of
-ambiguity were introduced during the conversion of some of the copious
-superscripts (`^') and subscripts (`_'). Divining _which_ has been
-left as an exercise to the reader. But at least we don't put our brave
-astronauts at risk by encoding the _greater-than-or-equal_ symbol as
-`<u>></u>'.
-
- If you modify `sicp.texi' to correct errors or improve the ASCII
-art, then update the `@set utfversion 2.neilvandyke4' line to reflect
-your delta. For example, if you started with Lytha's version `1', and
-your name is Bob, then you could name your successive versions
-`1.bob1', `1.bob2', ... `1.bobn'. Also update `utfversiondate'. If
-you want to distribute your version on the Web, then embedding the
-string "sicp.texi" somewhere in the file or Web page will make it
-easier for people to find with Web search engines.
-
- It is believed that the Unofficial Texinfo Format is in keeping with
-the spirit of the graciously freely-distributed HTML version. But you
-never know when someone's armada of lawyers might need something to do,
-and get their shorts all in a knot over some benign little thing, so
-think twice before you use your full name or distribute Info, DVI,
-PostScript, or PDF formats that might embed your account or machine
-name.
-
-Peath,
-
-Lytha Ayth
-
- Addendum: See also the SICP video lectures by Abelson and Sussman:
-`http://www.swiss.ai.mit.edu/classes/6.001/abelson-sussman-lectures/'
-
-\1f
-File: sicp.info, Node: Dedication, Next: Foreword, Prev: UTF, Up: Top
-
-Dedication
-**********
-
-This book is dedicated, in respect and admiration, to the spirit that
-lives in the computer.
-
- "I think that it's extraordinarily important that we in computer
- science keep fun in computing. When it started out, it was an
- awful lot of fun. Of course, the paying customers got shafted
- every now and then, and after a while we began to take their
- complaints seriously. We began to feel as if we really were
- responsible for the successful, error-free perfect use of these
- machines. I don't think we are. I think we're responsible for
- stretching them, setting them off in new directions, and keeping
- fun in the house. I hope the field of computer science never
- loses its sense of fun. Above all, I hope we don't become
- missionaries. Don't feel as if you're Bible salesmen. The world
- has too many of those already. What you know about computing
- other people will learn. Don't feel as if the key to successful
- computing is only in your hands. What's in your hands, I think
- and hope, is intelligence: the ability to see the machine as more
- than when you were first led up to it, that you can make it more."
-
- --Alan J. Perlis (April 1, 1922 February 7, 1990)
-
-\1f
-File: sicp.info, Node: Foreword, Next: Preface, Prev: Dedication, Up: Top
-
-Foreword
-********
-
-Educators, generals, dieticians, psychologists, and parents program.
-Armies, students, and some societies are programmed. An assault on
-large problems employs a succession of programs, most of which spring
-into existence en route. These programs are rife with issues that
-appear to be particular to the problem at hand. To appreciate
-programming as an intellectual activity in its own right you must turn
-to computer programming; you must read and write computer
-programs--many of them. It doesn't matter much what the programs are
-about or what applications they serve. What does matter is how well
-they perform and how smoothly they fit with other programs in the
-creation of still greater programs. The programmer must seek both
-perfection of part and adequacy of collection. In this book the use of
-"program" is focused on the creation, execution, and study of programs
-written in a dialect of Lisp for execution on a digital computer.
-Using Lisp we restrict or limit not what we may program, but only the
-notation for our program descriptions.
-
- Our traffic with the subject matter of this book involves us with
-three foci of phenomena: the human mind, collections of computer
-programs, and the computer. Every computer program is a model, hatched
-in the mind, of a real or mental process. These processes, arising
-from human experience and thought, are huge in number, intricate in
-detail, and at any time only partially understood. They are modeled to
-our permanent satisfaction rarely by our computer programs. Thus even
-though our programs are carefully handcrafted discrete collections of
-symbols, mosaics of interlocking functions, they continually evolve: we
-change them as our perception of the model deepens, enlarges,
-generalizes until the model ultimately attains a metastable place
-within still another model with which we struggle. The source of the
-exhilaration associated with computer programming is the continual
-unfolding within the mind and on the computer of mechanisms expressed
-as programs and the explosion of perception they generate. If art
-interprets our dreams, the computer executes them in the guise of
-programs!
-
- For all its power, the computer is a harsh taskmaster. Its programs
-must be correct, and what we wish to say must be said accurately in
-every detail. As in every other symbolic activity, we become convinced
-of program truth through argument. Lisp itself can be assigned a
-semantics (another model, by the way), and if a program's function can
-be specified, say, in the predicate calculus, the proof methods of
-logic can be used to make an acceptable correctness argument.
-Unfortunately, as programs get large and complicated, as they almost
-always do, the adequacy, consistency, and correctness of the
-specifications themselves become open to doubt, so that complete formal
-arguments of correctness seldom accompany large programs. Since large
-programs grow from small ones, it is crucial that we develop an arsenal
-of standard program structures of whose correctness we have become
-sure--we call them idioms--and learn to combine them into larger
-structures using organizational techniques of proven value. These
-techniques are treated at length in this book, and understanding them
-is essential to participation in the Promethean enterprise called
-programming. More than anything else, the uncovering and mastery of
-powerful organizational techniques accelerates our ability to create
-large, significant programs. Conversely, since writing large programs
-is very taxing, we are stimulated to invent new methods of reducing the
-mass of function and detail to be fitted into large programs.
-
- Unlike programs, computers must obey the laws of physics. If they
-wish to perform rapidly--a few nanoseconds per state change--they must
-transmit electrons only small distances (at most 11 over 2 feet). The
-heat generated by the huge number of devices so concentrated in space
-has to be removed. An exquisite engineering art has been developed
-balancing between multiplicity of function and density of devices. In
-any event, hardware always operates at a level more primitive than that
-at which we care to program. The processes that transform our Lisp
-programs to "machine" programs are themselves abstract models which we
-program. Their study and creation give a great deal of insight into
-the organizational programs associated with programming arbitrary
-models. Of course the computer itself can be so modeled. Think of it:
-the behavior of the smallest physical switching element is modeled by
-quantum mechanics described by differential equations whose detailed
-behavior is captured by numerical approximations represented in
-computer programs executing on computers composed of ...!
-
- It is not merely a matter of tactical convenience to separately
-identify the three foci. Even though, as they say, it's all in the
-head, this logical separation induces an acceleration of symbolic
-traffic between these foci whose richness, vitality, and potential is
-exceeded in human experience only by the evolution of life itself. At
-best, relationships between the foci are metastable. The computers are
-never large enough or fast enough. Each breakthrough in hardware
-technology leads to more massive programming enterprises, new
-organizational principles, and an enrichment of abstract models. Every
-reader should ask himself periodically "Toward what end, toward what
-end?"--but do not ask it too often lest you pass up the fun of
-programming for the constipation of bittersweet philosophy.
-
- Among the programs we write, some (but never enough) perform a
-precise mathematical function such as sorting or finding the maximum of
-a sequence of numbers, determining primality, or finding the square
-root. We call such programs algorithms, and a great deal is known of
-their optimal behavior, particularly with respect to the two important
-parameters of execution time and data storage requirements. A
-programmer should acquire good algorithms and idioms. Even though some
-programs resist precise specifications, it is the responsibility of the
-programmer to estimate, and always to attempt to improve, their
-performance.
-
- Lisp is a survivor, having been in use for about a quarter of a
-century. Among the active programming languages only Fortran has had a
-longer life. Both languages have supported the programming needs of
-important areas of application, Fortran for scientific and engineering
-computation and Lisp for artificial intelligence. These two areas
-continue to be important, and their programmers are so devoted to these
-two languages that Lisp and Fortran may well continue in active use for
-at least another quarter-century.
-
- Lisp changes. The Scheme dialect used in this text has evolved from
-the original Lisp and differs from the latter in several important
-ways, including static scoping for variable binding and permitting
-functions to yield functions as values. In its semantic structure
-Scheme is as closely akin to Algol 60 as to early Lisps. Algol 60,
-never to be an active language again, lives on in the genes of Scheme
-and Pascal. It would be difficult to find two languages that are the
-communicating coin of two more different cultures than those gathered
-around these two languages. Pascal is for building pyramids--imposing,
-breathtaking, static structures built by armies pushing heavy blocks
-into place. Lisp is for building organisms--imposing, breathtaking,
-dynamic structures built by squads fitting fluctuating myriads of
-simpler organisms into place. The organizing principles used are the
-same in both cases, except for one extraordinarily important
-difference: The discretionary exportable functionality entrusted to the
-individual Lisp programmer is more than an order of magnitude greater
-than that to be found within Pascal enterprises. Lisp programs inflate
-libraries with functions whose utility transcends the application that
-produced them. The list, Lisp's native data structure, is largely
-responsible for such growth of utility. The simple structure and
-natural applicability of lists are reflected in functions that are
-amazingly nonidiosyncratic. In Pascal the plethora of declarable data
-structures induces a specialization within functions that inhibits and
-penalizes casual cooperation. It is better to have 100 functions
-operate on one data structure than to have 10 functions operate on 10
-data structures. As a result the pyramid must stand unchanged for a
-millennium; the organism must evolve or perish.
-
- To illustrate this difference, compare the treatment of material and
-exercises within this book with that in any first-course text using
-Pascal. Do not labor under the illusion that this is a text digestible
-at MIT only, peculiar to the breed found there. It is precisely what a
-serious book on programming Lisp must be, no matter who the student is
-or where it is used.
-
- Note that this is a text about programming, unlike most Lisp books,
-which are used as a preparation for work in artificial intelligence.
-After all, the critical programming concerns of software engineering
-and artificial intelligence tend to coalesce as the systems under
-investigation become larger. This explains why there is such growing
-interest in Lisp outside of artificial intelligence.
-
- As one would expect from its goals, artificial intelligence research
-generates many significant programming problems. In other programming
-cultures this spate of problems spawns new languages. Indeed, in any
-very large programming task a useful organizing principle is to control
-and isolate traffic within the task modules via the invention of
-language. These languages tend to become less primitive as one
-approaches the boundaries of the system where we humans interact most
-often. As a result, such systems contain complex language-processing
-functions replicated many times. Lisp has such a simple syntax and
-semantics that parsing can be treated as an elementary task. Thus
-parsing technology plays almost no role in Lisp programs, and the
-construction of language processors is rarely an impediment to the rate
-of growth and change of large Lisp systems. Finally, it is this very
-simplicity of syntax and semantics that is responsible for the burden
-and freedom borne by all Lisp programmers. No Lisp program of any size
-beyond a few lines can be written without being saturated with
-discretionary functions. Invent and fit; have fits and reinvent! We
-toast the Lisp programmer who pens his thoughts within nests of
-parentheses.
-
-Alan J. Perlis
-New Haven, Connecticut
-
-\1f
-File: sicp.info, Node: Preface, Next: Preface 1e, Prev: Foreword, Up: Top
-
-Preface to the Second Edition
-*****************************
-
- Is it possible that software is not like anything else, that it is
- meant to be discarded: that the whole point is to always see it as
- a soap bubble?
-
- --Alan J. Perlis
-
- The material in this book has been the basis of MIT's entry-level
-computer science subject since 1980. We had been teaching this
-material for four years when the first edition was published, and
-twelve more years have elapsed until the appearance of this second
-edition. We are pleased that our work has been widely adopted and
-incorporated into other texts. We have seen our students take the
-ideas and programs in this book and build them in as the core of new
-computer systems and languages. In literal realization of an ancient
-Talmudic pun, our students have become our builders. We are lucky to
-have such capable students and such accomplished builders.
-
- In preparing this edition, we have incorporated hundreds of
-clarifications suggested by our own teaching experience and the
-comments of colleagues at MIT and elsewhere. We have redesigned most
-of the major programming systems in the book, including the
-generic-arithmetic system, the interpreters, the register-machine
-simulator, and the compiler; and we have rewritten all the program
-examples to ensure that any Scheme implementation conforming to the
-IEEE Scheme standard (IEEE 1990) will be able to run the code.
-
- This edition emphasizes several new themes. The most important of
-these is the central role played by different approaches to dealing
-with time in computational models: objects with state, concurrent
-programming, functional programming, lazy evaluation, and
-nondeterministic programming. We have included new sections on
-concurrency and nondeterminism, and we have tried to integrate this
-theme throughout the book.
-
- The first edition of the book closely followed the syllabus of our
-MIT one-semester subject. With all the new material in the second
-edition, it will not be possible to cover everything in a single
-semester, so the instructor will have to pick and choose. In our own
-teaching, we sometimes skip the section on logic programming (section
-*Note 4-4::), we have students use the register-machine simulator but
-we do not cover its implementation (section *Note 5-2::), and we give
-only a cursory overview of the compiler (section *Note 5-5::). Even
-so, this is still an intense course. Some instructors may wish to
-cover only the first three or four chapters, leaving the other material
-for subsequent courses.
-
- The World-Wide-Web site `http://mitpress.mit.edu/sicp/' provides
-support for users of this book. This includes programs from the book,
-sample programming assignments, supplementary materials, and
-downloadable implementations of the Scheme dialect of Lisp.
-
-\1f
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-
-Preface to the First Edition
-****************************
-
- A computer is like a violin. You can imagine a novice trying
- first a phonograph and then a violin. The latter, he says, sounds
- terrible. That is the argument we have heard from our humanists
- and most of our computer scientists. Computer programs are good,
- they say, for particular purposes, but they aren't flexible.
- Neither is a violin, or a typewriter, until you learn how to use
- it.
-
- --Marvin Minsky, "Why Programming Is a Good Medium for Expressing
- Poorly-Understood and Sloppily-Formulated Ideas"
-
- "The Structure and Interpretation of Computer Programs" is the
-entry-level subject in computer science at the Massachusetts Institute
-of Technology. It is required of all students at MIT who major in
-electrical engineering or in computer science, as one-fourth of the
-"common core curriculum," which also includes two subjects on circuits
-and linear systems and a subject on the design of digital systems. We
-have been involved in the development of this subject since 1978, and
-we have taught this material in its present form since the fall of 1980
-to between 600 and 700 students each year. Most of these students have
-had little or no prior formal training in computation, although many
-have played with computers a bit and a few have had extensive
-programming or hardware-design experience.
-
- Our design of this introductory computer-science subject reflects
-two major concerns. First, we want to establish the idea that a
-computer language is not just a way of getting a computer to perform
-operations but rather that it is a novel formal medium for expressing
-ideas about methodology. Thus, programs must be written for people to
-read, and only incidentally for machines to execute. Second, we
-believe that the essential material to be addressed by a subject at
-this level is not the syntax of particular programming-language
-constructs, nor clever algorithms for computing particular functions
-efficiently, nor even the mathematical analysis of algorithms and the
-foundations of computing, but rather the techniques used to control the
-intellectual complexity of large software systems.
-
- Our goal is that students who complete this subject should have a
-good feel for the elements of style and the aesthetics of programming.
-They should have command of the major techniques for controlling
-complexity in a large system. They should be capable of reading a
-50-page-long program, if it is written in an exemplary style. They
-should know what not to read, and what they need not understand at any
-moment. They should feel secure about modifying a program, retaining
-the spirit and style of the original author.
-
- These skills are by no means unique to computer programming. The
-techniques we teach and draw upon are common to all of engineering
-design. We control complexity by building abstractions that hide
-details when appropriate. We control complexity by establishing
-conventional interfaces that enable us to construct systems by
-combining standard, well-understood pieces in a "mix and match" way.
-We control complexity by establishing new languages for describing a
-design, each of which emphasizes particular aspects of the design and
-deemphasizes others.
-
- Underlying our approach to this subject is our conviction that
-"computer science" is not a science and that its significance has
-little to do with computers. The computer revolution is a revolution
-in the way we think and in the way we express what we think. The
-essence of this change is the emergence of what might best be called "procedural
-epistemology"--the study of the structure of knowledge from an
-imperative point of view, as opposed to the more declarative point of
-view taken by classical mathematical subjects. Mathematics provides a
-framework for dealing precisely with notions of "what is." Computation
-provides a framework for dealing precisely with notions of "how to."
-
- In teaching our material we use a dialect of the programming
-language Lisp. We never formally teach the language, because we don't
-have to. We just use it, and students pick it up in a few days. This
-is one great advantage of Lisp-like languages: They have very few ways
-of forming compound expressions, and almost no syntactic structure.
-All of the formal properties can be covered in an hour, like the rules
-of chess. After a short time we forget about syntactic details of the
-language (because there are none) and get on with the real
-issues--figuring out what we want to compute, how we will decompose
-problems into manageable parts, and how we will work on the parts.
-Another advantage of Lisp is that it supports (but does not enforce)
-more of the large-scale strategies for modular decomposition of
-programs than any other language we know. We can make procedural and
-data abstractions, we can use higher-order functions to capture common
-patterns of usage, we can model local state using assignment and data
-mutation, we can link parts of a program with streams and delayed
-evaluation, and we can easily implement embedded languages. All of
-this is embedded in an interactive environment with excellent support
-for incremental program design, construction, testing, and debugging.
-We thank all the generations of Lisp wizards, starting with John
-McCarthy, who have fashioned a fine tool of unprecedented power and
-elegance.
-
- Scheme, the dialect of Lisp that we use, is an attempt to bring
-together the power and elegance of Lisp and Algol. From Lisp we take
-the metalinguistic power that derives from the simple syntax, the
-uniform representation of programs as data objects, and the
-garbage-collected heap-allocated data. From Algol we take lexical
-scoping and block structure, which are gifts from the pioneers of
-programming-language design who were on the Algol committee. We wish
-to cite John Reynolds and Peter Landin for their insights into the
-relationship of Church's [lambda] calculus to the structure of
-programming languages. We also recognize our debt to the
-mathematicians who scouted out this territory decades before computers
-appeared on the scene. These pioneers include Alonzo Church, Barkley
-Rosser, Stephen Kleene, and Haskell Curry.
-
-\1f
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-
-Acknowledgements
-****************
-
-We would like to thank the many people who have helped us develop this
-book and this curriculum.
-
- Our subject is a clear intellectual descendant of "6.231," a
-wonderful subject on programming linguistics and the [lambda] calculus
-taught at MIT in the late 1960s by Jack Wozencraft and Arthur Evans, Jr.
-
- We owe a great debt to Robert Fano, who reorganized MIT's
-introductory curriculum in electrical engineering and computer science
-to emphasize the principles of engineering design. He led us in
-starting out on this enterprise and wrote the first set of subject
-notes from which this book evolved.
-
- Much of the style and aesthetics of programming that we try to teach
-were developed in conjunction with Guy Lewis Steele Jr., who
-collaborated with Gerald Jay Sussman in the initial development of the
-Scheme language. In addition, David Turner, Peter Henderson, Dan
-Friedman, David Wise, and Will Clinger have taught us many of the
-techniques of the functional programming community that appear in this
-book.
-
- Joel Moses taught us about structuring large systems. His
-experience with the Macsyma system for symbolic computation provided
-the insight that one should avoid complexities of control and
-concentrate on organizing the data to reflect the real structure of the
-world being modeled.
-
- Marvin Minsky and Seymour Papert formed many of our attitudes about
-programming and its place in our intellectual lives. To them we owe
-the understanding that computation provides a means of expression for
-exploring ideas that would otherwise be too complex to deal with
-precisely. They emphasize that a student's ability to write and modify
-programs provides a powerful medium in which exploring becomes a
-natural activity.
-
- We also strongly agree with Alan Perlis that programming is lots of
-fun and we had better be careful to support the joy of programming.
-Part of this joy derives from observing great masters at work. We are
-fortunate to have been apprentice programmers at the feet of Bill
-Gosper and Richard Greenblatt.
-
- It is difficult to identify all the people who have contributed to
-the development of our curriculum. We thank all the lecturers,
-recitation instructors, and tutors who have worked with us over the
-past fifteen years and put in many extra hours on our subject,
-especially Bill Siebert, Albert Meyer, Joe Stoy, Randy Davis, Louis
-Braida, Eric Grimson, Rod Brooks, Lynn Stein, and Peter Szolovits. We
-would like to specially acknowledge the outstanding teaching
-contributions of Franklyn Turbak, now at Wellesley; his work in
-undergraduate instruction set a standard that we can all aspire to. We
-are grateful to Jerry Saltzer and Jim Miller for helping us grapple
-with the mysteries of concurrency, and to Peter Szolovits and David
-McAllester for their contributions to the exposition of
-nondeterministic evaluation in *Note Chapter 4::.
-
- Many people have put in significant effort presenting this material
-at other universities. Some of the people we have worked closely with
-are Jacob Katzenelson at the Technion, Hardy Mayer at the University of
-California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue, and
-Jan Komorowski at the Norwegian University of Science and Technology.
-We are exceptionally proud of our colleagues who have received major
-teaching awards for their adaptations of this subject at other
-universities, including Kenneth Yip at Yale, Brian Harvey at the
-University of California at Berkeley, and Dan Huttenlocher at Cornell.
-
- Al Moye' arranged for us to teach this material to engineers at
-Hewlett-Packard, and for the production of videotapes of these
-lectures. We would like to thank the talented instructors--in
-particular Jim Miller, Bill Siebert, and Mike Eisenberg--who have
-designed continuing education courses incorporating these tapes and
-taught them at universities and industry all over the world.
-
- Many educators in other countries have put in significant work
-translating the first edition. Michel Briand, Pierre Chamard, and
-Andre' Pic produced a French edition; Susanne Daniels-Herold produced a
-German edition; and Fumio Motoyoshi produced a Japanese edition. We do
-not know who produced the Chinese edition, but we consider it an honor
-to have been selected as the subject of an "unauthorized" translation.
-
- It is hard to enumerate all the people who have made technical
-contributions to the development of the Scheme systems we use for
-instructional purposes. In addition to Guy Steele, principal wizards
-have included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas,
-and Stephen Adams. Others who have put in significant time are Richard
-Stallman, Alan Bawden, Kent Pitman, Jon Taft, Neil Mayle, John Lamping,
-Gwyn Osnos, Tracy Larrabee, George Carrette, Soma Chaudhuri, Bill
-Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss, Todd Cass, Patrick
-O'Donnell, Kevin Theobald, Daniel Weise, Kenneth Sinclair, Anthony
-Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth Shyu.
-
- Beyond the MIT implementation, we would like to thank the many people
-who worked on the IEEE Scheme standard, including William Clinger and
-Jonathan Rees, who edited the R^4RS, and Chris Haynes, David Bartley,
-Chris Hanson, and Jim Miller, who prepared the IEEE standard.
-
- Dan Friedman has been a long-time leader of the Scheme community.
-The community's broader work goes beyond issues of language design to
-encompass significant educational innovations, such as the high-school
-curriculum based on EdScheme by Schemer's Inc., and the wonderful books
-by Mike Eisenberg and by Brian Harvey and Matthew Wright.
-
- We appreciate the work of those who contributed to making this a
-real book, especially Terry Ehling, Larry Cohen, and Paul Bethge at the
-MIT Press. Ella Mazel found the wonderful cover image. For the second
-edition we are particularly grateful to Bernard and Ella Mazel for help
-with the book design, and to David Jones, TeX wizard extraordinaire.
-We also are indebted to those readers who made penetrating comments on
-the new draft: Jacob Katzenelson, Hardy Mayer, Jim Miller, and
-especially Brian Harvey, who did unto this book as Julie did unto his
-book `Simply Scheme'.
-
- Finally, we would like to acknowledge the support of the
-organizations that have encouraged this work over the years, including
-suppport from Hewlett-Packard, made possible by Ira Goldstein and Joel
-Birnbaum, and support from DARPA, made possible by Bob Kahn.
-
-\1f
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-
-1 Building Abstractions with Procedures
-***************************************
-
- The acts of the mind, wherein it exerts its power over simple
- ideas, are chiefly these three: 1. Combining several simple ideas
- into one compound one, and thus all complex ideas are made. 2.
- The second is bringing two ideas, whether simple or complex,
- together, and setting them by one another so as to take a view of
- them at once, without uniting them into one, by which it gets all
- its ideas of relations. 3. The third is separating them from all
- other ideas that accompany them in their real existence: this is
- called abstraction, and thus all its general ideas are made.
-
- --John Locke, _An Essay Concerning Human Understanding_ (1690)
-
- We are about to study the idea of a "computational process".
-Computational processes are abstract beings that inhabit computers. As
-they evolve, processes manipulate other abstract things called "data".
-The evolution of a process is directed by a pattern of rules called a "program".
-People create programs to direct processes. In effect, we conjure the
-spirits of the computer with our spells.
-
- A computational process is indeed much like a sorcerer's idea of a
-spirit. It cannot be seen or touched. It is not composed of matter at
-all. However, it is very real. It can perform intellectual work. It
-can answer questions. It can affect the world by disbursing money at a
-bank or by controlling a robot arm in a factory. The programs we use
-to conjure processes are like a sorcerer's spells. They are carefully
-composed from symbolic expressions in arcane and esoteric "programming
-languages" that prescribe the tasks we want our processes to perform.
-
- A computational process, in a correctly working computer, executes
-programs precisely and accurately. Thus, like the sorcerer's
-apprentice, novice programmers must learn to understand and to
-anticipate the consequences of their conjuring. Even small errors
-(usually called "bugs" or "glitches") in programs can have complex and
-unanticipated consequences.
-
- Fortunately, learning to program is considerably less dangerous than
-learning sorcery, because the spirits we deal with are conveniently
-contained in a secure way. Real-world programming, however, requires
-care, expertise, and wisdom. A small bug in a computer-aided design
-program, for example, can lead to the catastrophic collapse of an
-airplane or a dam or the self-destruction of an industrial robot.
-
- Master software engineers have the ability to organize programs so
-that they can be reasonably sure that the resulting processes will
-perform the tasks intended. They can visualize the behavior of their
-systems in advance. They know how to structure programs so that
-unanticipated problems do not lead to catastrophic consequences, and
-when problems do arise, they can "debug" their programs. Well-designed
-computational systems, like well-designed automobiles or nuclear
-reactors, are designed in a modular manner, so that the parts can be
-constructed, replaced, and debugged separately.
-
-Programming in Lisp
-...................
-
-We need an appropriate language for describing processes, and we will
-use for this purpose the programming language Lisp. Just as our
-everyday thoughts are usually expressed in our natural language (such
-as English, French, or Japanese), and descriptions of quantitative
-phenomena are expressed with mathematical notations, our procedural
-thoughts will be expressed in Lisp. Lisp was invented in the late
-1950s as a formalism for reasoning about the use of certain kinds of
-logical expressions, called "recursion equations", as a model for
-computation. The language was conceived by John McCarthy and is based
-on his paper "Recursive Functions of Symbolic Expressions and Their
-Computation by Machine" (McCarthy 1960).
-
- Despite its inception as a mathematical formalism, Lisp is a
-practical programming language. A Lisp "interpreter" is a machine that
-carries out processes described in the Lisp language. The first Lisp
-interpreter was implemented by McCarthy with the help of colleagues and
-students in the Artificial Intelligence Group of the MIT Research
-Laboratory of Electronics and in the MIT Computation Center.(1) Lisp,
-whose name is an acronym for LISt Processing, was designed to provide
-symbol-manipulating capabilities for attacking programming problems
-such as the symbolic differentiation and integration of algebraic
-expressions. It included for this purpose new data objects known as
-atoms and lists, which most strikingly set it apart from all other
-languages of the period.
-
- Lisp was not the product of a concerted design effort. Instead, it
-evolved informally in an experimental manner in response to users'
-needs and to pragmatic implementation considerations. Lisp's informal
-evolution has continued through the years, and the community of Lisp
-users has traditionally resisted attempts to promulgate any "official"
-definition of the language. This evolution, together with the
-flexibility and elegance of the initial conception, has enabled Lisp,
-which is the second oldest language in widespread use today (only
-Fortran is older), to continually adapt to encompass the most modern
-ideas about program design. Thus, Lisp is by now a family of dialects,
-which, while sharing most of the original features, may differ from one
-another in significant ways. The dialect of Lisp used in this book is
-called Scheme.(2)
-
- Because of its experimental character and its emphasis on symbol
-manipulation, Lisp was at first very inefficient for numerical
-computations, at least in comparison with Fortran. Over the years,
-however, Lisp compilers have been developed that translate programs
-into machine code that can perform numerical computations reasonably
-efficiently. And for special applications, Lisp has been used with
-great effectiveness.(3) Although Lisp has not yet overcome its old
-reputation as hopelessly inefficient, Lisp is now used in many
-applications where efficiency is not the central concern. For example,
-Lisp has become a language of choice for operating-system shell
-languages and for extension languages for editors and computer-aided
-design systems.
-
- If Lisp is not a mainstream language, why are we using it as the
-framework for our discussion of programming? Because the language
-possesses unique features that make it an excellent medium for studying
-important programming constructs and data structures and for relating
-them to the linguistic features that support them. The most
-significant of these features is the fact that Lisp descriptions of
-processes, called "procedures", can themselves be represented and
-manipulated as Lisp data. The importance of this is that there are
-powerful program-design techniques that rely on the ability to blur the
-traditional distinction between "passive" data and "active" processes.
-As we shall discover, Lisp's flexibility in handling procedures as data
-makes it one of the most convenient languages in existence for
-exploring these techniques. The ability to represent procedures as
-data also makes Lisp an excellent language for writing programs that
-must manipulate other programs as data, such as the interpreters and
-compilers that support computer languages. Above and beyond these
-considerations, programming in Lisp is great fun.
-
-* Menu:
-
-* 1-1:: The Elements of Programming
-* 1-2:: Procedures and the Processes They Generate
-* 1-3:: Formulating Abstractions with Higher-Order Procedures
-
- ---------- Footnotes ----------
-
- (1) The `Lisp 1 Programmer's Manual' appeared in 1960, and the `Lisp
-1.5 Programmer's Manual' (McCarthy 1965) was published in 1962. The
-early history of Lisp is described in McCarthy 1978.
-
- (2) The two dialects in which most major Lisp programs of the 1970s
-were written are MacLisp (Moon 1978; Pitman 1983), developed at the MIT
-Project MAC, and Interlisp (Teitelman 1974), developed at Bolt Beranek
-and Newman Inc. and the Xerox Palo Alto Research Center. Portable
-Standard Lisp (Hearn 1969; Griss 1981) was a Lisp dialect designed to
-be easily portable between different machines. MacLisp spawned a
-number of subdialects, such as Franz Lisp, which was developed at the
-University of California at Berkeley, and Zetalisp (Moon 1981), which
-was based on a special-purpose processor designed at the MIT Artificial
-Intelligence Laboratory to run Lisp very efficiently. The Lisp dialect
-used in this book, called Scheme (Steele 1975), was invented in 1975 by
-Guy Lewis Steele Jr. and Gerald Jay Sussman of the MIT Artificial
-Intelligence Laboratory and later reimplemented for instructional use
-at MIT. Scheme became an IEEE standard in 1990 (IEEE 1990). The
-Common Lisp dialect (Steele 1982, Steele 1990) was developed by the
-Lisp community to combine features from the earlier Lisp dialects to
-make an industrial standard for Lisp. Common Lisp became an ANSI
-standard in 1994 (ANSI 1994).
-
- (3) One such special application was a breakthrough computation of
-scientific importance--an integration of the motion of the Solar System
-that extended previous results by nearly two orders of magnitude, and
-demonstrated that the dynamics of the Solar System is chaotic. This
-computation was made possible by new integration algorithms, a
-special-purpose compiler, and a special-purpose computer all
-implemented with the aid of software tools written in Lisp (Abelson et
-al. 1992; Sussman and Wisdom 1992).
-
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-
-1.1 The Elements of Programming
-===============================
-
-A powerful programming language is more than just a means for
-instructing a computer to perform tasks. The language also serves as a
-framework within which we organize our ideas about processes. Thus,
-when we describe a language, we should pay particular attention to the
-means that the language provides for combining simple ideas to form
-more complex ideas. Every powerful language has three mechanisms for
-accomplishing this:
-
-"primitive expressions"
- which represent the simplest entities the language is concerned
- with,
-
-"means of combination"
- by which compound elements are built from simpler ones, and
-
-"means of abstraction"
- by which compound elements can be named and manipulated as units.
-
-
- In programming, we deal with two kinds of elements: procedures and
-data. (Later we will discover that they are really not so distinct.)
-Informally, data is "stuff" that we want to manipulate, and procedures
-are descriptions of the rules for manipulating the data. Thus, any
-powerful programming language should be able to describe primitive data
-and primitive procedures and should have methods for combining and
-abstracting procedures and data.
-
- In this chapter we will deal only with simple numerical data so that
-we can focus on the rules for building procedures.(1) In later chapters
-we will see that these same rules allow us to build procedures to
-manipulate compound data as well.
-
-* Menu:
-
-* 1-1-1:: Expressions
-* 1-1-2:: Naming and the Environment
-* 1-1-3:: Evaluating Combinations
-* 1-1-4:: Compound Procedures
-* 1-1-5:: The Substitution Model for Procedure Application
-* 1-1-6:: Conditional Expressions and Predicates
-* 1-1-7:: Example: Square Roots by Newton's Method
-* 1-1-8:: Procedures as Black-Box Abstractions
-
- ---------- Footnotes ----------
-
- (1) The characterization of numbers as "simple data" is a barefaced
-bluff. In fact, the treatment of numbers is one of the trickiest and
-most confusing aspects of any programming language. Some typical
-issues involved are these: Some computer systems distinguish "integers",
-such as 2, from "real numbers", such as 2.71. Is the real number 2.00
-different from the integer 2? Are the arithmetic operations used for
-integers the same as the operations used for real numbers? Does 6
-divided by 2 produce 3, or 3.0? How large a number can we represent?
-How many decimal places of accuracy can we represent? Is the range of
-integers the same as the range of real numbers? Above and beyond these
-questions, of course, lies a collection of issues concerning roundoff
-and truncation errors - the entire science of numerical analysis.
-Since our focus in this book is on large-scale program design rather
-than on numerical techniques, we are going to ignore these problems.
-The numerical examples in this chapter will exhibit the usual roundoff
-behavior that one observes when using arithmetic operations that
-preserve a limited number of decimal places of accuracy in noninteger
-operations.
-
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-
-1.1.1 Expressions
------------------
-
-One easy way to get started at programming is to examine some typical
-interactions with an interpreter for the Scheme dialect of Lisp.
-Imagine that you are sitting at a computer terminal. You type an "expression",
-and the interpreter responds by displaying the result of its "evaluating"
-that expression.
-
- One kind of primitive expression you might type is a number. (More
-precisely, the expression that you type consists of the numerals that
-represent the number in base 10.) If you present Lisp with a number
-
- 486
-
-the interpreter will respond by printing (1)
-
- 486
-
- Expressions representing numbers may be combined with an expression
-representing a primitive procedure (such as `+' or `*') to form a
-compound expression that represents the application of the procedure to
-those numbers. For example:
-
- (+ 137 349)
- 486
-
- (- 1000 334)
- 666
-
- (* 5 99)
- 495
-
- (/ 10 5)
- 2
-
- (+ 2.7 10)
- 12.7
-
- Expressions such as these, formed by delimiting a list of
-expressions within parentheses in order to denote procedure
-application, are called "combinations". The leftmost element in the
-list is called the "operator", and the other elements are called "operands".
-The value of a combination is obtained by applying the procedure
-specified by the operator to the "arguments" that are the values of the
-operands.
-
- The convention of placing the operator to the left of the operands
-is known as "prefix notation", and it may be somewhat confusing at
-first because it departs significantly from the customary mathematical
-convention. Prefix notation has several advantages, however. One of
-them is that it can accommodate procedures that may take an arbitrary
-number of arguments, as in the following examples:
-
- (+ 21 35 12 7)
- 75
-
- (* 25 4 12)
- 1200
-
- No ambiguity can arise, because the operator is always the leftmost
-element and the entire combination is delimited by the parentheses.
-
- A second advantage of prefix notation is that it extends in a
-straightforward way to allow combinations to be nested, that is, to
-have combinations whose elements are themselves combinations:
-
- (+ (* 3 5) (- 10 6))
- 19
-
- There is no limit (in principle) to the depth of such nesting and to
-the overall complexity of the expressions that the Lisp interpreter can
-evaluate. It is we humans who get confused by still relatively simple
-expressions such as
-
- (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))
-
-which the interpreter would readily evaluate to be 57. We can help
-ourselves by writing such an expression in the form
-
- (+ (* 3
- (+ (* 2 4)
- (+ 3 5)))
- (+ (- 10 7)
- 6))
-
-following a formatting convention known as "pretty-printing", in which
-each long combination is written so that the operands are aligned
-vertically. The resulting indentations display clearly the structure
-of the expression.(2)
-
- Even with complex expressions, the interpreter always operates in
-the same basic cycle: It reads an expression from the terminal,
-evaluates the expression, and prints the result. This mode of
-operation is often expressed by saying that the interpreter runs in a "read-eval-print
-loop". Observe in particular that it is not necessary to explicitly
-instruct the interpreter to print the value of the expression.(3)
-
- ---------- Footnotes ----------
-
- (1) Throughout this book, when we wish to emphasize the distinction
-between the input typed by the user and the response printed by the
-interpreter, we will show the latter in slanted characters.
-
- (2) Lisp systems typically provide features to aid the user in
-formatting expressions. Two especially useful features are one that
-automatically indents to the proper pretty-print position whenever a
-new line is started and one that highlights the matching left
-parenthesis whenever a right parenthesis is typed.
-
- (3) Lisp obeys the convention that every expression has a value.
-This convention, together with the old reputation of Lisp as an
-inefficient language, is the source of the quip by Alan Perlis
-(paraphrasing Oscar Wilde) that "Lisp programmers know the value of
-everything but the cost of nothing."
-
-\1f
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-
-1.1.2 Naming and the Environment
---------------------------------
-
-A critical aspect of a programming language is the means it provides
-for using names to refer to computational objects. We say that the
-name identifies a "variable" whose "value" is the object.
-
- In the Scheme dialect of Lisp, we name things with `define'. Typing
-
- (define size 2)
-
-causes the interpreter to associate the value 2 with the name
-`size'.(1) Once the name `size' has been associated with the number 2,
-we can refer to the value 2 by name:
-
- size
- 2
-
- (* 5 size)
- 10
-
- Here are further examples of the use of `define':
-
- (define pi 3.14159)
-
- (define radius 10)
-
- (* pi (* radius radius))
- 314.159
-
- (define circumference (* 2 pi radius))
-
- circumference
- 62.8318
-
- `Define' is our language's simplest means of abstraction, for it
-allows us to use simple names to refer to the results of compound
-operations, such as the `circumference' computed above. In general,
-computational objects may have very complex structures, and it would be
-extremely inconvenient to have to remember and repeat their details
-each time we want to use them. Indeed, complex programs are
-constructed by building, step by step, computational objects of
-increasing complexity. The interpreter makes this step-by-step program
-construction particularly convenient because name-object associations
-can be created incrementally in successive interactions. This feature
-encourages the incremental development and testing of programs and is
-largely responsible for the fact that a Lisp program usually consists
-of a large number of relatively simple procedures.
-
- It should be clear that the possibility of associating values with
-symbols and later retrieving them means that the interpreter must
-maintain some sort of memory that keeps track of the name-object pairs.
-This memory is called the "environment" (more precisely the "global
-environment", since we will see later that a computation may involve a
-number of different environments).(2)
-
- ---------- Footnotes ----------
-
- (1) In this book, we do not show the interpreter's response to
-evaluating definitions, since this is highly implementation-dependent.
-
- (2) *Note Chapter 3:: will show that this notion of environment is
-crucial, both for understanding how the interpreter works and for
-implementing interpreters.
-
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-
-1.1.3 Evaluating Combinations
------------------------------
-
-One of our goals in this chapter is to isolate issues about thinking
-procedurally. As a case in point, let us consider that, in evaluating
-combinations, the interpreter is itself following a procedure.
-
- To evaluate a combination, do the following:
-
- 1. Evaluate the subexpressions of the combination.
-
- 2. Apply the procedure that is the value of the leftmost
- subexpression (the operator) to the arguments that are the
- values of the other subexpressions (the operands).
-
-
- Even this simple rule illustrates some important points about
-processes in general. First, observe that the first step dictates that
-in order to accomplish the evaluation process for a combination we must
-first perform the evaluation process on each element of the
-combination. Thus, the evaluation rule is "recursive" in nature; that
-is, it includes, as one of its steps, the need to invoke the rule
-itself.(1)
-
- Notice how succinctly the idea of recursion can be used to express
-what, in the case of a deeply nested combination, would otherwise be
-viewed as a rather complicated process. For example, evaluating
-
- (* (+ 2 (* 4 6))
- (+ 3 5 7))
-
-requires that the evaluation rule be applied to four different
-combinations. We can obtain a picture of this process by representing
-the combination in the form of a tree, as shown in *Note Figure 1-1::.
-Each combination is represented by a node with branches corresponding
-to the operator and the operands of the combination stemming from it.
-The terminal nodes (that is, nodes with no branches stemming from them)
-represent either operators or numbers. Viewing evaluation in terms of
-the tree, we can imagine that the values of the operands percolate
-upward, starting from the terminal nodes and then combining at higher
-and higher levels. In general, we shall see that recursion is a very
-powerful technique for dealing with hierarchical, treelike objects. In
-fact, the "percolate values upward" form of the evaluation rule is an
-example of a general kind of process known as "tree accumulation".
-
- *Figure 1.1:* Tree representation, showing the value of each
- subcombination.
-
- 390
- /|\____________
- / | \
- * 26 15
- /|\ |
- / | \ // \\
- + 2 24 / | | \
- /|\ + 3 5 7
- / | \
- * 4 6
-
- Next, observe that the repeated application of the first step brings
-us to the point where we need to evaluate, not combinations, but
-primitive expressions such as numerals, built-in operators, or other
-names. We take care of the primitive cases by stipulating that
-
- * the values of numerals are the numbers that they name,
-
- * the values of built-in operators are the machine instruction
- sequences that carry out the corresponding operations, and
-
- * the values of other names are the objects associated with those
- names in the environment.
-
-
- We may regard the second rule as a special case of the third one by
-stipulating that symbols such as `+' and `*' are also included in the
-global environment, and are associated with the sequences of machine
-instructions that are their "values." The key point to notice is the
-role of the environment in determining the meaning of the symbols in
-expressions. In an interactive language such as Lisp, it is
-meaningless to speak of the value of an expression such as `(+ x 1)'
-without specifying any information about the environment that would
-provide a meaning for the symbol `x' (or even for the symbol `+'). As
-we shall see in *Note Chapter 3::, the general notion of the
-environment as providing a context in which evaluation takes place will
-play an important role in our understanding of program execution.
-
- Notice that the evaluation rule given above does not handle
-definitions. For instance, evaluating `(define x 3)' does not apply
-`define' to two arguments, one of which is the value of the symbol `x'
-and the other of which is 3, since the purpose of the `define' is
-precisely to associate `x' with a value. (That is, `(define x 3)' is
-not a combination.)
-
- Such exceptions to the general evaluation rule are called forms
-"special forms". `Define' is the only example of a special form that
-we have seen so far, but we will meet others shortly. Each special
-form has its own evaluation rule. The various kinds of expressions
-(each with its associated evaluation rule) constitute the syntax of the
-programming language. In comparison with most other programming
-languages, Lisp has a very simple syntax; that is, the evaluation rule
-for expressions can be described by a simple general rule together with
-specialized rules for a small number of special forms.(2)
-
- ---------- Footnotes ----------
-
- (1) It may seem strange that the evaluation rule says, as part of
-the first step, that we should evaluate the leftmost element of a
-combination, since at this point that can only be an operator such as
-`+' or `*' representing a built-in primitive procedure such as addition
-or multiplication. We will see later that it is useful to be able to
-work with combinations whose operators are themselves compound
-expressions.
-
- (2) Special syntactic forms that are simply convenient alternative
-surface structures for things that can be written in more uniform ways
-are sometimes called "syntactic sugar", to use a phrase coined by Peter
-Landin. In comparison with users of other languages, Lisp programmers,
-as a rule, are less concerned with matters of syntax. (By contrast,
-examine any Pascal manual and notice how much of it is devoted to
-descriptions of syntax.) This disdain for syntax is due partly to the
-flexibility of Lisp, which makes it easy to change surface syntax, and
-partly to the observation that many "convenient" syntactic constructs,
-which make the language less uniform, end up causing more trouble than
-they are worth when programs become large and complex. In the words of
-Alan Perlis, "Syntactic sugar causes cancer of the semicolon."
-
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-
-1.1.4 Compound Procedures
--------------------------
-
-We have identified in Lisp some of the elements that must appear in any
-powerful programming language:
-
- * Numbers and arithmetic operations are primitive data and
- procedures.
-
- * Nesting of combinations provides a means of combining operations.
-
- * Definitions that associate names with values provide a limited
- means of abstraction.
-
-
- Now we will learn about "procedure definitions", a much more powerful
-abstraction technique by which a compound operation can be given a name
-and then referred to as a unit.
-
- We begin by examining how to express the idea of "squaring." We
-might say, "To square something, multiply it by itself." This is
-expressed in our language as
-
- (define (square x) (* x x))
-
- We can understand this in the following way:
-
- (define (square x) (* x x))
- | | | | | |
- To square something, multiply it by itself.
-
- We have here a "compound procedure", which has been given the name
-`square'. The procedure represents the operation of multiplying
-something by itself. The thing to be multiplied is given a local name,
-`x', which plays the same role that a pronoun plays in natural
-language. Evaluating the definition creates this compound procedure
-and associates it with the name `square'.(1)
-
- The general form of a procedure definition is
-
- (define (<NAME> <FORMAL PARAMETERS>) <BODY>)
-
- The <NAME> is a symbol to be associated with the procedure
-definition in the environment.(2) The <FORMAL PARAMETERS> are the names
-used within the body of the procedure to refer to the corresponding
-arguments of the procedure. The <BODY> is an expression that will
-yield the value of the procedure application when the formal parameters
-are replaced by the actual arguments to which the procedure is
-applied.(3) The <NAME> and the <FORMAL PARAMETERS> are grouped within
-parentheses, just as they would be in an actual call to the procedure
-being defined.
-
- Having defined `square', we can now use it:
-
- (square 21)
- 441
-
- (square (+ 2 5))
- 49
-
- (square (square 3))
- 81
-
- We can also use `square' as a building block in defining other
-procedures. For example, x^2 + y^2 can be expressed as
-
- (+ (square x) (square y))
-
- We can easily define a procedure `sum-of-squares' that, given any two
-numbers as arguments, produces the sum of their squares:
-
- (define (sum-of-squares x y)
- (+ (square x) (square y)))
- (sum-of-squares 3 4)
- 25
-
- Now we can use `sum-of-squares' as a building block in constructing
-further procedures:
-
- (define (f a)
- (sum-of-squares (+ a 1) (* a 2)))
-
- (f 5)
- 136
-
- Compound procedures are used in exactly the same way as primitive
-procedures. Indeed, one could not tell by looking at the definition of
-`sum-of-squares' given above whether `square' was built into the
-interpreter, like `+' and `*', or defined as a compound procedure.
-
- ---------- Footnotes ----------
-
- (1) Observe that there are two different operations being combined
-here: we are creating the procedure, and we are giving it the name
-`square'. It is possible, indeed important, to be able to separate
-these two notions--to create procedures without naming them, and to
-give names to procedures that have already been created. We will see
-how to do this in section *Note 1-3-2::.
-
- (2) Throughout this book, we will describe the general syntax of
-expressions by using italic symbols delimited by angle brackets--e.g.,
-<NAME>--to denote the "slots" in the expression to be filled in when
-such an expression is actually used.
-
- (3) More generally, the body of the procedure can be a sequence of
-expressions. In this case, the interpreter evaluates each expression
-in the sequence in turn and returns the value of the final expression
-as the value of the procedure application.
-
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-
-1.1.5 The Substitution Model for Procedure Application
-------------------------------------------------------
-
-To evaluate a combination whose operator names a compound procedure, the
-interpreter follows much the same process as for combinations whose
-operators name primitive procedures, which we described in section
-*Note 1-1-3::. That is, the interpreter evaluates the elements of the
-combination and applies the procedure (which is the value of the
-operator of the combination) to the arguments (which are the values of
-the operands of the combination).
-
- We can assume that the mechanism for applying primitive procedures
-to arguments is built into the interpreter. For compound procedures,
-the application process is as follows:
-
- To apply a compound procedure to arguments, evaluate the body of
- the procedure with each formal parameter replaced by the
- corresponding argument.
-
- To illustrate this process, let's evaluate the combination
-
- (f 5)
-
-where `f' is the procedure defined in section *Note 1-1-4::. We begin
-by retrieving the body of `f':
-
- (sum-of-squares (+ a 1) (* a 2))
-
- Then we replace the formal parameter `a' by the argument 5:
-
- (sum-of-squares (+ 5 1) (* 5 2))
-
- Thus the problem reduces to the evaluation of a combination with two
-operands and an operator `sum-of-squares'. Evaluating this combination
-involves three subproblems. We must evaluate the operator to get the
-procedure to be applied, and we must evaluate the operands to get the
-arguments. Now `(+ 5 1)' produces 6 and `(* 5 2)' produces 10, so we
-must apply the `sum-of-squares' procedure to 6 and 10. These values
-are substituted for the formal parameters `x' and `y' in the body of
-`sum-of-squares', reducing the expression to
-
- (+ (square 6) (square 10))
-
- If we use the definition of `square', this reduces to
-
- (+ (* 6 6) (* 10 10))
-
-which reduces by multiplication to
-
- (+ 36 100)
-
-and finally to
-
- 136
-
- The process we have just described is called the "substitution model"
-for procedure application. It can be taken as a model that determines
-the "meaning" of procedure application, insofar as the procedures in
-this chapter are concerned. However, there are two points that should
-be stressed:
-
- * The purpose of the substitution is to help us think about procedure
- application, not to provide a description of how the interpreter
- really works. Typical interpreters do not evaluate procedure
- applications by manipulating the text of a procedure to substitute
- values for the formal parameters. In practice, the "substitution"
- is accomplished by using a local environment for the formal
- parameters. We will discuss this more fully in *Note Chapter 3::
- and *Note Chapter 4:: when we examine the implementation of an
- interpreter in detail.
-
- * Over the course of this book, we will present a sequence of
- increasingly elaborate models of how interpreters work,
- culminating with a complete implementation of an interpreter and
- compiler in *Note Chapter 5::. The substitution model is only the
- first of these models--a way to get started thinking formally
- about the evaluation process. In general, when modeling phenomena
- in science and engineering, we begin with simplified, incomplete
- models. As we examine things in greater detail, these simple
- models become inadequate and must be replaced by more refined
- models. The substitution model is no exception. In particular,
- when we address in *Note Chapter 3:: the use of procedures with
- "mutable data," we will see that the substitution model breaks
- down and must be replaced by a more complicated model of procedure
- application.(1)
-
-
-Applicative order versus normal order
-.....................................
-
-According to the description of evaluation given in section *Note
-1-1-3::, the interpreter first evaluates the operator and operands and
-then applies the resulting procedure to the resulting arguments. This
-is not the only way to perform evaluation. An alternative evaluation
-model would not evaluate the operands until their values were needed.
-Instead it would first substitute operand expressions for parameters
-until it obtained an expression involving only primitive operators, and
-would then perform the evaluation. If we used this method, the
-evaluation of `(f 5)' would proceed according to the sequence of
-expansions
-
- (sum-of-squares (+ 5 1) (* 5 2))
-
- (+ (square (+ 5 1)) (square (* 5 2)) )
-
- (+ (* (+ 5 1) (+ 5 1)) (* (* 5 2) (* 5 2)))
-
-followed by the reductions
-
- (+ (* 6 6) (* 10 10))
-
- (+ 36 100)
-
- 136
-
- This gives the same answer as our previous evaluation model, but the
-process is different. In particular, the evaluations of `(+ 5 1)' and
-`(* 5 2)' are each performed twice here, corresponding to the reduction
-of the expression `(* x x)' with `x' replaced respectively by `(+ 5 1)'
-and `(* 5 2)'.
-
- This alternative "fully expand and then reduce" evaluation method is
-known as "normal-order evaluation", in contrast to the "evaluate the
-arguments and then apply" method that the interpreter actually uses,
-which is called "applicative-order evaluation". It can be shown that,
-for procedure applications that can be modeled using substitution
-(including all the procedures in the first two chapters of this book)
-and that yield legitimate values, normal-order and applicative-order
-evaluation produce the same value. (See *Note Exercise 1-5:: for an
-instance of an "illegitimate" value where normal-order and
-applicative-order evaluation do not give the same result.)
-
- Lisp uses applicative-order evaluation, partly because of the
-additional efficiency obtained from avoiding multiple evaluations of
-expressions such as those illustrated with `(+ 5 1)' and `(* 5 2)'
-above and, more significantly, because normal-order evaluation becomes
-much more complicated to deal with when we leave the realm of
-procedures that can be modeled by substitution. On the other hand,
-normal-order evaluation can be an extremely valuable tool, and we will
-investigate some of its implications in *Note Chapter 3:: and *Note
-Chapter 4::.(2)
-
- ---------- Footnotes ----------
-
- (1) Despite the simplicity of the substitution idea, it turns out to
-be surprisingly complicated to give a rigorous mathematical definition
-of the substitution process. The problem arises from the possibility of
-confusion between the names used for the formal parameters of a
-procedure and the (possibly identical) names used in the expressions to
-which the procedure may be applied. Indeed, there is a long history of
-erroneous definitions of "substitution" in the literature of logic and
-programming semantics. See Stoy 1977 for a careful discussion of
-substitution.
-
- (2) In *Note Chapter 3:: we will introduce "stream processing",
-which is a way of handling apparently "infinite" data structures by
-incorporating a limited form of normal-order evaluation. In section
-*Note 4-2:: we will modify the Scheme interpreter to produce a
-normal-order variant of Scheme.
-
-\1f
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-
-1.1.6 Conditional Expressions and Predicates
---------------------------------------------
-
-The expressive power of the class of procedures that we can define at
-this point is very limited, because we have no way to make tests and to
-perform different operations depending on the result of a test. For
-instance, we cannot define a procedure that computes the absolute value
-of a number by testing whether the number is positive, negative, or
-zero and taking different actions in the different cases according to
-the rule
-
- /
- | x if x > 0
- |x| = < 0 if x = 0
- | -x if x < 0
- \
-
- This construct is called a "case analysis", and there is a special
-form in Lisp for notating such a case analysis. It is called `cond'
-(which stands for "conditional"), and it is used as follows:
-
- (define (abs x)
- (cond ((> x 0) x)
- ((= x 0) 0)
- ((< x 0) (- x))))
-
- The general form of a conditional expression is
-
- (cond (<P1> <E1>)
- (<P2> <E2>)
- ...
- (<PN> <EN>))
-
-consisting of the symbol `cond' followed by parenthesized pairs of
-expressions
-
- (<P> <E>)
-
-called "clauses". The first expression in each pair is a "predicate"--that
-is, an expression whose value is interpreted as either true or false.(1)
-
- Conditional expressions are evaluated as follows. The predicate
-<P1> is evaluated first. If its value is false, then <P2> is
-evaluated. If <P2>'s value is also false, then <P3> is evaluated.
-This process continues until a predicate is found whose value is true,
-in which case the interpreter returns the value of the corresponding expression
-"consequent expression" <E> of the clause as the value of the
-conditional expression. If none of the <P>'s is found to be true, the
-value of the `cond' is undefined.
-
- The word "predicate" is used for procedures that return true or
-false, as well as for expressions that evaluate to true or false. The
-absolute-value procedure `abs' makes use of the primitive predicates
-`>', `<', and `='.(2) These take two numbers as arguments and test
-whether the first number is, respectively, greater than, less than, or
-equal to the second number, returning true or false accordingly.
-
- Another way to write the absolute-value procedure is
-
- (define (abs x)
- (cond ((< x 0) (- x))
- (else x)))
-
-which could be expressed in English as "If x is less than zero return -
-x; otherwise return x." `Else' is a special symbol that can be used in
-place of the <P> in the final clause of a `cond'. This causes the
-`cond' to return as its value the value of the corresponding <E>
-whenever all previous clauses have been bypassed. In fact, any
-expression that always evaluates to a true value could be used as the
-<P> here.
-
- Here is yet another way to write the absolute-value procedure:
-
- (define (abs x)
- (if (< x 0)
- (- x)
- x))
-
- This uses the special form `if', a restricted type of conditional
-that can be used when there are precisely two cases in the case
-analysis. The general form of an `if' expression is
-
- (if <PREDICATE> <CONSEQUENT> <ALTERNATIVE>)
-
- To evaluate an `if' expression, the interpreter starts by evaluating
-the <PREDICATE> part of the expression. If the <PREDICATE> evaluates
-to a true value, the interpreter then evaluates the <CONSEQUENT> and
-returns its value. Otherwise it evaluates the <ALTERNATIVE> and returns
-its value.(3)
-
- In addition to primitive predicates such as `<', `=', and `>', there
-are logical composition operations, which enable us to construct
-compound predicates. The three most frequently used are these:
-
- * `(and <E1> ... <EN>)'
-
- The interpreter evaluates the expressions <E> one at a time, in
- left-to-right order. If any <E> evaluates to false, the value of
- the `and' expression is false, and the rest of the <E>'s are not
- evaluated. If all <E>'s evaluate to true values, the value of the
- `and' expression is the value of the last one.
-
- * `(or <E1> ... <EN>)'
-
- The interpreter evaluates the expressions <E> one at a time, in
- left-to-right order. If any <E> evaluates to a true value, that
- value is returned as the value of the `or' expression, and the
- rest of the <E>'s are not evaluated. If all <E>'s evaluate to
- false, the value of the `or' expression is false.
-
- * `(not <E>)'
-
- The value of a `not' expression is true when the expression <E>
- evaluates to false, and false otherwise.
-
-
- Notice that `and' and `or' are special forms, not procedures, because
-the subexpressions are not necessarily all evaluated. `Not' is an
-ordinary procedure.
-
- As an example of how these are used, the condition that a number x
-be in the range 5 < x < 10 may be expressed as
-
- (and (> x 5) (< x 10))
-
- As another example, we can define a predicate to test whether one
-number is greater than or equal to another as
-
- (define (>= x y)
- (or (> x y) (= x y)))
-
-or alternatively as
-
- (define (>= x y)
- (not (< x y)))
-
- *Exercise 1.1:* Below is a sequence of expressions. What is the
- result printed by the interpreter in response to each expression?
- Assume that the sequence is to be evaluated in the order in which
- it is presented.
-
- 10
-
- (+ 5 3 4)
-
- (- 9 1)
-
- (/ 6 2)
-
- (+ (* 2 4) (- 4 6))
-
- (define a 3)
-
- (define b (+ a 1))
-
- (+ a b (* a b))
-
- (= a b)
-
- (if (and (> b a) (< b (* a b)))
- b
- a)
-
- (cond ((= a 4) 6)
- ((= b 4) (+ 6 7 a))
- (else 25))
-
- (+ 2 (if (> b a) b a))
-
- (* (cond ((> a b) a)
- ((< a b) b)
- (else -1))
- (+ a 1))
-
- *Exercise 1.2:* Translate the following expression into prefix
- form.
-
- 5 + 4 + (2 - (3 - (6 + 4/5)))
- -----------------------------
- 3(6 - 2)(2 - 7)
-
- *Exercise 1.3:* Define a procedure that takes three numbers as
- arguments and returns the sum of the squares of the two larger
- numbers.
-
- *Exercise 1.4:* Observe that our model of evaluation allows for
- combinations whose operators are compound expressions. Use this
- observation to describe the behavior of the following procedure:
-
- (define (a-plus-abs-b a b)
- ((if (> b 0) + -) a b))
-
- *Exercise 1.5:* Ben Bitdiddle has invented a test to determine
- whether the interpreter he is faced with is using
- applicative-order evaluation or normal-order evaluation. He
- defines the following two procedures:
-
- (define (p) (p))
-
- (define (test x y)
- (if (= x 0)
- 0
- y))
-
- Then he evaluates the expression
-
- (test 0 (p))
-
- What behavior will Ben observe with an interpreter that uses
- applicative-order evaluation? What behavior will he observe with
- an interpreter that uses normal-order evaluation? Explain your
- answer. (Assume that the evaluation rule for the special form
- `if' is the same whether the interpreter is using normal or
- applicative order: The predicate expression is evaluated first,
- and the result determines whether to evaluate the consequent or
- the alternative expression.)
-
- ---------- Footnotes ----------
-
- (1) "Interpreted as either true or false" means this: In Scheme,
-there are two distinguished values that are denoted by the constants
-`#t' and `#f'. When the interpreter checks a predicate's value, it
-interprets `#f' as false. Any other value is treated as true. (Thus,
-providing `#t' is logically unnecessary, but it is convenient.) In
-this book we will use names `true' and `false', which are associated
-with the values `#t' and `#f' respectively.
-
- (2) `Abs' also uses the "minus" operator `-', which, when used with
-a single operand, as in `(- x)', indicates negation.
-
- (3) A minor difference between `if' and `cond' is that the <E> part
-of each `cond' clause may be a sequence of expressions. If the
-corresponding <P> is found to be true, the expressions <E> are
-evaluated in sequence and the value of the final expression in the
-sequence is returned as the value of the `cond'. In an `if'
-expression, however, the <CONSEQUENT> and <ALTERNATIVE> must be single
-expressions.
-
-\1f
-File: sicp.info, Node: 1-1-7, Next: 1-1-8, Prev: 1-1-6, Up: 1-1
-
-1.1.7 Example: Square Roots by Newton's Method
-----------------------------------------------
-
-Procedures, as introduced above, are much like ordinary mathematical
-functions. They specify a value that is determined by one or more
-parameters. But there is an important difference between mathematical
-functions and computer procedures. Procedures must be effective.
-
- As a case in point, consider the problem of computing square roots.
-We can define the square-root function as
-
- sqrt(x) = the y such that y >= 0 and y^2 = x
-
- This describes a perfectly legitimate mathematical function. We
-could use it to recognize whether one number is the square root of
-another, or to derive facts about square roots in general. On the
-other hand, the definition does not describe a procedure. Indeed, it
-tells us almost nothing about how to actually find the square root of a
-given number. It will not help matters to rephrase this definition in
-pseudo-Lisp:
-
- (define (sqrt x)
- (the y (and (>= y 0)
- (= (square y) x))))
-
- This only begs the question.
-
- The contrast between function and procedure is a reflection of the
-general distinction between describing properties of things and
-describing how to do things, or, as it is sometimes referred to, the
-distinction between declarative knowledge and imperative knowledge. In
-mathematics we are usually concerned with declarative (what is)
-descriptions, whereas in computer science we are usually concerned with
-imperative (how to) descriptions.(1)
-
- How does one compute square roots? The most common way is to use
-Newton's method of successive approximations, which says that whenever
-we have a guess y for the value of the square root of a number x, we
-can perform a simple manipulation to get a better guess (one closer to
-the actual square root) by averaging y with x/y.(2) For example, we can
-compute the square root of 2 as follows. Suppose our initial guess is
-1:
-
- Guess Quotient Average
- 1 (2/1) = 2 ((2 + 1)/2) = 1.5
- 1.5 (2/1.5) = 1.3333 ((1.3333 + 1.5)/2) = 1.4167
- 1.4167 (2/1.4167) = 1.4118 ((1.4167 + 1.4118)/2) = 1.4142
- 1.4142 ... ...
-
-Continuing this process, we obtain better and better approximations to
-the square root.
-
- Now let's formalize the process in terms of procedures. We start
-with a value for the radicand (the number whose square root we are
-trying to compute) and a value for the guess. If the guess is good
-enough for our purposes, we are done; if not, we must repeat the
-process with an improved guess. We write this basic strategy as a
-procedure:
-
- (define (sqrt-iter guess x)
- (if (good-enough? guess x)
- guess
- (sqrt-iter (improve guess x)
- x)))
-
- A guess is improved by averaging it with the quotient of the
-radicand and the old guess:
-
- (define (improve guess x)
- (average guess (/ x guess)))
-
-where
-
- (define (average x y)
- (/ (+ x y) 2))
-
- We also have to say what we mean by "good enough." The following
-will do for illustration, but it is not really a very good test. (See
-exercise *Note Exercise 1-7::.) The idea is to improve the answer
-until it is close enough so that its square differs from the radicand
-by less than a predetermined tolerance (here 0.001):(3)
-
- (define (good-enough? guess x)
- (< (abs (- (square guess) x)) 0.001))
-
- Finally, we need a way to get started. For instance, we can always
-guess that the square root of any number is 1:(4)
-
- (define (sqrt x)
- (sqrt-iter 1.0 x))
-
- If we type these definitions to the interpreter, we can use `sqrt'
-just as we can use any procedure:
-
- (sqrt 9)
- 3.00009155413138
-
- (sqrt (+ 100 37))
- 11.704699917758145
-
- (sqrt (+ (sqrt 2) (sqrt 3)))
- 1.7739279023207892
-
- (square (sqrt 1000))
- 1000.000369924366
-
- The `sqrt' program also illustrates that the simple procedural
-language we have introduced so far is sufficient for writing any purely
-numerical program that one could write in, say, C or Pascal. This
-might seem surprising, since we have not included in our language any
-iterative (looping) constructs that direct the computer to do something
-over and over again. `Sqrt-iter', on the other hand, demonstrates how
-iteration can be accomplished using no special construct other than the
-ordinary ability to call a procedure.(5)
-
- *Exercise 1.6:* Alyssa P. Hacker doesn't see why `if' needs to be
- provided as a special form. "Why can't I just define it as an
- ordinary procedure in terms of `cond'?" she asks. Alyssa's friend
- Eva Lu Ator claims this can indeed be done, and she defines a new
- version of `if':
-
- (define (new-if predicate then-clause else-clause)
- (cond (predicate then-clause)
- (else else-clause)))
-
- Eva demonstrates the program for Alyssa:
-
- (new-if (= 2 3) 0 5)
- 5
-
- (new-if (= 1 1) 0 5)
- 0
-
- Delighted, Alyssa uses `new-if' to rewrite the square-root program:
-
- (define (sqrt-iter guess x)
- (new-if (good-enough? guess x)
- guess
- (sqrt-iter (improve guess x)
- x)))
-
- What happens when Alyssa attempts to use this to compute square
- roots? Explain.
-
- *Exercise 1.7:* The `good-enough?' test used in computing square
- roots will not be very effective for finding the square roots of
- very small numbers. Also, in real computers, arithmetic operations
- are almost always performed with limited precision. This makes
- our test inadequate for very large numbers. Explain these
- statements, with examples showing how the test fails for small and
- large numbers. An alternative strategy for implementing
- `good-enough?' is to watch how `guess' changes from one iteration
- to the next and to stop when the change is a very small fraction
- of the guess. Design a square-root procedure that uses this kind
- of end test. Does this work better for small and large numbers?
-
- *Exercise 1.8:* Newton's method for cube roots is based on the
- fact that if y is an approximation to the cube root of x, then a
- better approximation is given by the value
-
- x/y^2 + 2y
- ----------
- 3
-
- Use this formula to implement a cube-root procedure analogous to
- the square-root procedure. (In section *Note 1-3-4:: we will see
- how to implement Newton's method in general as an abstraction of
- these square-root and cube-root procedures.)
-
- ---------- Footnotes ----------
-
- (1) Declarative and imperative descriptions are intimately related,
-as indeed are mathematics and computer science. For instance, to say
-that the answer produced by a program is "correct" is to make a
-declarative statement about the program. There is a large amount of
-research aimed at establishing techniques for proving that programs are
-correct, and much of the technical difficulty of this subject has to do
-with negotiating the transition between imperative statements (from
-which programs are constructed) and declarative statements (which can be
-used to deduce things). In a related vein, an important current area in
-programming-language design is the exploration of so-called very
-high-level languages, in which one actually programs in terms of
-declarative statements. The idea is to make interpreters sophisticated
-enough so that, given "what is" knowledge specified by the programmer,
-they can generate "how to" knowledge automatically. This cannot be
-done in general, but there are important areas where progress has been
-made. We shall revisit this idea in *Note Chapter 4::.
-
- (2) This square-root algorithm is actually a special case of
-Newton's method, which is a general technique for finding roots of
-equations. The square-root algorithm itself was developed by Heron of
-Alexandria in the first century A.D. We will see how to express the
-general Newton's method as a Lisp procedure in section *Note 1-3-4::.
-
- (3) We will usually give predicates names ending with question
-marks, to help us remember that they are predicates. This is just a
-stylistic convention. As far as the interpreter is concerned, the
-question mark is just an ordinary character.
-
- (4) Observe that we express our initial guess as 1.0 rather than 1.
-This would not make any difference in many Lisp implementations. MIT
-Scheme, however, distinguishes between exact integers and decimal
-values, and dividing two integers produces a rational number rather
-than a decimal. For example, dividing 10 by 6 yields 5/3, while
-dividing 10.0 by 6.0 yields 1.6666666666666667. (We will learn how to
-implement arithmetic on rational numbers in section *Note 2-1-1::.) If
-we start with an initial guess of 1 in our square-root program, and x
-is an exact integer, all subsequent values produced in the square-root
-computation will be rational numbers rather than decimals. Mixed
-operations on rational numbers and decimals always yield decimals, so
-starting with an initial guess of 1.0 forces all subsequent values to
-be decimals.
-
- (5) Readers who are worried about the efficiency issues involved in
-using procedure calls to implement iteration should note the remarks on
-"tail recursion" in section *Note 1-2-1::.
-
-\1f
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-
-1.1.8 Procedures as Black-Box Abstractions
-------------------------------------------
-
-`Sqrt' is our first example of a process defined by a set of mutually
-defined procedures. Notice that the definition of `sqrt-iter' is "recursive";
-that is, the procedure is defined in terms of itself. The idea of
-being able to define a procedure in terms of itself may be disturbing;
-it may seem unclear how such a "circular" definition could make sense
-at all, much less specify a well-defined process to be carried out by a
-computer. This will be addressed more carefully in section *Note
-1-2::. But first let's consider some other important points
-illustrated by the `sqrt' example.
-
- Observe that the problem of computing square roots breaks up
-naturally into a number of subproblems: how to tell whether a guess is
-good enough, how to improve a guess, and so on. Each of these tasks is
-accomplished by a separate procedure. The entire `sqrt' program can be
-viewed as a cluster of procedures (shown in *Note Figure 1-2::) that
-mirrors the decomposition of the problem into subproblems.
-
- *Figure 1.2:* Procedural decomposition of the `sqrt' program.
-
- sqrt
- |
- sqrt-iter
- / \
- good-enough improve
- / \ |
- square abs average
-
- The importance of this decomposition strategy is not simply that one
-is dividing the program into parts. After all, we could take any large
-program and divide it into parts--the first ten lines, the next ten
-lines, the next ten lines, and so on. Rather, it is crucial that each
-procedure accomplishes an identifiable task that can be used as a
-module in defining other procedures. For example, when we define the
-`good-enough?' procedure in terms of `square', we are able to regard
-the `square' procedure as a "black box." We are not at that moment
-concerned with _how_ the procedure computes its result, only with the
-fact that it computes the square. The details of how the square is
-computed can be suppressed, to be considered at a later time. Indeed,
-as far as the `good-enough?' procedure is concerned, `square' is not
-quite a procedure but rather an abstraction of a procedure, a so-called "procedural
-abstraction". At this level of abstraction, any procedure that
-computes the square is equally good.
-
- Thus, considering only the values they return, the following two
-procedures for squaring a number should be indistinguishable. Each
-takes a numerical argument and produces the square of that number as
-the value.(1)
-
- (define (square x) (* x x))
-
- (define (square x)
- (exp (double (log x))))
-
- (define (double x) (+ x x))
-
- So a procedure definition should be able to suppress detail. The
-users of the procedure may not have written the procedure themselves,
-but may have obtained it from another programmer as a black box. A
-user should not need to know how the procedure is implemented in order
-to use it.
-
-Local names
-...........
-
-One detail of a procedure's implementation that should not matter to
-the user of the procedure is the implementer's choice of names for the
-procedure's formal parameters. Thus, the following procedures should
-not be distinguishable:
-
- (define (square x) (* x x))
-
- (define (square y) (* y y))
-
- This principle--that the meaning of a procedure should be
-independent of the parameter names used by its author--seems on the
-surface to be self-evident, but its consequences are profound. The
-simplest consequence is that the parameter names of a procedure must be
-local to the body of the procedure. For example, we used `square' in
-the definition of `good-enough?' in our square-root procedure:
-
- (define (good-enough? guess x)
- (< (abs (- (square guess) x)) 0.001))
-
- The intention of the author of `good-enough?' is to determine if the
-square of the first argument is within a given tolerance of the second
-argument. We see that the author of `good-enough?' used the name
-`guess' to refer to the first argument and `x' to refer to the second
-argument. The argument of `square' is `guess'. If the author of
-`square' used `x' (as above) to refer to that argument, we see that the
-`x' in `good-enough?' must be a different `x' than the one in `square'.
-Running the procedure `square' must not affect the value of `x' that
-is used by `good-enough?', because that value of `x' may be needed by
-`good-enough?' after `square' is done computing.
-
- If the parameters were not local to the bodies of their respective
-procedures, then the parameter `x' in `square' could be confused with
-the parameter `x' in `good-enough?', and the behavior of `good-enough?'
-would depend upon which version of `square' we used. Thus, `square'
-would not be the black box we desired.
-
- A formal parameter of a procedure has a very special role in the
-procedure definition, in that it doesn't matter what name the formal
-parameter has. Such a name is called a "bound variable", and we say
-that the procedure definition "binds" its formal parameters. The
-meaning of a procedure definition is unchanged if a bound variable is
-consistently renamed throughout the definition.(2) If a variable is
-not bound, we say that it is "free". The set of expressions for which
-a binding defines a name is called the "scope" of that name. In a
-procedure definition, the bound variables declared as the formal
-parameters of the procedure have the body of the procedure as their
-scope.
-
- In the definition of `good-enough?' above, `guess' and `x' are bound
-variables but `<', `-', `abs', and `square' are free. The meaning of
-`good-enough?' should be independent of the names we choose for `guess'
-and `x' so long as they are distinct and different from `<', `-',
-`abs', and `square'. (If we renamed `guess' to `abs' we would have
-introduced a bug by "capturing" the variable `abs'. It would have
-changed from free to bound.) The meaning of `good-enough?' is not
-independent of the names of its free variables, however. It surely
-depends upon the fact (external to this definition) that the symbol
-`abs' names a procedure for computing the absolute value of a number.
-`Good-enough?' will compute a different function if we substitute `cos'
-for `abs' in its definition.
-
-Internal definitions and block structure
-........................................
-
-We have one kind of name isolation available to us so far: The formal
-parameters of a procedure are local to the body of the procedure. The
-square-root program illustrates another way in which we would like to
-control the use of names. The existing program consists of separate
-procedures:
-
- (define (sqrt x)
- (sqrt-iter 1.0 x))
-
- (define (sqrt-iter guess x)
- (if (good-enough? guess x)
- guess
- (sqrt-iter (improve guess x) x)))
-
- (define (good-enough? guess x)
- (< (abs (- (square guess) x)) 0.001))
-
- (define (improve guess x)
- (average guess (/ x guess)))
-
- The problem with this program is that the only procedure that is
-important to users of `sqrt' is `sqrt'. The other procedures
-(`sqrt-iter', `good-enough?', and `improve') only clutter up their
-minds. They may not define any other procedure called `good-enough?'
-as part of another program to work together with the square-root
-program, because `sqrt' needs it. The problem is especially severe in
-the construction of large systems by many separate programmers. For
-example, in the construction of a large library of numerical
-procedures, many numerical functions are computed as successive
-approximations and thus might have procedures named `good-enough?' and
-`improve' as auxiliary procedures. We would like to localize the
-subprocedures, hiding them inside `sqrt' so that `sqrt' could coexist
-with other successive approximations, each having its own private
-`good-enough?' procedure. To make this possible, we allow a procedure
-to have internal definitions that are local to that procedure. For
-example, in the square-root problem we can write
-
- (define (sqrt x)
- (define (good-enough? guess x)
- (< (abs (- (square guess) x)) 0.001))
- (define (improve guess x)
- (average guess (/ x guess)))
- (define (sqrt-iter guess x)
- (if (good-enough? guess x)
- guess
- (sqrt-iter (improve guess x) x)))
- (sqrt-iter 1.0 x))
-
- Such nesting of definitions, called "block structure", is basically
-the right solution to the simplest name-packaging problem. But there
-is a better idea lurking here. In addition to internalizing the
-definitions of the auxiliary procedures, we can simplify them. Since
-`x' is bound in the definition of `sqrt', the procedures
-`good-enough?', `improve', and `sqrt-iter', which are defined
-internally to `sqrt', are in the scope of `x'. Thus, it is not
-necessary to pass `x' explicitly to each of these procedures. Instead,
-we allow `x' to be a free variable in the internal definitions, as
-shown below. Then `x' gets its value from the argument with which the
-enclosing procedure `sqrt' is called. This discipline is called "lexical
-scoping".(3)
-
- (define (sqrt x)
- (define (good-enough? guess)
- (< (abs (- (square guess) x)) 0.001))
- (define (improve guess)
- (average guess (/ x guess)))
- (define (sqrt-iter guess)
- (if (good-enough? guess)
- guess
- (sqrt-iter (improve guess))))
- (sqrt-iter 1.0))
-
- We will use block structure extensively to help us break up large
-programs into tractable pieces.(4) The idea of block structure
-originated with the programming language Algol 60. It appears in most
-advanced programming languages and is an important tool for helping to
-organize the construction of large programs.
-
- ---------- Footnotes ----------
-
- (1) It is not even clear which of these procedures is a more
-efficient implementation. This depends upon the hardware available.
-There are machines for which the "obvious" implementation is the less
-efficient one. Consider a machine that has extensive tables of
-logarithms and antilogarithms stored in a very efficient manner.
-
- (2) The concept of consistent renaming is actually subtle and
-difficult to define formally. Famous logicians have made embarrassing
-errors here.
-
- (3) [Footnote 28] Lexical scoping dictates that free variables in a
-procedure are taken to refer to bindings made by enclosing procedure
-definitions; that is, they are looked up in the environment in which the
-procedure was defined. We will see how this works in detail in chapter
-3 when we study environments and the detailed behavior of the
-interpreter.
-
- (4) Embedded definitions must come first in a procedure body. The
-management is not responsible for the consequences of running programs
-that intertwine definition and use.
-
-\1f
-File: sicp.info, Node: 1-2, Next: 1-3, Prev: 1-1, Up: Chapter 1
-
-1.2 Procedures and the Processes They Generate
-==============================================
-
-We have now considered the elements of programming: We have used
-primitive arithmetic operations, we have combined these operations, and
-we have abstracted these composite operations by defining them as
-compound procedures. But that is not enough to enable us to say that
-we know how to program. Our situation is analogous to that of someone
-who has learned the rules for how the pieces move in chess but knows
-nothing of typical openings, tactics, or strategy. Like the novice
-chess player, we don't yet know the common patterns of usage in the
-domain. We lack the knowledge of which moves are worth making (which
-procedures are worth defining). We lack the experience to predict the
-consequences of making a move (executing a procedure).
-
- The ability to visualize the consequences of the actions under
-consideration is crucial to becoming an expert programmer, just as it
-is in any synthetic, creative activity. In becoming an expert
-photographer, for example, one must learn how to look at a scene and
-know how dark each region will appear on a print for each possible
-choice of exposure and development conditions. Only then can one
-reason backward, planning framing, lighting, exposure, and development
-to obtain the desired effects. So it is with programming, where we are
-planning the course of action to be taken by a process and where we
-control the process by means of a program. To become experts, we must
-learn to visualize the processes generated by various types of
-procedures. Only after we have developed such a skill can we learn to
-reliably construct programs that exhibit the desired behavior.
-
- A procedure is a pattern for the "local evolution" of a computational
-process. It specifies how each stage of the process is built upon the
-previous stage. We would like to be able to make statements about the
-overall, or "global", behavior of a process whose local evolution has
-been specified by a procedure. This is very difficult to do in
-general, but we can at least try to describe some typical patterns of
-process evolution.
-
- In this section we will examine some common "shapes" for processes
-generated by simple procedures. We will also investigate the rates at
-which these processes consume the important computational resources of
-time and space. The procedures we will consider are very simple.
-Their role is like that played by test patterns in photography: as
-oversimplified prototypical patterns, rather than practical examples in
-their own right.
-
-* Menu:
-
-* 1-2-1:: Linear Recursion and Iteration
-* 1-2-2:: Tree Recursion
-* 1-2-3:: Orders of Growth
-* 1-2-4:: Exponentiation
-* 1-2-5:: Greatest Common Divisors
-* 1-2-6:: Example: Testing for Primality
-
-\1f
-File: sicp.info, Node: 1-2-1, Next: 1-2-2, Prev: 1-2, Up: 1-2
-
-1.2.1 Linear Recursion and Iteration
-------------------------------------
-
- *Figure 1.3:* A linear recursive process for computing 6!.
-
- (factorial 6) ------------------------.
- (* 6 (factorial 5)) |
- (* 6 (* 5 (factorial 4))) |
- (* 6 (* 5 (* 4 (factorial 3)))) |
- (* 6 (* 5 (* 4 (* 3 (factorial 2))))) |
- (* 6 (* 5 (* 4 (* 3 (* 2 (factorial 1)))))) |
- (* 6 (* 5 (* 4 (* 3 (* 2 1))))) |
- (* 6 (* 5 (* 4 (* 3 2)))) |
- (* 6 (* 5 (* 4 6))) |
- (* 6 (* 5 24)) |
- (* 6 120) |
- 720 <-------------------------------'
-
-We begin by considering the factorial function, defined by
-
- n! = n * (n - 1) * (n - 2) ... 3 * 2 * 1
-
- There are many ways to compute factorials. One way is to make use
-of the observation that n! is equal to n times (n - 1)! for any positive
-integer n:
-
- n! = n * [(n - 1) * (n - 2) ... 3 * 2 * 1] = n * (n - 1)!
-
- Thus, we can compute n! by computing (n - 1)! and multiplying the
-result by n. If we add the stipulation that 1! is equal to 1, this
-observation translates directly into a procedure:
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* n (factorial (- n 1)))))
-
- We can use the substitution model of section *Note 1-1-5:: to watch
-this procedure in action computing 6!, as shown in *Note Figure 1-3::.
-
- Now let's take a different perspective on computing factorials. We
-could describe a rule for computing n! by specifying that we first
-multiply 1 by 2, then multiply the result by 3, then by 4, and so on
-until we reach n. More formally, we maintain a running product,
-together with a counter that counts from 1 up to n. We can describe
-the computation by saying that the counter and the product
-simultaneously change from one step to the next according to the rule
-
- product <- counter ... product
-
- counter <- counter + 1
-
-and stipulating that n! is the value of the product when the counter
-exceeds n.
-
- *Figure 1.4:* A linear iterative process for computing 6!.
-
- (factorial 6) -----.
- (fact-iter 1 1 6) |
- (fact-iter 1 2 6) |
- (fact-iter 2 3 6) |
- (fact-iter 6 4 6) |
- (fact-iter 24 5 6) |
- (fact-iter 120 6 6) |
- (fact-iter 720 7 6) V
- 720
-
- Once again, we can recast our description as a procedure for
-computing factorials:(1)
-
- (define (factorial n)
- (fact-iter 1 1 n))
-
- (define (fact-iter product counter max-count)
- (if (> counter max-count)
- product
- (fact-iter (* counter product)
- (+ counter 1)
- max-count)))
-
- As before, we can use the substitution model to visualize the
-process of computing 6!, as shown in *Note Figure 1-4::.
-
- Compare the two processes. From one point of view, they seem hardly
-different at all. Both compute the same mathematical function on the
-same domain, and each requires a number of steps proportional to n to
-compute n!. Indeed, both processes even carry out the same sequence of
-multiplications, obtaining the same sequence of partial products. On
-the other hand, when we consider the "shapes" of the two processes, we
-find that they evolve quite differently.
-
- Consider the first process. The substitution model reveals a shape
-of expansion followed by contraction, indicated by the arrow in *Note
-Figure 1-3::. The expansion occurs as the process builds up a chain of operations
-"deferred operations" (in this case, a chain of multiplications). The
-contraction occurs as the operations are actually performed. This type
-of process, characterized by a chain of deferred operations, is called
-a "recursive process". Carrying out this process requires that the
-interpreter keep track of the operations to be performed later on. In
-the computation of n!, the length of the chain of deferred
-multiplications, and hence the amount of information needed to keep
-track of it, grows linearly with n (is proportional to n), just like
-the number of steps. Such a process is called a "linear recursive
-process".
-
- By contrast, the second process does not grow and shrink. At each
-step, all we need to keep track of, for any n, are the current values
-of the variables `product', `counter', and `max-count'. We call this an "iterative
-process". In general, an iterative process is one whose state can be
-summarized by a fixed number of "state variables", together with a
-fixed rule that describes how the state variables should be updated as
-the process moves from state to state and an (optional) end test that
-specifies conditions under which the process should terminate. In
-computing n!, the number of steps required grows linearly with n. Such
-a process is called a "linear iterative process".
-
- The contrast between the two processes can be seen in another way.
-In the iterative case, the program variables provide a complete
-description of the state of the process at any point. If we stopped
-the computation between steps, all we would need to do to resume the
-computation is to supply the interpreter with the values of the three
-program variables. Not so with the recursive process. In this case
-there is some additional "hidden" information, maintained by the
-interpreter and not contained in the program variables, which indicates
-"where the process is" in negotiating the chain of deferred operations.
-The longer the chain, the more information must be maintained.(2)
-
- In contrasting iteration and recursion, we must be careful not to
-confuse the notion of a recursive "process" with the notion of a
-recursive "procedure". When we describe a procedure as recursive, we
-are referring to the syntactic fact that the procedure definition
-refers (either directly or indirectly) to the procedure itself. But
-when we describe a process as following a pattern that is, say,
-linearly recursive, we are speaking about how the process evolves, not
-about the syntax of how a procedure is written. It may seem disturbing
-that we refer to a recursive procedure such as `fact-iter' as
-generating an iterative process. However, the process really is
-iterative: Its state is captured completely by its three state
-variables, and an interpreter need keep track of only three variables
-in order to execute the process.
-
- One reason that the distinction between process and procedure may be
-confusing is that most implementations of common languages (including
-Ada, Pascal, and C) are designed in such a way that the interpretation
-of any recursive procedure consumes an amount of memory that grows with
-the number of procedure calls, even when the process described is, in
-principle, iterative. As a consequence, these languages can describe
-iterative processes only by resorting to special-purpose "looping
-constructs" such as `do', `repeat', `until', `for', and `while'. The
-implementation of Scheme we shall consider in *Note Chapter 5:: does
-not share this defect. It will execute an iterative process in
-constant space, even if the iterative process is described by a
-recursive procedure. An implementation with this property is called "tail-recursive".
-With a tail-recursive implementation, iteration can be expressed using
-the ordinary procedure call mechanism, so that special iteration
-constructs are useful only as syntactic sugar.(3)
-
- *Exercise 1.9:* Each of the following two procedures defines a
- method for adding two positive integers in terms of the procedures
- `inc', which increments its argument by 1, and `dec', which
- decrements its argument by 1.
-
- (define (+ a b)
- (if (= a 0)
- b
- (inc (+ (dec a) b))))
-
- (define (+ a b)
- (if (= a 0)
- b
- (+ (dec a) (inc b))))
-
- Using the substitution model, illustrate the process generated by
- each procedure in evaluating `(+ 4 5)'. Are these processes
- iterative or recursive?
-
- *Exercise 1.10:* The following procedure computes a mathematical
- function called Ackermann's function.
-
- (define (A x y)
- (cond ((= y 0) 0)
- ((= x 0) (* 2 y))
- ((= y 1) 2)
- (else (A (- x 1)
- (A x (- y 1))))))
-
- What are the values of the following expressions?
-
- (A 1 10)
-
- (A 2 4)
-
- (A 3 3)
-
- Consider the following procedures, where `A' is the procedure
- defined above:
-
- (define (f n) (A 0 n))
-
- (define (g n) (A 1 n))
-
- (define (h n) (A 2 n))
-
- (define (k n) (* 5 n n))
-
- Give concise mathematical definitions for the functions computed
- by the procedures `f', `g', and `h' for positive integer values of
- n. For example, `(k n)' computes 5n^2.
-
- ---------- Footnotes ----------
-
- (1) In a real program we would probably use the block structure
-introduced in the last section to hide the definition of `fact-iter':
-
- (define (factorial n)
- (define (iter product counter)
- (if (> counter n)
- product
- (iter (* counter product)
- (+ counter 1))))
- (iter 1 1))
-
- We avoided doing this here so as to minimize the number of things to
-think about at once.
-
- (2) When we discuss the implementation of procedures on register
-machines in *Note Chapter 5::, we will see that any iterative process
-can be realized "in hardware" as a machine that has a fixed set of
-registers and no auxiliary memory. In contrast, realizing a recursive
-process requires a machine that uses an auxiliary data structure known
-as a "stack".
-
- (3) Tail recursion has long been known as a compiler optimization
-trick. A coherent semantic basis for tail recursion was provided by
-Carl Hewitt (1977), who explained it in terms of the "message-passing"
-model of computation that we shall discuss in *Note Chapter 3::.
-Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see
-Steele 1975) constructed a tail-recursive interpreter for Scheme.
-Steele later showed how tail recursion is a consequence of the natural
-way to compile procedure calls (Steele 1977). The IEEE standard for
-Scheme requires that Scheme implementations be tail-recursive.
-
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-
-1.2.2 Tree Recursion
---------------------
-
-Another common pattern of computation is called "tree recursion". As
-an example, consider computing the sequence of Fibonacci numbers, in
-which each number is the sum of the preceding two:
-
- 0, 1, 1, 2, 3, 4, 8, 13, 21, ...
-
- In general, the Fibonacci numbers can be defined by the rule
-
- /
- | 0 if n = 0
- Fib(n) = < 1 if n = 1
- | Fib(n - 1) + Fib(n - 2) otherwise
- \
-
- We can immediately translate this definition into a recursive
-procedure for computing Fibonacci numbers:
-
- (define (fib n)
- (cond ((= n 0) 0)
- ((= n 1) 1)
- (else (+ (fib (- n 1))
- (fib (- n 2))))))
-
- *Figure 1.5:* The tree-recursive process generated in computing
- `(fib 5)'.
-
- ..<............ fib5 <..........
- ... ___________/ \___________ .
- ... / . ..... \ .
- .. fib4 . . . . . fib3 .
- .. ____/. \____ .. . __/ \__ .
- .. / . . .. \ . .. / . . \ .
- .. fib3 . . fib2 . . fib2 . . fib1 .
- .. / . \ . . / \ . . / \ ... . | .
- .. / . . \ . . / . \ . . / . \ . . 1 .
- . fib2 . . fib1. .fib1 . fib0 . .fib1. . fib0 . . .
- . / \ . . | . . | . . | . . | . . | . .>
- V / . \ . 1 . . 1 . . 0 . . 1 . . 0 ..
- . fib1 .. fib0.. . . . . . V . .. .
- . | . . | . .> .>. . . ..>. .>
- . 1 . . 0 .
- . . . .
- .>. ..
-
- Consider the pattern of this computation. To compute `(fib 5)', we
-compute `(fib 4)' and `(fib 3)'. To compute `(fib 4)', we compute
-`(fib 3)' and `(fib 2)'. In general, the evolved process looks like a
-tree, as shown in *Note Figure 1-5::. Notice that the branches split
-into two at each level (except at the bottom); this reflects the fact
-that the `fib' procedure calls itself twice each time it is invoked.
-
- This procedure is instructive as a prototypical tree recursion, but
-it is a terrible way to compute Fibonacci numbers because it does so
-much redundant computation. Notice in *Note Figure 1-5:: that the
-entire computation of `(fib 3)'--almost half the work--is duplicated.
-In fact, it is not hard to show that the number of times the procedure
-will compute `(fib 1)' or `(fib 0)' (the number of leaves in the above
-tree, in general) is precisely _Fib_(n + 1). To get an idea of how bad
-this is, one can show that the value of _Fib_(n) grows exponentially
-with n. More precisely (see *Note Exercise 1-13::), _Fib_(n) is the
-closest integer to [phi]^n /[sqrt](5), where
-
- [phi] = (1 + [sqrt]5)/2 ~= 1.6180
-
-is the "golden ratio", which satisfies the equation
-
- [phi]^2 = [phi] + 1
-
- Thus, the process uses a number of steps that grows exponentially
-with the input. On the other hand, the space required grows only
-linearly with the input, because we need keep track only of which nodes
-are above us in the tree at any point in the computation. In general,
-the number of steps required by a tree-recursive process will be
-proportional to the number of nodes in the tree, while the space
-required will be proportional to the maximum depth of the tree.
-
- We can also formulate an iterative process for computing the
-Fibonacci numbers. The idea is to use a pair of integers a and b,
-initialized to _Fib_(1) = 1 and _Fib_(0) = 0, and to repeatedly apply
-the simultaneous transformations
-
- a <- a + b
- b <- a
-
-It is not hard to show that, after applying this transformation n times,
-a and b will be equal, respectively, to _Fib_(n + 1) and _Fib_(n).
-Thus, we can compute Fibonacci numbers iteratively using the procedure
-
- (define (fib n)
- (fib-iter 1 0 n))
-
- (define (fib-iter a b count)
- (if (= count 0)
- b
- (fib-iter (+ a b) a (- count 1))))
-
- This second method for computing _Fib_(n) is a linear iteration. The
-difference in number of steps required by the two methods--one linear in
-n, one growing as fast as _Fib_(n) itself--is enormous, even for small
-inputs.
-
- One should not conclude from this that tree-recursive processes are
-useless. When we consider processes that operate on hierarchically
-structured data rather than numbers, we will find that tree recursion
-is a natural and powerful tool.(1) But even in numerical operations,
-tree-recursive processes can be useful in helping us to understand and
-design programs. For instance, although the first `fib' procedure is
-much less efficient than the second one, it is more straightforward,
-being little more than a translation into Lisp of the definition of the
-Fibonacci sequence. To formulate the iterative algorithm required
-noticing that the computation could be recast as an iteration with
-three state variables.
-
-Example: Counting change
-........................
-
-It takes only a bit of cleverness to come up with the iterative
-Fibonacci algorithm. In contrast, consider the following problem: How
-many different ways can we make change of $ 1.00, given half-dollars,
-quarters, dimes, nickels, and pennies? More generally, can we write a
-procedure to compute the number of ways to change any given amount of
-money?
-
- This problem has a simple solution as a recursive procedure.
-Suppose we think of the types of coins available as arranged in some
-order. Then the following relation holds:
-
- The number of ways to change amount a using n kinds of coins equals
-
- * the number of ways to change amount a using all but the first kind
- of coin, plus
-
- * the number of ways to change amount a - d using all n kinds of
- coins, where d is the denomination of the first kind of coin.
-
-
- To see why this is true, observe that the ways to make change can be
-divided into two groups: those that do not use any of the first kind of
-coin, and those that do. Therefore, the total number of ways to make
-change for some amount is equal to the number of ways to make change
-for the amount without using any of the first kind of coin, plus the
-number of ways to make change assuming that we do use the first kind of
-coin. But the latter number is equal to the number of ways to make
-change for the amount that remains after using a coin of the first kind.
-
- Thus, we can recursively reduce the problem of changing a given
-amount to the problem of changing smaller amounts using fewer kinds of
-coins. Consider this reduction rule carefully, and convince yourself
-that we can use it to describe an algorithm if we specify the following
-degenerate cases:(2)
-
- * If a is exactly 0, we should count that as 1 way to make change.
-
- * If a is less than 0, we should count that as 0 ways to make change.
-
- * If n is 0, we should count that as 0 ways to make change.
-
-
- We can easily translate this description into a recursive procedure:
-
- (define (count-change amount)
- (cc amount 5))
-
- (define (cc amount kinds-of-coins)
- (cond ((= amount 0) 1)
- ((or (< amount 0) (= kinds-of-coins 0)) 0)
- (else (+ (cc amount
- (- kinds-of-coins 1))
- (cc (- amount
- (first-denomination kinds-of-coins))
- kinds-of-coins)))))
-
- (define (first-denomination kinds-of-coins)
- (cond ((= kinds-of-coins 1) 1)
- ((= kinds-of-coins 2) 5)
- ((= kinds-of-coins 3) 10)
- ((= kinds-of-coins 4) 25)
- ((= kinds-of-coins 5) 50)))
-
- (The `first-denomination' procedure takes as input the number of
-kinds of coins available and returns the denomination of the first
-kind. Here we are thinking of the coins as arranged in order from
-largest to smallest, but any order would do as well.) We can now
-answer our original question about changing a dollar:
-
- (count-change 100)
- 292
-
- `Count-change' generates a tree-recursive process with redundancies
-similar to those in our first implementation of `fib'. (It will take
-quite a while for that 292 to be computed.) On the other hand, it is
-not obvious how to design a better algorithm for computing the result,
-and we leave this problem as a challenge. The observation that a
-tree-recursive process may be highly inefficient but often easy to
-specify and understand has led people to propose that one could get the
-best of both worlds by designing a "smart compiler" that could
-transform tree-recursive procedures into more efficient procedures that
-compute the same result.(3)
-
- *Exercise 1.11:* A function f is defined by the rule that f(n) = n
- if n<3 and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n>= 3.
- Write a procedure that computes f by means of a recursive process.
- Write a procedure that computes f by means of an iterative
- process.
-
- *Exercise 1.12:* The following pattern of numbers is called "Pascal's
- triangle".
-
- 1
- 1 1
- 1 2 1
- 1 3 3 1
- 1 4 6 4 1
-
- The numbers at the edge of the triangle are all 1, and each number
- inside the triangle is the sum of the two numbers above it.(4)
- Write a procedure that computes elements of Pascal's triangle by
- means of a recursive process.
-
- *Exercise 1.13:* Prove that _Fib_(n) is the closest integer to
- [phi]^n/[sqrt](5), where [phi] = (1 + [sqrt](5))/2. Hint: Let
- [illegiblesymbol] = (1 - [sqrt](5))/2. Use induction and the
- definition of the Fibonacci numbers (see section *Note 1-2-2::) to
- prove that _Fib_(n) = ([phi]^n - [illegiblesymbol]^n)/[sqrt](5).
-
- ---------- Footnotes ----------
-
- (1) An example of this was hinted at in section *Note 1-1-3::: The
-interpreter itself evaluates expressions using a tree-recursive process.
-
- (2) For example, work through in detail how the reduction rule
-applies to the problem of making change for 10 cents using pennies and
-nickels.
-
- (3) One approach to coping with redundant computations is to arrange
-matters so that we automatically construct a table of values as they
-are computed. Each time we are asked to apply the procedure to some
-argument, we first look to see if the value is already stored in the
-table, in which case we avoid performing the redundant computation.
-This strategy, known as "tabulation" or "memoization", can be
-implemented in a straightforward way. Tabulation can sometimes be used
-to transform processes that require an exponential number of steps
-(such as `count-change') into processes whose space and time
-requirements grow linearly with the input. See *Note Exercise 3-27::.
-
- (4) The elements of Pascal's triangle are called the "binomial
-coefficients", because the nth row consists of the coefficients of the
-terms in the expansion of (x + y)^n. This pattern for computing the
-coefficients appeared in Blaise Pascal's 1653 seminal work on
-probability theory, `Traite' du triangle arithme'tique'. According to
-Knuth (1973), the same pattern appears in the `Szu-yuen Yu"-chien'
-("The Precious Mirror of the Four Elements"), published by the Chinese
-mathematician Chu Shih-chieh in 1303, in the works of the
-twelfth-century Persian poet and mathematician Omar Khayyam, and in the
-works of the twelfth-century Hindu mathematician Bha'scara A'cha'rya.
-
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-
-1.2.3 Orders of Growth
-----------------------
-
-The previous examples illustrate that processes can differ considerably
-in the rates at which they consume computational resources. One
-convenient way to describe this difference is to use the notion of "order
-of growth" to obtain a gross measure of the resources required by a
-process as the inputs become larger.
-
- Let n be a parameter that measures the size of the problem, and let
-R(n) be the amount of resources the process requires for a problem of
-size n. In our previous examples we took n to be the number for which
-a given function is to be computed, but there are other possibilities.
-For instance, if our goal is to compute an approximation to the square
-root of a number, we might take n to be the number of digits accuracy
-required. For matrix multiplication we might take n to be the number
-of rows in the matrices. In general there are a number of properties
-of the problem with respect to which it will be desirable to analyze a
-given process. Similarly, R(n) might measure the number of internal
-storage registers used, the number of elementary machine operations
-performed, and so on. In computers that do only a fixed number of
-operations at a time, the time required will be proportional to the
-number of elementary machine operations performed.
-
- We say that R(n) has order of growth [theta](f(n)), written R(n) =
-[theta](f(n)) (pronounced "theta of f(n)"), if there are positive
-constants k_1 and k_2 independent of n such that
-
- k_1 f(n) <= R(n) <= k_2 f(n)
-
-for any sufficiently large value of n. (In other words, for large n,
-the value R(n) is sandwiched between k_1f(n) and k_2f(n).)
-
- For instance, with the linear recursive process for computing
-factorial described in section *Note 1-2-1:: the number of steps grows
-proportionally to the input n. Thus, the steps required for this
-process grows as [theta](n). We also saw that the space required grows
-as [theta](n). For the iterative factorial, the number of steps is
-still [theta](n) but the space is [theta](1)--that is, constant.(1) The
-tree-recursive Fibonacci computation requires [theta]([phi]^n) steps
-and space [theta](n), where [phi] is the golden ratio described in
-section *Note 1-2-2::.
-
- Orders of growth provide only a crude description of the behavior of
-a process. For example, a process requiring n^2 steps and a process
-requiring 1000n^2 steps and a process requiring 3n^2 + 10n + 17 steps
-all have [theta](n^2) order of growth. On the other hand, order of
-growth provides a useful indication of how we may expect the behavior
-of the process to change as we change the size of the problem. For a
-[theta](n) (linear) process, doubling the size will roughly double the
-amount of resources used. For an exponential process, each increment
-in problem size will multiply the resource utilization by a constant
-factor. In the remainder of section *Note 1-2:: we will examine two
-algorithms whose order of growth is logarithmic, so that doubling the
-problem size increases the resource requirement by a constant amount.
-
- *Exercise 1.14:* Draw the tree illustrating the process generated
- by the `count-change' procedure of section *Note 1-2-2:: in making
- change for 11 cents. What are the orders of growth of the space
- and number of steps used by this process as the amount to be
- changed increases?
-
- *Exercise 1.15:* The sine of an angle (specified in radians) can
- be computed by making use of the approximation `sin' xapprox x if
- x is sufficiently small, and the trigonometric identity
-
- x x
- sin x = 3 sin --- - 4 sin^3 ---
- 3 3
-
- to reduce the size of the argument of `sin'. (For purposes of this
- exercise an angle is considered "sufficiently small" if its
- magnitude is not greater than 0.1 radians.) These ideas are
- incorporated in the following procedures:
-
- (define (cube x) (* x x x))
-
- (define (p x) (- (* 3 x) (* 4 (cube x))))
-
- (define (sine angle)
- (if (not (> (abs angle) 0.1))
- angle
- (p (sine (/ angle 3.0)))))
-
- a. How many times is the procedure `p' applied when `(sine
- 12.15)' is evaluated?
-
- b. What is the order of growth in space and number of steps (as
- a function of a) used by the process generated by the `sine'
- procedure when `(sine a)' is evaluated?
-
-
- ---------- Footnotes ----------
-
- (1) These statements mask a great deal of oversimplification. For
-instance, if we count process steps as "machine operations" we are
-making the assumption that the number of machine operations needed to
-perform, say, a multiplication is independent of the size of the
-numbers to be multiplied, which is false if the numbers are
-sufficiently large. Similar remarks hold for the estimates of space.
-Like the design and description of a process, the analysis of a process
-can be carried out at various levels of abstraction.
-
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-
-1.2.4 Exponentiation
---------------------
-
-Consider the problem of computing the exponential of a given number.
-We would like a procedure that takes as arguments a base b and a
-positive integer exponent n and computes b^n. One way to do this is
-via the recursive definition
-
- b^n = b * b^(n - 1)
- b^0 = 1
-
-which translates readily into the procedure
-
- (define (expt b n)
- (if (= n 0)
- 1
- (* b (expt b (- n 1)))))
-
- This is a linear recursive process, which requires [theta](n) steps
-and [theta](n) space. Just as with factorial, we can readily formulate
-an equivalent linear iteration:
-
- (define (expt b n)
- (expt-iter b n 1))
-
- (define (expt-iter b counter product)
- (if (= counter 0)
- product
- (expt-iter b
- (- counter 1)
- (* b product))))
-
- This version requires [theta](n) steps and [theta](1) space.
-
- We can compute exponentials in fewer steps by using successive
-squaring. For instance, rather than computing b^8 as
-
- b * (b * (b * (b * (b * (b * (b * b))))))
-
-we can compute it using three multiplications:
-
- b^2 = b * b
- b^4 = b^2 * b^2
- b^8 = b^4 * b^4
-
- This method works fine for exponents that are powers of 2. We can
-also take advantage of successive squaring in computing exponentials in
-general if we use the rule
-
- b^n = (b^(b/2))^2 if n is even
- b^n = b * b^(n - 1) if n is odd
-
- We can express this method as a procedure:
-
- (define (fast-expt b n)
- (cond ((= n 0) 1)
- ((even? n) (square (fast-expt b (/ n 2))))
- (else (* b (fast-expt b (- n 1))))))
-
-where the predicate to test whether an integer is even is defined in
-terms of the primitive procedure `remainder' by
-
- (define (even? n)
- (= (remainder n 2) 0))
-
- The process evolved by `fast-expt' grows logarithmically with n in
-both space and number of steps. To see this, observe that computing
-b^(2n) using `fast-expt' requires only one more multiplication than
-computing b^n. The size of the exponent we can compute therefore
-doubles (approximately) with every new multiplication we are allowed.
-Thus, the number of multiplications required for an exponent of n grows
-about as fast as the logarithm of n to the base 2. The process has
-[theta](`log' n) growth.(1)
-
- The difference between [theta](`log' n) growth and [theta](n) growth
-becomes striking as n becomes large. For example, `fast-expt' for n =
-1000 requires only 14 multiplications.(2) It is also possible to use
-the idea of successive squaring to devise an iterative algorithm that
-computes exponentials with a logarithmic number of steps (see *Note
-Exercise 1-16::), although, as is often the case with iterative
-algorithms, this is not written down so straightforwardly as the
-recursive algorithm.(3)
-
- *Exercise 1.16:* Design a procedure that evolves an iterative
- exponentiation process that uses successive squaring and uses a
- logarithmic number of steps, as does `fast-expt'. (Hint: Using the
- observation that (b^(n/2))^2 = (b^2)^(n/2), keep, along with the
- exponent n and the base b, an additional state variable a, and
- define the state transformation in such a way that the product a
- b^n is unchanged from state to state. At the beginning of the
- process a is taken to be 1, and the answer is given by the value
- of a at the end of the process. In general, the technique of
- defining an "invariant quantity" that remains unchanged from state
- to state is a powerful way to think about the design of iterative
- algorithms.)
-
- *Exercise 1.17:* The exponentiation algorithms in this section are
- based on performing exponentiation by means of repeated
- multiplication. In a similar way, one can perform integer
- multiplication by means of repeated addition. The following
- multiplication procedure (in which it is assumed that our language
- can only add, not multiply) is analogous to the `expt' procedure:
-
- (define (* a b)
- (if (= b 0)
- 0
- (+ a (* a (- b 1)))))
-
- This algorithm takes a number of steps that is linear in `b'. Now
- suppose we include, together with addition, operations `double',
- which doubles an integer, and `halve', which divides an (even)
- integer by 2. Using these, design a multiplication procedure
- analogous to `fast-expt' that uses a logarithmic number of steps.
-
- *Exercise 1.18:* Using the results of *Note Exercise 1-16:: and
- *Note Exercise 1-17::, devise a procedure that generates an
- iterative process for multiplying two integers in terms of adding,
- doubling, and halving and uses a logarithmic number of steps.(4)
-
- *Exercise 1.19:* There is a clever algorithm for computing the
- Fibonacci numbers in a logarithmic number of steps. Recall the
- transformation of the state variables a and b in the `fib-iter'
- process of section *Note 1-2-2::: a <- a + b and b <- a. Call
- this transformation T, and observe that applying T over and over
- again n times, starting with 1 and 0, produces the pair _Fib_(n +
- 1) and _Fib_(n). In other words, the Fibonacci numbers are
- produced by applying T^n, the nth power of the transformation T,
- starting with the pair (1,0). Now consider T to be the special
- case of p = 0 and q = 1 in a family of transformations T_(pq),
- where T_(pq) transforms the pair (a,b) according to a <- bq + aq +
- ap and b <- bp + aq. Show that if we apply such a transformation
- T_(pq) twice, the effect is the same as using a single
- transformation T_(p'q') of the same form, and compute p' and q' in
- terms of p and q. This gives us an explicit way to square these
- transformations, and thus we can compute T^n using successive
- squaring, as in the `fast-expt' procedure. Put this all together
- to complete the following procedure, which runs in a logarithmic
- number of steps:(5)
-
- (define (fib n)
- (fib-iter 1 0 0 1 n))
-
- (define (fib-iter a b p q count)
- (cond ((= count 0) b)
- ((even? count)
- (fib-iter a
- b
- <??> ; compute p'
- <??> ; compute q'
- (/ count 2)))
- (else (fib-iter (+ (* b q) (* a q) (* a p))
- (+ (* b p) (* a q))
- p
- q
- (- count 1)))))
-
- ---------- Footnotes ----------
-
- (1) More precisely, the number of multiplications required is equal
-to 1 less than the log base 2 of n plus the number of ones in the
-binary representation of n. This total is always less than twice the
-log base 2 of n. The arbitrary constants k_1 and k_2 in the definition
-of order notation imply that, for a logarithmic process, the base to
-which logarithms are taken does not matter, so all such processes are
-described as [theta](`log' n).
-
- (2) You may wonder why anyone would care about raising numbers to
-the 1000th power. See section *Note 1-2-6::.
-
- (3) This iterative algorithm is ancient. It appears in the
-`Chandah-sutra' by A'cha'rya Pingala, written before 200 B.C. See Knuth
-1981, section 4.6.3, for a full discussion and analysis of this and
-other methods of exponentiation.
-
- (4) This algorithm, which is sometimes known as the "Russian peasant
-method" of multiplication, is ancient. Examples of its use are found
-in the Rhind Papyrus, one of the two oldest mathematical documents in
-existence, written about 1700 B.C. (and copied from an even older
-document) by an Egyptian scribe named A'h-mose.
-
- (5) This exercise was suggested to us by Joe Stoy, based on an
-example in Kaldewaij 1990.
-
-\1f
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-
-1.2.5 Greatest Common Divisors
-------------------------------
-
-The greatest common divisor (GCD) of two integers a and b is defined to
-be the largest integer that divides both a and b with no remainder.
-For example, the GCD of 16 and 28 is 4. In *Note Chapter 2::, when we
-investigate how to implement rational-number arithmetic, we will need
-to be able to compute GCDs in order to reduce rational numbers to
-lowest terms. (To reduce a rational number to lowest terms, we must
-divide both the numerator and the denominator by their GCD. For
-example, 16/28 reduces to 4/7.) One way to find the GCD of two
-integers is to factor them and search for common factors, but there is
-a famous algorithm that is much more efficient.
-
- The idea of the algorithm is based on the observation that, if r is
-the remainder when a is divided by b, then the common divisors of a and
-b are precisely the same as the common divisors of b and r. Thus, we
-can use the equation
-
- GCD(a,b) = GCD(b,r)
-
-to successively reduce the problem of computing a GCD to the problem of
-computing the GCD of smaller and smaller pairs of integers. For
-example,
-
- GCD(206,40) = GCD(40,6)
- = GCD(6,4)
- = GCD(4,2)
- = GCD(2,0)
- = 2
-
-reduces GCD(206,40) to GCD(2,0), which is 2. It is possible to show
-that starting with any two positive integers and performing repeated
-reductions will always eventually produce a pair where the second
-number is 0. Then the GCD is the other number in the pair. This
-method for computing the GCD is known as Algorithm "Euclid's
-Algorithm".(1)
-
- It is easy to express Euclid's Algorithm as a procedure:
-
- (define (gcd a b)
- (if (= b 0)
- a
- (gcd b (remainder a b))))
-
- This generates an iterative process, whose number of steps grows as
-the logarithm of the numbers involved.
-
- The fact that the number of steps required by Euclid's Algorithm has
-logarithmic growth bears an interesting relation to the Fibonacci
-numbers:
-
- *Lame''s Theorem:* If Euclid's Algorithm requires k steps to
- compute the GCD of some pair, then the smaller number in the pair
- must be greater than or equal to the kth Fibonacci number.(2)
-
- We can use this theorem to get an order-of-growth estimate for
-Euclid's Algorithm. Let n be the smaller of the two inputs to the
-procedure. If the process takes k steps, then we must have n>= _Fib_(k)
-approx [phi]^k/[sqrt](5). Therefore the number of steps k grows as the
-logarithm (to the base [phi]) of n. Hence, the order of growth is
-[theta](`log' n).
-
- *Exercise 1.20:* The process that a procedure generates is of
- course dependent on the rules used by the interpreter. As an
- example, consider the iterative `gcd' procedure given above.
- Suppose we were to interpret this procedure using normal-order
- evaluation, as discussed in section *Note 1-1-5::. (The
- normal-order-evaluation rule for `if' is described in *Note
- Exercise 1-5::.) Using the substitution method (for normal
- order), illustrate the process generated in evaluating `(gcd 206
- 40)' and indicate the `remainder' operations that are actually
- performed. How many `remainder' operations are actually performed
- in the normal-order evaluation of `(gcd 206 40)'? In the
- applicative-order evaluation?
-
- ---------- Footnotes ----------
-
- (1) Euclid's Algorithm is so called because it appears in Euclid's
-`Elements' (Book 7, ca. 300 B.C.). According to Knuth (1973), it can
-be considered the oldest known nontrivial algorithm. The ancient
-Egyptian method of multiplication (*Note Exercise 1-18::) is surely
-older, but, as Knuth explains, Euclid's algorithm is the oldest known
-to have been presented as a general algorithm, rather than as a set of
-illustrative examples.
-
- (2) This theorem was proved in 1845 by Gabriel Lame', a French
-mathematician and engineer known chiefly for his contributions to
-mathematical physics. To prove the theorem, we consider pairs (a_k
-,b_k), where a_k>= b_k, for which Euclid's Algorithm terminates in k
-steps. The proof is based on the claim that, if (a_(k+1), b_(k+1)) ->
-(a_k, b_k) -> (a_(k-1), b_(k-1)) are three successive pairs in the
-reduction process, then we must have b_(k+1)>= b_k + b_(k-1). To
-verify the claim, consider that a reduction step is defined by applying
-the transformation a_(k-1) = b_k, b_(k-1) = remainder of a_k divided by
-b_k. The second equation means that a_k = qb_k + b_(k-1) for some
-positive integer q. And since q must be at least 1 we have a_k = qb_k
-+ b_(k-1) >= b_k + b_(k-1). But in the previous reduction step we have
-b_(k+1) = a_k. Therefore, b_(k+1) = a_k>= b_k + b_(k-1). This verifies
-the claim. Now we can prove the theorem by induction on k, the number
-of steps that the algorithm requires to terminate. The result is true
-for k = 1, since this merely requires that b be at least as large as
-_Fib_(1) = 1. Now, assume that the result is true for all integers
-less than or equal to k and establish the result for k + 1. Let
-(a_(k+1), b_(k+1)) -> (a_k, b_k) -> (a_k-1), b_(k-1)) be successive
-pairs in the reduction process. By our induction hypotheses, we have
-b_(k-1) >= _Fib_(k - 1) and b_k >= _Fib_(k). Thus, applying the claim
-we just proved together with the definition of the Fibonacci numbers
-gives b_(k+1) >= b_k + b_(k-1) >= _Fib_(k) + _Fib_(k - 1) = _Fib_(k +
-1), which completes the proof of Lame''s Theorem.
-
-\1f
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-
-1.2.6 Example: Testing for Primality
-------------------------------------
-
-This section describes two methods for checking the primality of an
-integer n, one with order of growth [theta](_[sqrt]_(n)), and a
-"probabilistic" algorithm with order of growth [theta](`log' n). The
-exercises at the end of this section suggest programming projects based
-on these algorithms.
-
-Searching for divisors
-......................
-
-Since ancient times, mathematicians have been fascinated by problems
-concerning prime numbers, and many people have worked on the problem of
-determining ways to test if numbers are prime. One way to test if a
-number is prime is to find the number's divisors. The following
-program finds the smallest integral divisor (greater than 1) of a given
-number n. It does this in a straightforward way, by testing n for
-divisibility by successive integers starting with 2.
-
- (define (smallest-divisor n)
- (find-divisor n 2))
-
- (define (find-divisor n test-divisor)
- (cond ((> (square test-divisor) n) n)
- ((divides? test-divisor n) test-divisor)
- (else (find-divisor n (+ test-divisor 1)))))
-
- (define (divides? a b)
- (= (remainder b a) 0))
-
- We can test whether a number is prime as follows: n is prime if and
-only if n is its own smallest divisor.
-
- (define (prime? n)
- (= n (smallest-divisor n)))
-
- The end test for `find-divisor' is based on the fact that if n is not
-prime it must have a divisor less than or equal to _[sqrt]_(n).(1)
-This means that the algorithm need only test divisors between 1 and
-_[sqrt]_(n). Consequently, the number of steps required to identify n
-as prime will have order of growth [theta](_[sqrt]_(n)).
-
-The Fermat test
-...............
-
-The [theta](`log' n) primality test is based on a result from number
-theory known as Fermat's Little Theorem.(2)
-
- *Fermat's Little Theorem:* If n is a prime number and a is any
- positive integer less than n, then a raised to the nth power is
- congruent to a modulo n.
-
- (Two numbers are said to be "congruent modulo" n if they both have
-the same remainder when divided by n. The remainder of a number a when
-divided by n is also referred to as the "remainder of" a "modulo" n, or
-simply as a "modulo" n.)
-
- If n is not prime, then, in general, most of the numbers a< n will
-not satisfy the above relation. This leads to the following algorithm
-for testing primality: Given a number n, pick a random number a < n and
-compute the remainder of a^n modulo n. If the result is not equal to
-a, then n is certainly not prime. If it is a, then chances are good
-that n is prime. Now pick another random number a and test it with the
-same method. If it also satisfies the equation, then we can be even
-more confident that n is prime. By trying more and more values of a,
-we can increase our confidence in the result. This algorithm is known
-as the Fermat test.
-
- To implement the Fermat test, we need a procedure that computes the
-exponential of a number modulo another number:
-
- (define (expmod base exp m)
- (cond ((= exp 0) 1)
- ((even? exp)
- (remainder (square (expmod base (/ exp 2) m))
- m))
- (else
- (remainder (* base (expmod base (- exp 1) m))
- m))))
-
- This is very similar to the `fast-expt' procedure of section *Note
-1-2-4::. It uses successive squaring, so that the number of steps
-grows logarithmically with the exponent.(3)
-
- The Fermat test is performed by choosing at random a number a
-between 1 and n - 1 inclusive and checking whether the remainder modulo
-n of the nth power of a is equal to a. The random number a is chosen
-using the procedure `random', which we assume is included as a primitive
-in Scheme. `Random' returns a nonnegative integer less than its integer
-input. Hence, to obtain a random number between 1 and n - 1, we call
-`random' with an input of n - 1 and add 1 to the result:
-
- (define (fermat-test n)
- (define (try-it a)
- (= (expmod a n n) a))
- (try-it (+ 1 (random (- n 1)))))
-
- The following procedure runs the test a given number of times, as
-specified by a parameter. Its value is true if the test succeeds every
-time, and false otherwise.
-
- (define (fast-prime? n times)
- (cond ((= times 0) true)
- ((fermat-test n) (fast-prime? n (- times 1)))
- (else false)))
-
-Probabilistic methods
-.....................
-
-The Fermat test differs in character from most familiar algorithms, in
-which one computes an answer that is guaranteed to be correct. Here,
-the answer obtained is only probably correct. More precisely, if n
-ever fails the Fermat test, we can be certain that n is not prime. But
-the fact that n passes the test, while an extremely strong indication,
-is still not a guarantee that n is prime. What we would like to say is
-that for any number n, if we perform the test enough times and find
-that n always passes the test, then the probability of error in our
-primality test can be made as small as we like.
-
- Unfortunately, this assertion is not quite correct. There do exist
-numbers that fool the Fermat test: numbers n that are not prime and yet
-have the property that a^n is congruent to a modulo n for all integers
-a < n. Such numbers are extremely rare, so the Fermat test is quite
-reliable in practice.(4)
-
- There are variations of the Fermat test that cannot be fooled. In
-these tests, as with the Fermat method, one tests the primality of an
-integer n by choosing a random integer a<n and checking some condition
-that depends upon n and a. (See *Note Exercise 1-28:: for an example
-of such a test.) On the other hand, in contrast to the Fermat test,
-one can prove that, for any n, the condition does not hold for most of
-the integers a<n unless n is prime. Thus, if n passes the test for
-some random choice of a, the chances are better than even that n is
-prime. If n passes the test for two random choices of a, the chances
-are better than 3 out of 4 that n is prime. By running the test with
-more and more randomly chosen values of a we can make the probability
-of error as small as we like.
-
- The existence of tests for which one can prove that the chance of
-error becomes arbitrarily small has sparked interest in algorithms of
-this type, which have come to be known as "probabilistic algorithms".
-There is a great deal of research activity in this area, and
-probabilistic algorithms have been fruitfully applied to many fields.(5)
-
- *Exercise 1.21:* Use the `smallest-divisor' procedure to find the
- smallest divisor of each of the following numbers: 199, 1999,
- 19999.
-
- *Exercise 1.22:* Most Lisp implementations include a primitive
- called `runtime' that returns an integer that specifies the amount
- of time the system has been running (measured, for example, in
- microseconds). The following `timed-prime-test' procedure, when
- called with an integer n, prints n and checks to see if n is
- prime. If n is prime, the procedure prints three asterisks
- followed by the amount of time used in performing the test.
-
- (define (timed-prime-test n)
- (newline)
- (display n)
- (start-prime-test n (runtime)))
-
- (define (start-prime-test n start-time)
- (if (prime? n)
- (report-prime (- (runtime) start-time))))
-
- (define (report-prime elapsed-time)
- (display " *** ")
- (display elapsed-time))
-
- Using this procedure, write a procedure `search-for-primes' that
- checks the primality of consecutive odd integers in a specified
- range. Use your procedure to find the three smallest primes
- larger than 1000; larger than 10,000; larger than 100,000; larger
- than 1,000,000. Note the time needed to test each prime. Since
- the testing algorithm has order of growth of [theta](_[sqrt]_(n)),
- you should expect that testing for primes around 10,000 should
- take about _[sqrt]_(10) times as long as testing for primes around
- 1000. Do your timing data bear this out? How well do the data
- for 100,000 and 1,000,000 support the _[sqrt]_(n) prediction? Is
- your result compatible with the notion that programs on your
- machine run in time proportional to the number of steps required
- for the computation?
-
- *Exercise 1.23:* The `smallest-divisor' procedure shown at the
- start of this section does lots of needless testing: After it
- checks to see if the number is divisible by 2 there is no point in
- checking to see if it is divisible by any larger even numbers.
- This suggests that the values used for `test-divisor' should not
- be 2, 3, 4, 5, 6, ..., but rather 2, 3, 5, 7, 9, .... To
- implement this change, define a procedure `next' that returns 3 if
- its input is equal to 2 and otherwise returns its input plus 2.
- Modify the `smallest-divisor' procedure to use `(next
- test-divisor)' instead of `(+ test-divisor 1)'. With
- `timed-prime-test' incorporating this modified version of
- `smallest-divisor', run the test for each of the 12 primes found in
- *Note Exercise 1-22::. Since this modification halves the number
- of test steps, you should expect it to run about twice as fast.
- Is this expectation confirmed? If not, what is the observed ratio
- of the speeds of the two algorithms, and how do you explain the
- fact that it is different from 2?
-
- *Exercise 1.24:* Modify the `timed-prime-test' procedure of *Note
- Exercise 1-22:: to use `fast-prime?' (the Fermat method), and test
- each of the 12 primes you found in that exercise. Since the
- Fermat test has [theta](`log' n) growth, how would you expect the
- time to test primes near 1,000,000 to compare with the time needed
- to test primes near 1000? Do your data bear this out? Can you
- explain any discrepancy you find?
-
- *Exercise 1.25:* Alyssa P. Hacker complains that we went to a lot
- of extra work in writing `expmod'. After all, she says, since we
- already know how to compute exponentials, we could have simply
- written
-
- (define (expmod base exp m)
- (remainder (fast-expt base exp) m))
-
- Is she correct? Would this procedure serve as well for our fast
- prime tester? Explain.
-
- *Exercise 1.26:* Louis Reasoner is having great difficulty doing
- *Note Exercise 1-24::. His `fast-prime?' test seems to run more
- slowly than his `prime?' test. Louis calls his friend Eva Lu Ator
- over to help. When they examine Louis's code, they find that he
- has rewritten the `expmod' procedure to use an explicit
- multiplication, rather than calling `square':
-
- (define (expmod base exp m)
- (cond ((= exp 0) 1)
- ((even? exp)
- (remainder (* (expmod base (/ exp 2) m)
- (expmod base (/ exp 2) m))
- m))
- (else
- (remainder (* base (expmod base (- exp 1) m))
- m))))
-
- "I don't see what difference that could make," says Louis. "I
- do." says Eva. "By writing the procedure like that, you have
- transformed the [theta](`log' n) process into a [theta](n)
- process." Explain.
-
- *Exercise 1.27:* Demonstrate that the Carmichael numbers listed in
- *Note Footnote 1-47:: really do fool the Fermat test. That is,
- write a procedure that takes an integer n and tests whether a^n is
- congruent to a modulo n for every a<n, and try your procedure on
- the given Carmichael numbers.
-
- *Exercise 1.28:* One variant of the Fermat test that cannot be
- fooled is called the "Miller-Rabin test" (Miller 1976; Rabin
- 1980). This starts from an alternate form of Fermat's Little
- Theorem, which states that if n is a prime number and a is any
- positive integer less than n, then a raised to the (n - 1)st power
- is congruent to 1 modulo n. To test the primality of a number n
- by the Miller-Rabin test, we pick a random number a<n and raise a
- to the (n - 1)st power modulo n using the `expmod' procedure.
- However, whenever we perform the squaring step in `expmod', we
- check to see if we have discovered a "nontrivial square root of 1
- modulo n," that is, a number not equal to 1 or n - 1 whose square
- is equal to 1 modulo n. It is possible to prove that if such a
- nontrivial square root of 1 exists, then n is not prime. It is
- also possible to prove that if n is an odd number that is not
- prime, then, for at least half the numbers a<n, computing a^(n-1)
- in this way will reveal a nontrivial square root of 1 modulo n.
- (This is why the Miller-Rabin test cannot be fooled.) Modify the
- `expmod' procedure to signal if it discovers a nontrivial square
- root of 1, and use this to implement the Miller-Rabin test with a
- procedure analogous to `fermat-test'. Check your procedure by
- testing various known primes and non-primes. Hint: One convenient
- way to make `expmod' signal is to have it return 0.
-
- ---------- Footnotes ----------
-
- (1) If d is a divisor of n, then so is n/d. But d and n/d cannot
-both be greater than _[sqrt]_(n).
-
- (2) Pierre de Fermat (1601-1665) is considered to be the founder of
-modern number theory. He obtained many important number-theoretic
-results, but he usually announced just the results, without providing
-his proofs. Fermat's Little Theorem was stated in a letter he wrote in
-1640. The first published proof was given by Euler in 1736 (and an
-earlier, identical proof was discovered in the unpublished manuscripts
-of Leibniz). The most famous of Fermat's results--known as Fermat's
-Last Theorem--was jotted down in 1637 in his copy of the book
-`Arithmetic' (by the third-century Greek mathematician Diophantus) with
-the remark "I have discovered a truly remarkable proof, but this margin
-is too small to contain it." Finding a proof of Fermat's Last Theorem
-became one of the most famous challenges in number theory. A complete
-solution was finally given in 1995 by Andrew Wiles of Princeton
-University.
-
- (3) The reduction steps in the cases where the exponent e is greater
-than 1 are based on the fact that, for any integers x, y, and m, we can
-find the remainder of x times y modulo m by computing separately the
-remainders of x modulo m and y modulo m, multiplying these, and then
-taking the remainder of the result modulo m. For instance, in the case
-where e is even, we compute the remainder of b^(e/2) modulo m, square
-this, and take the remainder modulo m. This technique is useful
-because it means we can perform our computation without ever having to
-deal with numbers much larger than m. (Compare *Note Exercise 1-25::.)
-
- (4) [Footnote 1.47] Numbers that fool the Fermat test are called "Carmichael
-numbers", and little is known about them other than that they are
-extremely rare. There are 255 Carmichael numbers below 100,000,000.
-The smallest few are 561, 1105, 1729, 2465, 2821, and 6601. In testing
-primality of very large numbers chosen at random, the chance of
-stumbling upon a value that fools the Fermat test is less than the
-chance that cosmic radiation will cause the computer to make an error
-in carrying out a "correct" algorithm. Considering an algorithm to be
-inadequate for the first reason but not for the second illustrates the
-difference between mathematics and engineering.
-
- (5) One of the most striking applications of probabilistic prime
-testing has been to the field of cryptography. Although it is now
-computationally infeasible to factor an arbitrary 200-digit number, the
-primality of such a number can be checked in a few seconds with the
-Fermat test. This fact forms the basis of a technique for constructing
-"unbreakable codes" suggested by Rivest, Shamir, and Adleman (1977).
-The resulting "RSA algorithm" has become a widely used technique for
-enhancing the security of electronic communications. Because of this
-and related developments, the study of prime numbers, once considered
-the epitome of a topic in "pure" mathematics to be studied only for its
-own sake, now turns out to have important practical applications to
-cryptography, electronic funds transfer, and information retrieval.
-
-\1f
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-
-1.3 Formulating Abstractions with Higher-Order Procedures
-=========================================================
-
-We have seen that procedures are, in effect, abstractions that describe
-compound operations on numbers independent of the particular numbers.
-For example, when we
-
- (define (cube x) (* x x x))
-
-we are not talking about the cube of a particular number, but rather
-about a method for obtaining the cube of any number. Of course we
-could get along without ever defining this procedure, by always writing
-expressions such as
-
- (* 3 3 3)
- (* x x x)
- (* y y y)
-
-and never mentioning `cube' explicitly. This would place us at a
-serious disadvantage, forcing us to work always at the level of the
-particular operations that happen to be primitives in the language
-(multiplication, in this case) rather than in terms of higher-level
-operations. Our programs would be able to compute cubes, but our
-language would lack the ability to express the concept of cubing. One
-of the things we should demand from a powerful programming language is
-the ability to build abstractions by assigning names to common patterns
-and then to work in terms of the abstractions directly. Procedures
-provide this ability. This is why all but the most primitive
-programming languages include mechanisms for defining procedures.
-
- Yet even in numerical processing we will be severely limited in our
-ability to create abstractions if we are restricted to procedures whose
-parameters must be numbers. Often the same programming pattern will be
-used with a number of different procedures. To express such patterns
-as concepts, we will need to construct procedures that can accept
-procedures as arguments or return procedures as values. Procedures
-that manipulate procedures are called "higher-order procedures". This
-section shows how higher-order procedures can serve as powerful
-abstraction mechanisms, vastly increasing the expressive power of our
-language.
-
-* Menu:
-
-* 1-3-1:: Procedures as Arguments
-* 1-3-2:: Constructing Procedures Using `Lambda'
-* 1-3-3:: Procedures as General Methods
-* 1-3-4:: Procedures as Returned Values
-
-\1f
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-
-1.3.1 Procedures as Arguments
------------------------------
-
-Consider the following three procedures. The first computes the sum of
-the integers from `a' through `b':
-
- (define (sum-integers a b)
- (if (> a b)
- 0
- (+ a (sum-integers (+ a 1) b))))
-
- The second computes the sum of the cubes of the integers in the
-given range:
-
- (define (sum-cubes a b)
- (if (> a b)
- 0
- (+ (cube a) (sum-cubes (+ a 1) b))))
-
- The third computes the sum of a sequence of terms in the series
-
- 1 1 1
- ----- + ----- + ------ + ...
- 1 * 3 5 * 7 9 * 11
-
-which converges to [pi]/8 (very slowly):(1)
-
- (define (pi-sum a b)
- (if (> a b)
- 0
- (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b))))
-
- These three procedures clearly share a common underlying pattern.
-They are for the most part identical, differing only in the name of the
-procedure, the function of `a' used to compute the term to be added,
-and the function that provides the next value of `a'. We could
-generate each of the procedures by filling in slots in the same
-template:
-
- (define (<NAME> a b)
- (if (> a b)
- 0
- (+ (<TERM> a)
- (<NAME> (<NEXT> a) b))))
-
- The presence of such a common pattern is strong evidence that there
-is a useful abstraction waiting to be brought to the surface. Indeed,
-mathematicians long ago identified the abstraction of "summation of a
-series" and invented "sigma notation," for example
-
- b
- ---
- > f(n) = f(a) + ... + f(b)
- ---
- n=a
-
-to express this concept. The power of sigma notation is that it allows
-mathematicians to deal with the concept of summation itself rather than
-only with particular sums--for example, to formulate general results
-about sums that are independent of the particular series being summed.
-
- Similarly, as program designers, we would like our language to be
-powerful enough so that we can write a procedure that expresses the
-concept of summation itself rather than only procedures that compute
-particular sums. We can do so readily in our procedural language by
-taking the common template shown above and transforming the "slots"
-into formal parameters:
-
- (define (sum term a next b)
- (if (> a b)
- 0
- (+ (term a)
- (sum term (next a) next b))))
-
- Notice that `sum' takes as its arguments the lower and upper bounds
-`a' and `b' together with the procedures `term' and `next'. We can use
-`sum' just as we would any procedure. For example, we can use it
-(along with a procedure `inc' that increments its argument by 1) to
-define `sum-cubes':
-
- (define (inc n) (+ n 1))
-
- (define (sum-cubes a b)
- (sum cube a inc b))
-
- Using this, we can compute the sum of the cubes of the integers from
-1 to 10:
-
- (sum-cubes 1 10)
- 3025
-
- With the aid of an identity procedure to compute the term, we can
-define `sum-integers' in terms of `sum':
-
- (define (identity x) x)
-
- (define (sum-integers a b)
- (sum identity a inc b))
-
- Then we can add up the integers from 1 to 10:
-
- (sum-integers 1 10)
- 55
-
- We can also define `pi-sum' in the same way:(2)
-
- (define (pi-sum a b)
- (define (pi-term x)
- (/ 1.0 (* x (+ x 2))))
- (define (pi-next x)
- (+ x 4))
- (sum pi-term a pi-next b))
-
- Using these procedures, we can compute an approximation to [pi]:
-
- (* 8 (pi-sum 1 1000))
- 3.139592655589783
-
- Once we have `sum', we can use it as a building block in formulating
-further concepts. For instance, the definite integral of a function f
-between the limits a and b can be approximated numerically using the
-formula
-
- /b / / dx \ / dx \ / dx \ \
- | f = | f| a + -- | + f| a + dx + -- | + f| a + 2dx + -- | + ...| dx
- /a \ \ 2 / \ 2 / \ 2 / /
-
-for small values of dx. We can express this directly as a procedure:
-
- (define (integral f a b dx)
- (define (add-dx x) (+ x dx))
- (* (sum f (+ a (/ dx 2.0)) add-dx b)
- dx))
-
- (integral cube 0 1 0.01)
- .24998750000000042
-
- (integral cube 0 1 0.001)
- .249999875000001
-
-(The exact value of the integral of `cube' between 0 and 1 is 1/4.)
-
- *Exercise 1.29:* Simpson's Rule is a more accurate method of
- numerical integration than the method illustrated above. Using
- Simpson's Rule, the integral of a function f between a and b is
- approximated as
-
- h
- - (y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + ... + 2y_(n-2) + 4y_(n-1) + y_n)
- 3
-
- where h = (b - a)/n, for some even integer n, and y_k = f(a + kh).
- (Increasing n increases the accuracy of the approximation.)
- Define a procedure that takes as arguments f, a, b, and n and
- returns the value of the integral, computed using Simpson's Rule.
- Use your procedure to integrate `cube' between 0 and 1 (with n =
- 100 and n = 1000), and compare the results to those of the
- `integral' procedure shown above.
-
- *Exercise 1.30:* The `sum' procedure above generates a linear
- recursion. The procedure can be rewritten so that the sum is
- performed iteratively. Show how to do this by filling in the
- missing expressions in the following definition:
-
- (define (sum term a next b)
- (define (iter a result)
- (if <??>
- <??>
- (iter <??> <??>)))
- (iter <??> <??>))
-
- *Exercise 1.31:*
- a. The `sum' procedure is only the simplest of a vast number of
- similar abstractions that can be captured as higher-order
- procedures.(3) Write an analogous procedure called `product'
- that returns the product of the values of a function at
- points over a given range. Show how to define `factorial' in
- terms of `product'. Also use `product' to compute
- approximations to [pi] using the formula(4)
-
- pi 2 * 4 * 4 * 6 * 6 * 8 ...
- -- = -------------------------
- 4 3 * 3 * 5 * 5 * 7 * 7 ...
-
- b. If your `product' procedure generates a recursive process,
- write one that generates an iterative process. If it
- generates an iterative process, write one that generates a
- recursive process.
-
-
- *Exercise 1.32:*
- a. Show that `sum' and `product' (*Note Exercise 1-31::) are
- both special cases of a still more general notion called
- `accumulate' that combines a collection of terms, using some
- general accumulation function:
-
- (accumulate combiner null-value term a next b)
-
- `Accumulate' takes as arguments the same term and range
- specifications as `sum' and `product', together with a
- `combiner' procedure (of two arguments) that specifies how
- the current term is to be combined with the accumulation of
- the preceding terms and a `null-value' that specifies what
- base value to use when the terms run out. Write `accumulate'
- and show how `sum' and `product' can both be defined as
- simple calls to `accumulate'.
-
- b. If your `accumulate' procedure generates a recursive process,
- write one that generates an iterative process. If it
- generates an iterative process, write one that generates a
- recursive process.
-
-
- *Exercise 1.33:* You can obtain an even more general version of
- `accumulate' (*Note Exercise 1-32::) by introducing the notion of
- a "filter" on the terms to be combined. That is, combine only
- those terms derived from values in the range that satisfy a
- specified condition. The resulting `filtered-accumulate'
- abstraction takes the same arguments as accumulate, together with
- an additional predicate of one argument that specifies the filter.
- Write `filtered-accumulate' as a procedure. Show how to express
- the following using `filtered-accumulate':
-
- a. the sum of the squares of the prime numbers in the interval a
- to b (assuming that you have a `prime?' predicate already
- written)
-
- b. the product of all the positive integers less than n that are
- relatively prime to n (i.e., all positive integers i < n such
- that GCD(i,n) = 1).
-
-
- ---------- Footnotes ----------
-
- (1) This series, usually written in the equivalent form ([pi]/4) = 1
-- (1/3) + (1/5) - (1/7) + ..., is due to Leibniz. We'll see how to use
-this as the basis for some fancy numerical tricks in section *Note
-3-5-3::.
-
- (2) Notice that we have used block structure (section *Note 1-1-8::)
-to embed the definitions of `pi-next' and `pi-term' within `pi-sum',
-since these procedures are unlikely to be useful for any other purpose.
-We will see how to get rid of them altogether in section *Note 1-3-2::.
-
- (3) The intent of *Note Exercise 1-31:: through *Note Exercise
-1-33:: is to demonstrate the expressive power that is attained by using
-an appropriate abstraction to consolidate many seemingly disparate
-operations. However, though accumulation and filtering are elegant
-ideas, our hands are somewhat tied in using them at this point since we
-do not yet have data structures to provide suitable means of
-combination for these abstractions. We will return to these ideas in
-section *Note 2-2-3:: when we show how to use "sequences" as interfaces
-for combining filters and accumulators to build even more powerful
-abstractions. We will see there how these methods really come into
-their own as a powerful and elegant approach to designing programs.
-
- (4) This formula was discovered by the seventeenth-century English
-mathematician John Wallis.
-
-\1f
-File: sicp.info, Node: 1-3-2, Next: 1-3-3, Prev: 1-3-1, Up: 1-3
-
-1.3.2 Constructing Procedures Using `Lambda'
---------------------------------------------
-
-In using `sum' as in section *Note 1-3-1::, it seems terribly awkward to
-have to define trivial procedures such as `pi-term' and `pi-next' just
-so we can use them as arguments to our higher-order procedure. Rather
-than define `pi-next' and `pi-term', it would be more convenient to
-have a way to directly specify "the procedure that returns its input
-incremented by 4" and "the procedure that returns the reciprocal of its
-input times its input plus 2." We can do this by introducing the
-special form `lambda', which creates procedures. Using `lambda' we can
-describe what we want as
-
- (lambda (x) (+ x 4))
-
-and
-
- (lambda (x) (/ 1.0 (* x (+ x 2))))
-
- Then our `pi-sum' procedure can be expressed without defining any
-auxiliary procedures as
-
- (define (pi-sum a b)
- (sum (lambda (x) (/ 1.0 (* x (+ x 2))))
- a
- (lambda (x) (+ x 4))
- b))
-
- Again using `lambda', we can write the `integral' procedure without
-having to define the auxiliary procedure `add-dx':
-
- (define (integral f a b dx)
- (* (sum f
- (+ a (/ dx 2.0))
- (lambda (x) (+ x dx))
- b)
- dx))
-
- In general, `lambda' is used to create procedures in the same way as
-`define', except that no name is specified for the procedure:
-
- (lambda (<FORMAL-PARAMETERS>) <BODY>)
-
- The resulting procedure is just as much a procedure as one that is
-created using `define'. The only difference is that it has not been
-associated with any name in the environment. In fact,
-
- (define (plus4 x) (+ x 4))
-
-is equivalent to
-
- (define plus4 (lambda (x) (+ x 4)))
-
- We can read a `lambda' expression as follows:
-
- (lambda (x) (+ x 4))
- | | | | |
- the procedure of an argument x that adds x and 4
-
- Like any expression that has a procedure as its value, a `lambda'
-expression can be used as the operator in a combination such as
-
- ((lambda (x y z) (+ x y (square z))) 1 2 3)
- 12
-
-or, more generally, in any context where we would normally use a
-procedure name.(1)
-
-Using `let' to create local variables
-.....................................
-
-Another use of `lambda' is in creating local variables. We often need
-local variables in our procedures other than those that have been bound
-as formal parameters. For example, suppose we wish to compute the
-function
-
- f(x,y) = x(1 + xy)^2 + y(1 - y) + (1 + xy)(1 - y)
-
-which we could also express as
-
- a = 1 + xy
- b = 1 - y
- f(x,y) = xa^2 + yb + ab
-
- In writing a procedure to compute f, we would like to include as
-local variables not only x and y but also the names of intermediate
-quantities like a and b. One way to accomplish this is to use an
-auxiliary procedure to bind the local variables:
-
- (define (f x y)
- (define (f-helper a b)
- (+ (* x (square a))
- (* y b)
- (* a b)))
- (f-helper (+ 1 (* x y))
- (- 1 y)))
-
- Of course, we could use a `lambda' expression to specify an anonymous
-procedure for binding our local variables. The body of `f' then
-becomes a single call to that procedure:
-
- (define (f x y)
- ((lambda (a b)
- (+ (* x (square a))
- (* y b)
- (* a b)))
- (+ 1 (* x y))
- (- 1 y)))
-
- This construct is so useful that there is a special form called
-`let' to make its use more convenient. Using `let', the `f' procedure
-could be written as
-
- (define (f x y)
- (let ((a (+ 1 (* x y)))
- (b (- 1 y)))
- (+ (* x (square a))
- (* y b)
- (* a b))))
-
- The general form of a `let' expression is
-
- (let ((<VAR1> <EXP1>)
- (<VAR2> <EXP2>)
- ...
- (<VARN> <VARN>))
- <BODY>)
-
-which can be thought of as saying
-
- let <VAR_1> have the value <EXP_1> and
- <VAR_2> have the value <EXP_2> and
- ...
- <VAR_N> have the value <EXP_N>
- in <BODY>
-
- The first part of the `let' expression is a list of name-expression
-pairs. When the `let' is evaluated, each name is associated with the
-value of the corresponding expression. The body of the `let' is
-evaluated with these names bound as local variables. The way this
-happens is that the `let' expression is interpreted as an alternate
-syntax for
-
- ((lambda (<VAR_1> ... <VAR_N>)
- <BODY>)
- <EXP_1>
- ...
- <EXP_N>)
-
- No new mechanism is required in the interpreter in order to provide
-local variables. A `let' expression is simply syntactic sugar for the
-underlying `lambda' application.
-
- We can see from this equivalence that the scope of a variable
-specified by a `let' expression is the body of the `let'. This implies
-that:
-
- * `Let' allows one to bind variables as locally as possible to where
- they are to be used. For example, if the value of `x' is 5, the
- value of the expression
-
- (+ (let ((x 3))
- (+ x (* x 10)))
- x)
-
- is 38. Here, the `x' in the body of the `let' is 3, so the value
- of the `let' expression is 33. On the other hand, the `x' that is
- the second argument to the outermost `+' is still 5.
-
- * The variables' values are computed outside the `let'. This
- matters when the expressions that provide the values for the local
- variables depend upon variables having the same names as the local
- variables themselves. For example, if the value of `x' is 2, the
- expression
-
- (let ((x 3)
- (y (+ x 2)))
- (* x y))
-
- will have the value 12 because, inside the body of the `let', `x'
- will be 3 and `y' will be 4 (which is the outer `x' plus 2).
-
-
- Sometimes we can use internal definitions to get the same effect as
-with `let'. For example, we could have defined the procedure `f' above
-as
-
- (define (f x y)
- (define a (+ 1 (* x y)))
- (define b (- 1 y))
- (+ (* x (square a))
- (* y b)
- (* a b)))
-
- We prefer, however, to use `let' in situations like this and to use
-internal `define' only for internal procedures.(2)
-
- *Exercise 1.34:* Suppose we define the procedure
-
- (define (f g)
- (g 2))
-
- Then we have
-
- (f square)
- 4
-
- (f (lambda (z) (* z (+ z 1))))
- 6
-
- What happens if we (perversely) ask the interpreter to evaluate
- the combination `(f f)'? Explain.
-
- ---------- Footnotes ----------
-
- (1) It would be clearer and less intimidating to people learning
-Lisp if a name more obvious than `lambda', such as `make-procedure',
-were used. But the convention is firmly entrenched. The notation is
-adopted from the [lambda] calculus, a mathematical formalism introduced
-by the mathematical logician Alonzo Church (1941). Church developed
-the [lambda] calculus to provide a rigorous foundation for studying the
-notions of function and function application. The [lambda] calculus
-has become a basic tool for mathematical investigations of the
-semantics of programming languages.
-
- (2) Understanding internal definitions well enough to be sure a
-program means what we intend it to mean requires a more elaborate model
-of the evaluation process than we have presented in this chapter. The
-subtleties do not arise with internal definitions of procedures,
-however. We will return to this issue in section *Note 4-1-6::, after
-we learn more about evaluation.
-
-\1f
-File: sicp.info, Node: 1-3-3, Next: 1-3-4, Prev: 1-3-2, Up: 1-3
-
-1.3.3 Procedures as General Methods
------------------------------------
-
-We introduced compound procedures in section *Note 1-1-4:: as a
-mechanism for abstracting patterns of numerical operations so as to
-make them independent of the particular numbers involved. With
-higher-order procedures, such as the `integral' procedure of section
-*Note 1-3-1::, we began to see a more powerful kind of abstraction:
-procedures used to express general methods of computation, independent
-of the particular functions involved. In this section we discuss two
-more elaborate examples--general methods for finding zeros and fixed
-points of functions--and show how these methods can be expressed
-directly as procedures.
-
-Finding roots of equations by the half-interval method
-......................................................
-
-The "half-interval method" is a simple but powerful technique for
-finding roots of an equation f(x) = 0, where f is a continuous
-function. The idea is that, if we are given points a and b such that
-f(a) < 0 < f(b), then f must have at least one zero between a and b.
-To locate a zero, let x be the average of a and b and compute f(x). If
-f(x) > 0, then f must have a zero between a and x. If f(x) < 0, then f
-must have a zero between x and b. Continuing in this way, we can
-identify smaller and smaller intervals on which f must have a zero.
-When we reach a point where the interval is small enough, the process
-stops. Since the interval of uncertainty is reduced by half at each
-step of the process, the number of steps required grows as
-[theta](`log'( L/T)), where L is the length of the original interval
-and T is the error tolerance (that is, the size of the interval we will
-consider "small enough"). Here is a procedure that implements this
-strategy:
-
- (define (search f neg-point pos-point)
- (let ((midpoint (average neg-point pos-point)))
- (if (close-enough? neg-point pos-point)
- midpoint
- (let ((test-value (f midpoint)))
- (cond ((positive? test-value)
- (search f neg-point midpoint))
- ((negative? test-value)
- (search f midpoint pos-point))
- (else midpoint))))))
-
- We assume that we are initially given the function f together with
-points at which its values are negative and positive. We first compute
-the midpoint of the two given points. Next we check to see if the
-given interval is small enough, and if so we simply return the midpoint
-as our answer. Otherwise, we compute as a test value the value of f at
-the midpoint. If the test value is positive, then we continue the
-process with a new interval running from the original negative point to
-the midpoint. If the test value is negative, we continue with the
-interval from the midpoint to the positive point. Finally, there is
-the possibility that the test value is 0, in which case the midpoint is
-itself the root we are searching for.
-
- To test whether the endpoints are "close enough" we can use a
-procedure similar to the one used in section *Note 1-1-7:: for
-computing square roots:(1)
-
- (define (close-enough? x y)
- (< (abs (- x y)) 0.001))
-
- `Search' is awkward to use directly, because we can accidentally
-give it points at which f's values do not have the required sign, in
-which case we get a wrong answer. Instead we will use `search' via the
-following procedure, which checks to see which of the endpoints has a
-negative function value and which has a positive value, and calls the
-`search' procedure accordingly. If the function has the same sign on
-the two given points, the half-interval method cannot be used, in which
-case the procedure signals an error.(2)
-
- (define (half-interval-method f a b)
- (let ((a-value (f a))
- (b-value (f b)))
- (cond ((and (negative? a-value) (positive? b-value))
- (search f a b))
- ((and (negative? b-value) (positive? a-value))
- (search f b a))
- (else
- (error "Values are not of opposite sign" a b)))))
-
- The following example uses the half-interval method to approximate
-[pi] as the root between 2 and 4 of `sin' x = 0:
-
- (half-interval-method sin 2.0 4.0)
- 3.14111328125
-
- Here is another example, using the half-interval method to search
-for a root of the equation x^3 - 2x - 3 = 0 between 1 and 2:
-
- (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))
- 1.0
- 2.0)
- 1.89306640625
-
-Finding fixed points of functions
-.................................
-
-A number x is called a "fixed point" of a function f if x satisfies the
-equation f(x) = x. For some functions f we can locate a fixed point by
-beginning with an initial guess and applying f repeatedly,
-
- f(x), f(f(x), (f(f(f(x))))
-
-until the value does not change very much. Using this idea, we can
-devise a procedure `fixed-point' that takes as inputs a function and an
-initial guess and produces an approximation to a fixed point of the
-function. We apply the function repeatedly until we find two
-successive values whose difference is less than some prescribed
-tolerance:
-
- (define tolerance 0.00001)
-
- (define (fixed-point f first-guess)
- (define (close-enough? v1 v2)
- (< (abs (- v1 v2)) tolerance))
- (define (try guess)
- (let ((next (f guess)))
- (if (close-enough? guess next)
- next
- (try next))))
- (try first-guess))
-
- For example, we can use this method to approximate the fixed point
-of the cosine function, starting with 1 as an initial approximation:(3)
-
- (fixed-point cos 1.0)
- .7390822985224023
-
- Similarly, we can find a solution to the equation y = `sin' y +
-`cos' y:
-
- (fixed-point (lambda (y) (+ (sin y) (cos y)))
- 1.0)
- 1.2587315962971173
-
- The fixed-point process is reminiscent of the process we used for
-finding square roots in section *Note 1-1-7::. Both are based on the
-idea of repeatedly improving a guess until the result satisfies some
-criterion. In fact, we can readily formulate the square-root
-computation as a fixed-point search. Computing the square root of some
-number x requires finding a y such that y^2 = x. Putting this equation
-into the equivalent form y = x/y, we recognize that we are looking for
-a fixed point of the function(4) y |-> x/y, and we can therefore try to
-compute square roots as
-
- (define (sqrt x)
- (fixed-point (lambda (y) (/ x y))
- 1.0))
-
- Unfortunately, this fixed-point search does not converge. Consider
-an initial guess y_1. The next guess is y_2 = x/y_1 and the next guess
-is y_3 = x/y_2 = x/(x/y_1) = y_1. This results in an infinite loop in
-which the two guesses y_1 and y_2 repeat over and over, oscillating
-about the answer.
-
- One way to control such oscillations is to prevent the guesses from
-changing so much. Since the answer is always between our guess y and
-x/y, we can make a new guess that is not as far from y as x/y by
-averaging y with x/y, so that the next guess after y is (1/2)(y + x/y)
-instead of x/y. The process of making such a sequence of guesses is
-simply the process of looking for a fixed point of y |-> (1/2)(y + x/y):
-
- (define (sqrt x)
- (fixed-point (lambda (y) (average y (/ x y)))
- 1.0))
-
- (Note that y = (1/2)(y + x/y) is a simple transformation of the
-equation y = x/y; to derive it, add y to both sides of the equation and
-divide by 2.)
-
- With this modification, the square-root procedure works. In fact,
-if we unravel the definitions, we can see that the sequence of
-approximations to the square root generated here is precisely the same
-as the one generated by our original square-root procedure of section
-*Note 1-1-7::. This approach of averaging successive approximations to
-a solution, a technique we that we call "average damping", often aids
-the convergence of fixed-point searches.
-
- *Exercise 1.35:* Show that the golden ratio [phi] (section *Note
- 1-2-2::) is a fixed point of the transformation x |-> 1 + 1/x, and
- use this fact to compute [phi] by means of the `fixed-point'
- procedure.
-
- *Exercise 1.36:* Modify `fixed-point' so that it prints the
- sequence of approximations it generates, using the `newline' and
- `display' primitives shown in *Note Exercise 1-22::. Then find a
- solution to x^x = 1000 by finding a fixed point of x |->
- `log'(1000)/`log'(x). (Use Scheme's primitive `log' procedure,
- which computes natural logarithms.) Compare the number of steps
- this takes with and without average damping. (Note that you
- cannot start `fixed-point' with a guess of 1, as this would cause
- division by `log'(1) = 0.)
-
- *Exercise 1.37:*
- a. An infinite "continued fraction" is an expression of the form
-
- N_1
- f = ---------------------
- N_2
- D_1 + ---------------
- N_3
- D_2 + ---------
- D_3 + ...
-
- As an example, one can show that the infinite continued
- fraction expansion with the n_i and the D_i all equal to 1
- produces 1/[phi], where [phi] is the golden ratio (described
- in section *Note 1-2-2::). One way to approximate an
- infinite continued fraction is to truncate the expansion
- after a given number of terms. Such a truncation--a
- so-called finite continued fraction "k-term finite continued
- fraction"--has the form
-
- N_1
- -----------------
- N_2
- D_1 + -----------
- ... N_K
- + -----
- D_K
-
- Suppose that `n' and `d' are procedures of one argument (the
- term index i) that return the n_i and D_i of the terms of the
- continued fraction. Define a procedure `cont-frac' such that
- evaluating `(cont-frac n d k)' computes the value of the
- k-term finite continued fraction. Check your procedure by
- approximating 1/[phi] using
-
- (cont-frac (lambda (i) 1.0)
- (lambda (i) 1.0)
- k)
-
- for successive values of `k'. How large must you make `k' in
- order to get an approximation that is accurate to 4 decimal
- places?
-
- b. If your `cont-frac' procedure generates a recursive process,
- write one that generates an iterative process. If it
- generates an iterative process, write one that generates a
- recursive process.
-
-
- *Exercise 1.38:* In 1737, the Swiss mathematician Leonhard Euler
- published a memoir `De Fractionibus Continuis', which included a
- continued fraction expansion for e - 2, where e is the base of the
- natural logarithms. In this fraction, the n_i are all 1, and the
- D_i are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, .... Write
- a program that uses your `cont-frac' procedure from *Note Exercise
- 1-37:: to approximate e, based on Euler's expansion.
-
- *Exercise 1.39:* A continued fraction representation of the
- tangent function was published in 1770 by the German mathematician
- J.H. Lambert:
-
- x
- tan x = ---------------
- x^2
- 1 - -----------
- x^2
- 3 - -------
- 5 - ...
-
- where x is in radians. Define a procedure `(tan-cf x k)' that
- computes an approximation to the tangent function based on
- Lambert's formula. `K' specifies the number of terms to compute,
- as in *Note Exercise 1-37::.
-
- ---------- Footnotes ----------
-
- (1) We have used 0.001 as a representative "small" number to
-indicate a tolerance for the acceptable error in a calculation. The
-appropriate tolerance for a real calculation depends upon the problem
-to be solved and the limitations of the computer and the algorithm.
-This is often a very subtle consideration, requiring help from a
-numerical analyst or some other kind of magician.
-
- (2) This can be accomplished using `error', which takes as arguments
-a number of items that are printed as error messages.
-
- (3) Try this during a boring lecture: Set your calculator to radians
-mode and then repeatedly press the `cos' button until you obtain the
-fixed point.
-
- (4) |-> (pronounced "maps to") is the mathematician's way of writing
-`lambda'. y |-> x/y means `(lambda(y) (/ x y))', that is, the function
-whose value at y is x/y.
-
-\1f
-File: sicp.info, Node: 1-3-4, Prev: 1-3-3, Up: 1-3
-
-1.3.4 Procedures as Returned Values
------------------------------------
-
-The above examples demonstrate how the ability to pass procedures as
-arguments significantly enhances the expressive power of our
-programming language. We can achieve even more expressive power by
-creating procedures whose returned values are themselves procedures.
-
- We can illustrate this idea by looking again at the fixed-point
-example described at the end of section *Note 1-3-3::. We formulated a
-new version of the square-root procedure as a fixed-point search,
-starting with the observation that [sqrt]x is a fixed-point of the
-function y |-> x/y. Then we used average damping to make the
-approximations converge. Average damping is a useful general technique
-in itself. Namely, given a function f, we consider the function whose
-value at x is equal to the average of x and f(x).
-
- We can express the idea of average damping by means of the following
-procedure:
-
- (define (average-damp f)
- (lambda (x) (average x (f x))))
-
- `Average-damp' is a procedure that takes as its argument a procedure
-`f' and returns as its value a procedure (produced by the `lambda')
-that, when applied to a number `x', produces the average of `x' and `(f
-x)'. For example, applying `average-damp' to the `square' procedure
-produces a procedure whose value at some number x is the average of x
-and x^2. Applying this resulting procedure to 10 returns the average
-of 10 and 100, or 55:(1)
-
- ((average-damp square) 10)
- 55
-
- Using `average-damp', we can reformulate the square-root procedure as
-follows:
-
- (define (sqrt x)
- (fixed-point (average-damp (lambda (y) (/ x y)))
- 1.0))
-
- Notice how this formulation makes explicit the three ideas in the
-method: fixed-point search, average damping, and the function y |-> x/y.
-It is instructive to compare this formulation of the square-root method
-with the original version given in section *Note 1-1-7::. Bear in mind
-that these procedures express the same process, and notice how much
-clearer the idea becomes when we express the process in terms of these
-abstractions. In general, there are many ways to formulate a process
-as a procedure. Experienced programmers know how to choose procedural
-formulations that are particularly perspicuous, and where useful
-elements of the process are exposed as separate entities that can be
-reused in other applications. As a simple example of reuse, notice
-that the cube root of x is a fixed point of the function y |-> x/y^2,
-so we can immediately generalize our square-root procedure to one that
-extracts cube roots:(2)
-
- (define (cube-root x)
- (fixed-point (average-damp (lambda (y) (/ x (square y))))
- 1.0))
-
-Newton's method
-...............
-
-When we first introduced the square-root procedure, in section *Note
-1-1-7::, we mentioned that this was a special case of "Newton's
-method". If x |-> g(x) is a differentiable function, then a solution
-of the equation g(x) = 0 is a fixed point of the function x |-> f(x)
-where
-
- g(x)
- f(x) = x - -----
- Dg(x)
-
-and Dg(x) is the derivative of g evaluated at x. Newton's method is
-the use of the fixed-point method we saw above to approximate a
-solution of the equation by finding a fixed point of the function f.(3)
-
- For many functions g and for sufficiently good initial guesses for x,
-Newton's method converges very rapidly to a solution of g(x) = 0.(4)
-
- In order to implement Newton's method as a procedure, we must first
-express the idea of derivative. Note that "derivative," like average
-damping, is something that transforms a function into another function.
-For instance, the derivative of the function x |-> x^3 is the function
-x |-> 3x^2. In general, if g is a function and dx is a small number,
-then the derivative Dg of g is the function whose value at any number x
-is given (in the limit of small dx) by
-
- g(x + dx) - g(x)
- Dg(c) = ----------------
- dx
-
-Thus, we can express the idea of derivative (taking dx to be, say,
-0.00001) as the procedure
-
- (define (deriv g)
- (lambda (x)
- (/ (- (g (+ x dx)) (g x))
- dx)))
-
-along with the definition
-
- (define dx 0.00001)
-
- Like `average-damp', `deriv' is a procedure that takes a procedure as
-argument and returns a procedure as value. For example, to approximate
-the derivative of x |-> x^3 at 5 (whose exact value is 75) we can
-evaluate
-
- (define (cube x) (* x x x))
-
- ((deriv cube) 5)
- 75.00014999664018
-
- With the aid of `deriv', we can express Newton's method as a
-fixed-point process:
-
- (define (newton-transform g)
- (lambda (x)
- (- x (/ (g x) ((deriv g) x)))))
-
- (define (newtons-method g guess)
- (fixed-point (newton-transform g) guess))
-
- The `newton-transform' procedure expresses the formula at the
-beginning of this section, and `newtons-method' is readily defined in
-terms of this. It takes as arguments a procedure that computes the
-function for which we want to find a zero, together with an initial
-guess. For instance, to find the square root of x, we can use Newton's
-method to find a zero of the function y |-> y^2 - x starting with an
-initial guess of 1.(5)
-
- This provides yet another form of the square-root procedure:
-
- (define (sqrt x)
- (newtons-method (lambda (y) (- (square y) x))
- 1.0))
-
-Abstractions and first-class procedures
-.......................................
-
-We've seen two ways to express the square-root computation as an
-instance of a more general method, once as a fixed-point search and
-once using Newton's method. Since Newton's method was itself expressed
-as a fixed-point process, we actually saw two ways to compute square
-roots as fixed points. Each method begins with a function and finds a
-fixed point of some transformation of the function. We can express
-this general idea itself as a procedure:
-
- (define (fixed-point-of-transform g transform guess)
- (fixed-point (transform g) guess))
-
- This very general procedure takes as its arguments a procedure `g'
-that computes some function, a procedure that transforms `g', and an
-initial guess. The returned result is a fixed point of the transformed
-function.
-
- Using this abstraction, we can recast the first square-root
-computation from this section (where we look for a fixed point of the
-average-damped version of y |-> x/y) as an instance of this general
-method:
-
- (define (sqrt x)
- (fixed-point-of-transform (lambda (y) (/ x y))
- average-damp
- 1.0))
-
- Similarly, we can express the second square-root computation from
-this section (an instance of Newton's method that finds a fixed point
-of the Newton transform of y |-> y^2 - x) as
-
- (define (sqrt x)
- (fixed-point-of-transform (lambda (y) (- (square y) x))
- newton-transform
- 1.0))
-
- We began section *Note 1-3:: with the observation that compound
-procedures are a crucial abstraction mechanism, because they permit us
-to express general methods of computing as explicit elements in our
-programming language. Now we've seen how higher-order procedures
-permit us to manipulate these general methods to create further
-abstractions.
-
- As programmers, we should be alert to opportunities to identify the
-underlying abstractions in our programs and to build upon them and
-generalize them to create more powerful abstractions. This is not to
-say that one should always write programs in the most abstract way
-possible; expert programmers know how to choose the level of
-abstraction appropriate to their task. But it is important to be able
-to think in terms of these abstractions, so that we can be ready to
-apply them in new contexts. The significance of higher-order
-procedures is that they enable us to represent these abstractions
-explicitly as elements in our programming language, so that they can be
-handled just like other computational elements.
-
- In general, programming languages impose restrictions on the ways in
-which computational elements can be manipulated. Elements with the
-fewest restrictions are said to have "first-class" status. Some of the
-"rights and privileges" of first-class elements are:(6)
-
- * They may be named by variables.
-
- * They may be passed as arguments to procedures.
-
- * They may be returned as the results of procedures.
-
- * They may be included in data structures.(7)
-
-
- Lisp, unlike other common programming languages, awards procedures
-full first-class status. This poses challenges for efficient
-implementation, but the resulting gain in expressive power is
-enormous.(8)
-
- *Exercise 1.40:* Define a procedure `cubic' that can be used
- together with the `newtons-method' procedure in expressions of the
- form
-
- (newtons-method (cubic a b c) 1)
-
- to approximate zeros of the cubic x^3 + ax^2 + bx + c.
-
- *Exercise 1.41:* Define a procedure `double' that takes a
- procedure of one argument as argument and returns a procedure that
- applies the original procedure twice. For example, if `inc' is a
- procedure that adds 1 to its argument, then `(double inc)' should
- be a procedure that adds 2. What value is returned by
-
- (((double (double double)) inc) 5)
-
- *Exercise 1.42:* Let f and g be two one-argument functions. The "composition"
- f after g is defined to be the function x |-> f(g(x)). Define a
- procedure `compose' that implements composition. For example, if
- `inc' is a procedure that adds 1 to its argument,
-
- ((compose square inc) 6)
- 49
-
- *Exercise 1.43:* If f is a numerical function and n is a positive
- integer, then we can form the nth repeated application of f, which
- is defined to be the function whose value at x is
- f(f(...(f(x))...)). For example, if f is the function x |-> x +
- 1, then the nth repeated application of f is the function x |-> x
- + n. If f is the operation of squaring a number, then the nth
- repeated application of f is the function that raises its argument
- to the 2^nth power. Write a procedure that takes as inputs a
- procedure that computes f and a positive integer n and returns the
- procedure that computes the nth repeated application of f. Your
- procedure should be able to be used as follows:
-
- ((repeated square 2) 5)
- 625
-
- Hint: You may find it convenient to use `compose' from *Note
- Exercise 1-42::.
-
- *Exercise 1.44:* The idea of "smoothing" a function is an
- important concept in signal processing. If f is a function and dx
- is some small number, then the smoothed version of f is the
- function whose value at a point x is the average of f(x - dx),
- f(x), and f(x + dx). Write a procedure `smooth' that takes as
- input a procedure that computes f and returns a procedure that
- computes the smoothed f. It is sometimes valuable to repeatedly
- smooth a function (that is, smooth the smoothed function, and so
- on) to obtained the "n-fold smoothed function". Show how to
- generate the n-fold smoothed function of any given function using
- `smooth' and `repeated' from *Note Exercise 1-43::.
-
- *Exercise 1.45:* We saw in section *Note 1-3-3:: that attempting
- to compute square roots by naively finding a fixed point of y |->
- x/y does not converge, and that this can be fixed by average
- damping. The same method works for finding cube roots as fixed
- points of the average-damped y |-> x/y^2. Unfortunately, the
- process does not work for fourth roots--a single average damp is
- not enough to make a fixed-point search for y |-> x/y^3 converge.
- On the other hand, if we average damp twice (i.e., use the average
- damp of the average damp of y |-> x/y^3) the fixed-point search
- does converge. Do some experiments to determine how many average
- damps are required to compute nth roots as a fixed-point search
- based upon repeated average damping of y |-> x/y^(n-1). Use this
- to implement a simple procedure for computing nth roots using
- `fixed-point', `average-damp', and the `repeated' procedure of
- *Note Exercise 1-43::. Assume that any arithmetic operations you
- need are available as primitives.
-
- *Exercise 1.46:* Several of the numerical methods described in
- this chapter are instances of an extremely general computational
- strategy known as "iterative improvement". Iterative improvement
- says that, to compute something, we start with an initial guess
- for the answer, test if the guess is good enough, and otherwise
- improve the guess and continue the process using the improved
- guess as the new guess. Write a procedure `iterative-improve'
- that takes two procedures as arguments: a method for telling
- whether a guess is good enough and a method for improving a guess.
- `Iterative-improve' should return as its value a procedure that
- takes a guess as argument and keeps improving the guess until it
- is good enough. Rewrite the `sqrt' procedure of section *Note
- 1-1-7:: and the `fixed-point' procedure of section *Note 1-3-3::
- in terms of `iterative-improve'.
-
- ---------- Footnotes ----------
-
- (1) Observe that this is a combination whose operator is itself a
-combination. *Note Exercise 1-4:: already demonstrated the ability to
-form such combinations, but that was only a toy example. Here we begin
-to see the real need for such combinations--when applying a procedure
-that is obtained as the value returned by a higher-order procedure.
-
- (2) See *Note Exercise 1-45:: for a further generalization.
-
- (3) Elementary calculus books usually describe Newton's method in
-terms of the sequence of approximations x_(n+1) = x_n - g(x_n)/Dg(x_n).
-Having language for talking about processes and using the idea of
-fixed points simplifies the description of the method.
-
- (4) Newton's method does not always converge to an answer, but it can
-be shown that in favorable cases each iteration doubles the
-number-of-digits accuracy of the approximation to the solution. In
-such cases, Newton's method will converge much more rapidly than the
-half-interval method.
-
- (5) For finding square roots, Newton's method converges rapidly to
-the correct solution from any starting point.
-
- (6) The notion of first-class status of programming-language
-elements is due to the British computer scientist Christopher Strachey
-(1916-1975).
-
- (7) We'll see examples of this after we introduce data structures in
-*Note Chapter 2::.
-
- (8) The major implementation cost of first-class procedures is that
-allowing procedures to be returned as values requires reserving storage
-for a procedure's free variables even while the procedure is not
-executing. In the Scheme implementation we will study in section *Note
-4-1::, these variables are stored in the procedure's environment.
-
-\1f
-File: sicp.info, Node: Chapter 2, Next: Chapter 3, Prev: Chapter 1, Up: Top
-
-2 Building Abstractions with Data
-*********************************
-
- We now come to the decisive step of mathematical abstraction: we
- forget about what the symbols stand for. ...[The mathematician]
- need not be idle; there are many operations which he may carry out
- with these symbols, without ever having to look at the things they
- stand for.
-
- --Hermann Weyl, `The Mathematical Way of Thinking'
-
- We concentrated in *Note Chapter 1:: on computational processes and
-on the role of procedures in program design. We saw how to use
-primitive data (numbers) and primitive operations (arithmetic
-operations), how to combine procedures to form compound procedures
-through composition, conditionals, and the use of parameters, and how
-to abstract procedures by using `define'. We saw that a procedure can
-be regarded as a pattern for the local evolution of a process, and we
-classified, reasoned about, and performed simple algorithmic analyses of
-some common patterns for processes as embodied in procedures. We also
-saw that higher-order procedures enhance the power of our language by
-enabling us to manipulate, and thereby to reason in terms of, general
-methods of computation. This is much of the essence of programming.
-
- In this chapter we are going to look at more complex data. All the
-procedures in *Note Chapter 1:: operate on simple numerical data, and
-simple data are not sufficient for many of the problems we wish to
-address using computation. Programs are typically designed to model
-complex phenomena, and more often than not one must construct
-computational objects that have several parts in order to model
-real-world phenomena that have several aspects. Thus, whereas our
-focus in *Note Chapter 1:: was on building abstractions by combining
-procedures to form compound procedures, we turn in this chapter to
-another key aspect of any programming language: the means it provides
-for building abstractions by combining data objects to form "compound
-data".
-
- Why do we want compound data in a programming language? For the
-same reasons that we want compound procedures: to elevate the
-conceptual level at which we can design our programs, to increase the
-modularity of our designs, and to enhance the expressive power of our
-language. Just as the ability to define procedures enables us to deal
-with processes at a higher conceptual level than that of the primitive
-operations of the language, the ability to construct compound data
-objects enables us to deal with data at a higher conceptual level than
-that of the primitive data objects of the language.
-
- Consider the task of designing a system to perform arithmetic with
-rational numbers. We could imagine an operation `add-rat' that takes
-two rational numbers and produces their sum. In terms of simple data,
-a rational number can be thought of as two integers: a numerator and a
-denominator. Thus, we could design a program in which each rational
-number would be represented by two integers (a numerator and a
-denominator) and where `add-rat' would be implemented by two procedures
-(one producing the numerator of the sum and one producing the
-denominator). But this would be awkward, because we would then need to
-explicitly keep track of which numerators corresponded to which
-denominators. In a system intended to perform many operations on many
-rational numbers, such bookkeeping details would clutter the programs
-substantially, to say nothing of what they would do to our minds. It
-would be much better if we could "glue together" a numerator and
-denominator to form a pair--a "compound data object"--that our programs
-could manipulate in a way that would be consistent with regarding a
-rational number as a single conceptual unit.
-
- The use of compound data also enables us to increase the modularity
-of our programs. If we can manipulate rational numbers directly as
-objects in their own right, then we can separate the part of our
-program that deals with rational numbers per se from the details of how
-rational numbers may be represented as pairs of integers. The general
-technique of isolating the parts of a program that deal with how data
-objects are represented from the parts of a program that deal with how
-data objects are used is a powerful design methodology called "data
-abstraction". We will see how data abstraction makes programs much
-easier to design, maintain, and modify.
-
- The use of compound data leads to a real increase in the expressive
-power of our programming language. Consider the idea of forming a
-"linear combination" ax + by. We might like to write a procedure that
-would accept a, b, x, and y as arguments and return the value of ax +
-by. This presents no difficulty if the arguments are to be numbers,
-because we can readily define the procedure
-
- (define (linear-combination a b x y)
- (+ (* a x) (* b y)))
-
- But suppose we are not concerned only with numbers. Suppose we
-would like to express, in procedural terms, the idea that one can form
-linear combinations whenever addition and multiplication are
-defined--for rational numbers, complex numbers, polynomials, or
-whatever. We could express this as a procedure of the form
-
- (define (linear-combination a b x y)
- (add (mul a x) (mul b y)))
-
-where `add' and `mul' are not the primitive procedures `+' and `*' but
-rather more complex things that will perform the appropriate operations
-for whatever kinds of data we pass in as the arguments `a', `b', `x',
-and `y'. The key point is that the only thing `linear-combination'
-should need to know about `a', `b', `x', and `y' is that the procedures
-`add' and `mul' will perform the appropriate manipulations. From the
-perspective of the procedure `linear-combination', it is irrelevant
-what `a', `b', `x', and `y' are and even more irrelevant how they might
-happen to be represented in terms of more primitive data. This same
-example shows why it is important that our programming language provide
-the ability to manipulate compound objects directly: Without this,
-there is no way for a procedure such as `linear-combination' to pass
-its arguments along to `add' and `mul' without having to know their
-detailed structure.(1)
-
- We begin this chapter by implementing the rational-number arithmetic
-system mentioned above. This will form the background for our
-discussion of compound data and data abstraction. As with compound
-procedures, the main issue to be addressed is that of abstraction as a
-technique for coping with complexity, and we will see how data
-abstraction enables us to erect suitable "abstraction barriers" between
-different parts of a program.
-
- We will see that the key to forming compound data is that a
-programming language should provide some kind of "glue" so that data
-objects can be combined to form more complex data objects. There are
-many possible kinds of glue. Indeed, we will discover how to form
-compound data using no special "data" operations at all, only
-procedures. This will further blur the distinction between "procedure"
-and "data," which was already becoming tenuous toward the end of *Note
-Chapter 1::. We will also explore some conventional techniques for
-representing sequences and trees. One key idea in dealing with
-compound data is the notion of "closure"--that the glue we use for
-combining data objects should allow us to combine not only primitive
-data objects, but compound data objects as well. Another key idea is
-that compound data objects can serve as "conventional interfaces" for
-combining program modules in mix-and-match ways. We illustrate some of
-these ideas by presenting a simple graphics language that exploits
-closure.
-
- We will then augment the representational power of our language by
-introducing "symbolic expressions"--data whose elementary parts can be
-arbitrary symbols rather than only numbers. We explore various
-alternatives for representing sets of objects. We will find that, just
-as a given numerical function can be computed by many different
-computational processes, there are many ways in which a given data
-structure can be represented in terms of simpler objects, and the
-choice of representation can have significant impact on the time and
-space requirements of processes that manipulate the data. We will
-investigate these ideas in the context of symbolic differentiation, the
-representation of sets, and the encoding of information.
-
- Next we will take up the problem of working with data that may be
-represented differently by different parts of a program. This leads to
-the need to implement "generic operations", which must handle many
-different types of data. Maintaining modularity in the presence of
-generic operations requires more powerful abstraction barriers than can
-be erected with simple data abstraction alone. In particular, we
-introduce programming "data-directed programming" as a technique that
-allows individual data representations to be designed in isolation and
-then combined "additively" (i.e., without modification). To illustrate
-the power of this approach to system design, we close the chapter by
-applying what we have learned to the implementation of a package for
-performing symbolic arithmetic on polynomials, in which the
-coefficients of the polynomials can be integers, rational numbers,
-complex numbers, and even other polynomials.
-
-* Menu:
-
-* 2-1:: Introduction to Data Abstraction
-* 2-2:: Hierarchical Data and the Closure Property
-* 2-3:: Symbolic Data
-* 2-4:: Multiple Representations for Abstract Data
-* 2-5:: Systems with Generic Operations
-
- ---------- Footnotes ----------
-
- (1) The ability to directly manipulate procedures provides an
-analogous increase in the expressive power of a programming language.
-For example, in section *Note 1-3-1:: we introduced the `sum'
-procedure, which takes a procedure `term' as an argument and computes
-the sum of the values of `term' over some specified interval. In order
-to define `sum', it is crucial that we be able to speak of a procedure
-such as `term' as an entity in its own right, without regard for how
-`term' might be expressed with more primitive operations. Indeed, if
-we did not have the notion of "a procedure," it is doubtful that we
-would ever even think of the possibility of defining an operation such
-as `sum'. Moreover, insofar as performing the summation is concerned,
-the details of how `term' may be constructed from more primitive
-operations are irrelevant.
-
-\1f
-File: sicp.info, Node: 2-1, Next: 2-2, Prev: Chapter 2, Up: Chapter 2
-
-2.1 Introduction to Data Abstraction
-====================================
-
-In section *Note 1-1-8::, we noted that a procedure used as an element
-in creating a more complex procedure could be regarded not only as a
-collection of particular operations but also as a procedural
-abstraction. That is, the details of how the procedure was implemented
-could be suppressed, and the particular procedure itself could be
-replaced by any other procedure with the same overall behavior. In
-other words, we could make an abstraction that would separate the way
-the procedure would be used from the details of how the procedure would
-be implemented in terms of more primitive procedures. The analogous
-notion for compound data is called "data abstraction". Data
-abstraction is a methodology that enables us to isolate how a compound
-data object is used from the details of how it is constructed from more
-primitive data objects.
-
- The basic idea of data abstraction is to structure the programs that
-are to use compound data objects so that they operate on "abstract
-data." That is, our programs should use data in such a way as to make
-no assumptions about the data that are not strictly necessary for
-performing the task at hand. At the same time, a "concrete" data
-representation is defined independent of the programs that use the
-data. The interface between these two parts of our system will be a
-set of procedures, called "selectors" and "constructors", that
-implement the abstract data in terms of the concrete representation. To
-illustrate this technique, we will consider how to design a set of
-procedures for manipulating rational numbers.
-
-* Menu:
-
-* 2-1-1:: Example: Arithmetic Operations for Rational Numbers
-* 2-1-2:: Abstraction Barriers
-* 2-1-3:: What Is Meant by Data?
-* 2-1-4:: Extended Exercise: Interval Arithmetic
-
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-File: sicp.info, Node: 2-1-1, Next: 2-1-2, Prev: 2-1, Up: 2-1
-
-2.1.1 Example: Arithmetic Operations for Rational Numbers
----------------------------------------------------------
-
-Suppose we want to do arithmetic with rational numbers. We want to be
-able to add, subtract, multiply, and divide them and to test whether
-two rational numbers are equal.
-
- Let us begin by assuming that we already have a way of constructing
-a rational number from a numerator and a denominator. We also assume
-that, given a rational number, we have a way of extracting (or
-selecting) its numerator and its denominator. Let us further assume
-that the constructor and selectors are available as procedures:
-
- * `(make-rat <N> <D>)' returns therational number whose numerator is
- the integer `<N>' and whose denominator is the integer `<D>'.
-
- * `(numer <X>)' returns the numerator of the rational number `<X>'.
-
- * `(denom <X>)' returns the denominator of the rational number `<X>'.
-
-
- We are using here a powerful strategy of synthesis: "wishful
-thinking". We haven't yet said how a rational number is represented,
-or how the procedures `numer', `denom', and `make-rat' should be
-implemented. Even so, if we did have these three procedures, we could
-then add, subtract, multiply, divide, and test equality by using the
-following relations:
-
- n_1 n_2 n_1 d_2 + n_2 d_1
- --- + --- = -----------------
- d_1 d_2 d_1 d_2
-
- n_1 n_2 n_1 d_2 - n_2 d_1
- --- - --- = -----------------
- d_1 d_2 d_1 d_2
-
- n_1 n_2 n_1 n_2
- --- * --- = -------
- d_1 d_2 d_1 d_2
-
- n_1 / d_1 n_1 d_2
- --------- = -------
- n_2 / d_2 d_1 n_2
-
- n_1 n_2
- --- = --- if and only if n_1 d_2 = n_2 d_1
- d_1 d_2
-
-We can express these rules as procedures:
-
- (define (add-rat x y)
- (make-rat (+ (* (numer x) (denom y))
- (* (numer y) (denom x)))
- (* (denom x) (denom y))))
-
- (define (sub-rat x y)
- (make-rat (- (* (numer x) (denom y))
- (* (numer y) (denom x)))
- (* (denom x) (denom y))))
-
- (define (mul-rat x y)
- (make-rat (* (numer x) (numer y))
- (* (denom x) (denom y))))
-
- (define (div-rat x y)
- (make-rat (* (numer x) (denom y))
- (* (denom x) (numer y))))
-
- (define (equal-rat? x y)
- (= (* (numer x) (denom y))
- (* (numer y) (denom x))))
-
- Now we have the operations on rational numbers defined in terms of
-the selector and constructor procedures `numer', `denom', and
-`make-rat'. But we haven't yet defined these. What we need is some
-way to glue together a numerator and a denominator to form a rational
-number.
-
-Pairs
-.....
-
-To enable us to implement the concrete level of our data abstraction,
-our language provides a compound structure called a "pair", which can be
-constructed with the primitive procedure `cons'. This procedure takes
-two arguments and returns a compound data object that contains the two
-arguments as parts. Given a pair, we can extract the parts using the
-primitive procedures `car' and `cdr'.(1) Thus, we can use `cons',
-`car', and `cdr' as follows:
-
- (define x (cons 1 2))
-
- (car x)
- 1
-
- (cdr x)
- 2
-
- Notice that a pair is a data object that can be given a name and
-manipulated, just like a primitive data object. Moreover, `cons' can
-be used to form pairs whose elements are pairs, and so on:
-
- (define x (cons 1 2))
-
- (define y (cons 3 4))
-
- (define z (cons x y))
-
- (car (car z))
- 1
-
- (car (cdr z))
- 3
-
- In section *Note 2-2:: we will see how this ability to combine pairs
-means that pairs can be used as general-purpose building blocks to
-create all sorts of complex data structures. The single compound-data
-primitive "pair", implemented by the procedures `cons', `car', and
-`cdr', is the only glue we need. Data objects constructed from pairs
-are called "list-structured" data.
-
-Representing rational numbers
-.............................
-
-Pairs offer a natural way to complete the rational-number system.
-Simply represent a rational number as a pair of two integers: a
-numerator and a denominator. Then `make-rat', `numer', and `denom' are
-readily implemented as follows:(2)
-
- (define (make-rat n d) (cons n d))
-
- (define (numer x) (car x))
-
- (define (denom x) (cdr x))
-
- Also, in order to display the results of our computations, we can
-print rational numbers by printing the numerator, a slash, and the
-denominator:(3)
-
- (define (print-rat x)
- (newline)
- (display (numer x))
- (display "/")
- (display (denom x)))
-
- Now we can try our rational-number procedures:
-
- (define one-half (make-rat 1 2))
-
- (print-rat one-half)
- 1/2
-
- (define one-third (make-rat 1 3))
-
- (print-rat (add-rat one-half one-third))
- 5/6
-
- (print-rat (mul-rat one-half one-third))
- 1/6
-
- (print-rat (add-rat one-third one-third))
- 6/9
-
- As the final example shows, our rational-number implementation does
-not reduce rational numbers to lowest terms. We can remedy this by
-changing `make-rat'. If we have a `gcd' procedure like the one in
-section *Note 1-2-5:: that produces the greatest common divisor of two
-integers, we can use `gcd' to reduce the numerator and the denominator
-to lowest terms before constructing the pair:
-
- (define (make-rat n d)
- (let ((g (gcd n d)))
- (cons (/ n g) (/ d g))))
-
- Now we have
-
- (print-rat (add-rat one-third one-third))
- 2/3
-
-as desired. This modification was accomplished by changing the
-constructor `make-rat' without changing any of the procedures (such as
-`add-rat' and `mul-rat') that implement the actual operations.
-
- *Exercise 2.1:* Define a better version of `make-rat' that handles
- both positive and negative arguments. `Make-rat' should normalize
- the sign so that if the rational number is positive, both the
- numerator and denominator are positive, and if the rational number
- is negative, only the numerator is negative.
-
- ---------- Footnotes ----------
-
- (1) The name `cons' stands for "construct." The names `car' and
-`cdr' derive from the original implementation of Lisp on the IBM 704.
-That machine had an addressing scheme that allowed one to reference the
-"address" and "decrement" parts of a memory location. `Car' stands for
-"Contents of Address part of Register" and `cdr' (pronounced
-"could-er") stands for "Contents of Decrement part of Register."
-
- (2) Another way to define the selectors and constructor is
-
- (define make-rat cons)
- (define numer car)
- (define denom cdr)
-
- The first definition associates the name `make-rat' with the value
-of the expression `cons', which is the primitive procedure that
-constructs pairs. Thus `make-rat' and `cons' are names for the same
-primitive constructor.
-
- Defining selectors and constructors in this way is efficient:
-Instead of `make-rat' _calling_ `cons', `make-rat' _is_ `cons', so
-there is only one procedure called, not two, when `make-rat' is called.
-On the other hand, doing this defeats debugging aids that trace
-procedure calls or put breakpoints on procedure calls: You may want to
-watch `make-rat' being called, but you certainly don't want to watch
-every call to `cons'.
-
- We have chosen not to use this style of definition in this book.
-
- (3) `Display' is the Scheme primitive for printing data. The Scheme
-primitive `newline' starts a new line for printing. Neither of these
-procedures returns a useful value, so in the uses of `print-rat' below,
-we show only what `print-rat' prints, not what the interpreter prints
-as the value returned by `print-rat'.
-
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-
-2.1.2 Abstraction Barriers
---------------------------
-
-Before continuing with more examples of compound data and data
-abstraction, let us consider some of the issues raised by the
-rational-number example. We defined the rational-number operations in
-terms of a constructor `make-rat' and selectors `numer' and `denom'.
-In general, the underlying idea of data abstraction is to identify for
-each type of data object a basic set of operations in terms of which
-all manipulations of data objects of that type will be expressed, and
-then to use only those operations in manipulating the data.
-
- We can envision the structure of the rational-number system as shown
-in figure *Note Figure 2-1::. The horizontal lines represent barriers
-"abstraction barriers" that isolate different "levels" of the system.
-At each level, the barrier separates the programs (above) that use the
-data abstraction from the programs (below) that implement the data
-abstraction. Programs that use rational numbers manipulate them solely
-in terms of the procedures supplied "for public use" by the
-rational-number package: `add-rat', `sub-rat', `mul-rat', `div-rat',
-and `equal-rat?'. These, in turn, are implemented solely in terms of
-the constructor and selectors `make-rat', `numer', and `denom', which
-themselves are implemented in terms of pairs. The details of how pairs
-are implemented are irrelevant to the rest of the rational-number
-package so long as pairs can be manipulated by the use of `cons',
-`car', and `cdr'. In effect, procedures at each level are the
-interfaces that define the abstraction barriers and connect the
-different levels.
-
- *Figure 2.1:* Data-abstraction barriers in the rational-number
- package.
-
- +------------------------------------+
- --------| Programs that use rational numbers |--------
- +------------------------------------+
- Rational numbers in promblem domain
- +---------------------------+
- ------------| add-rat sub-rat ... |-------------
- +---------------------------+
- Rational numbers as numerators and denominators
- +------------------------+
- --------------| make-rat numer denom |--------------
- +------------------------+
- Rational numbers as pairs
- +----------------+
- ------------------| cons car cdr |------------------
- +----------------+
- However pairs are implemented
-
- This simple idea has many advantages. One advantage is that it
-makes programs much easier to maintain and to modify. Any complex data
-structure can be represented in a variety of ways with the primitive
-data structures provided by a programming language. Of course, the
-choice of representation influences the programs that operate on it;
-thus, if the representation were to be changed at some later time, all
-such programs might have to be modified accordingly. This task could
-be time-consuming and expensive in the case of large programs unless
-the dependence on the representation were to be confined by design to a
-very few program modules.
-
- For example, an alternate way to address the problem of reducing
-rational numbers to lowest terms is to perform the reduction whenever
-we access the parts of a rational number, rather than when we construct
-it. This leads to different constructor and selector procedures:
-
- (define (make-rat n d)
- (cons n d))
-
- (define (numer x)
- (let ((g (gcd (car x) (cdr x))))
- (/ (car x) g)))
-
- (define (denom x)
- (let ((g (gcd (car x) (cdr x))))
- (/ (cdr x) g)))
-
- The difference between this implementation and the previous one lies
-in when we compute the `gcd'. If in our typical use of rational
-numbers we access the numerators and denominators of the same rational
-numbers many times, it would be preferable to compute the `gcd' when
-the rational numbers are constructed. If not, we may be better off
-waiting until access time to compute the `gcd'. In any case, when we
-change from one representation to the other, the procedures `add-rat',
-`sub-rat', and so on do not have to be modified at all.
-
- Constraining the dependence on the representation to a few interface
-procedures helps us design programs as well as modify them, because it
-allows us to maintain the flexibility to consider alternate
-implementations. To continue with our simple example, suppose we are
-designing a rational-number package and we can't decide initially
-whether to perform the `gcd' at construction time or at selection time.
-The data-abstraction methodology gives us a way to defer that decision
-without losing the ability to make progress on the rest of the system.
-
- *Exercise 2.2:* Consider the problem of representing line segments
- in a plane. Each segment is represented as a pair of points: a
- starting point and an ending point. Define a constructor
- `make-segment' and selectors `start-segment' and `end-segment'
- that define the representation of segments in terms of points.
- Furthermore, a point can be represented as a pair of numbers: the
- x coordinate and the y coordinate. Accordingly, specify a
- constructor `make-point' and selectors `x-point' and `y-point'
- that define this representation. Finally, using your selectors
- and constructors, define a procedure `midpoint-segment' that takes
- a line segment as argument and returns its midpoint (the point
- whose coordinates are the average of the coordinates of the
- endpoints). To try your procedures, you'll need a way to print
- points:
-
- (define (print-point p)
- (newline)
- (display "(")
- (display (x-point p))
- (display ",")
- (display (y-point p))
- (display ")"))
-
- *Exercise 2.3:* Implement a representation for rectangles in a
- plane. (Hint: You may want to make use of *Note Exercise 2-2::.)
- In terms of your constructors and selectors, create procedures
- that compute the perimeter and the area of a given rectangle. Now
- implement a different representation for rectangles. Can you
- design your system with suitable abstraction barriers, so that the
- same perimeter and area procedures will work using either
- representation?
-
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-
-2.1.3 What Is Meant by Data?
-----------------------------
-
-We began the rational-number implementation in section *Note 2-1-1:: by
-implementing the rational-number operations `add-rat', `sub-rat', and
-so on in terms of three unspecified procedures: `make-rat', `numer',
-and `denom'. At that point, we could think of the operations as being
-defined in terms of data objects--numerators, denominators, and rational
-numbers--whose behavior was specified by the latter three procedures.
-
- But exactly what is meant by "data"? It is not enough to say
-"whatever is implemented by the given selectors and constructors."
-Clearly, not every arbitrary set of three procedures can serve as an
-appropriate basis for the rational-number implementation. We need to
-guarantee that, if we construct a rational number `x' from a pair of
-integers `n' and `d', then extracting the `numer' and the `denom' of
-`x' and dividing them should yield the same result as dividing `n' by
-`d'. In other words, `make-rat', `numer', and `denom' must satisfy the
-condition that, for any integer `n' and any non-zero integer `d', if
-`x' is (`make-rat n d'), then
-
- (numer x) n
- --------- = ---
- (denom x) d
-
- In fact, this is the only condition `make-rat', `numer', and `denom'
-must fulfill in order to form a suitable basis for a rational-number
-representation. In general, we can think of data as defined by some
-collection of selectors and constructors, together with specified
-conditions that these procedures must fulfill in order to be a valid
-representation.(1)
-
- This point of view can serve to define not only "high-level" data
-objects, such as rational numbers, but lower-level objects as well.
-Consider the notion of a pair, which we used in order to define our
-rational numbers. We never actually said what a pair was, only that
-the language supplied procedures `cons', `car', and `cdr' for operating
-on pairs. But the only thing we need to know about these three
-operations is that if we glue two objects together using `cons' we can
-retrieve the objects using `car' and `cdr'. That is, the operations
-satisfy the condition that, for any objects `x' and `y', if `z' is
-`(cons x y)' then `(car z)' is `x' and `(cdr z)' is `y'. Indeed, we
-mentioned that these three procedures are included as primitives in our
-language. However, any triple of procedures that satisfies the above
-condition can be used as the basis for implementing pairs. This point
-is illustrated strikingly by the fact that we could implement `cons',
-`car', and `cdr' without using any data structures at all but only
-using procedures. Here are the definitions:
-
- (define (cons x y)
- (define (dispatch m)
- (cond ((= m 0) x)
- ((= m 1) y)
- (else (error "Argument not 0 or 1 -- CONS" m))))
- dispatch)
-
- (define (car z) (z 0))
-
- (define (cdr z) (z 1))
-
- This use of procedures corresponds to nothing like our intuitive
-notion of what data should be. Nevertheless, all we need to do to show
-that this is a valid way to represent pairs is to verify that these
-procedures satisfy the condition given above.
-
- The subtle point to notice is that the value returned by `(cons x
-y)' is a procedure--namely the internally defined procedure `dispatch',
-which takes one argument and returns either `x' or `y' depending on
-whether the argument is 0 or 1. Correspondingly, `(car z)' is defined
-to apply `z' to 0. Hence, if `z' is the procedure formed by `(cons x
-y)', then `z' applied to 0 will yield `x'. Thus, we have shown that
-`(car (cons x y))' yields `x', as desired. Similarly, `(cdr (cons x
-y))' applies the procedure returned by `(cons x y)' to 1, which returns
-`y'. Therefore, this procedural implementation of pairs is a valid
-implementation, and if we access pairs using only `cons', `car', and
-`cdr' we cannot distinguish this implementation from one that uses
-"real" data structures.
-
- The point of exhibiting the procedural representation of pairs is
-not that our language works this way (Scheme, and Lisp systems in
-general, implement pairs directly, for efficiency reasons) but that it
-could work this way. The procedural representation, although obscure,
-is a perfectly adequate way to represent pairs, since it fulfills the
-only conditions that pairs need to fulfill. This example also
-demonstrates that the ability to manipulate procedures as objects
-automatically provides the ability to represent compound data. This
-may seem a curiosity now, but procedural representations of data will
-play a central role in our programming repertoire. This style of
-programming is often called "message passing", and we will be using it
-as a basic tool in *Note Chapter 3:: when we address the issues of
-modeling and simulation.
-
- *Exercise 2.4:* Here is an alternative procedural representation
- of pairs. For this representation, verify that `(car (cons x y))'
- yields `x' for any objects `x' and `y'.
-
- (define (cons x y)
- (lambda (m) (m x y)))
-
- (define (car z)
- (z (lambda (p q) p)))
-
- What is the corresponding definition of `cdr'? (Hint: To verify
- that this works, make use of the substitution model of section
- *Note 1-1-5::.)
-
- *Exercise 2.5:* Show that we can represent pairs of nonnegative
- integers using only numbers and arithmetic operations if we
- represent the pair a and b as the integer that is the product 2^a
- 3^b. Give the corresponding definitions of the procedures `cons',
- `car', and `cdr'.
-
- *Exercise 2.6:* In case representing pairs as procedures wasn't
- mind-boggling enough, consider that, in a language that can
- manipulate procedures, we can get by without numbers (at least
- insofar as nonnegative integers are concerned) by implementing 0
- and the operation of adding 1 as
-
- (define zero (lambda (f) (lambda (x) x)))
-
- (define (add-1 n)
- (lambda (f) (lambda (x) (f ((n f) x)))))
-
- This representation is known as "Church numerals", after its
- inventor, Alonzo Church, the logician who invented the [lambda]
- calculus.
-
- Define `one' and `two' directly (not in terms of `zero' and
- `add-1'). (Hint: Use substitution to evaluate `(add-1 zero)').
- Give a direct definition of the addition procedure `+' (not in
- terms of repeated application of `add-1').
-
- ---------- Footnotes ----------
-
- (1) Surprisingly, this idea is very difficult to formulate
-rigorously. There are two approaches to giving such a formulation. One,
-pioneered by C. A. R. Hoare (1972), is known as the method of models
-"abstract models". It formalizes the "procedures plus conditions"
-specification as outlined in the rational-number example above. Note
-that the condition on the rational-number representation was stated in
-terms of facts about integers (equality and division). In general,
-abstract models define new kinds of data objects in terms of previously
-defined types of data objects. Assertions about data objects can
-therefore be checked by reducing them to assertions about previously
-defined data objects. Another approach, introduced by Zilles at MIT,
-by Goguen, Thatcher, Wagner, and Wright at IBM (see Thatcher, Wagner,
-and Wright 1978), and by Guttag at Toronto (see Guttag 1977), is called "algebraic
-specification". It regards the "procedures" as elements of an abstract
-algebraic system whose behavior is specified by axioms that correspond
-to our "conditions," and uses the techniques of abstract algebra to
-check assertions about data objects. Both methods are surveyed in the
-paper by Liskov and Zilles (1975).
-
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-
-2.1.4 Extended Exercise: Interval Arithmetic
---------------------------------------------
-
-Alyssa P. Hacker is designing a system to help people solve engineering
-problems. One feature she wants to provide in her system is the
-ability to manipulate inexact quantities (such as measured parameters
-of physical devices) with known precision, so that when computations
-are done with such approximate quantities the results will be numbers
-of known precision.
-
- Electrical engineers will be using Alyssa's system to compute
-electrical quantities. It is sometimes necessary for them to compute
-the value of a parallel equivalent resistance R_p of two resistors R_1
-and R_2 using the formula
-
- 1
- R_p = -------------
- 1/R_1 + 1/R_2
-
- Resistance values are usually known only up to some tolerance
-guaranteed by the manufacturer of the resistor. For example, if you
-buy a resistor labeled "6.8 ohms with 10% tolerance" you can only be
-sure that the resistor has a resistance between 6.8 - 0.68 = 6.12 and
-6.8 + 0.68 = 7.48 ohms. Thus, if you have a 6.8-ohm 10% resistor in
-parallel with a 4.7-ohm 5% resistor, the resistance of the combination
-can range from about 2.58 ohms (if the two resistors are at the lower
-bounds) to about 2.97 ohms (if the two resistors are at the upper
-bounds).
-
- Alyssa's idea is to implement "interval arithmetic" as a set of
-arithmetic operations for combining "intervals" (objects that represent
-the range of possible values of an inexact quantity). The result of
-adding, subtracting, multiplying, or dividing two intervals is itself
-an interval, representing the range of the result.
-
- Alyssa postulates the existence of an abstract object called an
-"interval" that has two endpoints: a lower bound and an upper bound.
-She also presumes that, given the endpoints of an interval, she can
-construct the interval using the data constructor `make-interval'.
-Alyssa first writes a procedure for adding two intervals. She reasons
-that the minimum value the sum could be is the sum of the two lower
-bounds and the maximum value it could be is the sum of the two upper
-bounds:
-
- (define (add-interval x y)
- (make-interval (+ (lower-bound x) (lower-bound y))
- (+ (upper-bound x) (upper-bound y))))
-
- Alyssa also works out the product of two intervals by finding the
-minimum and the maximum of the products of the bounds and using them as
-the bounds of the resulting interval. (`Min' and `max' are primitives
-that find the minimum or maximum of any number of arguments.)
-
- (define (mul-interval x y)
- (let ((p1 (* (lower-bound x) (lower-bound y)))
- (p2 (* (lower-bound x) (upper-bound y)))
- (p3 (* (upper-bound x) (lower-bound y)))
- (p4 (* (upper-bound x) (upper-bound y))))
- (make-interval (min p1 p2 p3 p4)
- (max p1 p2 p3 p4))))
-
- To divide two intervals, Alyssa multiplies the first by the
-reciprocal of the second. Note that the bounds of the reciprocal
-interval are the reciprocal of the upper bound and the reciprocal of
-the lower bound, in that order.
-
- (define (div-interval x y)
- (mul-interval x
- (make-interval (/ 1.0 (upper-bound y))
- (/ 1.0 (lower-bound y)))))
-
- *Exercise 2.7:* Alyssa's program is incomplete because she has not
- specified the implementation of the interval abstraction. Here is
- a definition of the interval constructor:
-
- (define (make-interval a b) (cons a b))
-
- Define selectors `upper-bound' and `lower-bound' to complete the
- implementation.
-
- *Exercise 2.8:* Using reasoning analogous to Alyssa's, describe
- how the difference of two intervals may be computed. Define a
- corresponding subtraction procedure, called `sub-interval'.
-
- *Exercise 2.9:* The "width" of an interval is half of the
- difference between its upper and lower bounds. The width is a
- measure of the uncertainty of the number specified by the
- interval. For some arithmetic operations the width of the result
- of combining two intervals is a function only of the widths of the
- argument intervals, whereas for others the width of the
- combination is not a function of the widths of the argument
- intervals. Show that the width of the sum (or difference) of two
- intervals is a function only of the widths of the intervals being
- added (or subtracted). Give examples to show that this is not
- true for multiplication or division.
-
- *Exercise 2.10:* Ben Bitdiddle, an expert systems programmer,
- looks over Alyssa's shoulder and comments that it is not clear what
- it means to divide by an interval that spans zero. Modify
- Alyssa's code to check for this condition and to signal an error
- if it occurs.
-
- *Exercise 2.11:* In passing, Ben also cryptically comments: "By
- testing the signs of the endpoints of the intervals, it is
- possible to break `mul-interval' into nine cases, only one of which
- requires more than two multiplications." Rewrite this procedure
- using Ben's suggestion.
-
- After debugging her program, Alyssa shows it to a potential user,
- who complains that her program solves the wrong problem. He wants
- a program that can deal with numbers represented as a center value
- and an additive tolerance; for example, he wants to work with
- intervals such as 3.5 +/- 0.15 rather than [3.35, 3.65]. Alyssa
- returns to her desk and fixes this problem by supplying an
- alternate constructor and alternate selectors:
-
- (define (make-center-width c w)
- (make-interval (- c w) (+ c w)))
-
- (define (center i)
- (/ (+ (lower-bound i) (upper-bound i)) 2))
-
- (define (width i)
- (/ (- (upper-bound i) (lower-bound i)) 2))
-
- Unfortunately, most of Alyssa's users are engineers. Real
- engineering situations usually involve measurements with only a
- small uncertainty, measured as the ratio of the width of the
- interval to the midpoint of the interval. Engineers usually
- specify percentage tolerances on the parameters of devices, as in
- the resistor specifications given earlier.
-
- *Exercise 2.12:* Define a constructor `make-center-percent' that
- takes a center and a percentage tolerance and produces the desired
- interval. You must also define a selector `percent' that produces
- the percentage tolerance for a given interval. The `center'
- selector is the same as the one shown above.
-
- *Exercise 2.13:* Show that under the assumption of small
- percentage tolerances there is a simple formula for the approximate
- percentage tolerance of the product of two intervals in terms of
- the tolerances of the factors. You may simplify the problem by
- assuming that all numbers are positive.
-
- After considerable work, Alyssa P. Hacker delivers her finished
- system. Several years later, after she has forgotten all about
- it, she gets a frenzied call from an irate user, Lem E. Tweakit.
- It seems that Lem has noticed that the formula for parallel
- resistors can be written in two algebraically equivalent ways:
-
- R_1 R_2
- ---------
- R_1 + R_2
-
- and
-
- 1
- -------------
- 1/R_1 + 1/R_2
-
- He has written the following two programs, each of which computes
- the parallel-resistors formula differently:
-
- (define (par1 r1 r2)
- (div-interval (mul-interval r1 r2)
- (add-interval r1 r2)))
-
- (define (par2 r1 r2)
- (let ((one (make-interval 1 1)))
- (div-interval one
- (add-interval (div-interval one r1)
- (div-interval one r2)))))
-
- Lem complains that Alyssa's program gives different answers for
- the two ways of computing. This is a serious complaint.
-
- *Exercise 2.14:* Demonstrate that Lem is right. Investigate the
- behavior of the system on a variety of arithmetic expressions.
- Make some intervals A and B, and use them in computing the
- expressions A/A and A/B. You will get the most insight by using
- intervals whose width is a small percentage of the center value.
- Examine the results of the computation in center-percent form (see
- *Note Exercise 2-12::).
-
- *Exercise 2.15:* Eva Lu Ator, another user, has also noticed the
- different intervals computed by different but algebraically
- equivalent expressions. She says that a formula to compute with
- intervals using Alyssa's system will produce tighter error bounds
- if it can be written in such a form that no variable that
- represents an uncertain number is repeated. Thus, she says,
- `par2' is a "better" program for parallel resistances than `par1'.
- Is she right? Why?
-
- *Exercise 2.16:* Explain, in general, why equivalent algebraic
- expressions may lead to different answers. Can you devise an
- interval-arithmetic package that does not have this shortcoming,
- or is this task impossible? (Warning: This problem is very
- difficult.)
-
-\1f
-File: sicp.info, Node: 2-2, Next: 2-3, Prev: 2-1, Up: Chapter 2
-
-2.2 Hierarchical Data and the Closure Property
-==============================================
-
-As we have seen, pairs provide a primitive "glue" that we can use to
-construct compound data objects. *Note Figure 2-2:: shows a standard
-way to visualize a pair--in this case, the pair formed by `(cons 1 2)'.
-In this representation, which is called "box-and-pointer notation",
-each object is shown as a "pointer" to a box. The box for a primitive
-object contains a representation of the object. For example, the box
-for a number contains a numeral. The box for a pair is actually a
-double box, the left part containing (a pointer to) the `car' of the
-pair and the right part containing the `cdr'.
-
- *Figure 2.2:* Box-and-pointer representation of `(cons 1 2)'.
-
- +---+---+ +---+
- ---->| * | *-+---->| 2 |
- +-|-+---+ +---+
- |
- V
- +---+
- | 1 |
- +---+
-
- We have already seen that `cons' can be used to combine not only
-numbers but pairs as well. (You made use of this fact, or should have,
-in doing *Note Exercise 2-2:: and *Note Exercise 2-3::.) As a
-consequence, pairs provide a universal building block from which we can
-construct all sorts of data structures. *Note Figure 2-3:: shows two
-ways to use pairs to combine the numbers 1, 2, 3, and 4.
-
- *Figure 2.3:* Two ways to combine 1, 2, 3, and 4 using pairs.
-
- +---+---+ +---+---+ +---+---+ +---+
- ---->| * | *-+---->| * | * | ---->| * | *-+---->| 4 |
- +-|-+---+ +-|-+-|-+ +-|-+---+ +---+
- | | | |
- V V V V
- +---+---+ +---+ +---+ +---+---+ +---+---+
- | * | * | | 3 | | 4 | | * | *-+---->| * | * |
- +-|-+-|-+ +---+ +---+ +-|-+---+ +-|-+-|-+
- | | | | |
- V V V V V
- +---+ +---+ +---+ +---+ +---+
- | 1 | | 2 | | 1 | | 2 | | 3 |
- +---+ +---+ +---+ +---+ +---+
-
- (cons (cons 1 2) (cons (cons 1
- (cons 3 4)) (cons 2 3))
- 4)
-
- The ability to create pairs whose elements are pairs is the essence
-of list structure's importance as a representational tool. We refer to
-this ability as the "closure property" of `cons'. In general, an
-operation for combining data objects satisfies the closure property if
-the results of combining things with that operation can themselves be
-combined using the same operation.(1) Closure is the key to power in
-any means of combination because it permits us to create "hierarchical"
-structures--structures made up of parts, which themselves are made up
-of parts, and so on.
-
- From the outset of *Note Chapter 1::, we've made essential use of
-closure in dealing with procedures, because all but the very simplest
-programs rely on the fact that the elements of a combination can
-themselves be combinations. In this section, we take up the
-consequences of closure for compound data. We describe some
-conventional techniques for using pairs to represent sequences and
-trees, and we exhibit a graphics language that illustrates closure in a
-vivid way.(2)
-
-* Menu:
-
-* 2-2-1:: Representing Sequences
-* 2-2-2:: Hierarchical Structures
-* 2-2-3:: Sequences as Conventional Interfaces
-* 2-2-4:: Example: A Picture Language
-
- ---------- Footnotes ----------
-
- (1) The use of the word "closure" here comes from abstract algebra,
-where a set of elements is said to be closed under an operation if
-applying the operation to elements in the set produces an element that
-is again an element of the set. The Lisp community also
-(unfortunately) uses the word "closure" to describe a totally unrelated
-concept: A closure is an implementation technique for representing
-procedures with free variables. We do not use the word "closure" in
-this second sense in this book.
-
- (2) The notion that a means of combination should satisfy closure is
-a straightforward idea. Unfortunately, the data combiners provided in
-many popular programming languages do not satisfy closure, or make
-closure cumbersome to exploit. In Fortran or Basic, one typically
-combines data elements by assembling them into arrays--but one cannot
-form arrays whose elements are themselves arrays. Pascal and C admit
-structures whose elements are structures. However, this requires that
-the programmer manipulate pointers explicitly, and adhere to the
-restriction that each field of a structure can contain only elements of
-a prespecified form. Unlike Lisp with its pairs, these languages have
-no built-in general-purpose glue that makes it easy to manipulate
-compound data in a uniform way. This limitation lies behind Alan
-Perlis's comment in his foreword to this book: "In Pascal the plethora
-of declarable data structures induces a specialization within functions
-that inhibits and penalizes casual cooperation. It is better to have
-100 functions operate on one data structure than to have 10 functions
-operate on 10 data structures."
-
-\1f
-File: sicp.info, Node: 2-2-1, Next: 2-2-2, Prev: 2-2, Up: 2-2
-
-2.2.1 Representing Sequences
-----------------------------
-
- *Figure 2.4:* The sequence 1, 2, 3, 4 represented as a chain of
- pairs.
-
- +---+---+ +---+---+ +---+---+ +---+---+
- ---->| * | *-+---->| * | *-+---->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
- | | | |
- V V V V
- +---+ +---+ +---+ +---+
- | 1 | | 2 | | 3 | | 4 |
- +---+ +---+ +---+ +---+
-
-One of the useful structures we can build with pairs is a "sequence"--an
-ordered collection of data objects. There are, of course, many ways to
-represent sequences in terms of pairs. One particularly
-straightforward representation is illustrated in *Note Figure 2-4::,
-where the sequence 1, 2, 3, 4 is represented as a chain of pairs. The
-`car' of each pair is the corresponding item in the chain, and the
-`cdr' of the pair is the next pair in the chain. The `cdr' of the
-final pair signals the end of the sequence by pointing to a
-distinguished value that is not a pair, represented in box-and-pointer
-diagrams as a diagonal line and in programs as the value of the
-variable `nil'. The entire sequence is constructed by nested `cons'
-operations:
-
- (cons 1
- (cons 2
- (cons 3
- (cons 4 nil))))
-
- Such a sequence of pairs, formed by nested `cons'es, is called a "list",
-and Scheme provides a primitive called `list' to help in constructing
-lists.(1) The above sequence could be produced by `(list 1 2 3 4)'.
-In general,
-
- (list <A_1> <A_2> ... <A_N>)
-
-is equivalent to
-
- (cons <A_1>
- (cons <A_2>
- (cons ...
- (cons <A_N>
- nil)
- ...)))
-
- Lisp systems conventionally print lists by printing the sequence of
-elements, enclosed in parentheses. Thus, the data object in *Note
-Figure 2-4:: is printed as `(1 2 3 4)':
-
- (define one-through-four (list 1 2 3 4))
-
- one-through-four
- (1 2 3 4)
-
- Be careful not to confuse the expression `(list 1 2 3 4)' with the
-list `(1 2 3 4)', which is the result obtained when the expression is
-evaluated. Attempting to evaluate the expression `(1 2 3 4)' will
-signal an error when the interpreter tries to apply the procedure `1' to
-arguments `2', `3', and `4'.
-
- We can think of `car' as selecting the first item in the list, and of
-`cdr' as selecting the sublist consisting of all but the first item.
-Nested applications of `car' and `cdr' can be used to extract the
-second, third, and subsequent items in the list.(2) The constructor
-`cons' makes a list like the original one, but with an additional item
-at the beginning.
-
- (car one-through-four)
- 1
-
- (cdr one-through-four)
- (2 3 4)
-
- (car (cdr one-through-four))
- 2
-
- (cons 10 one-through-four)
- (10 1 2 3 4)
-
- (cons 5 one-through-four)
- (5 1 2 3 4)
-
- The value of `nil', used to terminate the chain of pairs, can be
-thought of as a sequence of no elements, the "empty list". The word "nil"
-is a contraction of the Latin word _nihil_, which means "nothing."(3)
-
-List operations
-...............
-
-The use of pairs to represent sequences of elements as lists is
-accompanied by conventional programming techniques for manipulating
-lists by successively "`cdr'ing down" the lists. For example, the
-procedure `list-ref' takes as arguments a list and a number n and
-returns the nth item of the list. It is customary to number the
-elements of the list beginning with 0. The method for computing
-`list-ref' is the following:
-
- * For n = 0, `list-ref' should return the `car' of the list.
-
- * Otherwise, `list-ref' should return the (n - 1)st item of the
- `cdr' of the list.
-
-
- (define (list-ref items n)
- (if (= n 0)
- (car items)
- (list-ref (cdr items) (- n 1))))
-
- (define squares (list 1 4 9 16 25))
-
- (list-ref squares 3)
- 16
-
- Often we `cdr' down the whole list. To aid in this, Scheme includes
-a primitive predicate `null?', which tests whether its argument is the
-empty list. The procedure `length', which returns the number of items
-in a list, illustrates this typical pattern of use:
-
- (define (length items)
- (if (null? items)
- 0
- (+ 1 (length (cdr items)))))
-
- (define odds (list 1 3 5 7))
-
- (length odds)
- 4
-
- The `length' procedure implements a simple recursive plan. The
-reduction step is:
-
- * The `length' of any list is 1 plus the `length' of the `cdr' of
- the list.
-
-
- This is applied successively until we reach the base case:
-
- * The `length' of the empty list is 0.
-
-
- We could also compute `length' in an iterative style:
-
- (define (length items)
- (define (length-iter a count)
- (if (null? a)
- count
- (length-iter (cdr a) (+ 1 count))))
- (length-iter items 0))
-
- Another conventional programming technique is to "`cons' up" an
-answer list while `cdr'ing down a list, as in the procedure `append',
-which takes two lists as arguments and combines their elements to make
-a new list:
-
- (append squares odds)
- (1 4 9 16 25 1 3 5 7)
-
- (append odds squares)
- (1 3 5 7 1 4 9 16 25)
-
- `Append' is also implemented using a recursive plan. To `append'
-lists `list1' and `list2', do the following:
-
- * If `list1' is the empty list, then the result is just `list2'.
-
- * Otherwise, `append' the `cdr' of `list1' and `list2', and `cons'
- the `car' of `list1' onto the result:
-
-
- (define (append list1 list2)
- (if (null? list1)
- list2
- (cons (car list1) (append (cdr list1) list2))))
-
- *Exercise 2.17:* Define a procedure `last-pair' that returns the
- list that contains only the last element of a given (nonempty)
- list:
-
- (last-pair (list 23 72 149 34))
- (34)
-
- *Exercise 2.18:* Define a procedure `reverse' that takes a list as
- argument and returns a list of the same elements in reverse order:
-
- (reverse (list 1 4 9 16 25))
- (25 16 9 4 1)
-
- *Exercise 2.19:* Consider the change-counting program of section
- *Note 1-2-2::. It would be nice to be able to easily change the
- currency used by the program, so that we could compute the number
- of ways to change a British pound, for example. As the program is
- written, the knowledge of the currency is distributed partly into
- the procedure `first-denomination' and partly into the procedure
- `count-change' (which knows that there are five kinds of U.S.
- coins). It would be nicer to be able to supply a list of coins to
- be used for making change.
-
- We want to rewrite the procedure `cc' so that its second argument
- is a list of the values of the coins to use rather than an integer
- specifying which coins to use. We could then have lists that
- defined each kind of currency:
-
- (define us-coins (list 50 25 10 5 1))
-
- (define uk-coins (list 100 50 20 10 5 2 1 0.5))
-
- We could then call `cc' as follows:
-
- (cc 100 us-coins)
- 292
-
- To do this will require changing the program `cc' somewhat. It
- will still have the same form, but it will access its second
- argument differently, as follows:
-
- (define (cc amount coin-values)
- (cond ((= amount 0) 1)
- ((or (< amount 0) (no-more? coin-values)) 0)
- (else
- (+ (cc amount
- (except-first-denomination coin-values))
- (cc (- amount
- (first-denomination coin-values))
- coin-values)))))
-
- Define the procedures `first-denomination',
- `except-first-denomination', and `no-more?' in terms of primitive
- operations on list structures. Does the order of the list
- `coin-values' affect the answer produced by `cc'? Why or why not?
-
- *Exercise 2.20:* The procedures `+', `*', and `list' take
- arbitrary numbers of arguments. One way to define such procedures
- is to use `define' with notation "dotted-tail notation". In a
- procedure definition, a parameter list that has a dot before the
- last parameter name indicates that, when the procedure is called,
- the initial parameters (if any) will have as values the initial
- arguments, as usual, but the final parameter's value will be a "list"
- of any remaining arguments. For instance, given the definition
-
- (define (f x y . z) <BODY>)
-
- the procedure `f' can be called with two or more arguments. If we
- evaluate
-
- (f 1 2 3 4 5 6)
-
- then in the body of `f', `x' will be 1, `y' will be 2, and `z'
- will be the list `(3 4 5 6)'. Given the definition
-
- (define (g . w) <BODY>)
-
- the procedure `g' can be called with zero or more arguments. If we
- evaluate
-
- (g 1 2 3 4 5 6)
-
- then in the body of `g', `w' will be the list `(1 2 3 4 5 6)'.(4)
-
- Use this notation to write a procedure `same-parity' that takes
- one or more integers and returns a list of all the arguments that
- have the same even-odd parity as the first argument. For example,
-
- (same-parity 1 2 3 4 5 6 7)
- (1 3 5 7)
-
- (same-parity 2 3 4 5 6 7)
- (2 4 6)
-
-Mapping over lists
-..................
-
-One extremely useful operation is to apply some transformation to each
-element in a list and generate the list of results. For instance, the
-following procedure scales each number in a list by a given factor:
-
- (define (scale-list items factor)
- (if (null? items)
- nil
- (cons (* (car items) factor)
- (scale-list (cdr items) factor))))
-
- (scale-list (list 1 2 3 4 5) 10)
- (10 20 30 40 50)
-
- We can abstract this general idea and capture it as a common pattern
-expressed as a higher-order procedure, just as in section *Note 1-3::.
-The higher-order procedure here is called `map'. `Map' takes as
-arguments a procedure of one argument and a list, and returns a list of
-the results produced by applying the procedure to each element in the
-list:(5)
-
- (define (map proc items)
- (if (null? items)
- nil
- (cons (proc (car items))
- (map proc (cdr items)))))
-
- (map abs (list -10 2.5 -11.6 17))
- (10 2.5 11.6 17)
-
- (map (lambda (x) (* x x))
- (list 1 2 3 4))
- (1 4 9 16)
-
- Now we can give a new definition of `scale-list' in terms of `map':
-
- (define (scale-list items factor)
- (map (lambda (x) (* x factor))
- items))
-
- `Map' is an important construct, not only because it captures a
-common pattern, but because it establishes a higher level of
-abstraction in dealing with lists. In the original definition of
-`scale-list', the recursive structure of the program draws attention to
-the element-by-element processing of the list. Defining `scale-list'
-in terms of `map' suppresses that level of detail and emphasizes that
-scaling transforms a list of elements to a list of results. The
-difference between the two definitions is not that the computer is
-performing a different process (it isn't) but that we think about the
-process differently. In effect, `map' helps establish an abstraction
-barrier that isolates the implementation of procedures that transform
-lists from the details of how the elements of the list are extracted
-and combined. Like the barriers shown in *Note Figure 2-1::, this
-abstraction gives us the flexibility to change the low-level details of
-how sequences are implemented, while preserving the conceptual
-framework of operations that transform sequences to sequences. Section
-*Note 2-2-3:: expands on this use of sequences as a framework for
-organizing programs.
-
- *Exercise 2.21:* The procedure `square-list' takes a list of
- numbers as argument and returns a list of the squares of those
- numbers.
-
- (square-list (list 1 2 3 4))
- (1 4 9 16)
-
- Here are two different definitions of `square-list'. Complete
- both of them by filling in the missing expressions:
-
- (define (square-list items)
- (if (null? items)
- nil
- (cons <??> <??>)))
-
- (define (square-list items)
- (map <??> <??>))
-
- *Exercise 2.22:* Louis Reasoner tries to rewrite the first
- `square-list' procedure of *Note Exercise 2-21:: so that it
- evolves an iterative process:
-
- (define (square-list items)
- (define (iter things answer)
- (if (null? things)
- answer
- (iter (cdr things)
- (cons (square (car things))
- answer))))
- (iter items nil))
-
- Unfortunately, defining `square-list' this way produces the answer
- list in the reverse order of the one desired. Why?
-
- Louis then tries to fix his bug by interchanging the arguments to
- `cons':
-
- (define (square-list items)
- (define (iter things answer)
- (if (null? things)
- answer
- (iter (cdr things)
- (cons answer
- (square (car things))))))
- (iter items nil))
-
- This doesn't work either. Explain.
-
- *Exercise 2.23:* The procedure `for-each' is similar to `map'. It
- takes as arguments a procedure and a list of elements. However,
- rather than forming a list of the results, `for-each' just applies
- the procedure to each of the elements in turn, from left to right.
- The values returned by applying the procedure to the elements are
- not used at all--`for-each' is used with procedures that perform
- an action, such as printing. For example,
-
- (for-each (lambda (x) (newline) (display x))
- (list 57 321 88))
- 57
- 321
- 88
-
- The value returned by the call to `for-each' (not illustrated
- above) can be something arbitrary, such as true. Give an
- implementation of `for-each'.
-
- ---------- Footnotes ----------
-
- (1) In this book, we use "list" to mean a chain of pairs terminated
-by the end-of-list marker. In contrast, the term "list structure"
-refers to any data structure made out of pairs, not just to lists.
-
- (2) Since nested applications of `car' and `cdr' are cumbersome to
-write, Lisp dialects provide abbreviations for them--for instance,
-
- (cadr (ARG)) = (car (cdr (ARG)))
-
- The names of all such procedures start with `c' and end with `r'.
-Each `a' between them stands for a `car' operation and each `d' for a
-`cdr' operation, to be applied in the same order in which they appear
-in the name. The names `car' and `cdr' persist because simple
-combinations like `cadr' are pronounceable.
-
- (3) It's remarkable how much energy in the standardization of Lisp
-dialects has been dissipated in arguments that are literally over
-nothing: Should `nil' be an ordinary name? Should the value of `nil'
-be a symbol? Should it be a list? Should it be a pair? In Scheme,
-`nil' is an ordinary name, which we use in this section as a variable
-whose value is the end-of-list marker (just as `true' is an ordinary
-variable that has a true value). Other dialects of Lisp, including
-Common Lisp, treat `nil' as a special symbol. The authors of this
-book, who have endured too many language standardization brawls, would
-like to avoid the entire issue. Once we have introduced quotation in
-section *Note 2-3::, we will denote the empty list as `'()' and
-dispense with the variable `nil' entirely.
-
- (4) To define `f' and `g' using `lambda' we would write
-
- (define f (lambda (x y . z) <BODY>))
- (define g (lambda w <BODY>))
-
- (5) [Footnote 12] Scheme standardly provides a `map' procedure that
-is more general than the one described here. This more general `map'
-takes a procedure of n arguments, together with n lists, and applies
-the procedure to all the first elements of the lists, all the second
-elements of the lists, and so on, returning a list of the results. For
-example:
-
- (map + (list 1 2 3) (list 40 50 60) (list 700 800 900))
- (741 852 963)
-
- (map (lambda (x y) (+ x (* 2 y)))
- (list 1 2 3)
- (list 4 5 6))
- (9 12 15)
-
-\1f
-File: sicp.info, Node: 2-2-2, Next: 2-2-3, Prev: 2-2-1, Up: 2-2
-
-2.2.2 Hierarchical Structures
------------------------------
-
-The representation of sequences in terms of lists generalizes naturally
-to represent sequences whose elements may themselves be sequences. For
-example, we can regard the object `((1 2) 3 4)' constructed by
-
- (cons (list 1 2) (list 3 4))
-
-as a list of three items, the first of which is itself a list, `(1 2)'.
-Indeed, this is suggested by the form in which the result is printed by
-the interpreter. *Note Figure 2-5:: shows the representation of this
-structure in terms of pairs.
-
- *Figure 2.5:* Structure formed by `(cons (list 1 2) (list 3 4))'.
-
- (3 4)
- |
- V
- ((1 2) 3 4) +---+---+ +---+---+ +---+---+
- ---->| * | *-+----------------->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- | | |
- V V V
- (1 2) +---+---+ +---+---+ +---+ +---+
- ---->| * | *-+---->| * | / | | 3 | | 4 |
- +-|-+---+ +-|-+---+ +---+ +---+
- | |
- V V
- +---+ +---+
- | 1 | | 2 |
- +---+ +---+
-
- Another way to think of sequences whose elements are sequences is as "trees".
-The elements of the sequence are the branches of the tree, and
-elements that are themselves sequences are subtrees. *Note Figure 2-6::
-shows the structure in *Note Figure 2-5:: viewed as a tree.
-
- *Figure 2.6:* The list structure in *Note Figure 2-5:: viewed as a
- tree.
-
- ((1 2) 3 4)
- /\\
- / | \
- (1 2) 3 4
- / \
- 1 2
-
- Recursion is a natural tool for dealing with tree structures, since
-we can often reduce operations on trees to operations on their
-branches, which reduce in turn to operations on the branches of the
-branches, and so on, until we reach the leaves of the tree. As an
-example, compare the `length' procedure of section *Note 2-2-1:: with
-the `count-leaves' procedure, which returns the total number of leaves
-of a tree:
-
- (define x (cons (list 1 2) (list 3 4)))
-
- (length x)
- 3
-
- (count-leaves x)
- 4
-
- (list x x)
- (((1 2) 3 4) ((1 2) 3 4))
-
- (length (list x x))
- 2
-
- (count-leaves (list x x))
- 8
-
- To implement `count-leaves', recall the recursive plan for computing
-`length':
-
- * `Length' of a list `x' is 1 plus `length' of the `cdr' of `x'.
-
- * `Length' of the empty list is 0.
-
-
- `Count-leaves' is similar. The value for the empty list is the same:
-
- * `Count-leaves' of the empty list is 0.
-
-
- But in the reduction step, where we strip off the `car' of the list,
-we must take into account that the `car' may itself be a tree whose
-leaves we need to count. Thus, the appropriate reduction step is
-
- * `Count-leaves' of a tree `x' is `count-leaves' of the `car' of `x'
- plus `count-leaves' of the `cdr' of `x'.
-
-
- Finally, by taking `car's we reach actual leaves, so we need another
-base case:
-
- * `Count-leaves' of a leaf is 1.
-
-
- To aid in writing recursive procedures on trees, Scheme provides the
-primitive predicate `pair?', which tests whether its argument is a
-pair. Here is the complete procedure:(1)
-
- (define (count-leaves x)
- (cond ((null? x) 0)
- ((not (pair? x)) 1)
- (else (+ (count-leaves (car x))
- (count-leaves (cdr x))))))
-
- *Exercise 2.24:* Suppose we evaluate the expression `(list 1 (list
- 2 (list 3 4)))'. Give the result printed by the interpreter, the
- corresponding box-and-pointer structure, and the interpretation of
- this as a tree (as in *Note Figure 2-6::).
-
- *Exercise 2.25:* Give combinations of `car's and `cdr's that will
- pick 7 from each of the following lists:
-
- (1 3 (5 7) 9)
-
- ((7))
-
- (1 (2 (3 (4 (5 (6 7))))))
-
- *Exercise 2.26:* Suppose we define `x' and `y' to be two lists:
-
- (define x (list 1 2 3))
-
- (define y (list 4 5 6))
-
- What result is printed by the interpreter in response to
- evaluating each of the following expressions:
-
- (append x y)
-
- (cons x y)
-
- (list x y)
-
- *Exercise 2.27:* Modify your `reverse' procedure of *Note Exercise
- 2-18:: to produce a `deep-reverse' procedure that takes a list as
- argument and returns as its value the list with its elements
- reversed and with all sublists deep-reversed as well. For example,
-
- (define x (list (list 1 2) (list 3 4)))
-
- x
- ((1 2) (3 4))
-
- (reverse x)
- ((3 4) (1 2))
-
- (deep-reverse x)
- ((4 3) (2 1))
-
- *Exercise 2.28:* Write a procedure `fringe' that takes as argument
- a tree (represented as a list) and returns a list whose elements
- are all the leaves of the tree arranged in left-to-right order.
- For example,
-
- (define x (list (list 1 2) (list 3 4)))
-
- (fringe x)
- (1 2 3 4)
-
- (fringe (list x x))
- (1 2 3 4 1 2 3 4)
-
- *Exercise 2.29:* A binary mobile consists of two branches, a left
- branch and a right branch. Each branch is a rod of a certain
- length, from which hangs either a weight or another binary mobile.
- We can represent a binary mobile using compound data by
- constructing it from two branches (for example, using `list'):
-
- (define (make-mobile left right)
- (list left right))
-
- A branch is constructed from a `length' (which must be a number)
- together with a `structure', which may be either a number
- (representing a simple weight) or another mobile:
-
- (define (make-branch length structure)
- (list length structure))
-
- a. Write the corresponding selectors `left-branch' and
- `right-branch', which return the branches of a mobile, and
- `branch-length' and `branch-structure', which return the
- components of a branch.
-
- b. Using your selectors, define a procedure `total-weight' that
- returns the total weight of a mobile.
-
- c. A mobile is said to be "balanced" if the torque applied by
- its top-left branch is equal to that applied by its top-right
- branch (that is, if the length of the left rod multiplied by
- the weight hanging from that rod is equal to the
- corresponding product for the right side) and if each of the
- submobiles hanging off its branches is balanced. Design a
- predicate that tests whether a binary mobile is balanced.
-
- d. Suppose we change the representation of mobiles so that the
- constructors are
-
- (define (make-mobile left right)
- (cons left right))
-
- (define (make-branch length structure)
- (cons length structure))
-
- How much do you need to change your programs to convert to
- the new representation?
-
-
-Mapping over trees
-..................
-
-Just as `map' is a powerful abstraction for dealing with sequences,
-`map' together with recursion is a powerful abstraction for dealing with
-trees. For instance, the `scale-tree' procedure, analogous to
-`scale-list' of section *Note 2-2-1::, takes as arguments a numeric
-factor and a tree whose leaves are numbers. It returns a tree of the
-same shape, where each number is multiplied by the factor. The
-recursive plan for `scale-tree' is similar to the one for
-`count-leaves':
-
- (define (scale-tree tree factor)
- (cond ((null? tree) nil)
- ((not (pair? tree)) (* tree factor))
- (else (cons (scale-tree (car tree) factor)
- (scale-tree (cdr tree) factor)))))
-
- (scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7))
- 10)
- (10 (20 (30 40) 50) (60 70))
-
- Another way to implement `scale-tree' is to regard the tree as a
-sequence of sub-trees and use `map'. We map over the sequence, scaling
-each sub-tree in turn, and return the list of results. In the base
-case, where the tree is a leaf, we simply multiply by the factor:
-
- (define (scale-tree tree factor)
- (map (lambda (sub-tree)
- (if (pair? sub-tree)
- (scale-tree sub-tree factor)
- (* sub-tree factor)))
- tree))
-
- Many tree operations can be implemented by similar combinations of
-sequence operations and recursion.
-
- *Exercise 2.30:* Define a procedure `square-tree' analogous to the
- `square-list' procedure of *Note Exercise 2-21::. That is,
- `square-list' should behave as follows:
-
- (square-tree
- (list 1
- (list 2 (list 3 4) 5)
- (list 6 7)))
- (1 (4 (9 16) 25) (36 49))
-
- Define `square-tree' both directly (i.e., without using any
- higher-order procedures) and also by using `map' and recursion.
-
- *Exercise 2.31:* Abstract your answer to *Note Exercise 2-30:: to
- produce a procedure `tree-map' with the property that
- `square-tree' could be defined as
-
- (define (square-tree tree) (tree-map square tree))
-
- *Exercise 2.32:* We can represent a set as a list of distinct
- elements, and we can represent the set of all subsets of the set as
- a list of lists. For example, if the set is `(1 2 3)', then the
- set of all subsets is `(() (3) (2) (2 3) (1) (1 3) (1 2) (1 2
- 3))'. Complete the following definition of a procedure that
- generates the set of subsets of a set and give a clear explanation
- of why it works:
-
- (define (subsets s)
- (if (null? s)
- (list nil)
- (let ((rest (subsets (cdr s))))
- (append rest (map <??> rest)))))
-
- ---------- Footnotes ----------
-
- (1) The order of the first two clauses in the `cond' matters, since
-the empty list satisfies `null?' and also is not a pair.
-
-\1f
-File: sicp.info, Node: 2-2-3, Next: 2-2-4, Prev: 2-2-2, Up: 2-2
-
-2.2.3 Sequences as Conventional Interfaces
-------------------------------------------
-
-In working with compound data, we've stressed how data abstraction
-permits us to design programs without becoming enmeshed in the details
-of data representations, and how abstraction preserves for us the
-flexibility to experiment with alternative representations. In this
-section, we introduce another powerful design principle for working
-with data structures--the use of "conventional interfaces".
-
- In section *Note 1-3:: we saw how program abstractions, implemented
-as higher-order procedures, can capture common patterns in programs
-that deal with numerical data. Our ability to formulate analogous
-operations for working with compound data depends crucially on the
-style in which we manipulate our data structures. Consider, for
-example, the following procedure, analogous to the `count-leaves'
-procedure of section *Note 2-2-2::, which takes a tree as argument and
-computes the sum of the squares of the leaves that are odd:
-
- (define (sum-odd-squares tree)
- (cond ((null? tree) 0)
- ((not (pair? tree))
- (if (odd? tree) (square tree) 0))
- (else (+ (sum-odd-squares (car tree))
- (sum-odd-squares (cdr tree))))))
-
- On the surface, this procedure is very different from the following
-one, which constructs a list of all the even Fibonacci numbers
-_Fib_(k), where k is less than or equal to a given integer n:
-
- (define (even-fibs n)
- (define (next k)
- (if (> k n)
- nil
- (let ((f (fib k)))
- (if (even? f)
- (cons f (next (+ k 1)))
- (next (+ k 1))))))
- (next 0))
-
- Despite the fact that these two procedures are structurally very
-different, a more abstract description of the two computations reveals
-a great deal of similarity. The first program
-
- * enumerates the leaves of a tree;
-
- * filters them, selecting the odd ones;
-
- * squares each of the selected ones; and
-
- * accumulates the results using `+', starting with 0.
-
-
- The second program
-
- * enumerates the integers from 0 to n;
-
- * computes the Fibonacci number for each integer;
-
- * filters them, selecting the even ones; and
-
- * accumulates the results using `cons', starting with the empty
- list.
-
-
- A signal-processing engineer would find it natural to conceptualize
-these processes in terms of signals flowing through a cascade of
-stages, each of which implements part of the program plan, as shown in
-*Note Figure 2-7::. In `sum-odd-squares', we begin with an "enumerator",
-which generates a "signal" consisting of the leaves of a given tree.
-This signal is passed through a "filter", which eliminates all but the
-odd elements. The resulting signal is in turn passed through a "map",
-which is a "transducer" that applies the `square' procedure to each
-element. The output of the map is then fed to an "accumulator", which
-combines the elements using `+', starting from an initial 0. The plan
-for `even-fibs' is analogous.
-
- *Figure 2.7:* The signal-flow plans for the procedures
- `sum-odd-squares' (top) and `even-fibs' (bottom) reveal the
- commonality between the two programs.
-
- +-------------+ +-------------+ +-------------+ +-------------+
- | enumerate: |-->| filter: |-->| map: |-->| accumulate: |
- | tree leaves | | odd? | | square | | +, 0 |
- +-------------+ +-------------+ +-------------+ +-------------+
-
- +-------------+ +-------------+ +-------------+ +-------------+
- | enumerate: |-->| map: |-->| filter: |-->| accumulate: |
- | integers | | fib | | even? | | cons, () |
- +-------------+ +-------------+ +-------------+ +-------------+
-
- Unfortunately, the two procedure definitions above fail to exhibit
-this signal-flow structure. For instance, if we examine the
-`sum-odd-squares' procedure, we find that the enumeration is
-implemented partly by the `null?' and `pair?' tests and partly by the
-tree-recursive structure of the procedure. Similarly, the accumulation
-is found partly in the tests and partly in the addition used in the
-recursion. In general, there are no distinct parts of either procedure
-that correspond to the elements in the signal-flow description. Our
-two procedures decompose the computations in a different way, spreading
-the enumeration over the program and mingling it with the map, the
-filter, and the accumulation. If we could organize our programs to
-make the signal-flow structure manifest in the procedures we write, this
-would increase the conceptual clarity of the resulting code.
-
-Sequence Operations
-...................
-
-The key to organizing programs so as to more clearly reflect the
-signal-flow structure is to concentrate on the "signals" that flow from
-one stage in the process to the next. If we represent these signals as
-lists, then we can use list operations to implement the processing at
-each of the stages. For instance, we can implement the mapping stages
-of the signal-flow diagrams using the `map' procedure from section
-*Note 2-2-1:::
-
- (map square (list 1 2 3 4 5))
- (1 4 9 16 25)
-
- Filtering a sequence to select only those elements that satisfy a
-given predicate is accomplished by
-
- (define (filter predicate sequence)
- (cond ((null? sequence) nil)
- ((predicate (car sequence))
- (cons (car sequence)
- (filter predicate (cdr sequence))))
- (else (filter predicate (cdr sequence)))))
-
- For example,
-
- (filter odd? (list 1 2 3 4 5))
- (1 3 5)
-
- Accumulations can be implemented by
-
- (define (accumulate op initial sequence)
- (if (null? sequence)
- initial
- (op (car sequence)
- (accumulate op initial (cdr sequence)))))
-
- (accumulate + 0 (list 1 2 3 4 5))
- 15
-
- (accumulate * 1 (list 1 2 3 4 5))
- 120
-
- (accumulate cons nil (list 1 2 3 4 5))
- (1 2 3 4 5)
-
- All that remains to implement signal-flow diagrams is to enumerate
-the sequence of elements to be processed. For `even-fibs', we need to
-generate the sequence of integers in a given range, which we can do as
-follows:
-
- (define (enumerate-interval low high)
- (if (> low high)
- nil
- (cons low (enumerate-interval (+ low 1) high))))
-
- (enumerate-interval 2 7)
- (2 3 4 5 6 7)
-
- To enumerate the leaves of a tree, we can use(1)
-
- (define (enumerate-tree tree)
- (cond ((null? tree) nil)
- ((not (pair? tree)) (list tree))
- (else (append (enumerate-tree (car tree))
- (enumerate-tree (cdr tree))))))
-
- (enumerate-tree (list 1 (list 2 (list 3 4)) 5))
- (1 2 3 4 5)
-
- Now we can reformulate `sum-odd-squares' and `even-fibs' as in the
-signal-flow diagrams. For `sum-odd-squares', we enumerate the sequence
-of leaves of the tree, filter this to keep only the odd numbers in the
-sequence, square each element, and sum the results:
-
- (define (sum-odd-squares tree)
- (accumulate +
- 0
- (map square
- (filter odd?
- (enumerate-tree tree)))))
-
- For `even-fibs', we enumerate the integers from 0 to n, generate the
-Fibonacci number for each of these integers, filter the resulting
-sequence to keep only the even elements, and accumulate the results
-into a list:
-
- (define (even-fibs n)
- (accumulate cons
- nil
- (filter even?
- (map fib
- (enumerate-interval 0 n)))))
-
- The value of expressing programs as sequence operations is that this
-helps us make program designs that are modular, that is, designs that
-are constructed by combining relatively independent pieces. We can
-encourage modular design by providing a library of standard components
-together with a conventional interface for connecting the components in
-flexible ways.
-
- Modular construction is a powerful strategy for controlling
-complexity in engineering design. In real signal-processing
-applications, for example, designers regularly build systems by
-cascading elements selected from standardized families of filters and
-transducers. Similarly, sequence operations provide a library of
-standard program elements that we can mix and match. For instance, we
-can reuse pieces from the `sum-odd-squares' and `even-fibs' procedures
-in a program that constructs a list of the squares of the first n + 1
-Fibonacci numbers:
-
- (define (list-fib-squares n)
- (accumulate cons
- nil
- (map square
- (map fib
- (enumerate-interval 0 n)))))
-
- (list-fib-squares 10)
- (0 1 1 4 9 25 64 169 441 1156 3025)
-
- We can rearrange the pieces and use them in computing the product of
-the odd integers in a sequence:
-
- (define (product-of-squares-of-odd-elements sequence)
- (accumulate *
- 1
- (map square
- (filter odd? sequence))))
-
- (product-of-squares-of-odd-elements (list 1 2 3 4 5))
- 225
-
- We can also formulate conventional data-processing applications in
-terms of sequence operations. Suppose we have a sequence of personnel
-records and we want to find the salary of the highest-paid programmer.
-Assume that we have a selector `salary' that returns the salary of a
-record, and a predicate `programmer?' that tests if a record is for a
-programmer. Then we can write
-
- (define (salary-of-highest-paid-programmer records)
- (accumulate max
- 0
- (map salary
- (filter programmer? records))))
-
- These examples give just a hint of the vast range of operations that
-can be expressed as sequence operations.(2)
-
- Sequences, implemented here as lists, serve as a conventional
-interface that permits us to combine processing modules. Additionally,
-when we uniformly represent structures as sequences, we have localized
-the data-structure dependencies in our programs to a small number of
-sequence operations. By changing these, we can experiment with
-alternative representations of sequences, while leaving the overall
-design of our programs intact. We will exploit this capability in
-section *Note 3-5::, when we generalize the sequence-processing
-paradigm to admit infinite sequences.
-
- *Exercise 2.33:* Fill in the missing expressions to complete the
- following definitions of some basic list-manipulation operations
- as accumulations:
-
- (define (map p sequence)
- (accumulate (lambda (x y) <??>) nil sequence))
-
- (define (append seq1 seq2)
- (accumulate cons <??> <??>))
-
- (define (length sequence)
- (accumulate <??> 0 sequence))
-
- *Exercise 2.34:* Evaluating a polynomial in x at a given value of
- x can be formulated as an accumulation. We evaluate the polynomial
-
- a_n r^n | a_(n-1) r^(n-1) + ... + a_1 r + a_0
-
- using a well-known algorithm called "Horner's rule", which
- structures the computation as
-
- (... (a_n r + a_(n-1)) r + ... + a_1) r + a_0
-
- In other words, we start with a_n, multiply by x, add a_(n-1),
- multiply by x, and so on, until we reach a_0.(3)
-
- Fill in the following template to produce a procedure that
- evaluates a polynomial using Horner's rule. Assume that the
- coefficients of the polynomial are arranged in a sequence, from
- a_0 through a_n.
-
- (define (horner-eval x coefficient-sequence)
- (accumulate (lambda (this-coeff higher-terms) <??>)
- 0
- coefficient-sequence))
-
- For example, to compute 1 + 3x + 5x^3 + x^(5) at x = 2 you would
- evaluate
-
- (horner-eval 2 (list 1 3 0 5 0 1))
-
- *Exercise 2.35:* Redefine `count-leaves' from section *Note
- 2-2-2:: as an accumulation:
-
- (define (count-leaves t)
- (accumulate <??> <??> (map <??> <??>)))
-
- *Exercise 2.36:* The procedure `accumulate-n' is similar to
- `accumulate' except that it takes as its third argument a sequence
- of sequences, which are all assumed to have the same number of
- elements. It applies the designated accumulation procedure to
- combine all the first elements of the sequences, all the second
- elements of the sequences, and so on, and returns a sequence of
- the results. For instance, if `s' is a sequence containing four
- sequences, `((1 2 3) (4 5 6) (7 8 9) (10 11 12)),' then the value
- of `(accumulate-n + 0 s)' should be the sequence `(22 26 30)'.
- Fill in the missing expressions in the following definition of
- `accumulate-n':
-
- (define (accumulate-n op init seqs)
- (if (null? (car seqs))
- nil
- (cons (accumulate op init <??>)
- (accumulate-n op init <??>))))
-
-Exercise 2.37
-.............
-
-Suppose we represent vectors v = (v_i) as sequences of numbers, and
-matrices m = (m_(ij)) as sequences of vectors (the rows of the matrix).
-For example, the matrix
-
- +- -+
- | 1 2 3 4 |
- | 4 5 6 6 |
- | 6 7 8 9 |
- +- -+
-
-is represented as the sequence `((1 2 3 4) (4 5 6 6) (6 7 8 9))'. With
-this representation, we can use sequence operations to concisely
-express the basic matrix and vector operations. These operations
-(which are described in any book on matrix algebra) are the following:
-
- __
- (dot-product v w) returns the sum >_i v_i w_i
-
- (matrix-*-vector m v) returns the vector t,
- __
- where t_i = >_j m_(ij) v_j
-
- (matrix-*-matrix m n) returns the matrix p,
- __
- where p_(ij) = >_k m_(ik) n_(kj)
-
- (transpose m) returns the matrix n,
- where n_(ij) = m_(ji)
-
- We can define the dot product as(4)
-
- (define (dot-product v w)
- (accumulate + 0 (map * v w)))
-
- Fill in the missing expressions in the following procedures for
-computing the other matrix operations. (The procedure `accumulate-n'
-is defined in *Note Exercise 2-36::.)
-
- (define (matrix-*-vector m v)
- (map <??> m))
-
- (define (transpose mat)
- (accumulate-n <??> <??> mat))
-
- (define (matrix-*-matrix m n)
- (let ((cols (transpose n)))
- (map <??> m)))
-
- *Exercise 2.38:* The `accumulate' procedure is also known as
- `fold-right', because it combines the first element of the
- sequence with the result of combining all the elements to the
- right. There is also a `fold-left', which is similar to
- `fold-right', except that it combines elements working in the
- opposite direction:
-
- (define (fold-left op initial sequence)
- (define (iter result rest)
- (if (null? rest)
- result
- (iter (op result (car rest))
- (cdr rest))))
- (iter initial sequence))
-
- What are the values of
-
- (fold-right / 1 (list 1 2 3))
-
- (fold-left / 1 (list 1 2 3))
-
- (fold-right list nil (list 1 2 3))
-
- (fold-left list nil (list 1 2 3))
-
- Give a property that `op' should satisfy to guarantee that
- `fold-right' and `fold-left' will produce the same values for any
- sequence.
-
- *Exercise 2.39:* Complete the following definitions of `reverse'
- (*Note Exercise 2-18::) in terms of `fold-right' and `fold-left'
- from *Note Exercise 2-38:::
-
- (define (reverse sequence)
- (fold-right (lambda (x y) <??>) nil sequence))
-
- (define (reverse sequence)
- (fold-left (lambda (x y) <??>) nil sequence))
-
-Nested Mappings
-...............
-
-We can extend the sequence paradigm to include many computations that
-are commonly expressed using nested loops.(5) Consider this problem:
-Given a positive integer n, find all ordered pairs of distinct positive
-integers i and j, where 1 <= j< i<= n, such that i + j is prime. For
-example, if n is 6, then the pairs are the following:
-
- i | 2 3 4 4 5 6 6
- j | 1 2 1 3 2 1 5
- ------+---------------
- i + j | 3 5 5 7 7 7 11
-
- A natural way to organize this computation is to generate the
-sequence of all ordered pairs of positive integers less than or equal
-to n, filter to select those pairs whose sum is prime, and then, for
-each pair (i, j) that passes through the filter, produce the triple
-(i,j,i + j).
-
- Here is a way to generate the sequence of pairs: For each integer i
-<= n, enumerate the integers j<i, and for each such i and j generate
-the pair (i,j). In terms of sequence operations, we map along the
-sequence `(enumerate-interval 1 n)'. For each i in this sequence, we
-map along the sequence `(enumerate-interval 1 (- i 1))'. For each j in
-this latter sequence, we generate the pair `(list i j)'. This gives us
-a sequence of pairs for each i. Combining all the sequences for all
-the i (by accumulating with `append') produces the required sequence of
-pairs:(6)
-
- (accumulate append
- nil
- (map (lambda (i)
- (map (lambda (j) (list i j))
- (enumerate-interval 1 (- i 1))))
- (enumerate-interval 1 n)))
-
- The combination of mapping and accumulating with `append' is so
-common in this sort of program that we will isolate it as a separate
-procedure:
-
- (define (flatmap proc seq)
- (accumulate append nil (map proc seq)))
-
- Now filter this sequence of pairs to find those whose sum is prime.
-The filter predicate is called for each element of the sequence; its
-argument is a pair and it must extract the integers from the pair.
-Thus, the predicate to apply to each element in the sequence is
-
- (define (prime-sum? pair)
- (prime? (+ (car pair) (cadr pair))))
-
- Finally, generate the sequence of results by mapping over the
-filtered pairs using the following procedure, which constructs a triple
-consisting of the two elements of the pair along with their sum:
-
- (define (make-pair-sum pair)
- (list (car pair) (cadr pair) (+ (car pair) (cadr pair))))
-
- Combining all these steps yields the complete procedure:
-
- (define (prime-sum-pairs n)
- (map make-pair-sum
- (filter prime-sum?
- (flatmap
- (lambda (i)
- (map (lambda (j) (list i j))
- (enumerate-interval 1 (- i 1))))
- (enumerate-interval 1 n)))))
-
- Nested mappings are also useful for sequences other than those that
-enumerate intervals. Suppose we wish to generate all the permutations
-of a set S; that is, all the ways of ordering the items in the set.
-For instance, the permutations of {1,2,3} are {1,2,3}, {1,3,2},
-{2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. Here is a plan for generating
-the permutations of S: For each item x in S, recursively generate the
-sequence of permutations of S - x,(7) and adjoin x to the front of each
-one. This yields, for each x in S, the sequence of permutations of S
-that begin with x. Combining these sequences for all x gives all the
-permutations of S:(8)
-
- (define (permutations s)
- (if (null? s) ; empty set?
- (list nil) ; sequence containing empty set
- (flatmap (lambda (x)
- (map (lambda (p) (cons x p))
- (permutations (remove x s))))
- s)))
-
- Notice how this strategy reduces the problem of generating
-permutations of S to the problem of generating the permutations of sets
-with fewer elements than S. In the terminal case, we work our way down
-to the empty list, which represents a set of no elements. For this, we
-generate `(list nil)', which is a sequence with one item, namely the
-set with no elements. The `remove' procedure used in `permutations'
-returns all the items in a given sequence except for a given item.
-This can be expressed as a simple filter:
-
- (define (remove item sequence)
- (filter (lambda (x) (not (= x item)))
- sequence))
-
- *Exercise 2.40:* Define a procedure `unique-pairs' that, given an
- integer n, generates the sequence of pairs (i,j) with 1 <= j< i <=
- n. Use `unique-pairs' to simplify the definition of
- `prime-sum-pairs' given above.
-
- *Exercise 2.41:* Write a procedure to find all ordered triples of
- distinct positive integers i, j, and k less than or equal to a
- given integer n that sum to a given integer s.
-
- *Figure 2.8:* A solution to the eight-queens puzzle.
-
- +---+---+---+---+---+---+---+---+
- | | | | | | Q | | |
- +---+---+---+---+---+---+---+---+
- | | | Q | | | | | |
- +---+---+---+---+---+---+---+---+
- | Q | | | | | | | |
- +---+---+---+---+---+---+---+---+
- | | | | | | | Q | |
- +---+---+---+---+---+---+---+---+
- | | | | | Q | | | |
- +---+---+---+---+---+---+---+---+
- | | | | | | | | Q |
- +---+---+---+---+---+---+---+---+
- | | Q | | | | | | |
- +---+---+---+---+---+---+---+---+
- | | | | Q | | | | |
- +---+---+---+---+---+---+---+---+
-
- *Exercise 2.42:* The "eight-queens puzzle" asks how to place eight
- queens on a chessboard so that no queen is in check from any other
- (i.e., no two queens are in the same row, column, or diagonal).
- One possible solution is shown in *Note Figure 2-8::. One way to
- solve the puzzle is to work across the board, placing a queen in
- each column. Once we have placed k - 1 queens, we must place the
- kth queen in a position where it does not check any of the queens
- already on the board. We can formulate this approach recursively:
- Assume that we have already generated the sequence of all possible
- ways to place k - 1 queens in the first k - 1 columns of the
- board. For each of these ways, generate an extended set of
- positions by placing a queen in each row of the kth column. Now
- filter these, keeping only the positions for which the queen in
- the kth column is safe with respect to the other queens. This
- produces the sequence of all ways to place k queens in the first k
- columns. By continuing this process, we will produce not only one
- solution, but all solutions to the puzzle.
-
- We implement this solution as a procedure `queens', which returns a
- sequence of all solutions to the problem of placing n queens on an
- n*n chessboard. `Queens' has an internal procedure `queen-cols'
- that returns the sequence of all ways to place queens in the first
- k columns of the board.
-
- (define (queens board-size)
- (define (queen-cols k)
- (if (= k 0)
- (list empty-board)
- (filter
- (lambda (positions) (safe? k positions))
- (flatmap
- (lambda (rest-of-queens)
- (map (lambda (new-row)
- (adjoin-position new-row k rest-of-queens))
- (enumerate-interval 1 board-size)))
- (queen-cols (- k 1))))))
- (queen-cols board-size))
-
- In this procedure `rest-of-queens' is a way to place k - 1 queens
- in the first k - 1 columns, and `new-row' is a proposed row in
- which to place the queen for the kth column. Complete the program
- by implementing the representation for sets of board positions,
- including the procedure `adjoin-position', which adjoins a new
- row-column position to a set of positions, and `empty-board',
- which represents an empty set of positions. You must also write
- the procedure `safe?', which determines for a set of positions,
- whether the queen in the kth column is safe with respect to the
- others. (Note that we need only check whether the new queen is
- safe--the other queens are already guaranteed safe with respect to
- each other.)
-
- *Exercise 2.43:* Louis Reasoner is having a terrible time doing
- *Note Exercise 2-42::. His `queens' procedure seems to work, but
- it runs extremely slowly. (Louis never does manage to wait long
- enough for it to solve even the 6*6 case.) When Louis asks Eva Lu
- Ator for help, she points out that he has interchanged the order
- of the nested mappings in the `flatmap', writing it as
-
- (flatmap
- (lambda (new-row)
- (map (lambda (rest-of-queens)
- (adjoin-position new-row k rest-of-queens))
- (queen-cols (- k 1))))
- (enumerate-interval 1 board-size))
-
- Explain why this interchange makes the program run slowly.
- Estimate how long it will take Louis's program to solve the
- eight-queens puzzle, assuming that the program in *Note Exercise
- 2-42:: solves the puzzle in time T.
-
- ---------- Footnotes ----------
-
- (1) This is, in fact, precisely the `fringe' procedure from *Note
-Exercise 2-28::. Here we've renamed it to emphasize that it is part of
-a family of general sequence-manipulation procedures.
-
- (2) Richard Waters (1979) developed a program that automatically
-analyzes traditional Fortran programs, viewing them in terms of maps,
-filters, and accumulations. He found that fully 90 percent of the code
-in the Fortran Scientific Subroutine Package fits neatly into this
-paradigm. One of the reasons for the success of Lisp as a programming
-language is that lists provide a standard medium for expressing ordered
-collections so that they can be manipulated using higher-order
-operations. The programming language APL owes much of its power and
-appeal to a similar choice. In APL all data are represented as arrays,
-and there is a universal and convenient set of generic operators for
-all sorts of array operations.
-
- (3) According to Knuth (1981), this rule was formulated by W. G.
-Horner early in the nineteenth century, but the method was actually used
-by Newton over a hundred years earlier. Horner's rule evaluates the
-polynomial using fewer additions and multiplications than does the
-straightforward method of first computing a_n x^n, then adding
-a_(n-1)x^(n-1), and so on. In fact, it is possible to prove that any
-algorithm for evaluating arbitrary polynomials must use at least as
-many additions and multiplications as does Horner's rule, and thus
-Horner's rule is an optimal algorithm for polynomial evaluation. This
-was proved (for the number of additions) by A. M. Ostrowski in a 1954
-paper that essentially founded the modern study of optimal algorithms.
-The analogous statement for multiplications was proved by V. Y. Pan in
-1966. The book by Borodin and Munro (1975) provides an overview of
-these and other results about optimal algorithms.
-
- (4) This definition uses the extended version of `map' described in
-*Note Footnote 12::.
-
- (5) This approach to nested mappings was shown to us by David
-Turner, whose languages KRC and Miranda provide elegant formalisms for
-dealing with these constructs. The examples in this section (see also
-*Note Exercise 2-42::) are adapted from Turner 1981. In section *Note
-3-5-3::, we'll see how this approach generalizes to infinite sequences.
-
- (6) We're representing a pair here as a list of two elements rather
-than as a Lisp pair. Thus, the "pair" (i,j) is represented as `(list i
-j)', not `(cons i j)'.
-
- (7) The set S - x is the set of all elements of S, excluding x.
-
- (8) Semicolons in Scheme code are used to introduce "comments".
-Everything from the semicolon to the end of the line is ignored by the
-interpreter. In this book we don't use many comments; we try to make
-our programs self-documenting by using descriptive names.
-
-\1f
-File: sicp.info, Node: 2-2-4, Prev: 2-2-3, Up: 2-2
-
-2.2.4 Example: A Picture Language
----------------------------------
-
-This section presents a simple language for drawing pictures that
-illustrates the power of data abstraction and closure, and also
-exploits higher-order procedures in an essential way. The language is
-designed to make it easy to experiment with patterns such as the ones
-in *Note Figure 2-9::, which are composed of repeated elements that are
-shifted and scaled.(1) In this language, the data objects being
-combined are represented as procedures rather than as list structure.
-Just as `cons', which satisfies the closure property, allowed us to
-easily build arbitrarily complicated list structure, the operations in
-this language, which also satisfy the closure property, allow us to
-easily build arbitrarily complicated patterns.
-
- *Figure 2.9:* Designs generated with the picture language.
-
- [two graphic images not included]
-
-The picture language
-....................
-
-When we began our study of programming in section *Note 1-1::, we
-emphasized the importance of describing a language by focusing on the
-language's primitives, its means of combination, and its means of
-abstraction. We'll follow that framework here.
-
- Part of the elegance of this picture language is that there is only
-one kind of element, called a "painter". A painter draws an image that
-is shifted and scaled to fit within a designated parallelogram-shaped
-frame. For example, there's a primitive painter we'll call `wave' that
-makes a crude line drawing, as shown in *Note Figure 2-10::. The
-actual shape of the drawing depends on the frame--all four images in
-*Note Figure 2-10:: are produced by the same `wave' painter, but with
-respect to four different frames. Painters can be more elaborate than
-this: The primitive painter called `rogers' paints a picture of MIT's
-founder, William Barton Rogers, as shown in *Note Figure 2-11::.(2) The
-four images in *Note Figure 2-11:: are drawn with respect to the same
-four frames as the `wave' images in *Note Figure 2-10::.
-
- To combine images, we use various operations that construct new
-painters from given painters. For example, the `beside' operation
-takes two painters and produces a new, compound painter that draws the
-first painter's image in the left half of the frame and the second
-painter's image in the right half of the frame. Similarly, `below'
-takes two painters and produces a compound painter that draws the first
-painter's image below the second painter's image. Some operations
-transform a single painter to produce a new painter. For example,
-`flip-vert' takes a painter and produces a painter that draws its image
-upside-down, and `flip-horiz' produces a painter that draws the
-original painter's image left-to-right reversed.
-
- *Figure 2.10:* Images produced by the `wave' painter, with respect
- to four different frames. The frames, shown with dotted lines,
- are not part of the images.
-
- [four graphic images not included]
-
- *Figure 2.11:* Images of William Barton Rogers, founder and first
- president of MIT, painted with respect to the same four frames as
- in *Note Figure 2-10:: (original image reprinted with the
- permission of the MIT Museum).
-
- [four graphic images not included]
-
- *Note Figure 2-12:: shows the drawing of a painter called `wave4'
-that is built up in two stages starting from `wave':
-
- (define wave2 (beside wave (flip-vert wave)))
- (define wave4 (below wave2 wave2))
-
- *Figure 2.12:* Creating a complex figure, starting from the `wave'
- painter of *Note Figure 2-10::.
-
- [two graphic images not included]
-
- (define wave2 (define wave4
- (beside wave (flip-vert wave))) (below wave2 wave2))
-
- In building up a complex image in this manner we are exploiting the
-fact that painters are closed under the language's means of
-combination. The `beside' or `below' of two painters is itself a
-painter; therefore, we can use it as an element in making more complex
-painters. As with building up list structure using `cons', the closure
-of our data under the means of combination is crucial to the ability to
-create complex structures while using only a few operations.
-
- Once we can combine painters, we would like to be able to abstract
-typical patterns of combining painters. We will implement the painter
-operations as Scheme procedures. This means that we don't need a
-special abstraction mechanism in the picture language: Since the means
-of combination are ordinary Scheme procedures, we automatically have
-the capability to do anything with painter operations that we can do
-with procedures. For example, we can abstract the pattern in `wave4' as
-
- (define (flipped-pairs painter)
- (let ((painter2 (beside painter (flip-vert painter))))
- (below painter2 painter2)))
-
-and define `wave4' as an instance of this pattern:
-
- (define wave4 (flipped-pairs wave))
-
- We can also define recursive operations. Here's one that makes
-painters split and branch towards the right as shown in figures *Note
-Figure 2-13:: and *Note Figure 2-14:::
-
- (define (right-split painter n)
- (if (= n 0)
- painter
- (let ((smaller (right-split painter (- n 1))))
- (beside painter (below smaller smaller)))))
-
- *Figure 2.13:* Recursive plans for `right-split' and
- `corner-split'.
-
- +-------------+-------------+ +------+------+-------------+
- | | | | up- | up- | |
- | | right-split | | split| split| corner-split|
- | | | | | | |
- | | n-1 | | n-1 | n-1 | n-1 |
- | | | | | | |
- | identity +-------------+ +------+------+-------------+
- | | | | | right-split |
- | | right-split | | | n-1 |
- | | | | identity +-------------+
- | | n-1 | | | right-split |
- | | | | | n-1 |
- +-------------+-------------+ +-------------+-------------+
-
- right-split n corner-split n
-
- We can produce balanced patterns by branching upwards as well as
-towards the right (see *Note Exercise 2-44:: and figures *Note Figure
-2-13:: and *Note Figure 2-14::):
-
- (define (corner-split painter n)
- (if (= n 0)
- painter
- (let ((up (up-split painter (- n 1)))
- (right (right-split painter (- n 1))))
- (let ((top-left (beside up up))
- (bottom-right (below right right))
- (corner (corner-split painter (- n 1))))
- (beside (below painter top-left)
- (below bottom-right corner))))))
-
- *Figure 2.14:* The recursive operations `right-split' and
- `corner-split' applied to the painters `wave' and `rogers'.
- Combining four `corner-split' figures produces symmetric
- `square-limit' designs as shown in *Note Figure 2-9::.
-
- [two graphic images not included]
-
- (right-split wave 4) (right-split rogers 4)
-
- [two graphic images not included]
-
- (corner-split wave 4) (corner-split rogers 4)
-
- By placing four copies of a `corner-split' appropriately, we obtain a
-pattern called `square-limit', whose application to `wave' and `rogers'
-is shown in *Note Figure 2-9:::
-
- (define (square-limit painter n)
- (let ((quarter (corner-split painter n)))
- (let ((half (beside (flip-horiz quarter) quarter)))
- (below (flip-vert half) half))))
-
- *Exercise 2.44:* Define the procedure `up-split' used by
- `corner-split'. It is similar to `right-split', except that it
- switches the roles of `below' and `beside'.
-
-Higher-order operations
-.......................
-
-In addition to abstracting patterns of combining painters, we can work
-at a higher level, abstracting patterns of combining painter
-operations. That is, we can view the painter operations as elements to
-manipulate and can write means of combination for these
-elements--procedures that take painter operations as arguments and
-create new painter operations.
-
- For example, `flipped-pairs' and `square-limit' each arrange four
-copies of a painter's image in a square pattern; they differ only in
-how they orient the copies. One way to abstract this pattern of
-painter combination is with the following procedure, which takes four
-one-argument painter operations and produces a painter operation that
-transforms a given painter with those four operations and arranges the
-results in a square. `Tl', `tr', `bl', and `br' are the
-transformations to apply to the top left copy, the top right copy, the
-bottom left copy, and the bottom right copy, respectively.
-
- (define (square-of-four tl tr bl br)
- (lambda (painter)
- (let ((top (beside (tl painter) (tr painter)))
- (bottom (beside (bl painter) (br painter))))
- (below bottom top))))
-
- Then `flipped-pairs' can be defined in terms of `square-of-four' as
-follows:(3)
-
- (define (flipped-pairs painter)
- (let ((combine4 (square-of-four identity flip-vert
- identity flip-vert)))
- (combine4 painter)))
-
-and `square-limit' can be expressed as(4)
-
- (define (square-limit painter n)
- (let ((combine4 (square-of-four flip-horiz identity
- rotate180 flip-vert)))
- (combine4 (corner-split painter n))))
-
- *Exercise 2.45:* `Right-split' and `up-split' can be expressed as
- instances of a general splitting operation. Define a procedure
- `split' with the property that evaluating
-
- (define right-split (split beside below))
- (define up-split (split below beside))
-
- produces procedures `right-split' and `up-split' with the same
- behaviors as the ones already defined.
-
-Frames
-......
-
-Before we can show how to implement painters and their means of
-combination, we must first consider frames. A frame can be described
-by three vectors--an origin vector and two edge vectors. The origin
-vector specifies the offset of the frame's origin from some absolute
-origin in the plane, and the edge vectors specify the offsets of the
-frame's corners from its origin. If the edges are perpendicular, the
-frame will be rectangular. Otherwise the frame will be a more general
-parallelogram.
-
- *Note Figure 2-15:: shows a frame and its associated vectors. In
-accordance with data abstraction, we need not be specific yet about how
-frames are represented, other than to say that there is a constructor
-`make-frame', which takes three vectors and produces a frame, and three
-corresponding selectors `origin-frame', `edge1-frame', and
-`edge2-frame' (see *Note Exercise 2-47::).
-
- *Figure 2.15:* A frame is described by three vectors - an origin
- and two edges.
-
- __
- __-- \
- __-- \
- __ __-- \ __
- |\ __-- \__-|
- \- __--
- frame \ __--
- edge2 \ __-- frame
- vector \ __-- edge1
- \_-- vector
- - <--+
- frame |
- origin +-- (0,0) point
- vector on display screen
-
- We will use coordinates in the unit square (0<= x,y<= 1) to specify
-images. With each frame, we associate a "frame coordinate map", which
-will be used to shift and scale images to fit the frame. The map
-transforms the unit square into the frame by mapping the vector v =
-(x,y) to the vector sum
-
- Origin(Frame) + r * Edge_1(Frame) + y * Edge_2(Frame)
-
-For example, (0,0) is mapped to the origin of the frame, (1,1) to the
-vertex diagonally opposite the origin, and (0.5,0.5) to the center of
-the frame. We can create a frame's coordinate map with the following
-procedure:(5)
-
- (define (frame-coord-map frame)
- (lambda (v)
- (add-vect
- (origin-frame frame)
- (add-vect (scale-vect (xcor-vect v)
- (edge1-frame frame))
- (scale-vect (ycor-vect v)
- (edge2-frame frame))))))
-
- Observe that applying `frame-coord-map' to a frame returns a
-procedure that, given a vector, returns a vector. If the argument
-vector is in the unit square, the result vector will be in the frame.
-For example,
-
- ((frame-coord-map a-frame) (make-vect 0 0))
-
-returns the same vector as
-
- (origin-frame a-frame)
-
- *Exercise 2.46:* A two-dimensional vector v running from the
- origin to a point can be represented as a pair consisting of an
- x-coordinate and a y-coordinate. Implement a data abstraction for
- vectors by giving a constructor `make-vect' and corresponding
- selectors `xcor-vect' and `ycor-vect'. In terms of your selectors
- and constructor, implement procedures `add-vect', `sub-vect', and
- `scale-vect' that perform the operations vector addition, vector
- subtraction, and multiplying a vector by a scalar:
-
- (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)
- (x_1, y_1) - (x_2, y_2) = (x_1 - x_2, y_1 - y_2)
- s * (x, y) = (sx, sy)
-
- *Exercise 2.47:* Here are two possible constructors for frames:
-
- (define (make-frame origin edge1 edge2)
- (list origin edge1 edge2))
-
- (define (make-frame origin edge1 edge2)
- (cons origin (cons edge1 edge2)))
-
- For each constructor supply the appropriate selectors to produce an
- implementation for frames.
-
-Painters
-........
-
-A painter is represented as a procedure that, given a frame as
-argument, draws a particular image shifted and scaled to fit the frame.
-That is to say, if `p' is a painter and `f' is a frame, then we
-produce `p''s image in `f' by calling `p' with `f' as argument.
-
- The details of how primitive painters are implemented depend on the
-particular characteristics of the graphics system and the type of image
-to be drawn. For instance, suppose we have a procedure `draw-line'
-that draws a line on the screen between two specified points. Then we
-can create painters for line drawings, such as the `wave' painter in
-*Note Figure 2-10::, from lists of line segments as follows:(6)
-
- (define (segments->painter segment-list)
- (lambda (frame)
- (for-each
- (lambda (segment)
- (draw-line
- ((frame-coord-map frame) (start-segment segment))
- ((frame-coord-map frame) (end-segment segment))))
- segment-list)))
-
- The segments are given using coordinates with respect to the unit
-square. For each segment in the list, the painter transforms the
-segment endpoints with the frame coordinate map and draws a line
-between the transformed points.
-
- Representing painters as procedures erects a powerful abstraction
-barrier in the picture language. We can create and intermix all sorts
-of primitive painters, based on a variety of graphics capabilities. The
-details of their implementation do not matter. Any procedure can serve
-as a painter, provided that it takes a frame as argument and draws
-something scaled to fit the frame.(7)
-
- *Exercise 2.48:* A directed line segment in the plane can be
- represented as a pair of vectors--the vector running from the
- origin to the start-point of the segment, and the vector running
- from the origin to the end-point of the segment. Use your vector
- representation from *Note Exercise 2-46:: to define a
- representation for segments with a constructor `make-segment' and
- selectors `start-segment' and `end-segment'.
-
- *Exercise 2.49:* Use `segments->painter' to define the following
- primitive painters:
-
- a. The painter that draws the outline of the designated frame.
-
- b. The painter that draws an "X" by connecting opposite corners
- of the frame.
-
- c. The painter that draws a diamond shape by connecting the
- midpoints of the sides of the frame.
-
- d. The `wave' painter.
-
-
-Transforming and combining painters
-...................................
-
-An operation on painters (such as `flip-vert' or `beside') works by
-creating a painter that invokes the original painters with respect to
-frames derived from the argument frame. Thus, for example, `flip-vert'
-doesn't have to know how a painter works in order to flip it--it just
-has to know how to turn a frame upside down: The flipped painter just
-uses the original painter, but in the inverted frame.
-
- Painter operations are based on the procedure `transform-painter',
-which takes as arguments a painter and information on how to transform
-a frame and produces a new painter. The transformed painter, when
-called on a frame, transforms the frame and calls the original painter
-on the transformed frame. The arguments to `transform-painter' are
-points (represented as vectors) that specify the corners of the new
-frame: When mapped into the frame, the first point specifies the new
-frame's origin and the other two specify the ends of its edge vectors.
-Thus, arguments within the unit square specify a frame contained within
-the original frame.
-
- (define (transform-painter painter origin corner1 corner2)
- (lambda (frame)
- (let ((m (frame-coord-map frame)))
- (let ((new-origin (m origin)))
- (painter
- (make-frame new-origin
- (sub-vect (m corner1) new-origin)
- (sub-vect (m corner2) new-origin)))))))
-
- Here's how to flip painter images vertically:
-
- (define (flip-vert painter)
- (transform-painter painter
- (make-vect 0.0 1.0) ; new `origin'
- (make-vect 1.0 1.0) ; new end of `edge1'
- (make-vect 0.0 0.0))) ; new end of `edge2'
-
- Using `transform-painter', we can easily define new transformations.
-For example, we can define a painter that shrinks its image to the
-upper-right quarter of the frame it is given:
-
- (define (shrink-to-upper-right painter)
- (transform-painter painter
- (make-vect 0.5 0.5)
- (make-vect 1.0 0.5)
- (make-vect 0.5 1.0)))
-
- Other transformations rotate images counterclockwise by 90 degrees(8)
-
- (define (rotate90 painter)
- (transform-painter painter
- (make-vect 1.0 0.0)
- (make-vect 1.0 1.0)
- (make-vect 0.0 0.0)))
-
-or squash images towards the center of the frame:(9)
-
- (define (squash-inwards painter)
- (transform-painter painter
- (make-vect 0.0 0.0)
- (make-vect 0.65 0.35)
- (make-vect 0.35 0.65)))
-
- Frame transformation is also the key to defining means of combining
-two or more painters. The `beside' procedure, for example, takes two
-painters, transforms them to paint in the left and right halves of an
-argument frame respectively, and produces a new, compound painter.
-When the compound painter is given a frame, it calls the first
-transformed painter to paint in the left half of the frame and calls
-the second transformed painter to paint in the right half of the frame:
-
- (define (beside painter1 painter2)
- (let ((split-point (make-vect 0.5 0.0)))
- (let ((paint-left
- (transform-painter painter1
- (make-vect 0.0 0.0)
- split-point
- (make-vect 0.0 1.0)))
- (paint-right
- (transform-painter painter2
- split-point
- (make-vect 1.0 0.0)
- (make-vect 0.5 1.0))))
- (lambda (frame)
- (paint-left frame)
- (paint-right frame)))))
-
- Observe how the painter data abstraction, and in particular the
-representation of painters as procedures, makes `beside' easy to
-implement. The `beside' procedure need not know anything about the
-details of the component painters other than that each painter will
-draw something in its designated frame.
-
- *Exercise 2.50:* Define the transformation `flip-horiz', which
- flips painters horizontally, and transformations that rotate
- painters counterclockwise by 180 degrees and 270 degrees.
-
- *Exercise 2.51:* Define the `below' operation for painters.
- `Below' takes two painters as arguments. The resulting painter,
- given a frame, draws with the first painter in the bottom of the
- frame and with the second painter in the top. Define `below' in
- two different ways--first by writing a procedure that is analogous
- to the `beside' procedure given above, and again in terms of
- `beside' and suitable rotation operations (from *Note Exercise
- 2-50::).
-
-Levels of language for robust design
-....................................
-
-The picture language exercises some of the critical ideas we've
-introduced about abstraction with procedures and data. The fundamental
-data abstractions, painters, are implemented using procedural
-representations, which enables the language to handle different basic
-drawing capabilities in a uniform way. The means of combination
-satisfy the closure property, which permits us to easily build up
-complex designs. Finally, all the tools for abstracting procedures are
-available to us for abstracting means of combination for painters.
-
- We have also obtained a glimpse of another crucial idea about
-languages and program design. This is the approach of "stratified
-design", the notion that a complex system should be structured as a
-sequence of levels that are described using a sequence of languages.
-Each level is constructed by combining parts that are regarded as
-primitive at that level, and the parts constructed at each level are
-used as primitives at the next level. The language used at each level
-of a stratified design has primitives, means of combination, and means
-of abstraction appropriate to that level of detail.
-
- Stratified design pervades the engineering of complex systems. For
-example, in computer engineering, resistors and transistors are
-combined (and described using a language of analog circuits) to produce
-parts such as and-gates and or-gates, which form the primitives of a
-language for digital-circuit design.(10) These parts are combined to
-build processors, bus structures, and memory systems, which are in turn
-combined to form computers, using languages appropriate to computer
-architecture. Computers are combined to form distributed systems, using
-languages appropriate for describing network interconnections, and so
-on.
-
- As a tiny example of stratification, our picture language uses
-primitive elements (primitive painters) that are created using a
-language that specifies points and lines to provide the lists of line
-segments for `segments->painter', or the shading details for a painter
-like `rogers'. The bulk of our description of the picture language
-focused on combining these primitives, using geometric combiners such
-as `beside' and `below'. We also worked at a higher level, regarding
-`beside' and `below' as primitives to be manipulated in a language
-whose operations, such as `square-of-four', capture common patterns of
-combining geometric combiners.
-
- Stratified design helps make programs "robust", that is, it makes it
-likely that small changes in a specification will require
-correspondingly small changes in the program. For instance, suppose we
-wanted to change the image based on `wave' shown in *Note Figure 2-9::.
-We could work at the lowest level to change the detailed appearance of
-the `wave' element; we could work at the middle level to change the way
-`corner-split' replicates the `wave'; we could work at the highest
-level to change how `square-limit' arranges the four copies of the
-corner. In general, each level of a stratified design provides a
-different vocabulary for expressing the characteristics of the system,
-and a different kind of ability to change it.
-
- *Exercise 2.52:* Make changes to the square limit of `wave' shown
- in *Note Figure 2-9:: by working at each of the levels described
- above. In particular:
-
- a. Add some segments to the primitive `wave' painter of *Note
- Exercise 2-49:: (to add a smile, for example).
-
- b. Change the pattern constructed by `corner-split' (for
- example, by using only one copy of the `up-split' and
- `right-split' images instead of two).
-
- c. Modify the version of `square-limit' that uses
- `square-of-four' so as to assemble the corners in a different
- pattern. (For example, you might make the big Mr. Rogers
- look outward from each corner of the square.)
-
-
- ---------- Footnotes ----------
-
- (1) The picture language is based on the language Peter Henderson
-created to construct images like M.C. Escher's "Square Limit" woodcut
-(see Henderson 1982). The woodcut incorporates a repeated scaled
-pattern, similar to the arrangements drawn using the `square-limit'
-procedure in this section.
-
- (2) William Barton Rogers (1804-1882) was the founder and first
-president of MIT. A geologist and talented teacher, he taught at
-William and Mary College and at the University of Virginia. In 1859 he
-moved to Boston, where he had more time for research, worked on a plan
-for establishing a "polytechnic institute," and served as
-Massachusetts's first State Inspector of Gas Meters.
-
- When MIT was established in 1861, Rogers was elected its first
-president. Rogers espoused an ideal of "useful learning" that was
-different from the university education of the time, with its
-overemphasis on the classics, which, as he wrote, "stand in the way of
-the broader, higher and more practical instruction and discipline of
-the natural and social sciences." This education was likewise to be
-different from narrow trade-school education. In Rogers's words:
-
- The world-enforced distinction between the practical and the
- scientific worker is utterly futile, and the whole experience of
- modern times has demonstrated its utter worthlessness.
-
- Rogers served as president of MIT until 1870, when he resigned due to
-ill health. In 1878 the second president of MIT, John Runkle, resigned
-under the pressure of a financial crisis brought on by the Panic of
-1873 and strain of fighting off attempts by Harvard to take over MIT.
-Rogers returned to hold the office of president until 1881.
-
- Rogers collapsed and died while addressing MIT's graduating class at
-the commencement exercises of 1882. Runkle quoted Rogers's last words
-in a memorial address delivered that same year:
-
- "As I stand here today and see what the Institute is, ... I call
- to mind the beginnings of science. I remember one hundred and
- fifty years ago Stephen Hales published a pamphlet on the subject
- of illuminating gas, in which he stated that his researches had
- demonstrated that 128 grains of bituminous coal - " "Bituminous
- coal," these were his last words on earth. Here he bent forward,
- as if consulting some notes on the table before him, then slowly
- regaining an erect position, threw up his hands, and was
- translated from the scene of his earthly labors and triumphs to
- "the tomorrow of death," where the mysteries of life are solved,
- and the disembodied spirit finds unending satisfaction in
- contemplating the new and still unfathomable mysteries of the
- infinite future.
-
- In the words of Francis A. Walker (MIT's third president):
-
- All his life he had borne himself most faithfully and heroically,
- and he died as so good a knight would surely have wished, in
- harness, at his post, and in the very part and act of public duty.
-
- (3) Equivalently, we could write
-
- (define flipped-pairs
- (square-of-four identity flip-vert identity flip-vert))
-
- (4) `Rotate180' rotates a painter by 180 degrees (see *Note Exercise
-2-50::). Instead of `rotate180' we could say `(compose flip-vert
-flip-horiz)', using the `compose' procedure from *Note Exercise 1-42::.
-
- (5) `Frame-coord-map' uses the vector operations described in *Note
-Exercise 2-46:: below, which we assume have been implemented using some
-representation for vectors. Because of data abstraction, it doesn't
-matter what this vector representation is, so long as the vector
-operations behave correctly.
-
- (6) `Segments->painter' uses the representation for line segments
-described in *Note Exercise 2-48:: below. It also uses the `for-each'
-procedure described in *Note Exercise 2-23::.
-
- (7) For example, the `rogers' painter of *Note Figure 2-11:: was
-constructed from a gray-level image. For each point in a given frame,
-the `rogers' painter determines the point in the image that is mapped
-to it under the frame coordinate map, and shades it accordingly. By
-allowing different types of painters, we are capitalizing on the
-abstract data idea discussed in section *Note 2-1-3::, where we argued
-that a rational-number representation could be anything at all that
-satisfies an appropriate condition. Here we're using the fact that a
-painter can be implemented in any way at all, so long as it draws
-something in the designated frame. Section *Note 2-1-3:: also showed
-how pairs could be implemented as procedures. Painters are our second
-example of a procedural representation for data.
-
- (8) `Rotate90' is a pure rotation only for square frames, because it
-also stretches and shrinks the image to fit into the rotated frame.
-
- (9) The diamond-shaped images in figures *Note Figure 2-10:: and
-*Note Figure 2-11:: were created with `squash-inwards' applied to
-`wave' and `rogers'.
-
- (10) Section *Note 3-3-4:: describes one such language.
-
-\1f
-File: sicp.info, Node: 2-3, Next: 2-4, Prev: 2-2, Up: Chapter 2
-
-2.3 Symbolic Data
-=================
-
-All the compound data objects we have used so far were constructed
-ultimately from numbers. In this section we extend the
-representational capability of our language by introducing the ability
-to work with arbitrary symbols as data.
-
-* Menu:
-
-* 2-3-1:: Quotation
-* 2-3-2:: Example: Symbolic Differentiation
-* 2-3-3:: Example: Representing Sets
-* 2-3-4:: Example: Huffman Encoding Trees
-
-\1f
-File: sicp.info, Node: 2-3-1, Next: 2-3-2, Prev: 2-3, Up: 2-3
-
-2.3.1 Quotation
----------------
-
-If we can form compound data using symbols, we can have lists such as
-
- (a b c d)
- (23 45 17)
- ((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4))
-
- Lists containing symbols can look just like the expressions of our
-language:
-
- (* (+ 23 45) (+ x 9))
-
- (define (fact n) (if (= n 1) 1 (* n (fact (- n 1)))))
-
- In order to manipulate symbols we need a new element in our
-language: the ability to "quote" a data object. Suppose we want to
-construct the list `(a b)'. We can't accomplish this with `(list a
-b)', because this expression constructs a list of the "values" of `a'
-and `b' rather than the symbols themselves. This issue is well known
-in the context of natural languages, where words and sentences may be
-regarded either as semantic entities or as character strings (syntactic
-entities). The common practice in natural languages is to use
-quotation marks to indicate that a word or a sentence is to be treated
-literally as a string of characters. For instance, the first letter of
-"John" is clearly "J." If we tell somebody "say your name aloud," we
-expect to hear that person's name. However, if we tell somebody "say
-`your name' aloud," we expect to hear the words "your name." Note that
-we are forced to nest quotation marks to describe what somebody else
-might say.(1)
-
- We can follow this same practice to identify lists and symbols that
-are to be treated as data objects rather than as expressions to be
-evaluated. However, our format for quoting differs from that of
-natural languages in that we place a quotation mark (traditionally, the
-single quote symbol `'') only at the beginning of the object to be
-quoted. We can get away with this in Scheme syntax because we rely on
-blanks and parentheses to delimit objects. Thus, the meaning of the
-single quote character is to quote the next object.(2)
-
- Now we can distinguish between symbols and their values:
-
- (define a 1)
-
- (define b 2)
-
- (list a b)
- (1 2)
-
- (list 'a 'b)
- (a b)
-
- (list 'a b)
- (a 2)
-
- Quotation also allows us to type in compound objects, using the
-conventional printed representation for lists:(3)
-
- (car '(a b c))
- a
-
- (cdr '(a b c))
- (b c)
-
- In keeping with this, we can obtain the empty list by evaluating
-`'()', and thus dispense with the variable `nil'.
-
- One additional primitive used in manipulating symbols is `eq?', which
-takes two symbols as arguments and tests whether they are the same.(4)
-Using `eq?', we can implement a useful procedure called `memq'. This
-takes two arguments, a symbol and a list. If the symbol is not
-contained in the list (i.e., is not `eq?' to any item in the list),
-then `memq' returns false. Otherwise, it returns the sublist of the
-list beginning with the first occurrence of the symbol:
-
- (define (memq item x)
- (cond ((null? x) false)
- ((eq? item (car x)) x)
- (else (memq item (cdr x)))))
-
- For example, the value of
-
- (memq 'apple '(pear banana prune))
-
-is false, whereas the value of
-
- (memq 'apple '(x (apple sauce) y apple pear))
-
-is `(apple pear)'.
-
- *Exercise 2.53:* What would the interpreter print in response to
- evaluating each of the following expressions?
-
- (list 'a 'b 'c)
-
- (list (list 'george))
-
- (cdr '((x1 x2) (y1 y2)))
-
- (cadr '((x1 x2) (y1 y2)))
-
- (pair? (car '(a short list)))
-
- (memq 'red '((red shoes) (blue socks)))
-
- (memq 'red '(red shoes blue socks))
-
- *Exercise 2.54:* Two lists are said to be `equal?' if they contain
- equal elements arranged in the same order. For example,
-
- (equal? '(this is a list) '(this is a list))
-
- is true, but
-
- (equal? '(this is a list) '(this (is a) list))
-
- is false. To be more precise, we can define `equal?' recursively
- in terms of the basic `eq?' equality of symbols by saying that `a'
- and `b' are `equal?' if they are both symbols and the symbols are
- `eq?', or if they are both lists such that `(car a)' is `equal?'
- to `(car b)' and `(cdr a)' is `equal?' to `(cdr b)'. Using this
- idea, implement `equal?' as a procedure.(5)
-
- *Exercise 2.55:* Eva Lu Ator types to the interpreter the
- expression
-
- (car ''abracadabra)
-
- To her surprise, the interpreter prints back `quote'. Explain.
-
- ---------- Footnotes ----------
-
- (1) Allowing quotation in a language wreaks havoc with the ability
-to reason about the language in simple terms, because it destroys the
-notion that equals can be substituted for equals. For example, three
-is one plus two, but the word "three" is not the phrase "one plus two."
-Quotation is powerful because it gives us a way to build expressions
-that manipulate other expressions (as we will see when we write an
-interpreter in *Note Chapter 4::). But allowing statements in a
-language that talk about other statements in that language makes it
-very difficult to maintain any coherent principle of what "equals can
-be substituted for equals" should mean. For example, if we know that
-the evening star is the morning star, then from the statement "the
-evening star is Venus" we can deduce "the morning star is Venus."
-However, given that "John knows that the evening star is Venus" we
-cannot infer that "John knows that the morning star is Venus."
-
- (2) The single quote is different from the double quote we have been
-using to enclose character strings to be printed. Whereas the single
-quote can be used to denote lists or symbols, the double quote is used
-only with character strings. In this book, the only use for character
-strings is as items to be printed.
-
- (3) Strictly, our use of the quotation mark violates the general
-rule that all compound expressions in our language should be delimited
-by parentheses and look like lists. We can recover this consistency by
-introducing a special form `quote', which serves the same purpose as
-the quotation mark. Thus, we would type `(quote a)' instead of `'a',
-and we would type `(quote (a b c))' instead of `'(a b c)'. This is
-precisely how the interpreter works. The quotation mark is just a
-single-character abbreviation for wrapping the next complete expression
-with `quote' to form `(quote <EXPRESSION>)'. This is important because
-it maintains the principle that any expression seen by the interpreter
-can be manipulated as a data object. For instance, we could construct
-the expression `(car '(a b c))', which is the same as `(car (quote (a b
-c)))', by evaluating `(list 'car (list 'quote '(a b c)))'.
-
- (4) We can consider two symbols to be "the same" if they consist of
-the same characters in the same order. Such a definition skirts a deep
-issue that we are not yet ready to address: the meaning of "sameness"
-in a programming language. We will return to this in *Note Chapter 3::
-(section *Note 3-1-3::).
-
- (5) In practice, programmers use `equal?' to compare lists that
-contain numbers as well as symbols. Numbers are not considered to be
-symbols. The question of whether two numerically equal numbers (as
-tested by `=') are also `eq?' is highly implementation-dependent. A
-better definition of `equal?' (such as the one that comes as a
-primitive in Scheme) would also stipulate that if `a' and `b' are both
-numbers, then `a' and `b' are `equal?' if they are numerically equal.
-
-\1f
-File: sicp.info, Node: 2-3-2, Next: 2-3-3, Prev: 2-3-1, Up: 2-3
-
-2.3.2 Example: Symbolic Differentiation
----------------------------------------
-
-As an illustration of symbol manipulation and a further illustration of
-data abstraction, consider the design of a procedure that performs
-symbolic differentiation of algebraic expressions. We would like the
-procedure to take as arguments an algebraic expression and a variable
-and to return the derivative of the expression with respect to the
-variable. For example, if the arguments to the procedure are ax^2 + bx
-+ c and x, the procedure should return 2ax + b. Symbolic
-differentiation is of special historical significance in Lisp. It was
-one of the motivating examples behind the development of a computer
-language for symbol manipulation. Furthermore, it marked the beginning
-of the line of research that led to the development of powerful systems
-for symbolic mathematical work, which are currently being used by a
-growing number of applied mathematicians and physicists.
-
- In developing the symbolic-differentiation program, we will follow
-the same strategy of data abstraction that we followed in developing
-the rational-number system of section *Note 2-1-1::. That is, we will
-first define a differentiation algorithm that operates on abstract
-objects such as "sums," "products," and "variables" without worrying
-about how these are to be represented. Only afterward will we address
-the representation problem.
-
-The differentiation program with abstract data
-..............................................
-
-In order to keep things simple, we will consider a very simple
-symbolic-differentiation program that handles expressions that are
-built up using only the operations of addition and multiplication with
-two arguments. Differentiation of any such expression can be carried
-out by applying the following reduction rules:
-
- dc
- -- = 0 for c a constant, or a variable different from x
- dx
-
- dx
- -- = 1
- dx
-
- d(u + v) du dv
- -------- = -- + --
- dx dx dx
-
- d(uv) / dv \ / du \
- ----- = u | -- | + v | -- |
- dx \ dx / \ dx /
-
- Observe that the latter two rules are recursive in nature. That is,
-to obtain the derivative of a sum we first find the derivatives of the
-terms and add them. Each of the terms may in turn be an expression
-that needs to be decomposed. Decomposing into smaller and smaller
-pieces will eventually produce pieces that are either constants or
-variables, whose derivatives will be either 0 or 1.
-
- To embody these rules in a procedure we indulge in a little wishful
-thinking, as we did in designing the rational-number implementation.
-If we had a means for representing algebraic expressions, we should be
-able to tell whether an expression is a sum, a product, a constant, or
-a variable. We should be able to extract the parts of an expression.
-For a sum, for example we want to be able to extract the addend (first
-term) and the augend (second term). We should also be able to
-construct expressions from parts. Let us assume that we already have
-procedures to implement the following selectors, constructors, and
-predicates:
-
- (variable? e) Is `e' a variable?
- (same-variable? v1 v2) Are `v1' and `v2' the same variable?
- (sum? e) Is `e' a sum?
- (addend e) Addend of the sum `e'.
- (augend e) Augend of the sum `e'.
- (make-sum a1 a2) Construct the sum of `a1' and `a2'.
- (product? e) Is `e' a product?
- (multiplier e) Multiplier of the product `e'.
- (multiplicand e) Multiplicand of the product `e'.
- (make-product m1 m2) Construct the product of `m1' and `m2'.
-
- Using these, and the primitive predicate `number?', which identifies
-numbers, we can express the differentiation rules as the following
-procedure:
-
- (define (deriv exp var)
- (cond ((number? exp) 0)
- ((variable? exp)
- (if (same-variable? exp var) 1 0))
- ((sum? exp)
- (make-sum (deriv (addend exp) var)
- (deriv (augend exp) var)))
- ((product? exp)
- (make-sum
- (make-product (multiplier exp)
- (deriv (multiplicand exp) var))
- (make-product (deriv (multiplier exp) var)
- (multiplicand exp))))
- (else
- (error "unknown expression type -- DERIV" exp))))
-
- This `deriv' procedure incorporates the complete differentiation
-algorithm. Since it is expressed in terms of abstract data, it will
-work no matter how we choose to represent algebraic expressions, as
-long as we design a proper set of selectors and constructors. This is
-the issue we must address next.
-
-Representing algebraic expressions
-..................................
-
-We can imagine many ways to use list structure to represent algebraic
-expressions. For example, we could use lists of symbols that mirror
-the usual algebraic notation, representing ax + b as the list `(a * x +
-b)' . However, one especially straightforward choice is to use the same
-parenthesized prefix notation that Lisp uses for combinations; that is,
-to represent ax + b as `(+ (* a x) b)'. Then our data representation
-for the differentiation problem is as follows:
-
- * The variables are symbols. They are identified by the primitive
- predicate `symbol?':
-
- (define (variable? x) (symbol? x))
-
- * Two variables are the same if the symbols representing them are
- `eq?':
-
- (define (same-variable? v1 v2)
- (and (variable? v1) (variable? v2) (eq? v1 v2)))
-
- * Sums and products are constructed as lists:
-
- (define (make-sum a1 a2) (list '+ a1 a2))
-
- (define (make-product m1 m2) (list '* m1 m2))
-
- * A sum is a list whose first element is the symbol `+':
-
- (define (sum? x)
- (and (pair? x) (eq? (car x) '+)))
-
- * The addend is the second item of the sum list:
-
- (define (addend s) (cadr s))
-
- * The augend is the third item of the sum list:
-
- (define (augend s) (caddr s))
-
- * A product is a list whose first element is the symbol `*':
-
- (define (product? x)
- (and (pair? x) (eq? (car x) '*)))
-
- * The multiplier is the second item of the product list:
-
- (define (multiplier p) (cadr p))
-
- * The multiplicand is the third item of the product list:
-
- (define (multiplicand p) (caddr p))
-
-
- Thus, we need only combine these with the algorithm as embodied by
-`deriv' in order to have a working symbolic-differentiation program.
-Let us look at some examples of its behavior:
-
- (deriv '(+ x 3) 'x)
- (+ 1 0)
-
- (deriv '(* x y) 'x)
- (+ (* x 0) (* 1 y))
-
- (deriv '(* (* x y) (+ x 3)) 'x)
- (+ (* (* x y) (+ 1 0))
- (* (+ (* x 0) (* 1 y))
- (+ x 3)))
-
- The program produces answers that are correct; however, they are
-unsimplified. It is true that
-
- d(xy)
- ----- = x * 0 + 1 * y
- dx
-
-but we would like the program to know that x * 0 = 0, 1 * y = y, and 0
-+ y = y. The answer for the second example should have been simply
-`y'. As the third example shows, this becomes a serious issue when the
-expressions are complex.
-
- Our difficulty is much like the one we encountered with the
-rational-number implementation: we haven't reduced answers to simplest
-form. To accomplish the rational-number reduction, we needed to change
-only the constructors and the selectors of the implementation. We can
-adopt a similar strategy here. We won't change `deriv' at all.
-Instead, we will change `make-sum' so that if both summands are
-numbers, `make-sum' will add them and return their sum. Also, if one
-of the summands is 0, then `make-sum' will return the other summand.
-
- (define (make-sum a1 a2)
- (cond ((=number? a1 0) a2)
- ((=number? a2 0) a1)
- ((and (number? a1) (number? a2)) (+ a1 a2))
- (else (list '+ a1 a2))))
-
- This uses the procedure `=number?', which checks whether an
-expression is equal to a given number:
-
- (define (=number? exp num)
- (and (number? exp) (= exp num)))
-
- Similarly, we will change `make-product' to build in the rules that 0
-times anything is 0 and 1 times anything is the thing itself:
-
- (define (make-product m1 m2)
- (cond ((or (=number? m1 0) (=number? m2 0)) 0)
- ((=number? m1 1) m2)
- ((=number? m2 1) m1)
- ((and (number? m1) (number? m2)) (* m1 m2))
- (else (list '* m1 m2))))
-
- Here is how this version works on our three examples:
-
- (deriv '(+ x 3) 'x)
- 1
-
- (deriv '(* x y) 'x)
- y
-
- (deriv '(* (* x y) (+ x 3)) 'x)
- (+ (* x y) (* y (+ x 3)))
-
- Although this is quite an improvement, the third example shows that
-there is still a long way to go before we get a program that puts
-expressions into a form that we might agree is "simplest." The problem
-of algebraic simplification is complex because, among other reasons, a
-form that may be simplest for one purpose may not be for another.
-
- *Exercise 2.56:* Show how to extend the basic differentiator to
- handle more kinds of expressions. For instance, implement the
- differentiation rule
-
- n_1 n_2
- --- = --- if and only if n_1 d_2 = n_2 d_1
- d_1 d_2
-
- by adding a new clause to the `deriv' program and defining
- appropriate procedures `exponentiation?', `base', `exponent', and
- `make-exponentiation'. (You may use the symbol `**' to denote
- exponentiation.) Build in the rules that anything raised to the
- power 0 is 1 and anything raised to the power 1 is the thing
- itself.
-
- *Exercise 2.57:* Extend the differentiation program to handle sums
- and products of arbitrary numbers of (two or more) terms. Then
- the last example above could be expressed as
-
- (deriv '(* x y (+ x 3)) 'x)
-
- Try to do this by changing only the representation for sums and
- products, without changing the `deriv' procedure at all. For
- example, the `addend' of a sum would be the first term, and the
- `augend' would be the sum of the rest of the terms.
-
- *Exercise 2.58:* Suppose we want to modify the differentiation
- program so that it works with ordinary mathematical notation, in
- which `+' and `*' are infix rather than prefix operators. Since
- the differentiation program is defined in terms of abstract data,
- we can modify it to work with different representations of
- expressions solely by changing the predicates, selectors, and
- constructors that define the representation of the algebraic
- expressions on which the differentiator is to operate.
-
- a. Show how to do this in order to differentiate algebraic
- expressions presented in infix form, such as `(x + (3 * (x +
- (y + 2))))'. To simplify the task, assume that `+' and `*'
- always take two arguments and that expressions are fully
- parenthesized.
-
- b. The problem becomes substantially harder if we allow standard
- algebraic notation, such as `(x + 3 * (x + y + 2))', which
- drops unnecessary parentheses and assumes that multiplication
- is done before addition. Can you design appropriate
- predicates, selectors, and constructors for this notation
- such that our derivative program still works?
-
-
-\1f
-File: sicp.info, Node: 2-3-3, Next: 2-3-4, Prev: 2-3-2, Up: 2-3
-
-2.3.3 Example: Representing Sets
---------------------------------
-
-In the previous examples we built representations for two kinds of
-compound data objects: rational numbers and algebraic expressions. In
-one of these examples we had the choice of simplifying (reducing) the
-expressions at either construction time or selection time, but other
-than that the choice of a representation for these structures in terms
-of lists was straightforward. When we turn to the representation of
-sets, the choice of a representation is not so obvious. Indeed, there
-are a number of possible representations, and they differ significantly
-from one another in several ways.
-
- Informally, a set is simply a collection of distinct objects. To
-give a more precise definition we can employ the method of data
-abstraction. That is, we define "set" by specifying the operations
-that are to be used on sets. These are `union-set',
-`intersection-set', `element-of-set?', and `adjoin-set'.
-`Element-of-set?' is a predicate that determines whether a given
-element is a member of a set. `Adjoin-set' takes an object and a set
-as arguments and returns a set that contains the elements of the
-original set and also the adjoined element. `Union-set' computes the
-union of two sets, which is the set containing each element that
-appears in either argument. `Intersection-set' computes the
-intersection of two sets, which is the set containing only elements
-that appear in both arguments. From the viewpoint of data abstraction,
-we are free to design any representation that implements these
-operations in a way consistent with the interpretations given above.(1)
-
-Sets as unordered lists
-.......................
-
-One way to represent a set is as a list of its elements in which no
-element appears more than once. The empty set is represented by the
-empty list. In this representation, `element-of-set?' is similar to
-the procedure `memq' of section *Note 2-3-1::. It uses `equal?'
-instead of `eq?' so that the set elements need not be symbols:
-
- (define (element-of-set? x set)
- (cond ((null? set) false)
- ((equal? x (car set)) true)
- (else (element-of-set? x (cdr set)))))
-
- Using this, we can write `adjoin-set'. If the object to be adjoined
-is already in the set, we just return the set. Otherwise, we use
-`cons' to add the object to the list that represents the set:
-
- (define (adjoin-set x set)
- (if (element-of-set? x set)
- set
- (cons x set)))
-
- For `intersection-set' we can use a recursive strategy. If we know
-how to form the intersection of `set2' and the `cdr' of `set1', we only
-need to decide whether to include the `car' of `set1' in this. But
-this depends on whether `(car set1)' is also in `set2'. Here is the
-resulting procedure:
-
- (define (intersection-set set1 set2)
- (cond ((or (null? set1) (null? set2)) '())
- ((element-of-set? (car set1) set2)
- (cons (car set1)
- (intersection-set (cdr set1) set2)))
- (else (intersection-set (cdr set1) set2))))
-
- In designing a representation, one of the issues we should be
-concerned with is efficiency. Consider the number of steps required by
-our set operations. Since they all use `element-of-set?', the speed of
-this operation has a major impact on the efficiency of the set
-implementation as a whole. Now, in order to check whether an object is
-a member of a set, `element-of-set?' may have to scan the entire set.
-(In the worst case, the object turns out not to be in the set.) Hence,
-if the set has n elements, `element-of-set?' might take up to n steps.
-Thus, the number of steps required grows as [theta](n). The number of
-steps required by `adjoin-set', which uses this operation, also grows
-as [theta](n). For `intersection-set', which does an `element-of-set?'
-check for each element of `set1', the number of steps required grows as
-the product of the sizes of the sets involved, or [theta](n^2) for two
-sets of size n. The same will be true of `union-set'.
-
- *Exercise 2.59:* Implement the `union-set' operation for the
- unordered-list representation of sets.
-
- *Exercise 2.60:* We specified that a set would be represented as a
- list with no duplicates. Now suppose we allow duplicates. For
- instance, the set {1,2,3} could be represented as the list `(2 3 2
- 1 3 2 2)'. Design procedures `element-of-set?', `adjoin-set',
- `union-set', and `intersection-set' that operate on this
- representation. How does the efficiency of each compare with the
- corresponding procedure for the non-duplicate representation? Are
- there applications for which you would use this representation in
- preference to the non-duplicate one?
-
-Sets as ordered lists
-.....................
-
-One way to speed up our set operations is to change the representation
-so that the set elements are listed in increasing order. To do this,
-we need some way to compare two objects so that we can say which is
-bigger. For example, we could compare symbols lexicographically, or we
-could agree on some method for assigning a unique number to an object
-and then compare the elements by comparing the corresponding numbers.
-To keep our discussion simple, we will consider only the case where the
-set elements are numbers, so that we can compare elements using `>' and
-`<'. We will represent a set of numbers by listing its elements in
-increasing order. Whereas our first representation above allowed us to
-represent the set {1,3,6,10} by listing the elements in any order, our
-new representation allows only the list `(1 3 6 10)'.
-
- One advantage of ordering shows up in `element-of-set?': In checking
-for the presence of an item, we no longer have to scan the entire set.
-If we reach a set element that is larger than the item we are looking
-for, then we know that the item is not in the set:
-
- (define (element-of-set? x set)
- (cond ((null? set) false)
- ((= x (car set)) true)
- ((< x (car set)) false)
- (else (element-of-set? x (cdr set)))))
-
- How many steps does this save? In the worst case, the item we are
-looking for may be the largest one in the set, so the number of steps
-is the same as for the unordered representation. On the other hand, if
-we search for items of many different sizes we can expect that
-sometimes we will be able to stop searching at a point near the
-beginning of the list and that other times we will still need to
-examine most of the list. On the average we should expect to have to
-examine about half of the items in the set. Thus, the average number
-of steps required will be about n/2. This is still [theta](n) growth,
-but it does save us, on the average, a factor of 2 in number of steps
-over the previous implementation.
-
- We obtain a more impressive speedup with `intersection-set'. In the
-unordered representation this operation required [theta](n^2) steps,
-because we performed a complete scan of `set2' for each element of
-`set1'. But with the ordered representation, we can use a more clever
-method. Begin by comparing the initial elements, `x1' and `x2', of the
-two sets. If `x1' equals `x2', then that gives an element of the
-intersection, and the rest of the intersection is the intersection of
-the `cdr's of the two sets. Suppose, however, that `x1' is less than
-`x2'. Since `x2' is the smallest element in `set2', we can immediately
-conclude that `x1' cannot appear anywhere in `set2' and hence is not in
-the intersection. Hence, the intersection is equal to the intersection
-of `set2' with the `cdr' of `set1'. Similarly, if `x2' is less than
-`x1', then the intersection is given by the intersection of `set1' with
-the `cdr' of `set2'. Here is the procedure:
-
- (define (intersection-set set1 set2)
- (if (or (null? set1) (null? set2))
- '()
- (let ((x1 (car set1)) (x2 (car set2)))
- (cond ((= x1 x2)
- (cons x1
- (intersection-set (cdr set1)
- (cdr set2))))
- ((< x1 x2)
- (intersection-set (cdr set1) set2))
- ((< x2 x1)
- (intersection-set set1 (cdr set2)))))))
-
- To estimate the number of steps required by this process, observe
-that at each step we reduce the intersection problem to computing
-intersections of smaller sets--removing the first element from `set1'
-or `set2' or both. Thus, the number of steps required is at most the
-sum of the sizes of `set1' and `set2', rather than the product of the
-sizes as with the unordered representation. This is [theta](n) growth
-rather than [theta](n^2)--a considerable speedup, even for sets of
-moderate size.
-
- *Exercise 2.61:* Give an implementation of `adjoin-set' using the
- ordered representation. By analogy with `element-of-set?' show
- how to take advantage of the ordering to produce a procedure that
- requires on the average about half as many steps as with the
- unordered representation.
-
- *Exercise 2.62:* Give a [theta](n) implementation of `union-set'
- for sets represented as ordered lists.
-
-Sets as binary trees
-....................
-
-We can do better than the ordered-list representation by arranging the
-set elements in the form of a tree. Each node of the tree holds one
-element of the set, called the "entry" at that node, and a link to each
-of two other (possibly empty) nodes. The "left" link points to
-elements smaller than the one at the node, and the "right" link to
-elements greater than the one at the node. *Note Figure 2-16:: shows
-some trees that represent the set {1,3,5,7,9,11}. The same set may be
-represented by a tree in a number of different ways. The only thing we
-require for a valid representation is that all elements in the left
-subtree be smaller than the node entry and that all elements in the
-right subtree be larger.
-
- *Figure 2.16:* Various binary trees that represent the set
- {1,3,5,7,9,11}.
-
- 7 3 5
- /\ /\ /\
- 3 9 1 7 3 9
- /\ \ /\ / /\
- 1 5 11 5 9 1 7 11
- \
- 11
-
- The advantage of the tree representation is this: Suppose we want to
-check whether a number x is contained in a set. We begin by comparing
-x with the entry in the top node. If x is less than this, we know that
-we need only search the left subtree; if x is greater, we need only
-search the right subtree. Now, if the tree is "balanced," each of
-these subtrees will be about half the size of the original. Thus, in
-one step we have reduced the problem of searching a tree of size n to
-searching a tree of size n/2. Since the size of the tree is halved at
-each step, we should expect that the number of steps needed to search a
-tree of size n grows as [theta](`log' n).(2) For large sets, this will
-be a significant speedup over the previous representations.
-
- We can represent trees by using lists. Each node will be a list of
-three items: the entry at the node, the left subtree, and the right
-subtree. A left or a right subtree of the empty list will indicate
-that there is no subtree connected there. We can describe this
-representation by the following procedures:(3)
-
- (define (entry tree) (car tree))
-
- (define (left-branch tree) (cadr tree))
-
- (define (right-branch tree) (caddr tree))
-
- (define (make-tree entry left right)
- (list entry left right))
-
- Now we can write the `element-of-set?' procedure using the strategy
-described above:
-
- (define (element-of-set? x set)
- (cond ((null? set) false)
- ((= x (entry set)) true)
- ((< x (entry set))
- (element-of-set? x (left-branch set)))
- ((> x (entry set))
- (element-of-set? x (right-branch set)))))
-
- Adjoining an item to a set is implemented similarly and also requires
-[theta](`log' n) steps. To adjoin an item `x', we compare `x' with the
-node entry to determine whether `x' should be added to the right or to
-the left branch, and having adjoined `x' to the appropriate branch we
-piece this newly constructed branch together with the original entry
-and the other branch. If `x' is equal to the entry, we just return the
-node. If we are asked to adjoin `x' to an empty tree, we generate a
-tree that has `x' as the entry and empty right and left branches. Here
-is the procedure:
-
- (define (adjoin-set x set)
- (cond ((null? set) (make-tree x '() '()))
- ((= x (entry set)) set)
- ((< x (entry set))
- (make-tree (entry set)
- (adjoin-set x (left-branch set))
- (right-branch set)))
- ((> x (entry set))
- (make-tree (entry set)
- (left-branch set)
- (adjoin-set x (right-branch set))))))
-
- The above claim that searching the tree can be performed in a
-logarithmic number of steps rests on the assumption that the tree is
-"balanced," i.e., that the left and the right subtree of every tree
-have approximately the same number of elements, so that each subtree
-contains about half the elements of its parent. But how can we be
-certain that the trees we construct will be balanced? Even if we start
-with a balanced tree, adding elements with `adjoin-set' may produce an
-unbalanced result. Since the position of a newly adjoined element
-depends on how the element compares with the items already in the set,
-we can expect that if we add elements "randomly" the tree will tend to
-be balanced on the average. But this is not a guarantee. For example,
-if we start with an empty set and adjoin the numbers 1 through 7 in
-sequence we end up with the highly unbalanced tree shown in *Note
-Figure 2-17::. In this tree all the left subtrees are empty, so it has
-no advantage over a simple ordered list. One way to solve this problem
-is to define an operation that transforms an arbitrary tree into a
-balanced tree with the same elements. Then we can perform this
-transformation after every few `adjoin-set' operations to keep our set
-in balance. There are also other ways to solve this problem, most of
-which involve designing new data structures for which searching and
-insertion both can be done in [theta](`log' n) steps.(4)
-
- *Figure 2.17:* Unbalanced tree produced by adjoining 1 through 7
- in sequence.
-
- 1
- \
- 2
- \
- 4
- \
- 5
- \
- 6
- \
- 7
-
- *Exercise 2.63:* Each of the following two procedures converts a
- binary tree to a list.
-
- (define (tree->list-1 tree)
- (if (null? tree)
- '()
- (append (tree->list-1 (left-branch tree))
- (cons (entry tree)
- (tree->list-1 (right-branch tree))))))
-
- (define (tree->list-2 tree)
- (define (copy-to-list tree result-list)
- (if (null? tree)
- result-list
- (copy-to-list (left-branch tree)
- (cons (entry tree)
- (copy-to-list (right-branch tree)
- result-list)))))
- (copy-to-list tree '()))
-
- a. Do the two procedures produce the same result for every tree?
- If not, how do the results differ? What lists do the two
- procedures produce for the trees in *Note Figure 2-16::?
-
- b. Do the two procedures have the same order of growth in the
- number of steps required to convert a balanced tree with n
- elements to a list? If not, which one grows more slowly?
-
-
- *Exercise 2.64:* The following procedure `list->tree' converts an
- ordered list to a balanced binary tree. The helper procedure
- `partial-tree' takes as arguments an integer n and list of at
- least n elements and constructs a balanced tree containing the
- first n elements of the list. The result returned by
- `partial-tree' is a pair (formed with `cons') whose `car' is the
- constructed tree and whose `cdr' is the list of elements not
- included in the tree.
-
- (define (list->tree elements)
- (car (partial-tree elements (length elements))))
-
- (define (partial-tree elts n)
- (if (= n 0)
- (cons '() elts)
- (let ((left-size (quotient (- n 1) 2)))
- (let ((left-result (partial-tree elts left-size)))
- (let ((left-tree (car left-result))
- (non-left-elts (cdr left-result))
- (right-size (- n (+ left-size 1))))
- (let ((this-entry (car non-left-elts))
- (right-result (partial-tree (cdr non-left-elts)
- right-size)))
- (let ((right-tree (car right-result))
- (remaining-elts (cdr right-result)))
- (cons (make-tree this-entry left-tree right-tree)
- remaining-elts))))))))
-
- a. Write a short paragraph explaining as clearly as you can how
- `partial-tree' works. Draw the tree produced by `list->tree'
- for the list `(1 3 5 7 9 11)'.
-
- b. What is the order of growth in the number of steps required by
- `list->tree' to convert a list of n elements?
-
-
- *Exercise 2.65:* Use the results of *Note Exercise 2-63:: and
- *Note Exercise 2-64:: to give [theta](n) implementations of
- `union-set' and `intersection-set' for sets implemented as
- (balanced) binary trees.(5)
-
-Sets and information retrieval
-..............................
-
-We have examined options for using lists to represent sets and have
-seen how the choice of representation for a data object can have a
-large impact on the performance of the programs that use the data.
-Another reason for concentrating on sets is that the techniques
-discussed here appear again and again in applications involving
-information retrieval.
-
- Consider a data base containing a large number of individual
-records, such as the personnel files for a company or the transactions
-in an accounting system. A typical data-management system spends a
-large amount of time accessing or modifying the data in the records and
-therefore requires an efficient method for accessing records. This is
-done by identifying a part of each record to serve as an identifying "key".
-A key can be anything that uniquely identifies the record. For a
-personnel file, it might be an employee's ID number. For an accounting
-system, it might be a transaction number. Whatever the key is, when we
-define the record as a data structure we should include a `key'
-selector procedure that retrieves the key associated with a given
-record.
-
- Now we represent the data base as a set of records. To locate the
-record with a given key we use a procedure `lookup', which takes as
-arguments a key and a data base and which returns the record that has
-that key, or false if there is no such record. `Lookup' is implemented
-in almost the same way as `element-of-set?'. For example, if the set
-of records is implemented as an unordered list, we could use
-
- (define (lookup given-key set-of-records)
- (cond ((null? set-of-records) false)
- ((equal? given-key (key (car set-of-records)))
- (car set-of-records))
- (else (lookup given-key (cdr set-of-records)))))
-
- Of course, there are better ways to represent large sets than as
-unordered lists. Information-retrieval systems in which records have
-to be "randomly accessed" are typically implemented by a tree-based
-method, such as the binary-tree representation discussed previously.
-In designing such a system the methodology of data abstraction can be a
-great help. The designer can create an initial implementation using a
-simple, straightforward representation such as unordered lists. This
-will be unsuitable for the eventual system, but it can be useful in
-providing a "quick and dirty" data base with which to test the rest of
-the system. Later on, the data representation can be modified to be
-more sophisticated. If the data base is accessed in terms of abstract
-selectors and constructors, this change in representation will not
-require any changes to the rest of the system.
-
- *Exercise 2.66:* Implement the `lookup' procedure for the case
- where the set of records is structured as a binary tree, ordered
- by the numerical values of the keys.
-
- ---------- Footnotes ----------
-
- (1) If we want to be more formal, we can specify "consistent with
-the interpretations given above" to mean that the operations satisfy a
-collection of rules such as these:
-
- * For any set `S' and any object `x', `(element-of-set? x
- (adjoin-set x S))' is true (informally: "Adjoining an object to a
- set produces a set that contains the object").
-
- * For any sets `S' and `T' and any object `x', `(element-of-set? x
- (union-set S T))' is equal to `(or (element-of-set? x S)
- (element-of-set? x T))' (informally: "The elements of `(union S
- T)' are the elements that are in `S' or in `T'").
-
- * For any object `x', `(element-of-set? x '())' is false
- (informally: "No object is an element of the empty set").
-
-
- (2) Halving the size of the problem at each step is the
-distinguishing characteristic of logarithmic growth, as we saw with the
-fast-exponentiation algorithm of section *Note 1-2-4:: and the
-half-interval search method of section *Note 1-3-3::.
-
- (3) We are representing sets in terms of trees, and trees in terms
-of lists--in effect, a data abstraction built upon a data abstraction.
-We can regard the procedures `entry', `left-branch', `right-branch',
-and `make-tree' as a way of isolating the abstraction of a "binary
-tree" from the particular way we might wish to represent such a tree in
-terms of list structure.
-
- (4) Examples of such structures include "B-trees" and "red-black
-trees". There is a large literature on data structures devoted to this
-problem. See Cormen, Leiserson, and Rivest 1990.
-
- (5) *Note Exercise 2-63:: through *Note Exercise 2-65:: are due to
-Paul Hilfinger.
-
-\1f
-File: sicp.info, Node: 2-3-4, Prev: 2-3-3, Up: 2-3
-
-2.3.4 Example: Huffman Encoding Trees
--------------------------------------
-
-This section provides practice in the use of list structure and data
-abstraction to manipulate sets and trees. The application is to
-methods for representing data as sequences of ones and zeros (bits).
-For example, the ASCII standard code used to represent text in
-computers encodes each character as a sequence of seven bits. Using
-seven bits allows us to distinguish 2^(7), or 128, possible different
-characters. In general, if we want to distinguish n different symbols,
-we will need to use `log'_2 n bits per symbol. If all our messages are
-made up of the eight symbols A, B, C, D, E, F, G, and H, we can choose
-a code with three bits per character, for example
-
- A 000 C 010 E 100 G 110
- B 001 D 011 F 101 H 111
-
-With this code, the message
-
- BACADAEAFABBAAAGAH
-
-is encoded as the string of 54 bits
-
- 001000010000011000100000101000001001000000000110000111
-
- Codes such as ASCII and the A-through-H code above are known as "fixed-length"
-codes, because they represent each symbol in the message with the same
-number of bits. It is sometimes advantageous to use "variable-length"
-codes, in which different symbols may be represented by different
-numbers of bits. For example, Morse code does not use the same number
-of dots and dashes for each letter of the alphabet. In particular, E,
-the most frequent letter, is represented by a single dot. In general,
-if our messages are such that some symbols appear very frequently and
-some very rarely, we can encode data more efficiently (i.e., using
-fewer bits per message) if we assign shorter codes to the frequent
-symbols. Consider the following alternative code for the letters A
-through H:
-
- A 0 C 1010 E 1100 G 1110
- B 100 D 1011 F 1101 H 1111
-
-With this code, the same message as above is encoded as the string
-
- 100010100101101100011010100100000111001111
-
- This string contains 42 bits, so it saves more than 20% in space in
-comparison with the fixed-length code shown above.
-
- One of the difficulties of using a variable-length code is knowing
-when you have reached the end of a symbol in reading a sequence of
-zeros and ones. Morse code solves this problem by using a special "separator
-code" (in this case, a pause) after the sequence of dots and dashes for
-each letter. Another solution is to design the code in such a way that
-no complete code for any symbol is the beginning (or "prefix") of the
-code for another symbol. Such a code is called a "prefix code". In
-the example above, A is encoded by 0 and B is encoded by 100, so no
-other symbol can have a code that begins with 0 or with 100.
-
- In general, we can attain significant savings if we use
-variable-length prefix codes that take advantage of the relative
-frequencies of the symbols in the messages to be encoded. One
-particular scheme for doing this is called the Huffman encoding method,
-after its discoverer, David Huffman. A Huffman code can be represented
-as a binary tree whose leaves are the symbols that are encoded. At
-each non-leaf node of the tree there is a set containing all the
-symbols in the leaves that lie below the node. In addition, each
-symbol at a leaf is assigned a weight (which is its relative
-frequency), and each non-leaf node contains a weight that is the sum of
-all the weights of the leaves lying below it. The weights are not used
-in the encoding or the decoding process. We will see below how they
-are used to help construct the tree.
-
- *Figure 2.18:* A Huffman encoding tree.
-
- {A B C D E F G H} 17
- *
- / \
- / \
- A 8 * {B C D E F G H} 9
- __________/ \_____________
- / \
- {B C D} 5 * * {E F G H} 4
- / \ ___/ \___
- / \ / \
- B 3 * {C D} 2 {E F} 2 * * {G H} 2
- / \ / \ / \
- / \ / \ / \
- C 1 D 1 E 1 F 1 G 1 H 1
-
- *Note Figure 2-18:: shows the Huffman tree for the A-through-H code
-given above. The weights at the leaves indicate that the tree was
-designed for messages in which A appears with relative frequency 8, B
-with relative frequency 3, and the other letters each with relative
-frequency 1.
-
- Given a Huffman tree, we can find the encoding of any symbol by
-starting at the root and moving down until we reach the leaf that holds
-the symbol. Each time we move down a left branch we add a 0 to the
-code, and each time we move down a right branch we add a 1. (We decide
-which branch to follow by testing to see which branch either is the
-leaf node for the symbol or contains the symbol in its set.) For
-example, starting from the root of the tree in *Note Figure 2-18::, we
-arrive at the leaf for D by following a right branch, then a left
-branch, then a right branch, then a right branch; hence, the code for D
-is 1011.
-
- To decode a bit sequence using a Huffman tree, we begin at the root
-and use the successive zeros and ones of the bit sequence to determine
-whether to move down the left or the right branch. Each time we come
-to a leaf, we have generated a new symbol in the message, at which
-point we start over from the root of the tree to find the next symbol.
-For example, suppose we are given the tree above and the sequence
-10001010. Starting at the root, we move down the right branch, (since
-the first bit of the string is 1), then down the left branch (since the
-second bit is 0), then down the left branch (since the third bit is
-also 0). This brings us to the leaf for B, so the first symbol of the
-decoded message is B. Now we start again at the root, and we make a
-left move because the next bit in the string is 0. This brings us to
-the leaf for A. Then we start again at the root with the rest of the
-string 1010, so we move right, left, right, left and reach C. Thus,
-the entire message is BAC.
-
-Generating Huffman trees
-........................
-
-Given an "alphabet" of symbols and their relative frequencies, how do we
-construct the "best" code? (In other words, which tree will encode
-messages with the fewest bits?) Huffman gave an algorithm for doing
-this and showed that the resulting code is indeed the best
-variable-length code for messages where the relative frequency of the
-symbols matches the frequencies with which the code was constructed.
-We will not prove this optimality of Huffman codes here, but we will
-show how Huffman trees are constructed.(1)
-
- The algorithm for generating a Huffman tree is very simple. The idea
-is to arrange the tree so that the symbols with the lowest frequency
-appear farthest away from the root. Begin with the set of leaf nodes,
-containing symbols and their frequencies, as determined by the initial
-data from which the code is to be constructed. Now find two leaves with
-the lowest weights and merge them to produce a node that has these two
-nodes as its left and right branches. The weight of the new node is the
-sum of the two weights. Remove the two leaves from the original set and
-replace them by this new node. Now continue this process. At each step,
-merge two nodes with the smallest weights, removing them from the set
-and replacing them with a node that has these two as its left and right
-branches. The process stops when there is only one node left, which is
-the root of the entire tree. Here is how the Huffman tree of *Note
-Figure 2-18:: was generated:
-
- Initial leaves {(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)}
- Merge {(A 8) (B 3) ({C D} 2) (E 1) (F 1) (G 1) (H 1)}
- Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) (G 1) (H 1)}
- Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) ({G H} 2)}
- Merge {(A 8) (B 3) ({C D} 2) ({E F G H} 4)}
- Merge {(A 8) ({B C D} 5) ({E F G H} 4)}
- Merge {(A 8) ({B C D E F G H} 9)}
- Final merge {({A B C D E F G H} 17)}
-
- The algorithm does not always specify a unique tree, because there
-may not be unique smallest-weight nodes at each step. Also, the choice
-of the order in which the two nodes are merged (i.e., which will be the
-right branch and which will be the left branch) is arbitrary.
-
-Representing Huffman trees
-..........................
-
-In the exercises below we will work with a system that uses Huffman
-trees to encode and decode messages and generates Huffman trees
-according to the algorithm outlined above. We will begin by discussing
-how trees are represented.
-
- Leaves of the tree are represented by a list consisting of the symbol
-`leaf', the symbol at the leaf, and the weight:
-
- (define (make-leaf symbol weight)
- (list 'leaf symbol weight))
-
- (define (leaf? object)
- (eq? (car object) 'leaf))
-
- (define (symbol-leaf x) (cadr x))
-
- (define (weight-leaf x) (caddr x))
-
- A general tree will be a list of a left branch, a right branch, a
-set of symbols, and a weight. The set of symbols will be simply a list
-of the symbols, rather than some more sophisticated set representation.
-When we make a tree by merging two nodes, we obtain the weight of the
-tree as the sum of the weights of the nodes, and the set of symbols as
-the union of the sets of symbols for the nodes. Since our symbol sets
-are represented as lists, we can form the union by using the `append'
-procedure we defined in section *Note 2-2-1:::
-
- (define (make-code-tree left right)
- (list left
- right
- (append (symbols left) (symbols right))
- (+ (weight left) (weight right))))
-
- If we make a tree in this way, we have the following selectors:
-
- (define (left-branch tree) (car tree))
-
- (define (right-branch tree) (cadr tree))
-
- (define (symbols tree)
- (if (leaf? tree)
- (list (symbol-leaf tree))
- (caddr tree)))
-
- (define (weight tree)
- (if (leaf? tree)
- (weight-leaf tree)
- (cadddr tree)))
-
- The procedures `symbols' and `weight' must do something slightly
-different depending on whether they are called with a leaf or a general
-tree. These are simple examples of "generic procedures" (procedures
-that can handle more than one kind of data), which we will have much
-more to say about in sections *Note 2-4:: and *Note 2-5::.
-
-The decoding procedure
-......................
-
-The following procedure implements the decoding algorithm. It takes as
-arguments a list of zeros and ones, together with a Huffman tree.
-
- (define (decode bits tree)
- (define (decode-1 bits current-branch)
- (if (null? bits)
- '()
- (let ((next-branch
- (choose-branch (car bits) current-branch)))
- (if (leaf? next-branch)
- (cons (symbol-leaf next-branch)
- (decode-1 (cdr bits) tree))
- (decode-1 (cdr bits) next-branch)))))
- (decode-1 bits tree))
-
- (define (choose-branch bit branch)
- (cond ((= bit 0) (left-branch branch))
- ((= bit 1) (right-branch branch))
- (else (error "bad bit -- CHOOSE-BRANCH" bit))))
-
- The procedure `decode-1' takes two arguments: the list of remaining
-bits and the current position in the tree. It keeps moving "down" the
-tree, choosing a left or a right branch according to whether the next
-bit in the list is a zero or a one. (This is done with the procedure
-`choose-branch'.) When it reaches a leaf, it returns the symbol at
-that leaf as the next symbol in the message by `cons'ing it onto the
-result of decoding the rest of the message, starting at the root of the
-tree. Note the error check in the final clause of `choose-branch',
-which complains if the procedure finds something other than a zero or a
-one in the input data.
-
-Sets of weighted elements
-.........................
-
-In our representation of trees, each non-leaf node contains a set of
-symbols, which we have represented as a simple list. However, the
-tree-generating algorithm discussed above requires that we also work
-with sets of leaves and trees, successively merging the two smallest
-items. Since we will be required to repeatedly find the smallest item
-in a set, it is convenient to use an ordered representation for this
-kind of set.
-
- We will represent a set of leaves and trees as a list of elements,
-arranged in increasing order of weight. The following `adjoin-set'
-procedure for constructing sets is similar to the one described in
-*Note Exercise 2-61::; however, items are compared by their weights,
-and the element being added to the set is never already in it.
-
- (define (adjoin-set x set)
- (cond ((null? set) (list x))
- ((< (weight x) (weight (car set))) (cons x set))
- (else (cons (car set)
- (adjoin-set x (cdr set))))))
-
- The following procedure takes a list of symbol-frequency pairs such
-as `((A 4) (B 2) (C 1) (D 1))' and constructs an initial ordered set of
-leaves, ready to be merged according to the Huffman algorithm:
-
- (define (make-leaf-set pairs)
- (if (null? pairs)
- '()
- (let ((pair (car pairs)))
- (adjoin-set (make-leaf (car pair) ; symbol
- (cadr pair)) ; frequency
- (make-leaf-set (cdr pairs))))))
-
- *Exercise 2.67:* Define an encoding tree and a sample message:
-
- (define sample-tree
- (make-code-tree (make-leaf 'A 4)
- (make-code-tree
- (make-leaf 'B 2)
- (make-code-tree (make-leaf 'D 1)
- (make-leaf 'C 1)))))
-
- (define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))
-
- Use the `decode' procedure to decode the message, and give the
- result.
-
- *Exercise 2.68:* The `encode' procedure takes as arguments a
- message and a tree and produces the list of bits that gives the
- encoded message.
-
- (define (encode message tree)
- (if (null? message)
- '()
- (append (encode-symbol (car message) tree)
- (encode (cdr message) tree))))
-
- `Encode-symbol' is a procedure, which you must write, that returns
- the list of bits that encodes a given symbol according to a given
- tree. You should design `encode-symbol' so that it signals an
- error if the symbol is not in the tree at all. Test your
- procedure by encoding the result you obtained in *Note Exercise
- 2-67:: with the sample tree and seeing whether it is the same as
- the original sample message.
-
- *Exercise 2.69:* The following procedure takes as its argument a
- list of symbol-frequency pairs (where no symbol appears in more
- than one pair) and generates a Huffman encoding tree according to
- the Huffman algorithm.
-
- (define (generate-huffman-tree pairs)
- (successive-merge (make-leaf-set pairs)))
-
- `Make-leaf-set' is the procedure given above that transforms the
- list of pairs into an ordered set of leaves. `Successive-merge'
- is the procedure you must write, using `make-code-tree' to
- successively merge the smallest-weight elements of the set until
- there is only one element left, which is the desired Huffman tree.
- (This procedure is slightly tricky, but not really complicated.
- If you find yourself designing a complex procedure, then you are
- almost certainly doing something wrong. You can take significant
- advantage of the fact that we are using an ordered set
- representation.)
-
- *Exercise 2.70:* The following eight-symbol alphabet with
- associated relative frequencies was designed to efficiently encode
- the lyrics of 1950s rock songs. (Note that the "symbols" of an
- "alphabet" need not be individual letters.)
-
- A 2 NA 16
- BOOM 1 SHA 3
- GET 2 YIP 9
- JOB 2 WAH 1
-
- Use `generate-huffman-tree' (*Note Exercise 2-69::) to generate a
- corresponding Huffman tree, and use `encode' (*Note Exercise
- 2-68::) to encode the following message:
-
- Get a job
-
- Sha na na na na na na na na
-
- Get a job
-
- Sha na na na na na na na na
-
- Wah yip yip yip yip yip yip yip yip yip
-
- Sha boom
-
- How many bits are required for the encoding? What is the smallest
- number of bits that would be needed to encode this song if we used
- a fixed-length code for the eight-symbol alphabet?
-
- *Exercise 2.71:* Suppose we have a Huffman tree for an alphabet of
- n symbols, and that the relative frequencies of the symbols are 1,
- 2, 4, ..., 2^(n-1). Sketch the tree for n=5; for n=10. In such a
- tree (for general n) how may bits are required to encode the most
- frequent symbol? the least frequent symbol?
-
- *Exercise 2.72:* Consider the encoding procedure that you designed
- in *Note Exercise 2-68::. What is the order of growth in the
- number of steps needed to encode a symbol? Be sure to include the
- number of steps needed to search the symbol list at each node
- encountered. To answer this question in general is difficult.
- Consider the special case where the relative frequencies of the n
- symbols are as described in *Note Exercise 2-71::, and give the
- order of growth (as a function of n) of the number of steps needed
- to encode the most frequent and least frequent symbols in the
- alphabet.
-
- ---------- Footnotes ----------
-
- (1) See Hamming 1980 for a discussion of the mathematical properties
-of Huffman codes.
-
-\1f
-File: sicp.info, Node: 2-4, Next: 2-5, Prev: 2-3, Up: Chapter 2
-
-2.4 Multiple Representations for Abstract Data
-==============================================
-
-We have introduced data abstraction, a methodology for structuring
-systems in such a way that much of a program can be specified
-independent of the choices involved in implementing the data objects
-that the program manipulates. For example, we saw in section *Note
-2-1-1:: how to separate the task of designing a program that uses
-rational numbers from the task of implementing rational numbers in
-terms of the computer language's primitive mechanisms for constructing
-compound data. The key idea was to erect an abstraction barrier - in
-this case, the selectors and constructors for rational numbers
-(`make-rat', `numer', `denom')--that isolates the way rational numbers
-are used from their underlying representation in terms of list
-structure. A similar abstraction barrier isolates the details of the
-procedures that perform rational arithmetic (`add-rat', `sub-rat',
-`mul-rat', and `div-rat') from the "higher-level" procedures that use
-rational numbers. The resulting program has the structure shown in
-*Note Figure 2-1::.
-
- These data-abstraction barriers are powerful tools for controlling
-complexity. By isolating the underlying representations of data
-objects, we can divide the task of designing a large program into
-smaller tasks that can be performed separately. But this kind of data
-abstraction is not yet powerful enough, because it may not always make
-sense to speak of "the underlying representation" for a data object.
-
- For one thing, there might be more than one useful representation
-for a data object, and we might like to design systems that can deal
-with multiple representations. To take a simple example, complex
-numbers may be represented in two almost equivalent ways: in
-rectangular form (real and imaginary parts) and in polar form
-(magnitude and angle). Sometimes rectangular form is more appropriate
-and sometimes polar form is more appropriate. Indeed, it is perfectly
-plausible to imagine a system in which complex numbers are represented
-in both ways, and in which the procedures for manipulating complex
-numbers work with either representation.
-
- More importantly, programming systems are often designed by many
-people working over extended periods of time, subject to requirements
-that change over time. In such an environment, it is simply not
-possible for everyone to agree in advance on choices of data
-representation. So in addition to the data-abstraction barriers that
-isolate representation from use, we need abstraction barriers that
-isolate different design choices from each other and permit different
-choices to coexist in a single program. Furthermore, since large
-programs are often created by combining pre-existing modules that were
-designed in isolation, we need conventions that permit programmers to
-incorporate modules into larger systems "additively", that is, without
-having to redesign or reimplement these modules.
-
- In this section, we will learn how to cope with data that may be
-represented in different ways by different parts of a program. This
-requires constructing "generic procedures"--procedures that can operate
-on data that may be represented in more than one way. Our main
-technique for building generic procedures will be to work in terms of
-data objects that have tags "type tags", that is, data objects that
-include explicit information about how they are to be processed. We
-will also discuss "data-directed" programming, a powerful and
-convenient implementation strategy for additively assembling systems
-with generic operations.
-
- We begin with the simple complex-number example. We will see how
-type tags and data-directed style enable us to design separate
-rectangular and polar representations for complex numbers while
-maintaining the notion of an abstract "complex-number" data object. We
-will accomplish this by defining arithmetic procedures for complex
-numbers (`add-complex', `sub-complex', `mul-complex', and
-`div-complex') in terms of generic selectors that access parts of a
-complex number independent of how the number is represented. The
-resulting complex-number system, as shown in *Note Figure 2-19::,
-contains two different kinds of abstraction barriers. The "horizontal"
-abstraction barriers play the same role as the ones in *Note Figure
-2-1::. They isolate "higher-level" operations from "lower-level"
-representations. In addition, there is a "vertical" barrier that gives
-us the ability to separately design and install alternative
-representations.
-
- *Figure 2.19:* Data-abstraction barriers in the complex-number
- system.
-
- Programs that use complex numbers
- +-------------------------------------------------+
- --| add-complex sub-complex mul-complex div-complex |--
- +-------------------------------------------------+
- Complex arithmetic package
- ---------------------------+---------------------------
- Rectangular | Polar
- representation | representation
- ---------------------------+---------------------------
- List structure and primitive machine arithmetic
-
- In section *Note 2-5:: we will show how to use type tags and
-data-directed style to develop a generic arithmetic package. This
-provides procedures (`add', `mul', and so on) that can be used to
-manipulate all sorts of "numbers" and can be easily extended when a new
-kind of number is needed. In section *Note 2-5-3::, we'll show how to
-use generic arithmetic in a system that performs symbolic algebra.
-
-* Menu:
-
-* 2-4-1:: Representations for Complex Numbers
-* 2-4-2:: Tagged data
-* 2-4-3:: Data-Directed Programming and Additivity
-
-\1f
-File: sicp.info, Node: 2-4-1, Next: 2-4-2, Prev: 2-4, Up: 2-4
-
-2.4.1 Representations for Complex Numbers
------------------------------------------
-
-We will develop a system that performs arithmetic operations on complex
-numbers as a simple but unrealistic example of a program that uses
-generic operations. We begin by discussing two plausible
-representations for complex numbers as ordered pairs: rectangular form
-(real part and imaginary part) and polar form (magnitude and angle).(1)
-Section *Note 2-4-2:: will show how both representations can be made
-to coexist in a single system through the use of type tags and generic
-operations.
-
- Like rational numbers, complex numbers are naturally represented as
-ordered pairs. The set of complex numbers can be thought of as a
-two-dimensional space with two orthogonal axes, the "real" axis and the
-"imaginary" axis. (See *Note Figure 2-20::.) From this point of view,
-the complex number z = x + iy (where i^2 = - 1) can be thought of as
-the point in the plane whose real coordinate is x and whose imaginary
-coordinate is y. Addition of complex numbers reduces in this
-representation to addition of coordinates:
-
- Real-part(z_1 + z_2) = Real-part(z_1) + Real-part(z_2)
-
- Imaginary-part(z_1 + z_2) = Imaginary-part(z_1) + Imaginary-part(z_2)
-
- When multiplying complex numbers, it is more natural to think in
-terms of representing a complex number in polar form, as a magnitude
-and an angle (r and A in *Note Figure 2-20::). The product of two
-complex numbers is the vector obtained by stretching one complex number
-by the length of the other and then rotating it through the angle of
-the other:
-
- Magnitude(z_1 * z_2) = Magnitude(z_1) * Magnitude(z_2)
-
- Angle(z_1 * z_2) = Angle(z_1) + Angle(z_2)
-
- *Figure 2.20:* Complex numbers as points in the plane.
-
- Imaginary
- ^
- |
- y |.........................* z = x + ?y = r e^(?A)
- | __-- .
- | __-- .
- | r __-- .
- | __-- .
- | __-- \ .
- |__-- A | .
- ----+----------+-------------------> Real
- x
-
- Thus, there are two different representations for complex numbers,
-which are appropriate for different operations. Yet, from the
-viewpoint of someone writing a program that uses complex numbers, the
-principle of data abstraction suggests that all the operations for
-manipulating complex numbers should be available regardless of which
-representation is used by the computer. For example, it is often
-useful to be able to find the magnitude of a complex number that is
-specified by rectangular coordinates. Similarly, it is often useful to
-be able to determine the real part of a complex number that is
-specified by polar coordinates.
-
- To design such a system, we can follow the same data-abstraction
-strategy we followed in designing the rational-number package in
-section *Note 2-1-1::. Assume that the operations on complex numbers
-are implemented in terms of four selectors: `real-part', `imag-part',
-`magnitude', and `angle'. Also assume that we have two procedures for
-constructing complex numbers: `make-from-real-imag' returns a complex
-number with specified real and imaginary parts, and `make-from-mag-ang'
-returns a complex number with specified magnitude and angle. These
-procedures have the property that, for any complex number `z', both
-
- (make-from-real-imag (real-part z) (imag-part z))
-
-and
-
- (make-from-mag-ang (magnitude z) (angle z))
-
-produce complex numbers that are equal to `z'.
-
- Using these constructors and selectors, we can implement arithmetic
-on complex numbers using the "abstract data" specified by the
-constructors and selectors, just as we did for rational numbers in
-section *Note 2-1-1::. As shown in the formulas above, we can add and
-subtract complex numbers in terms of real and imaginary parts while
-multiplying and dividing complex numbers in terms of magnitudes and
-angles:
-
- (define (add-complex z1 z2)
- (make-from-real-imag (+ (real-part z1) (real-part z2))
- (+ (imag-part z1) (imag-part z2))))
-
- (define (sub-complex z1 z2)
- (make-from-real-imag (- (real-part z1) (real-part z2))
- (- (imag-part z1) (imag-part z2))))
-
- (define (mul-complex z1 z2)
- (make-from-mag-ang (* (magnitude z1) (magnitude z2))
- (+ (angle z1) (angle z2))))
-
- (define (div-complex z1 z2)
- (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
- (- (angle z1) (angle z2))))
-
- To complete the complex-number package, we must choose a
-representation and we must implement the constructors and selectors in
-terms of primitive numbers and primitive list structure. There are two
-obvious ways to do this: We can represent a complex number in
-"rectangular form" as a pair (real part, imaginary part) or in "polar
-form" as a pair (magnitude, angle). Which shall we choose?
-
- In order to make the different choices concrete, imagine that there
-are two programmers, Ben Bitdiddle and Alyssa P. Hacker, who are
-independently designing representations for the complex-number system.
-Ben chooses to represent complex numbers in rectangular form. With
-this choice, selecting the real and imaginary parts of a complex number
-is straightforward, as is constructing a complex number with given real
-and imaginary parts. To find the magnitude and the angle, or to
-construct a complex number with a given magnitude and angle, he uses
-the trigonometric relations
-
- __________
- x = r cos A r = ./ x^2 + y^2
-
- y = r sin A A = arctan(y,x)
-
-which relate the real and imaginary parts (x, y) to the magnitude and
-the angle (r, A).(2) Ben's representation is therefore given by the
-following selectors and constructors:
-
- (define (real-part z) (car z))
-
- (define (imag-part z) (cdr z))
-
- (define (magnitude z)
- (sqrt (+ (square (real-part z)) (square (imag-part z)))))
-
- (define (angle z)
- (atan (imag-part z) (real-part z)))
-
- (define (make-from-real-imag x y) (cons x y))
-
- (define (make-from-mag-ang r a)
- (cons (* r (cos a)) (* r (sin a))))
-
- Alyssa, in contrast, chooses to represent complex numbers in polar
-form. For her, selecting the magnitude and angle is straightforward,
-but she has to use the trigonometric relations to obtain the real and
-imaginary parts. Alyssa's representation is:
-
- (define (real-part z)
- (* (magnitude z) (cos (angle z))))
-
- (define (imag-part z)
- (* (magnitude z) (sin (angle z))))
-
- (define (magnitude z) (car z))
-
- (define (angle z) (cdr z))
-
- (define (make-from-real-imag x y)
- (cons (sqrt (+ (square x) (square y)))
- (atan y x)))
-
- (define (make-from-mag-ang r a) (cons r a))
-
- The discipline of data abstraction ensures that the same
-implementation of `add-complex', `sub-complex', `mul-complex', and
-`div-complex' will work with either Ben's representation or Alyssa's
-representation.
-
- ---------- Footnotes ----------
-
- (1) In actual computational systems, rectangular form is preferable
-to polar form most of the time because of roundoff errors in conversion
-between rectangular and polar form. This is why the complex-number
-example is unrealistic. Nevertheless, it provides a clear illustration
-of the design of a system using generic operations and a good
-introduction to the more substantial systems to be developed later in
-this chapter.
-
- (2) The arctangent function referred to here, computed by Scheme's
-`atan' procedure, is defined so as to take two arguments y and x and to
-return the angle whose tangent is y/x. The signs of the arguments
-determine the quadrant of the angle.
-
-\1f
-File: sicp.info, Node: 2-4-2, Next: 2-4-3, Prev: 2-4-1, Up: 2-4
-
-2.4.2 Tagged data
------------------
-
-One way to view data abstraction is as an application of the "principle
-of least commitment." In implementing the complex-number system in
-section *Note 2-4-1::, we can use either Ben's rectangular
-representation or Alyssa's polar representation. The abstraction
-barrier formed by the selectors and constructors permits us to defer to
-the last possible moment the choice of a concrete representation for
-our data objects and thus retain maximum flexibility in our system
-design.
-
- The principle of least commitment can be carried to even further
-extremes. If we desire, we can maintain the ambiguity of
-representation even _after_ we have designed the selectors and
-constructors, and elect to use both Ben's representation _and_ Alyssa's
-representation. If both representations are included in a single
-system, however, we will need some way to distinguish data in polar
-form from data in rectangular form. Otherwise, if we were asked, for
-instance, to find the `magnitude' of the pair (3,4), we wouldn't know
-whether to answer 5 (interpreting the number in rectangular form) or 3
-(interpreting the number in polar form). A straightforward way to
-accomplish this distinction is to include a "type tag"--the symbol
-`rectangular' or `polar'--as part of each complex number. Then when we
-need to manipulate a complex number we can use the tag to decide which
-selector to apply.
-
- In order to manipulate tagged data, we will assume that we have
-procedures `type-tag' and `contents' that extract from a data object
-the tag and the actual contents (the polar or rectangular coordinates,
-in the case of a complex number). We will also postulate a procedure
-`attach-tag' that takes a tag and contents and produces a tagged data
-object. A straightforward way to implement this is to use ordinary
-list structure:
-
- (define (attach-tag type-tag contents)
- (cons type-tag contents))
-
- (define (type-tag datum)
- (if (pair? datum)
- (car datum)
- (error "Bad tagged datum -- TYPE-TAG" datum)))
-
- (define (contents datum)
- (if (pair? datum)
- (cdr datum)
- (error "Bad tagged datum -- CONTENTS" datum)))
-
- Using these procedures, we can define predicates `rectangular?' and
-`polar?', which recognize polar and rectangular numbers, respectively:
-
- (define (rectangular? z)
- (eq? (type-tag z) 'rectangular))
-
- (define (polar? z)
- (eq? (type-tag z) 'polar))
-
- With type tags, Ben and Alyssa can now modify their code so that
-their two different representations can coexist in the same system.
-Whenever Ben constructs a complex number, he tags it as rectangular.
-Whenever Alyssa constructs a complex number, she tags it as polar. In
-addition, Ben and Alyssa must make sure that the names of their
-procedures do not conflict. One way to do this is for Ben to append
-the suffix `rectangular' to the name of each of his representation
-procedures and for Alyssa to append `polar' to the names of hers. Here
-is Ben's revised rectangular representation from section *Note 2-4-1:::
-
- (define (real-part-rectangular z) (car z))
-
- (define (imag-part-rectangular z) (cdr z))
-
- (define (magnitude-rectangular z)
- (sqrt (+ (square (real-part-rectangular z))
- (square (imag-part-rectangular z)))))
-
- (define (angle-rectangular z)
- (atan (imag-part-rectangular z)
- (real-part-rectangular z)))
-
- (define (make-from-real-imag-rectangular x y)
- (attach-tag 'rectangular (cons x y)))
-
- (define (make-from-mag-ang-rectangular r a)
- (attach-tag 'rectangular
- (cons (* r (cos a)) (* r (sin a)))))
-
-and here is Alyssa's revised polar representation:
-
- (define (real-part-polar z)
- (* (magnitude-polar z) (cos (angle-polar z))))
-
- (define (imag-part-polar z)
- (* (magnitude-polar z) (sin (angle-polar z))))
-
- (define (magnitude-polar z) (car z))
-
- (define (angle-polar z) (cdr z))
-
- (define (make-from-real-imag-polar x y)
- (attach-tag 'polar
- (cons (sqrt (+ (square x) (square y)))
- (atan y x))))
-
- (define (make-from-mag-ang-polar r a)
- (attach-tag 'polar (cons r a)))
-
- Each generic selector is implemented as a procedure that checks the
-tag of its argument and calls the appropriate procedure for handling
-data of that type. For example, to obtain the real part of a complex
-number, `real-part' examines the tag to determine whether to use Ben's
-`real-part-rectangular' or Alyssa's `real-part-polar'. In either case,
-we use `contents' to extract the bare, untagged datum and send this to
-the rectangular or polar procedure as required:
-
- (define (real-part z)
- (cond ((rectangular? z)
- (real-part-rectangular (contents z)))
- ((polar? z)
- (real-part-polar (contents z)))
- (else (error "Unknown type -- REAL-PART" z))))
-
- (define (imag-part z)
- (cond ((rectangular? z)
- (imag-part-rectangular (contents z)))
- ((polar? z)
- (imag-part-polar (contents z)))
- (else (error "Unknown type -- IMAG-PART" z))))
-
- (define (magnitude z)
- (cond ((rectangular? z)
- (magnitude-rectangular (contents z)))
- ((polar? z)
- (magnitude-polar (contents z)))
- (else (error "Unknown type -- MAGNITUDE" z))))
-
- (define (angle z)
- (cond ((rectangular? z)
- (angle-rectangular (contents z)))
- ((polar? z)
- (angle-polar (contents z)))
- (else (error "Unknown type -- ANGLE" z))))
-
- To implement the complex-number arithmetic operations, we can use
-the same procedures `add-complex', `sub-complex', `mul-complex', and
-`div-complex' from section *Note 2-4-1::, because the selectors they
-call are generic, and so will work with either representation. For
-example, the procedure `add-complex' is still
-
- (define (add-complex z1 z2)
- (make-from-real-imag (+ (real-part z1) (real-part z2))
- (+ (imag-part z1) (imag-part z2))))
-
- Finally, we must choose whether to construct complex numbers using
-Ben's representation or Alyssa's representation. One reasonable choice
-is to construct rectangular numbers whenever we have real and imaginary
-parts and to construct polar numbers whenever we have magnitudes and
-angles:
-
- (define (make-from-real-imag x y)
- (make-from-real-imag-rectangular x y))
-
- (define (make-from-mag-ang r a)
- (make-from-mag-ang-polar r a))
-
- *Figure 2.21:* Structure of the generic complex-arithmetic system.
-
- +--------------------------------------------------+
- ----| add-complex sub-complex mul-complex- div-complex |----
- +--------------------------------------------------+
- Complex arithmetic package
- +-----------------------+
- | real-part imag-part |
- -----------------| |------------------
- | magnitude angle |
- +-----------+-----------+
- Rectangular | Polar
- representation | representation
- -----------------------------+------------------------------
- List structure and primitive machine arithmetic
-
- The resulting complex-number system has the structure shown in *Note
-Figure 2-21::. The system has been decomposed into three relatively
-independent parts: the complex-number-arithmetic operations, Alyssa's
-polar implementation, and Ben's rectangular implementation. The polar
-and rectangular implementations could have been written by Ben and
-Alyssa working separately, and both of these can be used as underlying
-representations by a third programmer implementing the
-complex-arithmetic procedures in terms of the abstract
-constructor/selector interface.
-
- Since each data object is tagged with its type, the selectors
-operate on the data in a generic manner. That is, each selector is
-defined to have a behavior that depends upon the particular type of
-data it is applied to. Notice the general mechanism for interfacing
-the separate representations: Within a given representation
-implementation (say, Alyssa's polar package) a complex number is an
-untyped pair (magnitude, angle). When a generic selector operates on a
-number of `polar' type, it strips off the tag and passes the contents on
-to Alyssa's code. Conversely, when Alyssa constructs a number for
-general use, she tags it with a type so that it can be appropriately
-recognized by the higher-level procedures. This discipline of
-stripping off and attaching tags as data objects are passed from level
-to level can be an important organizational strategy, as we shall see
-in section *Note 2-5::.
-
-\1f
-File: sicp.info, Node: 2-4-3, Prev: 2-4-2, Up: 2-4
-
-2.4.3 Data-Directed Programming and Additivity
-----------------------------------------------
-
-The general strategy of checking the type of a datum and calling an
-appropriate procedure is called "dispatching on type". This is a
-powerful strategy for obtaining modularity in system design. Oh the
-other hand, implementing the dispatch as in section *Note 2-4-2:: has
-two significant weaknesses. One weakness is that the generic interface
-procedures (`real-part', `imag-part', `magnitude', and `angle') must
-know about all the different representations. For instance, suppose we
-wanted to incorporate a new representation for complex numbers into our
-complex-number system. We would need to identify this new
-representation with a type, and then add a clause to each of the
-generic interface procedures to check for the new type and apply the
-appropriate selector for that representation.
-
- Another weakness of the technique is that even though the individual
-representations can be designed separately, we must guarantee that no
-two procedures in the entire system have the same name. This is why
-Ben and Alyssa had to change the names of their original procedures
-from section *Note 2-4-1::.
-
- The issue underlying both of these weaknesses is that the technique
-for implementing generic interfaces is not "additive". The person
-implementing the generic selector procedures must modify those
-procedures each time a new representation is installed, and the people
-interfacing the individual representations must modify their code to
-avoid name conflicts. In each of these cases, the changes that must be
-made to the code are straightforward, but they must be made
-nonetheless, and this is a source of inconvenience and error. This is
-not much of a problem for the complex-number system as it stands, but
-suppose there were not two but hundreds of different representations
-for complex numbers. And suppose that there were many generic
-selectors to be maintained in the abstract-data interface. Suppose, in
-fact, that no one programmer knew all the interface procedures or all
-the representations. The problem is real and must be addressed in such
-programs as large-scale data-base-management systems.
-
- What we need is a means for modularizing the system design even
-further. This is provided by the programming technique known as programming
-"data-directed programming". To understand how data-directed
-programming works, begin with the observation that whenever we deal
-with a set of generic operations that are common to a set of different
-types we are, in effect, dealing with a two-dimensional table that
-contains the possible operations on one axis and the possible types on
-the other axis. The entries in the table are the procedures that
-implement each operation for each type of argument presented. In the
-complex-number system developed in the previous section, the
-correspondence between operation name, data type, and actual procedure
-was spread out among the various conditional clauses in the generic
-interface procedures. But the same information could have been
-organized in a table, as shown in *Note Figure 2-22::.
-
- Data-directed programming is the technique of designing programs to
-work with such a table directly. Previously, we implemented the
-mechanism that interfaces the complex-arithmetic code with the two
-representation packages as a set of procedures that each perform an
-explicit dispatch on type. Here we will implement the interface as a
-single procedure that looks up the combination of the operation name
-and argument type in the table to find the correct procedure to apply,
-and then applies it to the contents of the argument. If we do this,
-then to add a new representation package to the system we need not
-change any existing procedures; we need only add new entries to the
-table.
-
- *Figure 2.22:* Table of operations for the complex-number system.
-
- | Types
- Operations | Polar | Rectangular
- ===========+=================+======================
- real-part | real-part-polar | real-part-rectangular
- imag-part | imag-part-polar | imag-part-rectangular
- magnitude | magnitude-polar | magnitude-rectangular
- angle | angle-polar | angle-rectangular
-
- To implement this plan, assume that we have two procedures, `put' and
-`get', for manipulating the operation-and-type table:
-
- * `(put <OP> <TYPE> <ITEM>)' installs the `<ITEM>' in the table,
- indexed by the `<OP>' and the `<TYPE>'.
-
- * `(get <OP> <TYPE>)' looks up the `<OP>', `<TYPE>' entry in the
- table and returns the item found there. If no item is found,
- `get' returns false.
-
-
- For now, we can assume that `put' and `get' are included in our
-language. In *Note Chapter 3:: (section *Note 3-3-3::, *Note Exercise
-3-24::) we will see how to implement these and other operations for
-manipulating tables.
-
- Here is how data-directed programming can be used in the
-complex-number system. Ben, who developed the rectangular
-representation, implements his code just as he did originally. He
-defines a collection of procedures, or a "package", and interfaces
-these to the rest of the system by adding entries to the table that
-tell the system how to operate on rectangular numbers. This is
-accomplished by calling the following procedure:
-
- (define (install-rectangular-package)
- ;; internal procedures
- (define (real-part z) (car z))
- (define (imag-part z) (cdr z))
- (define (make-from-real-imag x y) (cons x y))
- (define (magnitude z)
- (sqrt (+ (square (real-part z))
- (square (imag-part z)))))
- (define (angle z)
- (atan (imag-part z) (real-part z)))
- (define (make-from-mag-ang r a)
- (cons (* r (cos a)) (* r (sin a))))
-
- ;; interface to the rest of the system
- (define (tag x) (attach-tag 'rectangular x))
- (put 'real-part '(rectangular) real-part)
- (put 'imag-part '(rectangular) imag-part)
- (put 'magnitude '(rectangular) magnitude)
- (put 'angle '(rectangular) angle)
- (put 'make-from-real-imag 'rectangular
- (lambda (x y) (tag (make-from-real-imag x y))))
- (put 'make-from-mag-ang 'rectangular
- (lambda (r a) (tag (make-from-mag-ang r a))))
- 'done)
-
- Notice that the internal procedures here are the same procedures
-from section *Note 2-4-1:: that Ben wrote when he was working in
-isolation. No changes are necessary in order to interface them to the
-rest of the system. Moreover, since these procedure definitions are
-internal to the installation procedure, Ben needn't worry about name
-conflicts with other procedures outside the rectangular package. To
-interface these to the rest of the system, Ben installs his `real-part'
-procedure under the operation name `real-part' and the type
-`(rectangular)', and similarly for the other selectors.(1) The
-interface also defines the constructors to be used by the external
-system.(2) These are identical to Ben's internally defined
-constructors, except that they attach the tag.
-
- Alyssa's polar package is analogous:
-
- (define (install-polar-package)
- ;; internal procedures
- (define (magnitude z) (car z))
- (define (angle z) (cdr z))
- (define (make-from-mag-ang r a) (cons r a))
- (define (real-part z)
- (* (magnitude z) (cos (angle z))))
- (define (imag-part z)
- (* (magnitude z) (sin (angle z))))
- (define (make-from-real-imag x y)
- (cons (sqrt (+ (square x) (square y)))
- (atan y x)))
-
- ;; interface to the rest of the system
- (define (tag x) (attach-tag 'polar x))
- (put 'real-part '(polar) real-part)
- (put 'imag-part '(polar) imag-part)
- (put 'magnitude '(polar) magnitude)
- (put 'angle '(polar) angle)
- (put 'make-from-real-imag 'polar
- (lambda (x y) (tag (make-from-real-imag x y))))
- (put 'make-from-mag-ang 'polar
- (lambda (r a) (tag (make-from-mag-ang r a))))
- 'done)
-
- Even though Ben and Alyssa both still use their original procedures
-defined with the same names as each other's (e.g., `real-part'), these
-definitions are now internal to different procedures (see section *Note
-1-1-8::), so there is no name conflict.
-
- The complex-arithmetic selectors access the table by means of a
-general "operation" procedure called `apply-generic', which applies a
-generic operation to some arguments. `Apply-generic' looks in the
-table under the name of the operation and the types of the arguments
-and applies the resulting procedure if one is present:(3)
-
- (define (apply-generic op . args)
- (let ((type-tags (map type-tag args)))
- (let ((proc (get op type-tags)))
- (if proc
- (apply proc (map contents args))
- (error
- "No method for these types -- APPLY-GENERIC"
- (list op type-tags))))))
-
- Using `apply-generic', we can define our generic selectors as
-follows:
-
- (define (real-part z) (apply-generic 'real-part z))
- (define (imag-part z) (apply-generic 'imag-part z))
- (define (magnitude z) (apply-generic 'magnitude z))
- (define (angle z) (apply-generic 'angle z))
-
- Observe that these do not change at all if a new representation is
-added to the system.
-
- We can also extract from the table the constructors to be used by
-the programs external to the packages in making complex numbers from
-real and imaginary parts and from magnitudes and angles. As in section
-*Note 2-4-2::, we construct rectangular numbers whenever we have real
-and imaginary parts, and polar numbers whenever we have magnitudes and
-angles:
-
- (define (make-from-real-imag x y)
- ((get 'make-from-real-imag 'rectangular) x y))
-
- (define (make-from-mag-ang r a)
- ((get 'make-from-mag-ang 'polar) r a))
-
- *Exercise 2.73:* Section *Note 2-3-2:: described a program that
- performs symbolic differentiation:
-
- (define (deriv exp var)
- (cond ((number? exp) 0)
- ((variable? exp) (if (same-variable? exp var) 1 0))
- ((sum? exp)
- (make-sum (deriv (addend exp) var)
- (deriv (augend exp) var)))
- ((product? exp)
- (make-sum
- (make-product (multiplier exp)
- (deriv (multiplicand exp) var))
- (make-product (deriv (multiplier exp) var)
- (multiplicand exp))))
- <MORE RULES CAN BE ADDED HERE>
- (else (error "unknown expression type -- DERIV" exp))))
-
- We can regard this program as performing a dispatch on the type of
- the expression to be differentiated. In this situation the "type
- tag" of the datum is the algebraic operator symbol (such as `+')
- and the operation being performed is `deriv'. We can transform
- this program into data-directed style by rewriting the basic
- derivative procedure as
-
- (define (deriv exp var)
- (cond ((number? exp) 0)
- ((variable? exp) (if (same-variable? exp var) 1 0))
- (else ((get 'deriv (operator exp)) (operands exp)
- var))))
-
- (define (operator exp) (car exp))
-
- (define (operands exp) (cdr exp))
-
- a. Explain what was done above. Why can't we assimilate the
- predicates `number?' and `same-variable?' into the
- data-directed dispatch?
-
- b. Write the procedures for derivatives of sums and products,
- and the auxiliary code required to install them in the table
- used by the program above.
-
- c. Choose any additional differentiation rule that you like,
- such as the one for exponents (*Note Exercise 2-56::), and
- install it in this data-directed system.
-
- d. In this simple algebraic manipulator the type of an
- expression is the algebraic operator that binds it together.
- Suppose, however, we indexed the procedures in the opposite
- way, so that the dispatch line in `deriv' looked like
-
- ((get (operator exp) 'deriv) (operands exp) var)
-
- What corresponding changes to the derivative system are
- required?
-
-
- *Exercise 2.74:* Insatiable Enterprises, Inc., is a highly
- decentralized conglomerate company consisting of a large number of
- independent divisions located all over the world. The company's
- computer facilities have just been interconnected by means of a
- clever network-interfacing scheme that makes the entire network
- appear to any user to be a single computer. Insatiable's
- president, in her first attempt to exploit the ability of the
- network to extract administrative information from division files,
- is dismayed to discover that, although all the division files have
- been implemented as data structures in Scheme, the particular data
- structure used varies from division to division. A meeting of
- division managers is hastily called to search for a strategy to
- integrate the files that will satisfy headquarters' needs while
- preserving the existing autonomy of the divisions.
-
- Show how such a strategy can be implemented with data-directed
- programming. As an example, suppose that each division's
- personnel records consist of a single file, which contains a set
- of records keyed on employees' names. The structure of the set
- varies from division to division. Furthermore, each employee's
- record is itself a set (structured differently from division to
- division) that contains information keyed under identifiers such
- as `address' and `salary'. In particular:
-
- a. Implement for headquarters a `get-record' procedure that
- retrieves a specified employee's record from a specified
- personnel file. The procedure should be applicable to any
- division's file. Explain how the individual divisions' files
- should be structured. In particular, what type information
- must be supplied?
-
- b. Implement for headquarters a `get-salary' procedure that
- returns the salary information from a given employee's record
- from any division's personnel file. How should the record be
- structured in order to make this operation work?
-
- c. Implement for headquarters a `find-employee-record'
- procedure. This should search all the divisions' files for
- the record of a given employee and return the record. Assume
- that this procedure takes as arguments an employee's name and
- a list of all the divisions' files.
-
- d. When Insatiable takes over a new company, what changes must
- be made in order to incorporate the new personnel information
- into the central system?
-
-
-Message passing
-...............
-
-The key idea of data-directed programming is to handle generic
-operations in programs by dealing explicitly with operation-and-type
-tables, such as the table in *Note Figure 2-22::. The style of
-programming we used in section *Note 2-4-2:: organized the required
-dispatching on type by having each operation take care of its own
-dispatching. In effect, this decomposes the operation-and-type table
-into rows, with each generic operation procedure representing a row of
-the table.
-
- An alternative implementation strategy is to decompose the table
-into columns and, instead of using "intelligent operations" that
-dispatch on data types, to work with "intelligent data objects" that
-dispatch on operation names. We can do this by arranging things so
-that a data object, such as a rectangular number, is represented as a
-procedure that takes as input the required operation name and performs
-the operation indicated. In such a discipline, `make-from-real-imag'
-could be written as
-
- (define (make-from-real-imag x y)
- (define (dispatch op)
- (cond ((eq? op 'real-part) x)
- ((eq? op 'imag-part) y)
- ((eq? op 'magnitude)
- (sqrt (+ (square x) (square y))))
- ((eq? op 'angle) (atan y x))
- (else
- (error "Unknown op -- MAKE-FROM-REAL-IMAG" op))))
- dispatch)
-
- The corresponding `apply-generic' procedure, which applies a generic
-operation to an argument, now simply feeds the operation's name to the
-data object and lets the object do the work:(4)
-
- (define (apply-generic op arg) (arg op))
-
- Note that the value returned by `make-from-real-imag' is a
-procedure--the internal `dispatch' procedure. This is the procedure
-that is invoked when `apply-generic' requests an operation to be
-performed.
-
- This style of programming is called "message passing". The name
-comes from the image that a data object is an entity that receives the
-requested operation name as a "message." We have already seen an
-example of message passing in section *Note 2-1-3::, where we saw how
-`cons', `car', and `cdr' could be defined with no data objects but only
-procedures. Here we see that message passing is not a mathematical
-trick but a useful technique for organizing systems with generic
-operations. In the remainder of this chapter we will continue to use
-data-directed programming, rather than message passing, to discuss
-generic arithmetic operations. In *Note Chapter 3:: we will return to
-message passing, and we will see that it can be a powerful tool for
-structuring simulation programs.
-
- *Exercise 2.75:* Implement the constructor `make-from-mag-ang' in
- message-passing style. This procedure should be analogous to the
- `make-from-real-imag' procedure given above.
-
- *Exercise 2.76:* As a large system with generic operations
- evolves, new types of data objects or new operations may be needed.
- For each of the three strategies--generic operations with explicit
- dispatch, data-directed style, and message-passing-style--describe
- the changes that must be made to a system in order to add new
- types or new operations. Which organization would be most
- appropriate for a system in which new types must often be added?
- Which would be most appropriate for a system in which new
- operations must often be added?
-
- ---------- Footnotes ----------
-
- (1) We use the list `(rectangular)' rather than the symbol
-`rectangular' to allow for the possibility of operations with multiple
-arguments, not all of the same type.
-
- (2) The type the constructors are installed under needn't be a list
-because a constructor is always used to make an object of one
-particular type.
-
- (3) `Apply-generic' uses the dotted-tail notation described in *Note
-Exercise 2-20::, because different generic operations may take
-different numbers of arguments. In `apply-generic', `op' has as its
-value the first argument to `apply-generic' and `args' has as its value
-a list of the remaining arguments.
-
- `Apply-generic' also uses the primitive procedure `apply', which
-takes two arguments, a procedure and a list. `Apply' applies the
-procedure, using the elements in the list as arguments. For example,
-
- (apply + (list 1 2 3 4))
-
-returns 10.
-
- (4) One limitation of this organization is it permits only generic
-procedures of one argument.
-
-\1f
-File: sicp.info, Node: 2-5, Prev: 2-4, Up: Chapter 2
-
-2.5 Systems with Generic Operations
-===================================
-
-In the previous section, we saw how to design systems in which data
-objects can be represented in more than one way. The key idea is to
-link the code that specifies the data operations to the several
-representations by means of generic interface procedures. Now we will
-see how to use this same idea not only to define operations that are
-generic over different representations but also to define operations
-that are generic over different kinds of arguments. We have already
-seen several different packages of arithmetic operations: the primitive
-arithmetic (`+', `-', `*', `/') built into our language, the
-rational-number arithmetic (`add-rat', `sub-rat', `mul-rat', `div-rat')
-of section *Note 2-1-1::, and the complex-number arithmetic that we
-implemented in section *Note 2-4-3::. We will now use data-directed
-techniques to construct a package of arithmetic operations that
-incorporates all the arithmetic packages we have already constructed.
-
- *Note Figure 2-23:: shows the structure of the system we shall
-build. Notice the abstraction barriers. From the perspective of
-someone using "numbers," there is a single procedure `add' that
-operates on whatever numbers are supplied. `Add' is part of a generic
-interface that allows the separate ordinary-arithmetic,
-rational-arithmetic, and complex-arithmetic packages to be accessed
-uniformly by programs that use numbers. Any individual arithmetic
-package (such as the complex package) may itself be accessed through
-generic procedures (such as `add-complex') that combine packages
-designed for different representations (such as rectangular and polar).
-Moreover, the structure of the system is additive, so that one can
-design the individual arithmetic packages separately and combine them
-to produce a generic arithmetic system.
-
- *Figure 2.23:* Generic arithmetic system.
-
- Programs that use numbers
- +-----------------+
- ---------------------------| add sub mul div |-------------------
- +-----------------+
- Generic arithmetic package
- +-----------------+ +-------------------------+
- | add-rat sub-rat | | add-complex sub-complex | +---------+
- -| |-+-| |-+-| + - * / |-
- | mul-rat div-rat | | | mul-complex div-complex | | +---------+
- +-----------------+ | +-------------------------+ |
- Rational | Complex artithmetic | Ordinary
- arithmetic +--------------+--------------+ arithmetic
- | Rectangular | Polar |
- ---------------------+--------------+--------------+-------------
-
-* Menu:
-
-* 2-5-1:: Generic Arithmetic Operations
-* 2-5-2:: Combining Data of Different Types
-* 2-5-3:: Example: Symbolic Algebra
-
-\1f
-File: sicp.info, Node: 2-5-1, Next: 2-5-2, Prev: 2-5, Up: 2-5
-
-2.5.1 Generic Arithmetic Operations
------------------------------------
-
-The task of designing generic arithmetic operations is analogous to
-that of designing the generic complex-number operations. We would
-like, for instance, to have a generic addition procedure `add' that
-acts like ordinary primitive addition `+' on ordinary numbers, like
-`add-rat' on rational numbers, and like `add-complex' on complex
-numbers. We can implement `add', and the other generic arithmetic
-operations, by following the same strategy we used in section *Note
-2-4-3:: to implement the generic selectors for complex numbers. We
-will attach a type tag to each kind of number and cause the generic
-procedure to dispatch to an appropriate package according to the data
-type of its arguments.
-
- The generic arithmetic procedures are defined as follows:
-
- (define (add x y) (apply-generic 'add x y))
- (define (sub x y) (apply-generic 'sub x y))
- (define (mul x y) (apply-generic 'mul x y))
- (define (div x y) (apply-generic 'div x y))
-
- We begin by installing a package for handling "ordinary" numbers,
-that is, the primitive numbers of our language. We will tag these with
-the symbol `scheme-number'. The arithmetic operations in this package
-are the primitive arithmetic procedures (so there is no need to define
-extra procedures to handle the untagged numbers). Since these
-operations each take two arguments, they are installed in the table
-keyed by the list `(scheme-number scheme-number)':
-
- (define (install-scheme-number-package)
- (define (tag x)
- (attach-tag 'scheme-number x))
- (put 'add '(scheme-number scheme-number)
- (lambda (x y) (tag (+ x y))))
- (put 'sub '(scheme-number scheme-number)
- (lambda (x y) (tag (- x y))))
- (put 'mul '(scheme-number scheme-number)
- (lambda (x y) (tag (* x y))))
- (put 'div '(scheme-number scheme-number)
- (lambda (x y) (tag (/ x y))))
- (put 'make 'scheme-number
- (lambda (x) (tag x)))
- 'done)
-
- Users of the Scheme-number package will create (tagged) ordinary
-numbers by means of the procedure:
-
- (define (make-scheme-number n)
- ((get 'make 'scheme-number) n))
-
- Now that the framework of the generic arithmetic system is in place,
-we can readily include new kinds of numbers. Here is a package that
-performs rational arithmetic. Notice that, as a benefit of additivity,
-we can use without modification the rational-number code from section
-*Note 2-1-1:: as the internal procedures in the package:
-
- (define (install-rational-package)
- ;; internal procedures
- (define (numer x) (car x))
- (define (denom x) (cdr x))
- (define (make-rat n d)
- (let ((g (gcd n d)))
- (cons (/ n g) (/ d g))))
- (define (add-rat x y)
- (make-rat (+ (* (numer x) (denom y))
- (* (numer y) (denom x)))
- (* (denom x) (denom y))))
- (define (sub-rat x y)
- (make-rat (- (* (numer x) (denom y))
- (* (numer y) (denom x)))
- (* (denom x) (denom y))))
- (define (mul-rat x y)
- (make-rat (* (numer x) (numer y))
- (* (denom x) (denom y))))
- (define (div-rat x y)
- (make-rat (* (numer x) (denom y))
- (* (denom x) (numer y))))
-
- ;; interface to rest of the system
- (define (tag x) (attach-tag 'rational x))
- (put 'add '(rational rational)
- (lambda (x y) (tag (add-rat x y))))
- (put 'sub '(rational rational)
- (lambda (x y) (tag (sub-rat x y))))
- (put 'mul '(rational rational)
- (lambda (x y) (tag (mul-rat x y))))
- (put 'div '(rational rational)
- (lambda (x y) (tag (div-rat x y))))
-
- (put 'make 'rational
- (lambda (n d) (tag (make-rat n d))))
- 'done)
-
- (define (make-rational n d)
- ((get 'make 'rational) n d))
-
- We can install a similar package to handle complex numbers, using
-the tag `complex'. In creating the package, we extract from the table
-the operations `make-from-real-imag' and `make-from-mag-ang' that were
-defined by the rectangular and polar packages. Additivity permits us
-to use, as the internal operations, the same `add-complex',
-`sub-complex', `mul-complex', and `div-complex' procedures from section
-*Note 2-4-1::.
-
- (define (install-complex-package)
- ;; imported procedures from rectangular and polar packages
- (define (make-from-real-imag x y)
- ((get 'make-from-real-imag 'rectangular) x y))
- (define (make-from-mag-ang r a)
- ((get 'make-from-mag-ang 'polar) r a))
-
- ;; internal procedures
- (define (add-complex z1 z2)
- (make-from-real-imag (+ (real-part z1) (real-part z2))
- (+ (imag-part z1) (imag-part z2))))
- (define (sub-complex z1 z2)
- (make-from-real-imag (- (real-part z1) (real-part z2))
- (- (imag-part z1) (imag-part z2))))
- (define (mul-complex z1 z2)
- (make-from-mag-ang (* (magnitude z1) (magnitude z2))
- (+ (angle z1) (angle z2))))
- (define (div-complex z1 z2)
- (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
- (- (angle z1) (angle z2))))
-
- ;; interface to rest of the system
- (define (tag z) (attach-tag 'complex z))
- (put 'add '(complex complex)
- (lambda (z1 z2) (tag (add-complex z1 z2))))
- (put 'sub '(complex complex)
- (lambda (z1 z2) (tag (sub-complex z1 z2))))
- (put 'mul '(complex complex)
- (lambda (z1 z2) (tag (mul-complex z1 z2))))
- (put 'div '(complex complex)
- (lambda (z1 z2) (tag (div-complex z1 z2))))
- (put 'make-from-real-imag 'complex
- (lambda (x y) (tag (make-from-real-imag x y))))
- (put 'make-from-mag-ang 'complex
- (lambda (r a) (tag (make-from-mag-ang r a))))
- 'done)
-
- Programs outside the complex-number package can construct complex
-numbers either from real and imaginary parts or from magnitudes and
-angles. Notice how the underlying procedures, originally defined in
-the rectangular and polar packages, are exported to the complex
-package, and exported from there to the outside world.
-
- (define (make-complex-from-real-imag x y)
- ((get 'make-from-real-imag 'complex) x y))
-
- (define (make-complex-from-mag-ang r a)
- ((get 'make-from-mag-ang 'complex) r a))
-
- What we have here is a two-level tag system. A typical complex
-number, such as 3 + 4i in rectangular form, would be represented as
-shown in *Note Figure 2-24::. The outer tag (`complex') is used to
-direct the number to the complex package. Once within the complex
-package, the next tag (`rectangular') is used to direct the number to
-the rectangular package. In a large and complicated system there might
-be many levels, each interfaced with the next by means of generic
-operations. As a data object is passed "downward," the outer tag that
-is used to direct it to the appropriate package is stripped off (by
-applying `contents') and the next level of tag (if any) becomes visible
-to be used for further dispatching.
-
- *Figure 2.24:* Representation of 3 + 4i in rectangular form.
-
- +---+---+ +---+---+ +---+---+
- ---->| * | *-+---->| * | *-+---->| * | * |
- +-|-+---+ +-|-+---+ +-|-+-|-+
- | | | |
- V V V V
- +---------+ +-------------+ +---+ +---+
- | complex | | rectangular | | 3 | | 4 |
- +---------+ +-------------+ +---+ +---+
-
- In the above packages, we used `add-rat', `add-complex', and the
-other arithmetic procedures exactly as originally written. Once these
-definitions are internal to different installation procedures, however,
-they no longer need names that are distinct from each other: we could
-simply name them `add', `sub', `mul', and `div' in both packages.
-
- *Exercise 2.77:* Louis Reasoner tries to evaluate the expression
- `(magnitude z)' where `z' is the object shown in *Note Figure
- 2-24::. To his surprise, instead of the answer 5 he gets an error
- message from `apply-generic', saying there is no method for the
- operation `magnitude' on the types `(complex)'. He shows this
- interaction to Alyssa P. Hacker, who says "The problem is that the
- complex-number selectors were never defined for `complex' numbers,
- just for `polar' and `rectangular' numbers. All you have to do to
- make this work is add the following to the `complex' package:"
-
- (put 'real-part '(complex) real-part)
- (put 'imag-part '(complex) imag-part)
- (put 'magnitude '(complex) magnitude)
- (put 'angle '(complex) angle)
-
- Describe in detail why this works. As an example, trace through
- all the procedures called in evaluating the expression `(magnitude
- z)' where `z' is the object shown in *Note Figure 2-24::. In
- particular, how many times is `apply-generic' invoked? What
- procedure is dispatched to in each case?
-
- *Exercise 2.78:* The internal procedures in the `scheme-number'
- package are essentially nothing more than calls to the primitive
- procedures `+', `-', etc. It was not possible to use the
- primitives of the language directly because our type-tag system
- requires that each data object have a type attached to it. In
- fact, however, all Lisp implementations do have a type system,
- which they use internally. Primitive predicates such as `symbol?'
- and `number?' determine whether data objects have particular
- types. Modify the definitions of `type-tag', `contents', and
- `attach-tag' from section *Note 2-4-2:: so that our generic system
- takes advantage of Scheme's internal type system. That is to say,
- the system should work as before except that ordinary numbers
- should be represented simply as Scheme numbers rather than as
- pairs whose `car' is the symbol `scheme-number'.
-
- *Exercise 2.79:* Define a generic equality predicate `equ?' that
- tests the equality of two numbers, and install it in the generic
- arithmetic package. This operation should work for ordinary
- numbers, rational numbers, and complex numbers.
-
- *Exercise 2.80:* Define a generic predicate `=zero?' that tests if
- its argument is zero, and install it in the generic arithmetic
- package. This operation should work for ordinary numbers, rational
- numbers, and complex numbers.
-
-\1f
-File: sicp.info, Node: 2-5-2, Next: 2-5-3, Prev: 2-5-1, Up: 2-5
-
-2.5.2 Combining Data of Different Types
----------------------------------------
-
-We have seen how to define a unified arithmetic system that encompasses
-ordinary numbers, complex numbers, rational numbers, and any other type
-of number we might decide to invent, but we have ignored an important
-issue. The operations we have defined so far treat the different data
-types as being completely independent. Thus, there are separate
-packages for adding, say, two ordinary numbers, or two complex numbers.
-What we have not yet considered is the fact that it is meaningful to
-define operations that cross the type boundaries, such as the addition
-of a complex number to an ordinary number. We have gone to great pains
-to introduce barriers between parts of our programs so that they can be
-developed and understood separately. We would like to introduce the
-cross-type operations in some carefully controlled way, so that we can
-support them without seriously violating our module boundaries.
-
- One way to handle cross-type operations is to design a different
-procedure for each possible combination of types for which the
-operation is valid. For example, we could extend the complex-number
-package so that it provides a procedure for adding complex numbers to
-ordinary numbers and installs this in the table using the tag `(complex
-scheme-number)':(1)
-
- ;; to be included in the complex package
- (define (add-complex-to-schemenum z x)
- (make-from-real-imag (+ (real-part z) x)
- (imag-part z)))
-
- (put 'add '(complex scheme-number)
- (lambda (z x) (tag (add-complex-to-schemenum z x))))
-
- This technique works, but it is cumbersome. With such a system, the
-cost of introducing a new type is not just the construction of the
-package of procedures for that type but also the construction and
-installation of the procedures that implement the cross-type
-operations. This can easily be much more code than is needed to define
-the operations on the type itself. The method also undermines our
-ability to combine separate packages additively, or least to limit the
-extent to which the implementors of the individual packages need to
-take account of other packages. For instance, in the example above, it
-seems reasonable that handling mixed operations on complex numbers and
-ordinary numbers should be the responsibility of the complex-number
-package. Combining rational numbers and complex numbers, however,
-might be done by the complex package, by the rational package, or by
-some third package that uses operations extracted from these two
-packages. Formulating coherent policies on the division of
-responsibility among packages can be an overwhelming task in designing
-systems with many packages and many cross-type operations.
-
-Coercion
-........
-
-In the general situation of completely unrelated operations acting on
-completely unrelated types, implementing explicit cross-type operations,
-cumbersome though it may be, is the best that one can hope for.
-Fortunately, we can usually do better by taking advantage of additional
-structure that may be latent in our type system. Often the different
-data types are not completely independent, and there may be ways by
-which objects of one type may be viewed as being of another type. This
-process is called "coercion". For example, if we are asked to
-arithmetically combine an ordinary number with a complex number, we can
-view the ordinary number as a complex number whose imaginary part is
-zero. This transforms the problem to that of combining two complex
-numbers, which can be handled in the ordinary way by the
-complex-arithmetic package.
-
- In general, we can implement this idea by designing coercion
-procedures that transform an object of one type into an equivalent
-object of another type. Here is a typical coercion procedure, which
-transforms a given ordinary number to a complex number with that real
-part and zero imaginary part:
-
- (define (scheme-number->complex n)
- (make-complex-from-real-imag (contents n) 0))
-
- We install these coercion procedures in a special coercion table,
-indexed under the names of the two types:
-
- (put-coercion 'scheme-number 'complex scheme-number->complex)
-
- (We assume that there are `put-coercion' and `get-coercion'
-procedures available for manipulating this table.) Generally some of
-the slots in the table will be empty, because it is not generally
-possible to coerce an arbitrary data object of each type into all other
-types. For example, there is no way to coerce an arbitrary complex
-number to an ordinary number, so there will be no general
-`complex->scheme-number' procedure included in the table.
-
- Once the coercion table has been set up, we can handle coercion in a
-uniform manner by modifying the `apply-generic' procedure of section
-*Note 2-4-3::. When asked to apply an operation, we first check
-whether the operation is defined for the arguments' types, just as
-before. If so, we dispatch to the procedure found in the
-operation-and-type table. Otherwise, we try coercion. For simplicity,
-we consider only the case where there are two arguments.(2) We check
-the coercion table to see if objects of the first type can be coerced
-to the second type. If so, we coerce the first argument and try the
-operation again. If objects of the first type cannot in general be
-coerced to the second type, we try the coercion the other way around to
-see if there is a way to coerce the second argument to the type of the
-first argument. Finally, if there is no known way to coerce either
-type to the other type, we give up. Here is the procedure:
-
- (define (apply-generic op . args)
- (let ((type-tags (map type-tag args)))
- (let ((proc (get op type-tags)))
- (if proc
- (apply proc (map contents args))
- (if (= (length args) 2)
- (let ((type1 (car type-tags))
- (type2 (cadr type-tags))
- (a1 (car args))
- (a2 (cadr args)))
- (let ((t1->t2 (get-coercion type1 type2))
- (t2->t1 (get-coercion type2 type1)))
- (cond (t1->t2
- (apply-generic op (t1->t2 a1) a2))
- (t2->t1
- (apply-generic op a1 (t2->t1 a2)))
- (else
- (error "No method for these types"
- (list op type-tags))))))
- (error "No method for these types"
- (list op type-tags)))))))
-
- This coercion scheme has many advantages over the method of defining
-explicit cross-type operations, as outlined above. Although we still
-need to write coercion procedures to relate the types (possibly n^2
-procedures for a system with n types), we need to write only one
-procedure for each pair of types rather than a different procedure for
-each collection of types and each generic operation.(3) What we are
-counting on here is the fact that the appropriate transformation
-between types depends only on the types themselves, not on the
-operation to be applied.
-
- On the other hand, there may be applications for which our coercion
-scheme is not general enough. Even when neither of the objects to be
-combined can be converted to the type of the other it may still be
-possible to perform the operation by converting both objects to a third
-type. In order to deal with such complexity and still preserve
-modularity in our programs, it is usually necessary to build systems
-that take advantage of still further structure in the relations among
-types, as we discuss next.
-
-Hierarchies of types
-....................
-
-The coercion scheme presented above relied on the existence of natural
-relations between pairs of types. Often there is more "global"
-structure in how the different types relate to each other. For
-instance, suppose we are building a generic arithmetic system to handle
-integers, rational numbers, real numbers, and complex numbers. In such
-a system, it is quite natural to regard an integer as a special kind of
-rational number, which is in turn a special kind of real number, which
-is in turn a special kind of complex number. What we actually have is
-a so-called "hierarchy of types", in which, for example, integers are a "subtype"
-of rational numbers (i.e., any operation that can be applied to a
-rational number can automatically be applied to an integer).
-Conversely, we say that rational numbers form a "supertype" of
-integers. The particular hierarchy we have here is of a very simple
-kind, in which each type has at most one supertype and at most one
-subtype. Such a structure, called a "tower", is illustrated in *Note
-Figure 2-25::.
-
- *Figure 2.25:* A tower of types.
-
- complex
- ^
- |
- real
- ^
- |
- rational
- ^
- |
- integer
-
- If we have a tower structure, then we can greatly simplify the
-problem of adding a new type to the hierarchy, for we need only specify
-how the new type is embedded in the next supertype above it and how it
-is the supertype of the type below it. For example, if we want to add
-an integer to a complex number, we need not explicitly define a special
-coercion procedure `integer->complex'. Instead, we define how an
-integer can be transformed into a rational number, how a rational
-number is transformed into a real number, and how a real number is
-transformed into a complex number. We then allow the system to
-transform the integer into a complex number through these steps and
-then add the two complex numbers.
-
- We can redesign our `apply-generic' procedure in the following way:
-For each type, we need to supply a `raise' procedure, which "raises"
-objects of that type one level in the tower. Then when the system is
-required to operate on objects of different types it can successively
-raise the lower types until all the objects are at the same level in
-the tower. (*Note Exercise 2-83:: and *Note Exercise 2-84:: concern
-the details of implementing such a strategy.)
-
- Another advantage of a tower is that we can easily implement the
-notion that every type "inherits" all operations defined on a
-supertype. For instance, if we do not supply a special procedure for
-finding the real part of an integer, we should nevertheless expect that
-`real-part' will be defined for integers by virtue of the fact that
-integers are a subtype of complex numbers. In a tower, we can arrange
-for this to happen in a uniform way by modifying `apply-generic'. If
-the required operation is not directly defined for the type of the
-object given, we raise the object to its supertype and try again. We
-thus crawl up the tower, transforming our argument as we go, until we
-either find a level at which the desired operation can be performed or
-hit the top (in which case we give up).
-
- Yet another advantage of a tower over a more general hierarchy is
-that it gives us a simple way to "lower" a data object to the simplest
-representation. For example, if we add 2 + 3i to 4 - 3i, it would be
-nice to obtain the answer as the integer 6 rather than as the complex
-number 6 + 0i. *Note Exercise 2-85:: discusses a way to implement such
-a lowering operation. (The trick is that we need a general way to
-distinguish those objects that can be lowered, such as 6 + 0i, from
-those that cannot, such as 6 + 2i.)
-
- *Figure 2.26:* Relations among types of geometric figures.
-
- polygon
- / \
- / \
- triangle quadrilateral
- / \ / \
- / \ / \
- isosceles right trapezoid kite
- triangle triangle | |
- | \ | | |
- | \ | | |
- equilateral isosceles parallelogram |
- triangle right | \ |
- triangle | \ |
- rectangle rhombus
- \ /
- \ /
- square
-
-Inadequacies of hierarchies
-...........................
-
-If the data types in our system can be naturally arranged in a tower,
-this greatly simplifies the problems of dealing with generic operations
-on different types, as we have seen. Unfortunately, this is usually
-not the case. *Note Figure 2-26:: illustrates a more complex
-arrangement of mixed types, this one showing relations among different
-types of geometric figures. We see that, in general, a type may have
-more than one subtype. Triangles and quadrilaterals, for instance, are
-both subtypes of polygons. In addition, a type may have more than one
-supertype. For example, an isosceles right triangle may be regarded
-either as an isosceles triangle or as a right triangle. This
-multiple-supertypes issue is particularly thorny, since it means that
-there is no unique way to "raise" a type in the hierarchy. Finding the
-"correct" supertype in which to apply an operation to an object may
-involve considerable searching through the entire type network on the
-part of a procedure such as `apply-generic'. Since there generally are
-multiple subtypes for a type, there is a similar problem in coercing a
-value "down" the type hierarchy. Dealing with large numbers of
-interrelated types while still preserving modularity in the design of
-large systems is very difficult, and is an area of much current
-research.(4)
-
- *Exercise 2.81:* Louis Reasoner has noticed that `apply-generic'
- may try to coerce the arguments to each other's type even if they
- already have the same type. Therefore, he reasons, we need to put
- procedures in the coercion table to "coerce" arguments of each
- type to their own type. For example, in addition to the
- `scheme-number->complex' coercion shown above, he would do:
-
- (define (scheme-number->scheme-number n) n)
- (define (complex->complex z) z)
- (put-coercion 'scheme-number 'scheme-number
- scheme-number->scheme-number)
- (put-coercion 'complex 'complex complex->complex)
-
- a. With Louis's coercion procedures installed, what happens if
- `apply-generic' is called with two arguments of type
- `scheme-number' or two arguments of type `complex' for an
- operation that is not found in the table for those types?
- For example, assume that we've defined a generic
- exponentiation operation:
-
- (define (exp x y) (apply-generic 'exp x y))
-
- and have put a procedure for exponentiation in the
- Scheme-number package but not in any other package:
-
- ;; following added to Scheme-number package
- (put 'exp '(scheme-number scheme-number)
- (lambda (x y) (tag (expt x y)))) ; using primitive `expt'
-
- What happens if we call `exp' with two complex numbers as
- arguments?
-
- b. Is Louis correct that something had to be done about coercion
- with arguments of the same type, or does `apply-generic' work
- correctly as is?
-
- c. Modify `apply-generic' so that it doesn't try coercion if the
- two arguments have the same type.
-
-
- *Exercise 2.82:* Show how to generalize `apply-generic' to handle
- coercion in the general case of multiple arguments. One strategy
- is to attempt to coerce all the arguments to the type of the first
- argument, then to the type of the second argument, and so on.
- Give an example of a situation where this strategy (and likewise
- the two-argument version given above) is not sufficiently general.
- (Hint: Consider the case where there are some suitable mixed-type
- operations present in the table that will not be tried.)
-
- *Exercise 2.83:* Suppose you are designing a generic arithmetic
- system for dealing with the tower of types shown in *Note Figure
- 2-25::: integer, rational, real, complex. For each type (except
- complex), design a procedure that raises objects of that type one
- level in the tower. Show how to install a generic `raise'
- operation that will work for each type (except complex).
-
- *Exercise 2.84:* Using the `raise' operation of *Note Exercise
- 2-83::, modify the `apply-generic' procedure so that it coerces
- its arguments to have the same type by the method of successive
- raising, as discussed in this section. You will need to devise a
- way to test which of two types is higher in the tower. Do this in
- a manner that is "compatible" with the rest of the system and will
- not lead to problems in adding new levels to the tower.
-
- *Exercise 2.85:* This section mentioned a method for "simplifying"
- a data object by lowering it in the tower of types as far as
- possible. Design a procedure `drop' that accomplishes this for the
- tower described in *Note Exercise 2-83::. The key is to decide,
- in some general way, whether an object can be lowered. For
- example, the complex number 1.5 + 0i can be lowered as far as
- `real', the complex number 1 + 0i can be lowered as far as
- `integer', and the complex number 2 + 3i cannot be lowered at all.
- Here is a plan for determining whether an object can be lowered:
- Begin by defining a generic operation `project' that "pushes" an
- object down in the tower. For example, projecting a complex
- number would involve throwing away the imaginary part. Then a
- number can be dropped if, when we `project' it and `raise' the
- result back to the type we started with, we end up with something
- equal to what we started with. Show how to implement this idea in
- detail, by writing a `drop' procedure that drops an object as far
- as possible. You will need to design the various projection
- operations(5) and install `project' as a generic operation in the
- system. You will also need to make use of a generic equality
- predicate, such as described in *Note Exercise 2-79::. Finally,
- use `drop' to rewrite `apply-generic' from *Note Exercise 2-84::
- so that it "simplifies" its answers.
-
- *Exercise 2.86:* Suppose we want to handle complex numbers whose
- real parts, imaginary parts, magnitudes, and angles can be either
- ordinary numbers, rational numbers, or other numbers we might wish
- to add to the system. Describe and implement the changes to the
- system needed to accommodate this. You will have to define
- operations such as `sine' and `cosine' that are generic over
- ordinary numbers and rational numbers.
-
- ---------- Footnotes ----------
-
- (1) We also have to supply an almost identical procedure to handle
-the types `(scheme-number complex)'.
-
- (2) See *Note Exercise 2-82:: for generalizations.
-
- (3) If we are clever, we can usually get by with fewer than n^2
-coercion procedures. For instance, if we know how to convert from type
-1 to type 2 and from type 2 to type 3, then we can use this knowledge to
-convert from type 1 to type 3. This can greatly decrease the number of
-coercion procedures we need to supply explicitly when we add a new type
-to the system. If we are willing to build the required amount of
-sophistication into our system, we can have it search the "graph" of
-relations among types and automatically generate those coercion
-procedures that can be inferred from the ones that are supplied
-explicitly.
-
- (4) This statement, which also appears in the first edition of this
-book, is just as true now as it was when we wrote it twelve years ago.
-Developing a useful, general framework for expressing the relations
-among different types of entities (what philosophers call "ontology")
-seems intractably difficult. The main difference between the confusion
-that existed ten years ago and the confusion that exists now is that
-now a variety of inadequate ontological theories have been embodied in
-a plethora of correspondingly inadequate programming languages. For
-example, much of the complexity of object-oriented programming
-languages--and the subtle and confusing differences among contemporary
-object-oriented languages--centers on the treatment of generic
-operations on interrelated types. Our own discussion of computational
-objects in *Note Chapter 3:: avoids these issues entirely. Readers
-familiar with object-oriented programming will notice that we have much
-to say in *Note Chapter 3:: about local state, but we do not even
-mention "classes" or "inheritance." In fact, we suspect that these
-problems cannot be adequately addressed in terms of computer-language
-design alone, without also drawing on work in knowledge representation
-and automated reasoning.
-
- (5) A real number can be projected to an integer using the `round'
-primitive, which returns the closest integer to its argument.
-
-\1f
-File: sicp.info, Node: 2-5-3, Prev: 2-5-2, Up: 2-5
-
-2.5.3 Example: Symbolic Algebra
--------------------------------
-
-The manipulation of symbolic algebraic expressions is a complex process
-that illustrates many of the hardest problems that occur in the design
-of large-scale systems. An algebraic expression, in general, can be
-viewed as a hierarchical structure, a tree of operators applied to
-operands. We can construct algebraic expressions by starting with a
-set of primitive objects, such as constants and variables, and
-combining these by means of algebraic operators, such as addition and
-multiplication. As in other languages, we form abstractions that
-enable us to refer to compound objects in simple terms. Typical
-abstractions in symbolic algebra are ideas such as linear combination,
-polynomial, rational function, or trigonometric function. We can
-regard these as compound "types," which are often useful for directing
-the processing of expressions. For example, we could describe the
-expression
-
- x^2 sin (y^2 + 1) + r cos 2y + cos(y^3 - 2y^2)
-
-as a polynomial in x with coefficients that are trigonometric functions
-of polynomials in y whose coefficients are integers.
-
- We will not attempt to develop a complete algebraic-manipulation
-system here. Such systems are exceedingly complex programs, embodying
-deep algebraic knowledge and elegant algorithms. What we will do is
-look at a simple but important part of algebraic manipulation: the
-arithmetic of polynomials. We will illustrate the kinds of decisions
-the designer of such a system faces, and how to apply the ideas of
-abstract data and generic operations to help organize this effort.
-
-Arithmetic on polynomials
-.........................
-
-Our first task in designing a system for performing arithmetic on
-polynomials is to decide just what a polynomial is. Polynomials are
-normally defined relative to certain variables (the "indeterminates" of
-the polynomial). For simplicity, we will restrict ourselves to
-polynomials having just one indeterminate ("univariate
-polynomials").(1) We will define a polynomial to be a sum of terms,
-each of which is either a coefficient, a power of the indeterminate, or
-a product of a coefficient and a power of the indeterminate. A
-coefficient is defined as an algebraic expression that is not dependent
-upon the indeterminate of the polynomial. For example,
-
- 5x^2 + 3r + 7
-
-is a simple polynomial in x, and
-
- (y^2 + 1)r^3 + (2y)x + 1
-
-is a polynomial in x whose coefficients are polynomials in y.
-
- Already we are skirting some thorny issues. Is the first of these
-polynomials the same as the polynomial 5y^2 + 3y + 7, or not? A
-reasonable answer might be "yes, if we are considering a polynomial
-purely as a mathematical function, but no, if we are considering a
-polynomial to be a syntactic form." The second polynomial is
-algebraically equivalent to a polynomial in y whose coefficients are
-polynomials in x. Should our system recognize this, or not?
-Furthermore, there are other ways to represent a polynomial--for
-example, as a product of factors, or (for a univariate polynomial) as
-the set of roots, or as a listing of the values of the polynomial at a
-specified set of points.(2) We can finesse these questions by deciding
-that in our algebraic-manipulation system a "polynomial" will be a
-particular syntactic form, not its underlying mathematical meaning.
-
- Now we must consider how to go about doing arithmetic on
-polynomials. In this simple system, we will consider only addition and
-multiplication. Moreover, we will insist that two polynomials to be
-combined must have the same indeterminate.
-
- We will approach the design of our system by following the familiar
-discipline of data abstraction. We will represent polynomials using a
-data structure called a "poly", which consists of a variable and a
-collection of terms. We assume that we have selectors `variable' and
-`term-list' that extract those parts from a poly and a constructor
-`make-poly' that assembles a poly from a given variable and a term
-list. A variable will be just a symbol, so we can use the
-`same-variable?' procedure of section *Note 2-3-2:: to compare
-variables. The following procedures define addition and multiplication
-of polys:
-
- (define (add-poly p1 p2)
- (if (same-variable? (variable p1) (variable p2))
- (make-poly (variable p1)
- (add-terms (term-list p1)
- (term-list p2)))
- (error "Polys not in same var -- ADD-POLY"
- (list p1 p2))))
-
- (define (mul-poly p1 p2)
- (if (same-variable? (variable p1) (variable p2))
- (make-poly (variable p1)
- (mul-terms (term-list p1)
- (term-list p2)))
- (error "Polys not in same var -- MUL-POLY"
- (list p1 p2))))
-
- To incorporate polynomials into our generic arithmetic system, we
-need to supply them with type tags. We'll use the tag `polynomial',
-and install appropriate operations on tagged polynomials in the
-operation table. We'll embed all our code in an installation procedure
-for the polynomial package, similar to the ones in section *Note
-2-5-1:::
-
- (define (install-polynomial-package)
- ;; internal procedures
- ;; representation of poly
- (define (make-poly variable term-list)
- (cons variable term-list))
- (define (variable p) (car p))
- (define (term-list p) (cdr p))
- <_procedures `same-variable?' and `variable?' from section 2.3.2_>
-
- ;; representation of terms and term lists
- <_procedures `adjoin-term' ... `coeff' from text below_>
-
- ;; continued on next page
-
- (define (add-poly p1 p2) ...)
- <_procedures used by `add-poly'_>
- (define (mul-poly p1 p2) ...)
- <_procedures used by `mul-poly'_>
-
- ;; interface to rest of the system
- (define (tag p) (attach-tag 'polynomial p))
- (put 'add '(polynomial polynomial)
- (lambda (p1 p2) (tag (add-poly p1 p2))))
- (put 'mul '(polynomial polynomial)
- (lambda (p1 p2) (tag (mul-poly p1 p2))))
- (put 'make 'polynomial
- (lambda (var terms) (tag (make-poly var terms))))
- 'done)
-
- Polynomial addition is performed termwise. Terms of the same order
-(i.e., with the same power of the indeterminate) must be combined.
-This is done by forming a new term of the same order whose coefficient
-is the sum of the coefficients of the addends. Terms in one addend for
-which there are no terms of the same order in the other addend are
-simply accumulated into the sum polynomial being constructed.
-
- In order to manipulate term lists, we will assume that we have a
-constructor `the-empty-termlist' that returns an empty term list and a
-constructor `adjoin-term' that adjoins a new term to a term list. We
-will also assume that we have a predicate `empty-termlist?' that tells
-if a given term list is empty, a selector `first-term' that extracts
-the highest-order term from a term list, and a selector `rest-terms'
-that returns all but the highest-order term. To manipulate terms, we
-will suppose that we have a constructor `make-term' that constructs a
-term with given order and coefficient, and selectors `order' and
-`coeff' that return, respectively, the order and the coefficient of the
-term. These operations allow us to consider both terms and term lists
-as data abstractions, whose concrete representations we can worry about
-separately.
-
- Here is the procedure that constructs the term list for the sum of
-two polynomials:(3)
-
- (define (add-terms L1 L2)
- (cond ((empty-termlist? L1) L2)
- ((empty-termlist? L2) L1)
- (else
- (let ((t1 (first-term L1)) (t2 (first-term L2)))
- (cond ((> (order t1) (order t2))
- (adjoin-term
- t1 (add-terms (rest-terms L1) L2)))
- ((< (order t1) (order t2))
- (adjoin-term
- t2 (add-terms L1 (rest-terms L2))))
- (else
- (adjoin-term
- (make-term (order t1)
- (add (coeff t1) (coeff t2)))
- (add-terms (rest-terms L1)
- (rest-terms L2)))))))))
-
- The most important point to note here is that we used the generic
-addition procedure `add' to add together the coefficients of the terms
-being combined. This has powerful consequences, as we will see below.
-
- In order to multiply two term lists, we multiply each term of the
-first list by all the terms of the other list, repeatedly using
-`mul-term-by-all-terms', which multiplies a given term by all terms in
-a given term list. The resulting term lists (one for each term of the
-first list) are accumulated into a sum. Multiplying two terms forms a
-term whose order is the sum of the orders of the factors and whose
-coefficient is the product of the coefficients of the factors:
-
- (define (mul-terms L1 L2)
- (if (empty-termlist? L1)
- (the-empty-termlist)
- (add-terms (mul-term-by-all-terms (first-term L1) L2)
- (mul-terms (rest-terms L1) L2))))
-
- (define (mul-term-by-all-terms t1 L)
- (if (empty-termlist? L)
- (the-empty-termlist)
- (let ((t2 (first-term L)))
- (adjoin-term
- (make-term (+ (order t1) (order t2))
- (mul (coeff t1) (coeff t2)))
- (mul-term-by-all-terms t1 (rest-terms L))))))
-
- This is really all there is to polynomial addition and
-multiplication. Notice that, since we operate on terms using the
-generic procedures `add' and `mul', our polynomial package is
-automatically able to handle any type of coefficient that is known
-about by the generic arithmetic package. If we include a coercion
-mechanism such as one of those discussed in section *Note 2-5-2::, then
-we also are automatically able to handle operations on polynomials of
-different coefficient types, such as
-
- / 2 \
- [3x^2 + (2 + 3i)x + 7] * | x^4 + --- x^2 + (5 + 3i) |
- \ 3 /
-
- Because we installed the polynomial addition and multiplication
-procedures `add-poly' and `mul-poly' in the generic arithmetic system
-as the `add' and `mul' operations for type `polynomial', our system is
-also automatically able to handle polynomial operations such as
-
- [(y + 1)x^2 + (y^2 + 1)x + (y - 1)] * [(y - 2)x + (y^3 + 7)]
-
- The reason is that when the system tries to combine coefficients, it
-will dispatch through `add' and `mul'. Since the coefficients are
-themselves polynomials (in y), these will be combined using `add-poly'
-and `mul-poly'. The result is a kind of "data-directed recursion" in
-which, for example, a call to `mul-poly' will result in recursive calls
-to `mul-poly' in order to multiply the coefficients. If the
-coefficients of the coefficients were themselves polynomials (as might
-be used to represent polynomials in three variables), the data
-direction would ensure that the system would follow through another
-level of recursive calls, and so on through as many levels as the
-structure of the data dictates.(4)
-
-Representing term lists
-.......................
-
-Finally, we must confront the job of implementing a good representation
-for term lists. A term list is, in effect, a set of coefficients keyed
-by the order of the term. Hence, any of the methods for representing
-sets, as discussed in section *Note 2-3-3::, can be applied to this
-task. On the other hand, our procedures `add-terms' and `mul-terms'
-always access term lists sequentially from highest to lowest order.
-Thus, we will use some kind of ordered list representation.
-
- How should we structure the list that represents a term list? One
-consideration is the "density" of the polynomials we intend to
-manipulate. A polynomial is said to be "dense" if it has nonzero
-coefficients in terms of most orders. If it has many zero terms it is
-said to be "sparse". For example,
-
- A : x^5 + 2x^4 + 3x^2 - 2x - 5
-
-is a dense polynomial, whereas
-
- B : x^100 + 2x^2 + 1
-
-is sparse.
-
- The term lists of dense polynomials are most efficiently represented
-as lists of the coefficients. For example, A above would be nicely
-represented as `(1 2 0 3 -2 -5)'. The order of a term in this
-representation is the length of the sublist beginning with that term's
-coefficient, decremented by 1.(5) This would be a terrible
-representation for a sparse polynomial such as B: There would be a
-giant list of zeros punctuated by a few lonely nonzero terms. A more
-reasonable representation of the term list of a sparse polynomial is as
-a list of the nonzero terms, where each term is a list containing the
-order of the term and the coefficient for that order. In such a
-scheme, polynomial B is efficiently represented as `((100 1) (2 2) (0
-1))'. As most polynomial manipulations are performed on sparse
-polynomials, we will use this method. We will assume that term lists
-are represented as lists of terms, arranged from highest-order to
-lowest-order term. Once we have made this decision, implementing the
-selectors and constructors for terms and term lists is
-straightforward:(6)
-
- (define (adjoin-term term term-list)
- (if (=zero? (coeff term))
- term-list
- (cons term term-list)))
-
- (define (the-empty-termlist) '())
- (define (first-term term-list) (car term-list))
- (define (rest-terms term-list) (cdr term-list))
- (define (empty-termlist? term-list) (null? term-list))
-
- (define (make-term order coeff) (list order coeff))
- (define (order term) (car term))
- (define (coeff term) (cadr term))
-
-where `=zero?' is as defined in *Note Exercise 2-80::. (See also *Note
-Exercise 2-87:: below.)
-
- Users of the polynomial package will create (tagged) polynomials by
-means of the procedure:
-
- (define (make-polynomial var terms)
- ((get 'make 'polynomial) var terms))
-
- *Exercise 2.87:* Install `=zero?' for polynomials in the generic
- arithmetic package. This will allow `adjoin-term' to work for
- polynomials with coefficients that are themselves polynomials.
-
- *Exercise 2.88:* Extend the polynomial system to include
- subtraction of polynomials. (Hint: You may find it helpful to
- define a generic negation operation.)
-
- *Exercise 2.89:* Define procedures that implement the term-list
- representation described above as appropriate for dense
- polynomials.
-
- *Exercise 2.90:* Suppose we want to have a polynomial system that
- is efficient for both sparse and dense polynomials. One way to do
- this is to allow both kinds of term-list representations in our
- system. The situation is analogous to the complex-number example
- of section *Note 2-4::, where we allowed both rectangular and
- polar representations. To do this we must distinguish different
- types of term lists and make the operations on term lists generic.
- Redesign the polynomial system to implement this generalization.
- This is a major effort, not a local change.
-
- *Exercise 2.91:* A univariate polynomial can be divided by another
- one to produce a polynomial quotient and a polynomial remainder.
- For example,
-
- x^5 - 1
- ------- = x^3 + x, remainder x - 1
- x^2 - 1
-
- Division can be performed via long division. That is, divide the
- highest-order term of the dividend by the highest-order term of
- the divisor. The result is the first term of the quotient. Next,
- multiply the result by the divisor, subtract that from the
- dividend, and produce the rest of the answer by recursively
- dividing the difference by the divisor. Stop when the order of the
- divisor exceeds the order of the dividend and declare the dividend
- to be the remainder. Also, if the dividend ever becomes zero,
- return zero as both quotient and remainder.
-
- We can design a `div-poly' procedure on the model of `add-poly' and
- `mul-poly'. The procedure checks to see if the two polys have the
- same variable. If so, `div-poly' strips off the variable and
- passes the problem to `div-terms', which performs the division
- operation on term lists. `Div-poly' finally reattaches the
- variable to the result supplied by `div-terms'. It is convenient
- to design `div-terms' to compute both the quotient and the
- remainder of a division. `Div-terms' can take two term lists as
- arguments and return a list of the quotient term list and the
- remainder term list.
-
- Complete the following definition of `div-terms' by filling in the
- missing expressions. Use this to implement `div-poly', which
- takes two polys as arguments and returns a list of the quotient
- and remainder polys.
-
- (define (div-terms L1 L2)
- (if (empty-termlist? L1)
- (list (the-empty-termlist) (the-empty-termlist))
- (let ((t1 (first-term L1))
- (t2 (first-term L2)))
- (if (> (order t2) (order t1))
- (list (the-empty-termlist) L1)
- (let ((new-c (div (coeff t1) (coeff t2)))
- (new-o (- (order t1) (order t2))))
- (let ((rest-of-result
- <COMPUTE REST OF RESULT RECURSIVELY>
- ))
- <FORM COMPLETE RESULT>
- ))))))
-
-Hierarchies of types in symbolic algebra
-........................................
-
-Our polynomial system illustrates how objects of one type (polynomials)
-may in fact be complex objects that have objects of many different
-types as parts. This poses no real difficulty in defining generic
-operations. We need only install appropriate generic operations for
-performing the necessary manipulations of the parts of the compound
-types. In fact, we saw that polynomials form a kind of "recursive data
-abstraction," in that parts of a polynomial may themselves be
-polynomials. Our generic operations and our data-directed programming
-style can handle this complication without much trouble.
-
- On the other hand, polynomial algebra is a system for which the data
-types cannot be naturally arranged in a tower. For instance, it is
-possible to have polynomials in x whose coefficients are polynomials in
-y. It is also possible to have polynomials in y whose coefficients are
-polynomials in x. Neither of these types is "above" the other in any
-natural way, yet it is often necessary to add together elements from
-each set. There are several ways to do this. One possibility is to
-convert one polynomial to the type of the other by expanding and
-rearranging terms so that both polynomials have the same principal
-variable. One can impose a towerlike structure on this by ordering the
-variables and thus always converting any polynomial to a "canonical
-form" with the highest-priority variable dominant and the
-lower-priority variables buried in the coefficients. This strategy
-works fairly well, except that the conversion may expand a polynomial
-unnecessarily, making it hard to read and perhaps less efficient to
-work with. The tower strategy is certainly not natural for this domain
-or for any domain where the user can invent new types dynamically using
-old types in various combining forms, such as trigonometric functions,
-power series, and integrals.
-
- It should not be surprising that controlling coercion is a serious
-problem in the design of large-scale algebraic-manipulation systems.
-Much of the complexity of such systems is concerned with relationships
-among diverse types. Indeed, it is fair to say that we do not yet
-completely understand coercion. In fact, we do not yet completely
-understand the concept of a data type. Nevertheless, what we know
-provides us with powerful structuring and modularity principles to
-support the design of large systems.
-
- *Exercise 2.92:* By imposing an ordering on variables, extend the
- polynomial package so that addition and multiplication of
- polynomials works for polynomials in different variables. (This
- is not easy!)
-
-Extended exercise: Rational functions
-.....................................
-
-We can extend our generic arithmetic system to include functions
-"rational functions". These are "fractions" whose numerator and
-denominator are polynomials, such as
-
- x + 1
- -------
- x^3 - 1
-
- The system should be able to add, subtract, multiply, and divide
-rational functions, and to perform such computations as
-
- x + 1 x x^3 + 2x^2 + 3x + 1
- ------- + ------- = -------------------
- x^3 - 1 x^2 - 1 x^4 + x^3 - x - 1
-
-(Here the sum has been simplified by removing common factors. Ordinary
-"cross multiplication" would have produced a fourth-degree polynomial
-over a fifth-degree polynomial.)
-
- If we modify our rational-arithmetic package so that it uses generic
-operations, then it will do what we want, except for the problem of
-reducing fractions to lowest terms.
-
- *Exercise 2.93:* Modify the rational-arithmetic package to use
- generic operations, but change `make-rat' so that it does not
- attempt to reduce fractions to lowest terms. Test your system by
- calling `make-rational' on two polynomials to produce a rational
- function
-
- (define p1 (make-polynomial 'x '((2 1)(0 1))))
- (define p2 (make-polynomial 'x '((3 1)(0 1))))
- (define rf (make-rational p2 p1))
-
- Now add `rf' to itself, using `add'. You will observe that this
- addition procedure does not reduce fractions to lowest terms.
-
- We can reduce polynomial fractions to lowest terms using the same
- idea we used with integers: modifying `make-rat' to divide both
- the numerator and the denominator by their greatest common
- divisor. The notion of "greatest common divisor" makes sense for
- polynomials. In fact, we can compute the GCD of two polynomials
- using essentially the same Euclid's Algorithm that works for
- integers.(7) The integer version is
-
- (define (gcd a b)
- (if (= b 0)
- a
- (gcd b (remainder a b))))
-
- Using this, we could make the obvious modification to define a GCD
- operation that works on term lists:
-
- (define (gcd-terms a b)
- (if (empty-termlist? b)
- a
- (gcd-terms b (remainder-terms a b))))
-
- where `remainder-terms' picks out the remainder component of the
- list returned by the term-list division operation `div-terms' that
- was implemented in *Note Exercise 2-91::.
-
- *Exercise 2.94:* Using `div-terms', implement the procedure
- `remainder-terms' and use this to define `gcd-terms' as above.
- Now write a procedure `gcd-poly' that computes the polynomial GCD
- of two polys. (The procedure should signal an error if the two
- polys are not in the same variable.) Install in the system a
- generic operation `greatest-common-divisor' that reduces to
- `gcd-poly' for polynomials and to ordinary `gcd' for ordinary
- numbers. As a test, try
-
- (define p1 (make-polynomial 'x '((4 1) (3 -1) (2 -2) (1 2))))
- (define p2 (make-polynomial 'x '((3 1) (1 -1))))
- (greatest-common-divisor p1 p2)
-
- and check your result by hand.
-
- *Exercise 2.95:* Define P_1, P_2, and P_3 to be the polynomials
-
- P_1 : x^2 - 2x + 1
-
- P_2 : 11x^2 + 7
-
- P_3 : 13x + 5
-
- Now define Q_1 to be the product of P_1 and P_2 and Q_2 to be the
- product of P_1 and P_3, and use `greatest-common-divisor' (*Note
- Exercise 2-94::) to compute the GCD of Q_1 and Q_2. Note that the
- answer is not the same as P_1. This example introduces noninteger
- operations into the computation, causing difficulties with the GCD
- algorithm.(8) To understand what is happening, try tracing
- `gcd-terms' while computing the GCD or try performing the division
- by hand.
-
- We can solve the problem exhibited in *Note Exercise 2-95:: if we
- use the following modification of the GCD algorithm (which really
- works only in the case of polynomials with integer coefficients).
- Before performing any polynomial division in the GCD computation,
- we multiply the dividend by an integer constant factor, chosen to
- guarantee that no fractions will arise during the division
- process. Our answer will thus differ from the actual GCD by an
- integer constant factor, but this does not matter in the case of
- reducing rational functions to lowest terms; the GCD will be used
- to divide both the numerator and denominator, so the integer
- constant factor will cancel out.
-
- More precisely, if P and Q are polynomials, let O_1 be the order of
- P (i.e., the order of the largest term of P) and let O_2 be the
- order of Q. Let c be the leading coefficient of Q. Then it can be
- shown that, if we multiply P by the "integerizing factor" c^(1+O_1
- -O_2), the resulting polynomial can be divided by Q by using the
- `div-terms' algorithm without introducing any fractions. The
- operation of multiplying the dividend by this constant and then
- dividing is sometimes called the "pseudodivision" of P by Q. The
- remainder of the division is called the "pseudoremainder".
-
- *Exercise 2.96:*
- a. Implement the procedure `pseudoremainder-terms', which is
- just like `remainder-terms' except that it multiplies the
- dividend by the integerizing factor described above before
- calling `div-terms'. Modify `gcd-terms' to use
- `pseudoremainder-terms', and verify that
- `greatest-common-divisor' now produces an answer with integer
- coefficients on the example in *Note Exercise 2-95::.
-
- b. The GCD now has integer coefficients, but they are larger
- than those of P_1. Modify `gcd-terms' so that it removes
- common factors from the coefficients of the answer by
- dividing all the coefficients by their (integer) greatest
- common divisor.
-
-
- Thus, here is how to reduce a rational function to lowest terms:
-
- * Compute the GCD of the numerator and denominator, using the
- version of `gcd-terms' from *Note Exercise 2-96::.
-
- * When you obtain the GCD, multiply both numerator and
- denominator by the same integerizing factor before dividing
- through by the GCD, so that division by the GCD will not
- introduce any noninteger coefficients. As the factor you can
- use the leading coefficient of the GCD raised to the power 1
- + O_1 - O_2, where O_2 is the order of the GCD and O_1 is the
- maximum of the orders of the numerator and denominator. This
- will ensure that dividing the numerator and denominator by
- the GCD will not introduce any fractions.
-
- * The result of this operation will be a numerator and
- denominator with integer coefficients. The coefficients will
- normally be very large because of all of the integerizing
- factors, so the last step is to remove the redundant factors
- by computing the (integer) greatest common divisor of all the
- coefficients of the numerator and the denominator and
- dividing through by this factor.
-
-
- *Exercise 2.97:*
- a. Implement this algorithm as a procedure `reduce-terms' that
- takes two term lists `n' and `d' as arguments and returns a
- list `nn', `dd', which are `n' and `d' reduced to lowest
- terms via the algorithm given above. Also write a procedure
- `reduce-poly', analogous to `add-poly', that checks to see if
- the two polys have the same variable. If so, `reduce-poly'
- strips off the variable and passes the problem to
- `reduce-terms', then reattaches the variable to the two term
- lists supplied by `reduce-terms'.
-
- b. Define a procedure analogous to `reduce-terms' that does what
- the original `make-rat' did for integers:
-
- (define (reduce-integers n d)
- (let ((g (gcd n d)))
- (list (/ n g) (/ d g))))
-
- and define `reduce' as a generic operation that calls
- `apply-generic' to dispatch to either `reduce-poly' (for
- `polynomial' arguments) or `reduce-integers' (for
- `scheme-number' arguments). You can now easily make the
- rational-arithmetic package reduce fractions to lowest terms
- by having `make-rat' call `reduce' before combining the given
- numerator and denominator to form a rational number. The
- system now handles rational expressions in either integers or
- polynomials. To test your program, try the example at the
- beginning of this extended exercise:
-
- (define p1 (make-polynomial 'x '((1 1)(0 1))))
- (define p2 (make-polynomial 'x '((3 1)(0 -1))))
- (define p3 (make-polynomial 'x '((1 1))))
- (define p4 (make-polynomial 'x '((2 1)(0 -1))))
-
- (define rf1 (make-rational p1 p2))
- (define rf2 (make-rational p3 p4))
-
- (add rf1 rf2)
-
- See if you get the correct answer, correctly reduced to
- lowest terms.
-
- The GCD computation is at the heart of any system that does
- operations on rational functions. The algorithm used above,
- although mathematically straightforward, is extremely slow.
- The slowness is due partly to the large number of division
- operations and partly to the enormous size of the
- intermediate coefficients generated by the pseudodivisions.
- One of the active areas in the development of
- algebraic-manipulation systems is the design of better
- algorithms for computing polynomial GCDs.(9)
-
-
- ---------- Footnotes ----------
-
- (1) On the other hand, we will allow polynomials whose coefficients
-are themselves polynomials in other variables. This will give us
-essentially the same representational power as a full multivariate
-system, although it does lead to coercion problems, as discussed below.
-
- (2) For univariate polynomials, giving the value of a polynomial at
-a given set of points can be a particularly good representation. This
-makes polynomial arithmetic extremely simple. To obtain, for example,
-the sum of two polynomials represented in this way, we need only add
-the values of the polynomials at corresponding points. To transform
-back to a more familiar representation, we can use the Lagrange
-interpolation formula, which shows how to recover the coefficients of a
-polynomial of degree n given the values of the polynomial at n + 1
-points.
-
- (3) This operation is very much like the ordered `union-set'
-operation we developed in exercise *Note Exercise 2-62::. In fact, if
-we think of the terms of the polynomial as a set ordered according to
-the power of the indeterminate, then the program that produces the term
-list for a sum is almost identical to `union-set'.
-
- (4) To make this work completely smoothly, we should also add to our
-generic arithmetic system the ability to coerce a "number" to a
-polynomial by regarding it as a polynomial of degree zero whose
-coefficient is the number. This is necessary if we are going to
-perform operations such as
-
- [x^2 + (y + 1)x + 5] + [x^2 + 2x + 1]
-
-which requires adding the coefficient y + 1 to the coefficient 2.
-
- (5) In these polynomial examples, we assume that we have implemented
-the generic arithmetic system using the type mechanism suggested in
-*Note Exercise 2-78::. Thus, coefficients that are ordinary numbers
-will be represented as the numbers themselves rather than as pairs
-whose `car' is the symbol `scheme-number'.
-
- (6) Although we are assuming that term lists are ordered, we have
-implemented `adjoin-term' to simply `cons' the new term onto the
-existing term list. We can get away with this so long as we guarantee
-that the procedures (such as `add-terms') that use `adjoin-term' always
-call it with a higher-order term than appears in the list. If we did
-not want to make such a guarantee, we could have implemented
-`adjoin-term' to be similar to the `adjoin-set' constructor for the
-ordered-list representation of sets (*Note Exercise 2-61::).
-
- (7) The fact that Euclid's Algorithm works for polynomials is
-formalized in algebra by saying that polynomials form a kind of
-algebraic domain called a "Euclidean ring". A Euclidean ring is a
-domain that admits addition, subtraction, and commutative
-multiplication, together with a way of assigning to each element x of
-the ring a positive integer "measure" m(x) with the properties that
-m(xy)>= m(x) for any nonzero x and y and that, given any x and y, there
-exists a q such that y = qx + r and either r = 0 or m(r)< m(x). From
-an abstract point of view, this is what is needed to prove that
-Euclid's Algorithm works. For the domain of integers, the measure m of
-an integer is the absolute value of the integer itself. For the domain
-of polynomials, the measure of a polynomial is its degree.
-
- (8) In an implementation like MIT Scheme, this produces a polynomial
-that is indeed a divisor of Q_1 and Q_2, but with rational
-coefficients. In many other Scheme systems, in which division of
-integers can produce limited-precision decimal numbers, we may fail to
-get a valid divisor.
-
- (9) One extremely efficient and elegant method for computing
-polynomial GCDs was discovered by Richard Zippel (1979). The method is
-a probabilistic algorithm, as is the fast test for primality that we
-discussed in *Note Chapter 1::. Zippel's book (1993) describes this
-method, together with other ways to compute polynomial GCDs.
-
-\1f
-File: sicp.info, Node: Chapter 3, Next: Chapter 4, Prev: Chapter 2, Up: Top
-
-3 Modularity, Objects, and State
-********************************
-
- [greek not included here]
-
- (Even while it changes, it stands still.)
-
- --Heraclitus
-
- Plus c,a change, plus c'est la me*me chose.
-
- --Alphonse Karr
-
- The preceding chapters introduced the basic elements from which
-programs are made. We saw how primitive procedures and primitive data
-are combined to construct compound entities, and we learned that
-abstraction is vital in helping us to cope with the complexity of large
-systems. But these tools are not sufficient for designing programs.
-Effective program synthesis also requires organizational principles
-that can guide us in formulating the overall design of a program. In
-particular, we need strategies to help us structure large systems so
-that they will be "modular", that is, so that they can be divided
-"naturally" into coherent parts that can be separately developed and
-maintained.
-
- One powerful design strategy, which is particularly appropriate to
-the construction of programs for modeling physical systems, is to base
-the structure of our programs on the structure of the system being
-modeled. For each object in the system, we construct a corresponding
-computational object. For each system action, we define a symbolic
-operation in our computational model. Our hope in using this strategy
-is that extending the model to accommodate new objects or new actions
-will require no strategic changes to the program, only the addition of
-the new symbolic analogs of those objects or actions. If we have been
-successful in our system organization, then to add a new feature or
-debug an old one we will have to work on only a localized part of the
-system.
-
- To a large extent, then, the way we organize a large program is
-dictated by our perception of the system to be modeled. In this
-chapter we will investigate two prominent organizational strategies
-arising from two rather different "world views" of the structure of
-systems. The first organizational strategy concentrates on "objects",
-viewing a large system as a collection of distinct objects whose
-behaviors may change over time. An alternative organizational strategy
-concentrates on the "streams" of information that flow in the system,
-much as an electrical engineer views a signal-processing system.
-
- Both the object-based approach and the stream-processing approach
-raise significant linguistic issues in programming. With objects, we
-must be concerned with how a computational object can change and yet
-maintain its identity. This will force us to abandon our old
-substitution model of computation (section *Note 1-1-5::) in favor of a
-more mechanistic but less theoretically tractable "environment model"
-of computation. The difficulties of dealing with objects, change, and
-identity are a fundamental consequence of the need to grapple with time
-in our computational models. These difficulties become even greater
-when we allow the possibility of concurrent execution of programs. The
-stream approach can be most fully exploited when we decouple simulated
-time in our model from the order of the events that take place in the
-computer during evaluation. We will accomplish this using a technique
-known as "delayed evaluation".
-
-* Menu:
-
-* 3-1:: Assignment and Local State
-* 3-2:: The Environment Model of Evaluation
-* 3-3:: Modeling with Mutable Data
-* 3-4:: Concurrency: Time Is of the Essence
-* 3-5:: Streams
-
-\1f
-File: sicp.info, Node: 3-1, Next: 3-2, Prev: Chapter 3, Up: Chapter 3
-
-3.1 Assignment and Local State
-==============================
-
-We ordinarily view the world as populated by independent objects, each
-of which has a state that changes over time. An object is said to
-"have state" if its behavior is influenced by its history. A bank
-account, for example, has state in that the answer to the question "Can
-I withdraw $100?" depends upon the history of deposit and withdrawal
-transactions. We can characterize an object's state by one or more "state
-variables", which among them maintain enough information about history
-to determine the object's current behavior. In a simple banking
-system, we could characterize the state of an account by a current
-balance rather than by remembering the entire history of account
-transactions.
-
- In a system composed of many objects, the objects are rarely
-completely independent. Each may influence the states of others
-through interactions, which serve to couple the state variables of one
-object to those of other objects. Indeed, the view that a system is
-composed of separate objects is most useful when the state variables of
-the system can be grouped into closely coupled subsystems that are only
-loosely coupled to other subsystems.
-
- This view of a system can be a powerful framework for organizing
-computational models of the system. For such a model to be modular, it
-should be decomposed into computational objects that model the actual
-objects in the system. Each computational object must have its own "local
-state variables" describing the actual object's state. Since the
-states of objects in the system being modeled change over time, the
-state variables of the corresponding computational objects must also
-change. If we choose to model the flow of time in the system by the
-elapsed time in the computer, then we must have a way to construct
-computational objects whose behaviors change as our programs run. In
-particular, if we wish to model state variables by ordinary symbolic
-names in the programming language, then the language must provide an operator
-"assignment operator" to enable us to change the value associated with
-a name.
-
-* Menu:
-
-* 3-1-1:: Local State Variables
-* 3-1-2:: The Benefits of Introducing Assignment
-* 3-1-3:: The Costs of Introducing Assignment
-
-\1f
-File: sicp.info, Node: 3-1-1, Next: 3-1-2, Prev: 3-1, Up: 3-1
-
-3.1.1 Local State Variables
----------------------------
-
-To illustrate what we mean by having a computational object with
-time-varying state, let us model the situation of withdrawing money
-from a bank account. We will do this using a procedure `withdraw',
-which takes as argument an `amount' to be withdrawn. If there is
-enough money in the account to accommodate the withdrawal, then
-`withdraw' should return the balance remaining after the withdrawal.
-Otherwise, `withdraw' should return the message _Insufficient funds_.
-For example, if we begin with $100 in the account, we should obtain the
-following sequence of responses using `withdraw':
-
- (withdraw 25)
- 75
-
- (withdraw 25)
- 50
-
- (withdraw 60)
- "Insufficient funds"
-
- (withdraw 15)
- 35
-
- Observe that the expression `(withdraw 25)', evaluated twice, yields
-different values. This is a new kind of behavior for a procedure.
-Until now, all our procedures could be viewed as specifications for
-computing mathematical functions. A call to a procedure computed the
-value of the function applied to the given arguments, and two calls to
-the same procedure with the same arguments always produced the same
-result.(1)
-
- To implement `withdraw', we can use a variable `balance' to indicate
-the balance of money in the account and define `withdraw' as a procedure
-that accesses `balance'. The `withdraw' procedure checks to see if
-`balance' is at least as large as the requested `amount'. If so,
-`withdraw' decrements `balance' by `amount' and returns the new value
-of `balance'. Otherwise, `withdraw' returns the _Insufficient funds_
-message. Here are the definitions of `balance' and `withdraw':
-
- (define balance 100)
-
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
-
- Decrementing `balance' is accomplished by the expression
-
- (set! balance (- balance amount))
-
- This uses the `set!' special form, whose syntax is
-
- (set! <NAME> <NEW-VALUE>)
-
- Here <NAME> is a symbol and <NEW-VALUE> is any expression. `Set!'
-changes <NAME> so that its value is the result obtained by evaluating
-<NEW-VALUE>. In the case at hand, we are changing `balance' so that
-its new value will be the result of subtracting `amount' from the
-previous value of `balance'.(2)
-
- `Withdraw' also uses the `begin' special form to cause two
-expressions to be evaluated in the case where the `if' test is true:
-first decrementing `balance' and then returning the value of `balance'.
-In general, evaluating the expression
-
- (begin <EXP_1> <EXP_2> ... <EXP_K>)
-
-causes the expressions <EXP_1> through <EXP_K> to be evaluated in
-sequence and the value of the final expression <EXP_K> to be returned
-as the value of the entire `begin' form.(3)
-
- Although `withdraw' works as desired, the variable `balance' presents
-a problem. As specified above, `balance' is a name defined in the
-global environment and is freely accessible to be examined or modified
-by any procedure. It would be much better if we could somehow make
-`balance' internal to `withdraw', so that `withdraw' would be the only
-procedure that could access `balance' directly and any other procedure
-could access `balance' only indirectly (through calls to `withdraw').
-This would more accurately model the notion that `balance' is a local
-state variable used by `withdraw' to keep track of the state of the
-account.
-
- We can make `balance' internal to `withdraw' by rewriting the
-definition as follows:
-
- (define new-withdraw
- (let ((balance 100))
- (lambda (amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))))
-
- What we have done here is use `let' to establish an environment with
-a local variable `balance', bound to the initial value 100. Within this
-local environment, we use `lambda' to create a procedure that takes
-`amount' as an argument and behaves like our previous `withdraw'
-procedure. This procedure--returned as the result of evaluating the
-`let' expression--is `new-withdraw', which behaves in precisely the
-same way as `withdraw' but whose variable `balance' is not accessible
-by any other procedure.(4)
-
- Combining `set!' with local variables is the general programming
-technique we will use for constructing computational objects with local
-state. Unfortunately, using this technique raises a serious problem:
-When we first introduced procedures, we also introduced the
-substitution model of evaluation (section *Note 1-1-5::) to provide an
-interpretation of what procedure application means. We said that
-applying a procedure should be interpreted as evaluating the body of
-the procedure with the formal parameters replaced by their values. The
-trouble is that, as soon as we introduce assignment into our language,
-substitution is no longer an adequate model of procedure application.
-(We will see why this is so in section *Note 3-1-3::.) As a
-consequence, we technically have at this point no way to understand why
-the `new-withdraw' procedure behaves as claimed above. In order to
-really understand a procedure such as `new-withdraw', we will need to
-develop a new model of procedure application. In section *Note 3-2::
-we will introduce such a model, together with an explanation of `set!'
-and local variables. First, however, we examine some variations on the
-theme established by `new-withdraw'.
-
- The following procedure, `make-withdraw', creates "withdrawal
-processors." The formal parameter `balance' in `make-withdraw'
-specifies the initial amount of money in the account.(5)
-
- (define (make-withdraw balance)
- (lambda (amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds")))
-
- `Make-withdraw' can be used as follows to create two objects `W1' and
-`W2':
-
- (define W1 (make-withdraw 100))
- (define W2 (make-withdraw 100))
-
- (W1 50)
- 50
-
- (W2 70)
- 30
-
- (W2 40)
- "Insufficient funds"
-
- (W1 40)
- 10
-
- Observe that `W1' and `W2' are completely independent objects, each
-with its own local state variable `balance'. Withdrawals from one do
-not affect the other.
-
- We can also create objects that handle deposits as well as
-withdrawals, and thus we can represent simple bank accounts. Here is a
-procedure that returns a "bank-account object" with a specified initial
-balance:
-
- (define (make-account balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (define (dispatch m)
- (cond ((eq? m 'withdraw) withdraw)
- ((eq? m 'deposit) deposit)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch)
-
- Each call to `make-account' sets up an environment with a local state
-variable `balance'. Within this environment, `make-account' defines
-procedures `deposit' and `withdraw' that access `balance' and an
-additional procedure `dispatch' that takes a "message" as input and
-returns one of the two local procedures. The `dispatch' procedure
-itself is returned as the value that represents the bank-account
-object. This is precisely the "message-passing" style of programming
-that we saw in section *Note 2-4-3::, although here we are using it in
-conjunction with the ability to modify local variables.
-
- `Make-account' can be used as follows:
-
- (define acc (make-account 100))
-
- ((acc 'withdraw) 50)
- 50
-
- ((acc 'withdraw) 60)
- "Insufficient funds"
-
- ((acc 'deposit) 40)
- 90
-
- ((acc 'withdraw) 60)
- 30
-
- Each call to `acc' returns the locally defined `deposit' or
-`withdraw' procedure, which is then applied to the specified `amount'.
-As was the case with `make-withdraw', another call to `make-account'
-
- (define acc2 (make-account 100))
-
-will produce a completely separate account object, which maintains its
-own local `balance'.
-
- *Exercise 3.1:* An "accumulator" is a procedure that is called
- repeatedly with a single numeric argument and accumulates its
- arguments into a sum. Each time it is called, it returns the
- currently accumulated sum. Write a procedure `make-accumulator'
- that generates accumulators, each maintaining an independent sum.
- The input to `make-accumulator' should specify the initial value
- of the sum; for example
-
- (define A (make-accumulator 5))
-
- (A 10)
- 15
-
- (A 10)
- 25
-
- *Exercise 3.2:* In software-testing applications, it is useful to
- be able to count the number of times a given procedure is called
- during the course of a computation. Write a procedure
- `make-monitored' that takes as input a procedure, `f', that itself
- takes one input. The result returned by `make-monitored' is a
- third procedure, say `mf', that keeps track of the number of times
- it has been called by maintaining an internal counter. If the
- input to `mf' is the special symbol `how-many-calls?', then `mf'
- returns the value of the counter. If the input is the special
- symbol `reset-count', then `mf' resets the counter to zero. For
- any other input, `mf' returns the result of calling `f' on that
- input and increments the counter. For instance, we could make a
- monitored version of the `sqrt' procedure:
-
- (define s (make-monitored sqrt))
-
- (s 100)
- 10
-
- (s 'how-many-calls?)
- 1
-
- *Exercise 3.3:* Modify the `make-account' procedure so that it
- creates password-protected accounts. That is, `make-account'
- should take a symbol as an additional argument, as in
-
- (define acc (make-account 100 'secret-password))
-
- The resulting account object should process a request only if it
- is accompanied by the password with which the account was created,
- and should otherwise return a complaint:
-
- ((acc 'secret-password 'withdraw) 40)
- 60
-
- ((acc 'some-other-password 'deposit) 50)
- "Incorrect password"
-
- *Exercise 3.4:* Modify the `make-account' procedure of *Note
- Exercise 3-3:: by adding another local state variable so that, if
- an account is accessed more than seven consecutive times with an
- incorrect password, it invokes the procedure `call-the-cops'.
-
- ---------- Footnotes ----------
-
- (1) Actually, this is not quite true. One exception was the
-random-number generator in section *Note 1-2-6::. Another exception
-involved the operation/type tables we introduced in section *Note
-2-4-3::, where the values of two calls to `get' with the same arguments
-depended on intervening calls to `put'. On the other hand, until we
-introduce assignment, we have no way to create such procedures
-ourselves.
-
- (2) The value of a `set!' expression is implementation-dependent.
-`Set!' should be used only for its effect, not for its value.
-
- The name `set!' reflects a naming convention used in Scheme:
-Operations that change the values of variables (or that change data
-structures, as we will see in section *Note 3-3::) are given names that
-end with an exclamation point. This is similar to the convention of
-designating predicates by names that end with a question mark.
-
- (3) We have already used `begin' implicitly in our programs, because
-in Scheme the body of a procedure can be a sequence of expressions.
-Also, the <CONSEQUENT> part of each clause in a `cond' expression can
-be a sequence of expressions rather than a single expression.
-
- (4) In programming-language jargon, the variable `balance' is said
-to be "encapsulated" within the `new-withdraw' procedure.
-Encapsulation reflects the general system-design principle known as the "hiding
-principle": One can make a system more modular and robust by protecting
-parts of the system from each other; that is, by providing information
-access only to those parts of the system that have a "need to know."
-
- (5) In contrast with `new-withdraw' above, we do not have to use
-`let' to make `balance' a local variable, since formal parameters are
-already local. This will be clearer after the discussion of the
-environment model of evaluation in section *Note 3-2::. (See also
-*Note Exercise 3-10::.)
-
-\1f
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-
-3.1.2 The Benefits of Introducing Assignment
---------------------------------------------
-
-As we shall see, introducing assignment into our programming language
-leads us into a thicket of difficult conceptual issues. Nevertheless,
-viewing systems as collections of objects with local state is a
-powerful technique for maintaining a modular design. As a simple
-example, consider the design of a procedure `rand' that, whenever it is
-called, returns an integer chosen at random.
-
- It is not at all clear what is meant by "chosen at random." What we
-presumably want is for successive calls to `rand' to produce a sequence
-of numbers that has statistical properties of uniform distribution. We
-will not discuss methods for generating suitable sequences here.
-Rather, let us assume that we have a procedure `rand-update' that has
-the property that if we start with a given number x_1 and form
-
- x_2 = (rand-update x_1)
- x_3 = (rand-update x_2)
-
-then the sequence of values x_1, x_2, x_3, ..., will have the desired
-statistical properties.(1)
-
- We can implement `rand' as a procedure with a local state variable
-`x' that is initialized to some fixed value `random-init'. Each call
-to `rand' computes `rand-update' of the current value of `x', returns
-this as the random number, and also stores this as the new value of `x'.
-
- (define rand
- (let ((x random-init))
- (lambda ()
- (set! x (rand-update x))
- x)))
-
- Of course, we could generate the same sequence of random numbers
-without using assignment by simply calling `rand-update' directly.
-However, this would mean that any part of our program that used random
-numbers would have to explicitly remember the current value of `x' to
-be passed as an argument to `rand-update'. To realize what an
-annoyance this would be, consider using random numbers to implement a
-technique called simulation "Monte Carlo simulation".
-
- The Monte Carlo method consists of choosing sample experiments at
-random from a large set and then making deductions on the basis of the
-probabilities estimated from tabulating the results of those
-experiments. For example, we can approximate [pi] using the fact that
-6/[pi]^2 is the probability that two integers chosen at random will
-have no factors in common; that is, that their greatest common divisor
-will be 1.(2) To obtain the approximation to [pi], we perform a large
-number of experiments. In each experiment we choose two integers at
-random and perform a test to see if their GCD is 1. The fraction of
-times that the test is passed gives us our estimate of 6/[pi]^2, and
-from this we obtain our approximation to [pi].
-
- The heart of our program is a procedure `monte-carlo', which takes as
-arguments the number of times to try an experiment, together with the
-experiment, represented as a no-argument procedure that will return
-either true or false each time it is run. `Monte-carlo' runs the
-experiment for the designated number of trials and returns a number
-telling the fraction of the trials in which the experiment was found to
-be true.
-
- (define (estimate-pi trials)
- (sqrt (/ 6 (monte-carlo trials cesaro-test))))
-
- (define (cesaro-test)
- (= (gcd (rand) (rand)) 1))
-
- (define (monte-carlo trials experiment)
- (define (iter trials-remaining trials-passed)
- (cond ((= trials-remaining 0)
- (/ trials-passed trials))
- ((experiment)
- (iter (- trials-remaining 1) (+ trials-passed 1)))
- (else
- (iter (- trials-remaining 1) trials-passed))))
- (iter trials 0))
-
- Now let us try the same computation using `rand-update' directly
-rather than `rand', the way we would be forced to proceed if we did not
-use assignment to model local state:
-
- (define (estimate-pi trials)
- (sqrt (/ 6 (random-gcd-test trials random-init))))
-
- (define (random-gcd-test trials initial-x)
- (define (iter trials-remaining trials-passed x)
- (let ((x1 (rand-update x)))
- (let ((x2 (rand-update x1)))
- (cond ((= trials-remaining 0)
- (/ trials-passed trials))
- ((= (gcd x1 x2) 1)
- (iter (- trials-remaining 1)
- (+ trials-passed 1)
- x2))
- (else
- (iter (- trials-remaining 1)
- trials-passed
- x2))))))
- (iter trials 0 initial-x))
-
- While the program is still simple, it betrays some painful breaches
-of modularity. In our first version of the program, using `rand', we
-can express the Monte Carlo method directly as a general `monte-carlo'
-procedure that takes as an argument an arbitrary `experiment' procedure.
-In our second version of the program, with no local state for the
-random-number generator, `random-gcd-test' must explicitly manipulate
-the random numbers `x1' and `x2' and recycle `x2' through the iterative
-loop as the new input to `rand-update'. This explicit handling of the
-random numbers intertwines the structure of accumulating test results
-with the fact that our particular experiment uses two random numbers,
-whereas other Monte Carlo experiments might use one random number or
-three. Even the top-level procedure `estimate-pi' has to be concerned
-with supplying an initial random number. The fact that the
-random-number generator's insides are leaking out into other parts of
-the program makes it difficult for us to isolate the Monte Carlo idea
-so that it can be applied to other tasks. In the first version of the
-program, assignment encapsulates the state of the random-number
-generator within the `rand' procedure, so that the details of
-random-number generation remain independent of the rest of the program.
-
- The general phenomenon illustrated by the Monte Carlo example is
-this: From the point of view of one part of a complex process, the
-other parts appear to change with time. They have hidden time-varying
-local state. If we wish to write computer programs whose structure
-reflects this decomposition, we make computational objects (such as
-bank accounts and random-number generators) whose behavior changes with
-time. We model state with local state variables, and we model the
-changes of state with assignments to those variables.
-
- It is tempting to conclude this discussion by saying that, by
-introducing assignment and the technique of hiding state in local
-variables, we are able to structure systems in a more modular fashion
-than if all state had to be manipulated explicitly, by passing
-additional parameters. Unfortunately, as we shall see, the story is
-not so simple.
-
- *Exercise 3.5:* "Monte Carlo integration" is a method of
- estimating definite integrals by means of Monte Carlo simulation.
- Consider computing the area of a region of space described by a
- predicate P(x, y) that is true for points (x, y) in the region and
- false for points not in the region. For example, the region
- contained within a circle of radius 3 centered at (5, 7) is
- described by the predicate that tests whether (x - 5)^2 + (y -
- 7)^2 <= 3^2. To estimate the area of the region described by such
- a predicate, begin by choosing a rectangle that contains the
- region. For example, a rectangle with diagonally opposite corners
- at (2, 4) and (8, 10) contains the circle above. The desired
- integral is the area of that portion of the rectangle that lies in
- the region. We can estimate the integral by picking, at random,
- points (x,y) that lie in the rectangle, and testing P(x, y) for
- each point to determine whether the point lies in the region. If
- we try this with many points, then the fraction of points that
- fall in the region should give an estimate of the proportion of
- the rectangle that lies in the region. Hence, multiplying this
- fraction by the area of the entire rectangle should produce an
- estimate of the integral.
-
- Implement Monte Carlo integration as a procedure
- `estimate-integral' that takes as arguments a predicate `P', upper
- and lower bounds `x1', `x2', `y1', and `y2' for the rectangle, and
- the number of trials to perform in order to produce the estimate.
- Your procedure should use the same `monte-carlo' procedure that
- was used above to estimate [pi]. Use your `estimate-integral' to
- produce an estimate of [pi] by measuring the area of a unit circle.
-
- You will find it useful to have a procedure that returns a number
- chosen at random from a given range. The following
- `random-in-range' procedure implements this in terms of the
- `random' procedure used in section *Note 1-2-6::, which returns a
- nonnegative number less than its input.(3)
-
- (define (random-in-range low high)
- (let ((range (- high low)))
- (+ low (random range))))
-
- *Exercise 3.6:* It is useful to be able to reset a random-number
- generator to produce a sequence starting from a given value.
- Design a new `rand' procedure that is called with an argument that
- is either the symbol `generate' or the symbol `reset' and behaves
- as follows: `(rand 'generate)' produces a new random number;
- `((rand 'reset) <NEW-VALUE>)' resets the internal state variable
- to the designated <NEW-VALUE>. Thus, by resetting the state, one
- can generate repeatable sequences. These are very handy to have
- when testing and debugging programs that use random numbers.
-
- ---------- Footnotes ----------
-
- (1) One common way to implement `rand-update' is to use the rule
-that x is updated to ax + b modulo m, where a, b, and m are
-appropriately chosen integers. Chapter 3 of Knuth 1981 includes an
-extensive discussion of techniques for generating sequences of random
-numbers and establishing their statistical properties. Notice that the
-`rand-update' procedure computes a mathematical function: Given the
-same input twice, it produces the same output. Therefore, the number
-sequence produced by `rand-update' certainly is not "random," if by
-"random" we insist that each number in the sequence is unrelated to the
-preceding number. The relation between "real randomness" and so-called "pseudo-random"
-sequences, which are produced by well-determined computations and yet
-have suitable statistical properties, is a complex question involving
-difficult issues in mathematics and philosophy. Kolmogorov,
-Solomonoff, and Chaitin have made great progress in clarifying these
-issues; a discussion can be found in Chaitin 1975.
-
- (2) This theorem is due to E. Cesa`ro. See section 4.5.2 of Knuth
-1981 for a discussion and a proof.
-
- (3) MIT Scheme provides such a procedure. If `random' is given an
-exact integer (as in section *Note 1-2-6::) it returns an exact
-integer, but if it is given a decimal value (as in this exercise) it
-returns a decimal value.
-
-\1f
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-
-3.1.3 The Costs of Introducing Assignment
------------------------------------------
-
-As we have seen, the `set!' operation enables us to model objects that
-have local state. However, this advantage comes at a price. Our
-programming language can no longer be interpreted in terms of the
-substitution model of procedure application that we introduced in
-section *Note 1-1-5::. Moreover, no simple model with "nice"
-mathematical properties can be an adequate framework for dealing with
-objects and assignment in programming languages.
-
- So long as we do not use assignments, two evaluations of the same
-procedure with the same arguments will produce the same result, so that
-procedures can be viewed as computing mathematical functions.
-Programming without any use of assignments, as we did throughout the
-first two chapters of this book, is accordingly known as "functional
-programming".
-
- To understand how assignment complicates matters, consider a
-simplified version of the `make-withdraw' procedure of section *Note
-3-1-1:: that does not bother to check for an insufficient amount:
-
- (define (make-simplified-withdraw balance)
- (lambda (amount)
- (set! balance (- balance amount))
- balance))
-
- (define W (make-simplified-withdraw 25))
-
- (W 20)
- 5
-
- (W 10)
- - 5
-
- Compare this procedure with the following `make-decrementer'
-procedure, which does not use `set!':
-
- (define (make-decrementer balance)
- (lambda (amount)
- (- balance amount)))
-
- `Make-decrementer' returns a procedure that subtracts its input from
-a designated amount `balance', but there is no accumulated effect over
-successive calls, as with `make-simplified-withdraw':
-
- (define D (make-decrementer 25))
-
- (D 20)
- 5
-
- (D 10)
- 15
-
- We can use the substitution model to explain how `make-decrementer'
-works. For instance, let us analyze the evaluation of the expression
-
- ((make-decrementer 25) 20)
-
- We first simplify the operator of the combination by substituting 25
-for `balance' in the body of `make-decrementer'. This reduces the
-expression to
-
- ((lambda (amount) (- 25 amount)) 20)
-
- Now we apply the operator by substituting 20 for `amount' in the
-body of the `lambda' expression:
-
- (- 25 20)
-
- The final answer is 5.
-
- Observe, however, what happens if we attempt a similar substitution
-analysis with `make-simplified-withdraw':
-
- ((make-simplified-withdraw 25) 20)
-
- We first simplify the operator by substituting 25 for `balance' in
-the body of `make-simplified-withdraw'. This reduces the expression
-to(1)
-
- ((lambda (amount) (set! balance (- 25 amount)) 25) 20)
-
- Now we apply the operator by substituting 20 for `amount' in the
-body of the `lambda' expression:
-
- (set! balance (- 25 20)) 25
-
- If we adhered to the substitution model, we would have to say that
-the meaning of the procedure application is to first set `balance' to 5
-and then return 25 as the value of the expression. This gets the wrong
-answer. In order to get the correct answer, we would have to somehow
-distinguish the first occurrence of `balance' (before the effect of the
-`set!') from the second occurrence of `balance' (after the effect of
-the `set!'), and the substitution model cannot do this.
-
- The trouble here is that substitution is based ultimately on the
-notion that the symbols in our language are essentially names for
-values. But as soon as we introduce `set!' and the idea that the value
-of a variable can change, a variable can no longer be simply a name.
-Now a variable somehow refers to a place where a value can be stored,
-and the value stored at this place can change. In section *Note 3-2::
-we will see how environments play this role of "place" in our
-computational model.
-
-Sameness and change
-...................
-
-The issue surfacing here is more profound than the mere breakdown of a
-particular model of computation. As soon as we introduce change into
-our computational models, many notions that were previously
-straightforward become problematical. Consider the concept of two
-things being "the same."
-
- Suppose we call `make-decrementer' twice with the same argument to
-create two procedures:
-
- (define D1 (make-decrementer 25))
-
- (define D2 (make-decrementer 25))
-
- Are `D1' and `D2' the same? An acceptable answer is yes, because
-`D1' and `D2' have the same computational behavior--each is a procedure
-that subtracts its input from 25. In fact, `D1' could be substituted
-for `D2' in any computation without changing the result.
-
- Contrast this with making two calls to `make-simplified-withdraw':
-
- (define W1 (make-simplified-withdraw 25))
-
- (define W2 (make-simplified-withdraw 25))
-
- Are `W1' and `W2' the same? Surely not, because calls to `W1' and
-`W2' have distinct effects, as shown by the following sequence of
-interactions:
-
- (W1 20)
- 5
-
- (W1 20)
- - 15
-
- (W2 20)
- 5
-
- Even though `W1' and `W2' are "equal" in the sense that they are
-both created by evaluating the same expression,
-`(make-simplified-withdraw 25)', it is not true that `W1' could be
-substituted for `W2' in any expression without changing the result of
-evaluating the expression.
-
- A language that supports the concept that "equals can be substituted
-for equals" in an expresssion without changing the value of the
-expression is said to be "referentially transparent". Referential
-transparency is violated when we include `set!' in our computer
-language. This makes it tricky to determine when we can simplify
-expressions by substituting equivalent expressions. Consequently,
-reasoning about programs that use assignment becomes drastically more
-difficult.
-
- Once we forgo referential transparency, the notion of what it means
-for computational objects to be "the same" becomes difficult to capture
-in a formal way. Indeed, the meaning of "same" in the real world that
-our programs model is hardly clear in itself. In general, we can
-determine that two apparently identical objects are indeed "the same
-one" only by modifying one object and then observing whether the other
-object has changed in the same way. But how can we tell if an object
-has "changed" other than by observing the "same" object twice and
-seeing whether some property of the object differs from one observation
-to the next? Thus, we cannot determine "change" without some _a
-priori_ notion of "sameness," and we cannot determine sameness without
-observing the effects of change.
-
- As an example of how this issue arises in programming, consider the
-situation where Peter and Paul have a bank account with $100 in it.
-There is a substantial difference between modeling this as
-
- (define peter-acc (make-account 100))
- (define paul-acc (make-account 100))
-
-and modeling it as
-
- (define peter-acc (make-account 100))
- (define paul-acc peter-acc)
-
- In the first situation, the two bank accounts are distinct.
-Transactions made by Peter will not affect Paul's account, and vice
-versa. In the second situation, however, we have defined `paul-acc' to
-be _the same thing_ as `peter-acc'. In effect, Peter and Paul now have
-a joint bank account, and if Peter makes a withdrawal from `peter-acc'
-Paul will observe less money in `paul-acc'. These two similar but
-distinct situations can cause confusion in building computational
-models. With the shared account, in particular, it can be especially
-confusing that there is one object (the bank account) that has two
-different names (`peter-acc' and `paul-acc'); if we are searching for
-all the places in our program where `paul-acc' can be changed, we must
-remember to look also at things that change `peter-acc'.(2)
-
- With reference to the above remarks on "sameness" and "change,"
-observe that if Peter and Paul could only examine their bank balances,
-and could not perform operations that changed the balance, then the
-issue of whether the two accounts are distinct would be moot. In
-general, so long as we never modify data objects, we can regard a
-compound data object to be precisely the totality of its pieces. For
-example, a rational number is determined by giving its numerator and
-its denominator. But this view is no longer valid in the presence of
-change, where a compound data object has an "identity" that is
-something different from the pieces of which it is composed. A bank
-account is still "the same" bank account even if we change the balance
-by making a withdrawal; conversely, we could have two different bank
-accounts with the same state information. This complication is a
-consequence, not of our programming language, but of our perception of
-a bank account as an object. We do not, for example, ordinarily regard
-a rational number as a changeable object with identity, such that we
-could change the numerator and still have "the same" rational number.
-
-Pitfalls of imperative programming
-..................................
-
-In contrast to functional programming, programming that makes extensive
-use of assignment is known as "imperative programming". In addition to
-raising complications about computational models, programs written in
-imperative style are susceptible to bugs that cannot occur in functional
-programs. For example, recall the iterative factorial program from
-section *Note 1-2-1:::
-
- (define (factorial n)
- (define (iter product counter)
- (if (> counter n)
- product
- (iter (* counter product)
- (+ counter 1))))
- (iter 1 1))
-
- Instead of passing arguments in the internal iterative loop, we
-could adopt a more imperative style by using explicit assignment to
-update the values of the variables `product' and `counter':
-
- (define (factorial n)
- (let ((product 1)
- (counter 1))
- (define (iter)
- (if (> counter n)
- product
- (begin (set! product (* counter product))
- (set! counter (+ counter 1))
- (iter))))
- (iter)))
-
- This does not change the results produced by the program, but it
-does introduce a subtle trap. How do we decide the order of the
-assignments? As it happens, the program is correct as written. But
-writing the assignments in the opposite order
-
- (set! counter (+ counter 1))
- (set! product (* counter product))
-
-would have produced a different, incorrect result. In general,
-programming with assignment forces us to carefully consider the
-relative orders of the assignments to make sure that each statement is
-using the correct version of the variables that have been changed.
-This issue simply does not arise in functional programs.(3)
-
- The complexity of imperative programs becomes even worse if we
-consider applications in which several processes execute concurrently.
-We will return to this in section *Note 3-4::. First, however, we will
-address the issue of providing a computational model for expressions
-that involve assignment, and explore the uses of objects with local
-state in designing simulations.
-
- *Exercise 3.7:* Consider the bank account objects created by
- `make-account', with the password modification described in *Note
- Exercise 3-3::. Suppose that our banking system requires the
- ability to make joint accounts. Define a procedure `make-joint'
- that accomplishes this. `Make-joint' should take three arguments.
- The first is a password-protected account. The second argument
- must match the password with which the account was defined in
- order for the `make-joint' operation to proceed. The third
- argument is a new password. `Make-joint' is to create an
- additional access to the original account using the new password.
- For example, if `peter-acc' is a bank account with password
- `open-sesame', then
-
- (define paul-acc
- (make-joint peter-acc 'open-sesame 'rosebud))
-
- will allow one to make transactions on `peter-acc' using the name
- `paul-acc' and the password `rosebud'. You may wish to modify your
- solution to *Note Exercise 3-3:: to accommodate this new feature
-
- *Exercise 3.8:* When we defined the evaluation model in section
- *Note 1-1-3::, we said that the first step in evaluating an
- expression is to evaluate its subexpressions. But we never
- specified the order in which the subexpressions should be
- evaluated (e.g., left to right or right to left). When we
- introduce assignment, the order in which the arguments to a
- procedure are evaluated can make a difference to the result.
- Define a simple procedure `f' such that evaluating `(+ (f 0) (f
- 1))' will return 0 if the arguments to `+' are evaluated from left
- to right but will return 1 if the arguments are evaluated from
- right to left.
-
- ---------- Footnotes ----------
-
- (1) We don't substitute for the occurrence of `balance' in the
-`set!' expression because the <NAME> in a `set!' is not evaluated. If
-we did substitute for it, we would get `(set! 25 (- 25 amount))', which
-makes no sense.
-
- (2) The phenomenon of a single computational object being accessed
-by more than one name is known as "aliasing". The joint bank account
-situation illustrates a very simple example of an alias. In section
-*Note 3-3:: we will see much more complex examples, such as "distinct"
-compound data structures that share parts. Bugs can occur in our
-programs if we forget that a change to an object may also, as a "side
-effect," change a "different" object because the two "different"
-objects are actually a single object appearing under different aliases.
-These so-called "side-effect bugs" are so difficult to locate and to
-analyze that some people have proposed that programming languages be
-designed in such a way as to not allow side effects or aliasing
-(Lampson et al. 1981; Morris, Schmidt, and Wadler 1980).
-
- (3) In view of this, it is ironic that introductory programming is
-most often taught in a highly imperative style. This may be a vestige
-of a belief, common throughout the 1960s and 1970s, that programs that
-call procedures must inherently be less efficient than programs that
-perform assignments. (Steele (1977) debunks this argument.)
-Alternatively it may reflect a view that step-by-step assignment is
-easier for beginners to visualize than procedure call. Whatever the
-reason, it often saddles beginning programmers with "should I set this
-variable before or after that one" concerns that can complicate
-programming and obscure the important ideas.
-
-\1f
-File: sicp.info, Node: 3-2, Next: 3-3, Prev: 3-1, Up: Chapter 3
-
-3.2 The Environment Model of Evaluation
-=======================================
-
-When we introduced compound procedures in *Note Chapter 1::, we used the
-substitution model of evaluation (section *Note 1-1-5::) to define what
-is meant by applying a procedure to arguments:
-
- * To apply a compound procedure to arguments, evaluate the body of
- the procedure with each formal parameter replaced by the
- corresponding argument.
-
-
- Once we admit assignment into our programming language, such a
-definition is no longer adequate. In particular, section *Note 3-1-3::
-argued that, in the presence of assignment, a variable can no longer be
-considered to be merely a name for a value. Rather, a variable must
-somehow designate a "place" in which values can be stored. In our new
-model of evaluation, these places will be maintained in structures
-called "environments".
-
- An environment is a sequence of "frames". Each frame is a table
-(possibly empty) of "bindings", which associate variable names with
-their corresponding values. (A single frame may contain at most one
-binding for any variable.) Each frame also has a pointer to its environment
-"enclosing environment", unless, for the purposes of discussion, the
-frame is considered to be "global". The "value of a variable" with
-respect to an environment is the value given by the binding of the
-variable in the first frame in the environment that contains a binding
-for that variable. If no frame in the sequence specifies a binding for
-the variable, then the variable is said to be "unbound" in the
-environment.
-
- *Figure 3.1:* A simple environment structure.
-
- +--------+
- | I |
- | x: 3 |
- | y: 5 |
- +--------+
- ^ ^
- | |
- C | | D
- +---------+ | | +----------+
- | II | | | | III |
- | z: 6 +---+ +---+ m: 1 |
- | x: 7 | | y: 2 |
- +---------+ +----------+
-
- *Note Figure 3-1:: shows a simple environment structure consisting
-of three frames, labeled I, II, and III. In the diagram, A, B, C, and
-D are pointers to environments. C and D point to the same environment.
-The variables `z' and `x' are bound in frame II, while `y' and `x' are
-bound in frame I. The value of `x' in environment D is 3. The value
-of `x' with respect to environment B is also 3. This is determined as
-follows: We examine the first frame in the sequence (frame III) and do
-not find a binding for `x', so we proceed to the enclosing environment
-D and find the binding in frame I. On the other hand, the value of `x'
-in environment A is 7, because the first frame in the sequence (frame
-II) contains a binding of `x' to 7. With respect to environment A, the
-binding of `x' to 7 in frame II is said to "shadow" the binding of `x'
-to 3 in frame I.
-
- The environment is crucial to the evaluation process, because it
-determines the context in which an expression should be evaluated.
-Indeed, one could say that expressions in a programming language do
-not, in themselves, have any meaning. Rather, an expression acquires a
-meaning only with respect to some environment in which it is evaluated.
-Even the interpretation of an expression as straightforward as `(+ 1
-1)' depends on an understanding that one is operating in a context in
-which `+' is the symbol for addition. Thus, in our model of evaluation
-we will always speak of evaluating an expression with respect to some
-environment. To describe interactions with the interpreter, we will
-suppose that there is a global environment, consisting of a single frame
-(with no enclosing environment) that includes values for the symbols
-associated with the primitive procedures. For example, the idea that
-`+' is the symbol for addition is captured by saying that the symbol
-`+' is bound in the global environment to the primitive addition
-procedure.
-
-* Menu:
-
-* 3-2-1:: The Rules for Evaluation
-* 3-2-2:: Applying Simple Procedures
-* 3-2-3:: Frames as the Repository of Local State
-* 3-2-4:: Internal Definitions
-
-\1f
-File: sicp.info, Node: 3-2-1, Next: 3-2-2, Prev: 3-2, Up: 3-2
-
-3.2.1 The Rules for Evaluation
-------------------------------
-
-The overall specification of how the interpreter evaluates a combination
-remains the same as when we first introduced it in section *Note
-1-1-3:::
-
- * To evaluate a combination:
-
-
- 1. Evaluate the subexpressions of the combination.(1)
-
- 2. Apply the value of the operator subexpression to the values of the
- operand subexpressions.
-
-
- The environment model of evaluation replaces the substitution model
-in specifying what it means to apply a compound procedure to arguments.
-
- In the environment model of evaluation, a procedure is always a pair
-consisting of some code and a pointer to an environment. Procedures
-are created in one way only: by evaluating a `lambda' expression. This
-produces a procedure whose code is obtained from the text of the
-`lambda' expression and whose environment is the environment in which
-the `lambda' expression was evaluated to produce the procedure. For
-example, consider the procedure definition
-
- (define (square x)
- (* x x))
-
-evaluated in the global environment. The procedure definition syntax
-is just syntactic sugar for an underlying implicit `lambda' expression.
-It would have been equivalent to have used
-
- (define square
- (lambda (x) (* x x)))
-
-which evaluates `(lambda (x) (* x x))' and binds `square' to the
-resulting value, all in the global environment.
-
- *Note Figure 3-2:: shows the result of evaluating this `define'
-expression. The procedure object is a pair whose code specifies that
-the procedure has one formal parameter, namely `x', and a procedure
-body `(* x x)'. The environment part of the procedure is a pointer to
-the global environment, since that is the environment in which the
-`lambda' expression was evaluated to produce the procedure. A new
-binding, which associates the procedure object with the symbol
-`square', has been added to the global frame. In general, `define'
-creates definitions by adding bindings to frames.
-
- *Figure 3.2:* Environment structure produced by evaluating
- `(define (square x) (* x x))' in the global environment.
-
- +----------------------+
- | other variables |
- global --->| |
- env | square: --+ |
- +-----------|----------+
- | ^
- (define (square x) | |
- (* x x)) V |
- .---.---. |
- | O | O-+---+
- `-|-^---'
- |
- V
- parameters: x
- body: (* x x)
-
- Now that we have seen how procedures are created, we can describe how
-procedures are applied. The environment model specifies: To apply a
-procedure to arguments, create a new environment containing a frame
-that binds the parameters to the values of the arguments. The
-enclosing environment of this frame is the environment specified by the
-procedure. Now, within this new environment, evaluate the procedure
-body.
-
- To show how this rule is followed, *Note Figure 3-3:: illustrates
-the environment structure created by evaluating the expression `(square
-5)' in the global environment, where `square' is the procedure
-generated in *Note Figure 3-2::. Applying the procedure results in the
-creation of a new environment, labeled E1 in the figure, that begins
-with a frame in which `x', the formal parameter for the procedure, is
-bound to the argument 5. The pointer leading upward from this frame
-shows that the frame's enclosing environment is the global environment.
-The global environment is chosen here, because this is the environment
-that is indicated as part of the `square' procedure object. Within E1,
-we evaluate the body of the procedure, `(* x x)'. Since the value of
-`x' in E1 is 5, the result is `(* 5 5)', or 25.
-
- *Figure 3.3:* Environment created by evaluating `(square 5)' in
- the global environment.
-
- +------------------------------------+
- | other variables |
- global -->| |
- env | square: --+ |
- +-----------|---------------------+--+
- | ^ ^
- (square 5) | | |
- V | |
- .---.---. | +---+--+
- | O | O-+---+ E1 -->| x: 5 |
- `-|-^---' +------+
- |
- V
- parameters: x
- body: (* x x)
-
- The environment model of procedure application can be summarized by
-two rules:
-
- * A procedure object is applied to a set of arguments by
- constructing a frame, binding the formal parameters of the
- procedure to the arguments of the call, and then evaluating the
- body of the procedure in the context of the new environment
- constructed. The new frame has as its enclosing environment the
- environment part of the procedure object being applied.
-
- * A procedure is created by evaluating a `lambda' expression
- relative to a given environment. The resulting procedure object
- is a pair consisting of the text of the `lambda' expression and a
- pointer to the environment in which the procedure was created.
-
-
- We also specify that defining a symbol using `define' creates a
-binding in the current environment frame and assigns to the symbol the
-indicated value.(2) Finally, we specify the behavior of `set!', the
-operation that forced us to introduce the environment model in the
-first place. Evaluating the expression `(set! <VARIABLE> <VALUE>)' in
-some environment locates the binding of the variable in the environment
-and changes that binding to indicate the new value. That is, one finds
-the first frame in the environment that contains a binding for the
-variable and modifies that frame. If the variable is unbound in the
-environment, then `set!' signals an error.
-
- These evaluation rules, though considerably more complex than the
-substitution model, are still reasonably straightforward. Moreover,
-the evaluation model, though abstract, provides a correct description
-of how the interpreter evaluates expressions. In *Note Chapter 4:: we
-shall see how this model can serve as a blueprint for implementing a
-working interpreter. The following sections elaborate the details of
-the model by analyzing some illustrative programs.
-
- ---------- Footnotes ----------
-
- (1) ssignment introduces a subtlety into step 1 of the evaluation
-rule. As shown in *Note Exercise 3-8::, the presence of assignment
-allows us to write expressions that will produce different values
-depending on the order in which the subexpressions in a combination are
-evaluated. Thus, to be precise, we should specify an evaluation order
-in step 1 (e.g., left to right or right to left). However, this order
-should always be considered to be an implementation detail, and one
-should never write programs that depend on some particular order. For
-instance, a sophisticated compiler might optimize a program by varying
-the order in which subexpressions are evaluated.
-
- (2) If there is already a binding for the variable in the current
-frame, then the binding is changed. This is convenient because it
-allows redefinition of symbols; however, it also means that `define'
-can be used to change values, and this brings up the issues of
-assignment without explicitly using `set!'. Because of this, some
-people prefer redefinitions of existing symbols to signal errors or
-warnings.
-
-\1f
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-
-3.2.2 Applying Simple Procedures
---------------------------------
-
-When we introduced the substitution model in section *Note 1-1-5:: we
-showed how the combination `(f 5)' evaluates to 136, given the
-following procedure definitions:
-
- (define (square x)
- (* x x))
-
- (define (sum-of-squares x y)
- (+ (square x) (square y)))
-
- (define (f a)
- (sum-of-squares (+ a 1) (* a 2)))
-
- We can analyze the same example using the environment model. *Note
-Figure 3-4:: shows the three procedure objects created by evaluating
-the definitions of `f', `square', and `sum-of-squares' in the global
-environment. Each procedure object consists of some code, together
-with a pointer to the global environment.
-
- *Figure 3.4:* Procedure objects in the global frame.
-
- +--------------------------------------------+
- | sum-of-squares: |
- global -->| square: |
- env | f: --+ |
- +------|--------------+--------------+-------+
- | ^ | ^ | ^
- | | | | | |
- V | V | V |
- .---.---. | .---.---. | .---.---. |
- | O | O-+-+ | O | O-+-+ | O | O-+-+
- `-|-^---' `-|-^---' `-|-^---'
- | | |
- V V V
- parameters: a parameters: x parameters: x, y
- body: (sum-of-squares body: (* x x) body: (+ (square x)
- (+ a 1) (square y))
- (* a 2))
-
- In *Note Figure 3-5:: we see the environment structure created by
-evaluating the expression `(f 5)'. The call to `f' creates a new
-environment E1 beginning with a frame in which `a', the formal
-parameter of `f', is bound to the argument 5. In E1, we evaluate the
-body of `f':
-
- (sum-of-squares (+ a 1) (* a 2))
-
- *Figure 3.5:* Environments created by evaluating `(f 5)' using the
- procedures in *Note Figure 3-4::.
-
- +-----------------------------------------------------+
- global -->| |
- env +-----------------------------------------------------+
- ^ ^ ^ ^
- (f 5) | | | |
- +------+ +-------+ +------+ +-------+
- E1 -->| a: 5 | E2 ->| x: 6 | E3 -->| x: 6 | E4 -->| x: 10 |
- | | | y: 10 | | | | |
- +------+ +-------+ +------+ +-------+
- (sum-of-squares (+ (square x) (* x x) (* x x)
- (+ a 1) (square u))
- (+ a 2))
-
- To evaluate this combination, we first evaluate the subexpressions.
-The first subexpression, `sum-of-squares', has a value that is a
-procedure object. (Notice how this value is found: We first look in
-the first frame of E1, which contains no binding for `sum-of-squares'.
-Then we proceed to the enclosing environment, i.e. the global
-environment, and find the binding shown in *Note Figure 3-4::.) The
-other two subexpressions are evaluated by applying the primitive
-operations `+' and `*' to evaluate the two combinations `(+ a 1)' and
-`(* a 2)' to obtain 6 and 10, respectively.
-
- Now we apply the procedure object `sum-of-squares' to the arguments
-6 and 10. This results in a new environment E2 in which the formal
-parameters `x' and `y' are bound to the arguments. Within E2 we
-evaluate the combination `(+ (square x) (square y))'. This leads us to
-evaluate `(square x)', where `square' is found in the global frame and
-`x' is 6. Once again, we set up a new environment, E3, in which `x' is
-bound to 6, and within this we evaluate the body of `square', which is
-`(* x x)'. Also as part of applying `sum-of-squares', we must evaluate
-the subexpression `(square y)', where `y' is 10. This second call to
-`square' creates another environment, E4, in which `x', the formal
-parameter of `square', is bound to 10. And within E4 we must evaluate
-`(* x x)'.
-
- The important point to observe is that each call to `square' creates
-a new environment containing a binding for `x'. We can see here how the
-different frames serve to keep separate the different local variables
-all named `x'. Notice that each frame created by `square' points to
-the global environment, since this is the environment indicated by the
-`square' procedure object.
-
- After the subexpressions are evaluated, the results are returned.
-The values generated by the two calls to `square' are added by
-`sum-of-squares', and this result is returned by `f'. Since our focus
-here is on the environment structures, we will not dwell on how these
-returned values are passed from call to call; however, this is also an
-important aspect of the evaluation process, and we will return to it in
-detail in *Note Chapter 5::.
-
- *Exercise 3.9:* In section *Note 1-2-1:: we used the substitution
- model to analyze two procedures for computing factorials, a
- recursive version
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* n (factorial (- n 1)))))
-
- and an iterative version
-
- (define (factorial n)
- (fact-iter 1 1 n))
-
- (define (fact-iter product counter max-count)
- (if (> counter max-count)
- product
- (fact-iter (* counter product)
- (+ counter 1)
- max-count)))
-
- Show the environment structures created by evaluating `(factorial
- 6)' using each version of the `factorial' procedure.(1)
-
- ---------- Footnotes ----------
-
- (1) The environment model will not clarify our claim in section
-*Note 1-2-1:: that the interpreter can execute a procedure such as
-`fact-iter' in a constant amount of space using tail recursion. We
-will discuss tail recursion when we deal with the control structure of
-the interpreter in section *Note 5-4::.
-
-\1f
-File: sicp.info, Node: 3-2-3, Next: 3-2-4, Prev: 3-2-2, Up: 3-2
-
-3.2.3 Frames as the Repository of Local State
----------------------------------------------
-
-We can turn to the environment model to see how procedures and
-assignment can be used to represent objects with local state. As an
-example, consider the "withdrawal processor" from section *Note 3-1-1::
-created by calling the procedure
-
- (define (make-withdraw balance)
- (lambda (amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds")))
-
- Let us describe the evaluation of
-
- (define W1 (make-withdraw 100))
-
-followed by
-
- (W1 50)
- 50
-
- *Note Figure 3-6:: shows the result of defining the `make-withdraw'
-procedure in the global environment. This produces a procedure object
-that contains a pointer to the global environment. So far, this is no
-different from the examples we have already seen, except that the body
-of the procedure is itself a `lambda' expression.
-
- *Figure 3.6:* Result of defining `make-withdraw' in the global
- environment.
-
- +---------------------------+
- global -->| make-withdraw: --+ |
- env +------------------|--------+
- | ^
- V |
- .---.---. |
- | O | O-+--+
- `-|-^---'
- |
- V
- parameters: balance
- body: (lambda (amount)
- (if (>= balance amount)
- (begin (set! balance
- (- balance amount))
- balance)
- "Insufficient funds"))
-
- The interesting part of the computation happens when we apply the
-procedure `make-withdraw' to an argument:
-
- (define W1 (make-withdraw 100))
-
- We begin, as usual, by setting up an environment E1 in which the
-formal parameter `balance' is bound to the argument 100. Within this
-environment, we evaluate the body of `make-withdraw', namely the
-`lambda' expression. This constructs a new procedure object, whose code
-is as specified by the `lambda' and whose environment is E1, the
-environment in which the `lambda' was evaluated to produce the
-procedure. The resulting procedure object is the value returned by the
-call to `make-withdraw'. This is bound to `W1' in the global
-environment, since the `define' itself is being evaluated in the global
-environment. *Note Figure 3-7:: shows the resulting environment
-structure.
-
- *Figure 3.7:* Result of evaluating `(define W1 (make-withdraw
- 100))'.
-
- +-----------------------------------------------+
- | make-withdraw: -----------------------+ |
- global -->| | |
- | W1: --+ | |
- +-------|-------------------------------|-------+
- | ^ | ^
- | | V |
- | +-------+------+ .---.---. |
- | E1 -->| balance: 100 | | O | O-+-+
- | +--------------+ `-|-^---'
- V ^ |
- .---.---. | V
- +-+-O | O-+------------+ parameters: balance
- | `---^---' body: ...
- V
- parameters: amount
- body: (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds")
-
- Now we can analyze what happens when `W1' is applied to an argument:
-
- (W1 50)
- 50
-
- We begin by constructing a frame in which `amount', the formal
-parameter of `W1', is bound to the argument 50. The crucial point to
-observe is that this frame has as its enclosing environment not the
-global environment, but rather the environment E1, because this is the
-environment that is specified by the `W1' procedure object. Within
-this new environment, we evaluate the body of the procedure:
-
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds")
-
- The resulting environment structure is shown in *Note Figure 3-8::.
-The expression being evaluated references both `amount' and `balance'.
-`Amount' will be found in the first frame in the environment, while
-`balance' will be found by following the enclosing-environment pointer
-to E1.
-
- *Figure 3.8:* Environments created by applying the procedure
- object `W1'.
-
- +---------------------------------------------------+
- | make-withdraw: ... |
- global -->| |
- env | W1: --+ |
- +-------|-------------------------------------------+
- | ^
- | |
- | +-------+------+ Here is the balance
- | E1 -->| balance: 100 | that will be changed
- | +--------------+ by the set!.
- V ^ ^
- .---.---. | +----+
- | O | O-+-----------+ |
- `-|-^---' +------+-----+
- | | amount: 50 |
- V +------------+
- parameters: amount (if (>= balance amount)
- body: ... (begin (set! balance
- (- balance amount))
- balance)
- "Insufficient funds")
-
- When the `set!' is executed, the binding of `balance' in E1 is
-changed. At the completion of the call to `W1', `balance' is 50, and
-the frame that contains `balance' is still pointed to by the procedure
-object `W1'. The frame that binds `amount' (in which we executed the
-code that changed `balance') is no longer relevant, since the procedure
-call that constructed it has terminated, and there are no pointers to
-that frame from other parts of the environment. The next time `W1' is
-called, this will build a new frame that binds `amount' and whose
-enclosing environment is E1. We see that E1 serves as the "place" that
-holds the local state variable for the procedure object `W1'. *Note
-Figure 3-9:: shows the situation after the call to `W1'.
-
- *Figure 3.9:* Environments after the call to `W1'.
-
- +------------------------------------+
- | make-withdraw: ... |
- global --->| |
- env | W1: --+ |
- +-------|----------------------------+
- | ^
- | |
- | +------+------+
- | E1 --->| balance: 50 |
- | +-------------+
- V ^
- .---.---. |
- | O | O-+---------------+
- `-|-^---'
- |
- V
- parameters: amount
- body: ...
-
- Observe what happens when we create a second "withdraw" object by
-making another call to `make-withdraw':
-
- (define W2 (make-withdraw 100))
-
- This produces the environment structure of *Note Figure 3-10::,
-which shows that `W2' is a procedure object, that is, a pair with some
-code and an environment. The environment E2 for `W2' was created by
-the call to `make-withdraw'. It contains a frame with its own local
-binding for `balance'. On the other hand, `W1' and `W2' have the same
-code: the code specified by the `lambda' expression in the body of
-`make-withdraw'.(1) We see here why `W1' and `W2' behave as independent
-objects. Calls to `W1' reference the state variable `balance' stored
-in E1, whereas calls to `W2' reference the `balance' stored in E2.
-Thus, changes to the local state of one object do not affect the other
-object.
-
- *Figure 3.10:* Using `(define W2 (make-withdraw 100))' to create a
- second object.
-
- +-------------------------------------------------+
- | make-withdraw: ... |
- global ->| W2: ---------------------------+ |
- env | W1: --+ | |
- +-------|------------------------|----------------+
- | ^ | ^
- | | | |
- | +------+------+ | +------+-------+
- | E1 ->| balance: 50 | | E2 ->| balance: 100 |
- | +-------------+ | +--------------+
- V ^ V ^
- .---.---. | .---.---. |
- | O | O-+----------+ | O | O-+----------+
- `-|-^---' `-|-^---'
- | +----------------------+
- V V
- parameters: amount
- body: ...
-
- *Exercise 3.10:* In the `make-withdraw' procedure, the local
- variable `balance' is created as a parameter of `make-withdraw'.
- We could also create the local state variable explicitly, using
- `let', as follows:
-
- (define (make-withdraw initial-amount)
- (let ((balance initial-amount))
- (lambda (amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))))
-
- Recall from section *Note 1-3-2:: that `let' is simply syntactic
- sugar for a procedure call:
-
- (let ((<VAR> <EXP>)) <BODY>)
-
- is interpreted as an alternate syntax for
-
- ((lambda (<VAR>) <BODY>) <EXP>)
-
- Use the environment model to analyze this alternate version of
- `make-withdraw', drawing figures like the ones above to illustrate
- the interactions
-
- (define W1 (make-withdraw 100))
-
- (W1 50)
-
- (define W2 (make-withdraw 100))
-
- Show that the two versions of `make-withdraw' create objects with
- the same behavior. How do the environment structures differ for
- the two versions?
-
- ---------- Footnotes ----------
-
- (1) Whether `W1' and `W2' share the same physical code stored in the
-computer, or whether they each keep a copy of the code, is a detail of
-the implementation. For the interpreter we implement in *Note Chapter
-4::, the code is in fact shared.
-
-\1f
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-
-3.2.4 Internal Definitions
---------------------------
-
-Section *Note 1-1-8:: introduced the idea that procedures can have
-internal definitions, thus leading to a block structure as in the
-following procedure to compute square roots:
-
- (define (sqrt x)
- (define (good-enough? guess)
- (< (abs (- (square guess) x)) 0.001))
- (define (improve guess)
- (average guess (/ x guess)))
- (define (sqrt-iter guess)
- (if (good-enough? guess)
- guess
- (sqrt-iter (improve guess))))
- (sqrt-iter 1.0))
-
- Now we can use the environment model to see why these internal
-definitions behave as desired. *Note Figure 3-11:: shows the point in
-the evaluation of the expression `(sqrt 2)' where the internal
-procedure `good-enough?' has been called for the first time with
-`guess' equal to 1.
-
- *Figure 3.11:* `Sqrt' procedure with internal definitions.
-
- +--------------------------------------------------+
- global -->| sqrt: --+ |
- env | | |
- +---------|----------------------------------------+
- V ^ ^
- .---.---. | |
- +----------+-O | O-+---+ +----------+------------+
- | `---^---' | x: 2 |
- V E1 -->| good-enough?: -+ |
- parameters: x | improve: ... | |
- body: (define good-enough? ...) | sqrt-iter: ... | |
- (define improve ...) +----------------|------+
- (define sqrt-iter ...) ^ ^ | ^
- (sqrt-iter 1.0) | | V |
- +---------++ | .---.---. |
- E2 -->| guess: 1 | | | O | O-+-+
- +----------+ | `-|-^---'
- call to sqrt-iter | |
- | V
- +---------++ parameters: guess
- E3 -->| guess: 1 | body: (< (abs ...)
- +----------+ ...)
- call to good-enough?
-
- Observe the structure of the environment. `Sqrt' is a symbol in the
-global environment that is bound to a procedure object whose associated
-environment is the global environment. When `sqrt' was called, a new
-environment E1 was formed, subordinate to the global environment, in
-which the parameter `x' is bound to 2. The body of `sqrt' was then
-evaluated in E1. Since the first expression in the body of `sqrt' is
-
- (define (good-enough? guess)
- (< (abs (- (square guess) x)) 0.001))
-
-evaluating this expression defined the procedure `good-enough?' in the
-environment E1. To be more precise, the symbol `good-enough?' was added
-to the first frame of E1, bound to a procedure object whose associated
-environment is E1. Similarly, `improve' and `sqrt-iter' were defined
-as procedures in E1. For conciseness, *Note Figure 3-11:: shows only
-the procedure object for `good-enough?'.
-
- After the local procedures were defined, the expression `(sqrt-iter
-1.0)' was evaluated, still in environment E1. So the procedure object
-bound to `sqrt-iter' in E1 was called with 1 as an argument. This
-created an environment E2 in which `guess', the parameter of
-`sqrt-iter', is bound to 1. `Sqrt-iter' in turn called `good-enough?'
-with the value of `guess' (from E2) as the argument for `good-enough?'.
-This set up another environment, E3, in which `guess' (the parameter of
-`good-enough?') is bound to 1. Although `sqrt-iter' and `good-enough?'
-both have a parameter named `guess', these are two distinct local
-variables located in different frames. Also, E2 and E3 both have E1 as
-their enclosing environment, because the `sqrt-iter' and `good-enough?'
-procedures both have E1 as their environment part. One consequence of
-this is that the symbol `x' that appears in the body of `good-enough?'
-will reference the binding of `x' that appears in E1, namely the value
-of `x' with which the original `sqrt' procedure was called.
-
- The environment model thus explains the two key properties that make
-local procedure definitions a useful technique for modularizing
-programs:
-
- * The names of the local procedures do not interfere with names
- external to the enclosing procedure, because the local procedure
- names will be bound in the frame that the procedure creates when
- it is run, rather than being bound in the global environment.
-
- * The local procedures can access the arguments of the enclosing
- procedure, simply by using parameter names as free variables.
- This is because the body of the local procedure is evaluated in an
- environment that is subordinate to the evaluation environment for
- the enclosing procedure.
-
-
- *Exercise 3.11:* In section *Note 3-2-3:: we saw how the
- environment model described the behavior of procedures with local
- state. Now we have seen how internal definitions work. A typical
- message-passing procedure contains both of these aspects.
- Consider the bank account procedure of section *Note 3-1-1:::
-
- (define (make-account balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (define (dispatch m)
- (cond ((eq? m 'withdraw) withdraw)
- ((eq? m 'deposit) deposit)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch)
-
- Show the environment structure generated by the sequence of
- interactions
-
- (define acc (make-account 50))
-
- ((acc 'deposit) 40)
- 90
-
- ((acc 'withdraw) 60)
- 30
-
- Where is the local state for `acc' kept? Suppose we define another
- account
-
- (define acc2 (make-account 100))
-
- How are the local states for the two accounts kept distinct?
- Which parts of the environment structure are shared between `acc'
- and `acc2'?
-
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-File: sicp.info, Node: 3-3, Next: 3-4, Prev: 3-2, Up: Chapter 3
-
-3.3 Modeling with Mutable Data
-==============================
-
-Chapter 2 dealt with compound data as a means for constructing
-computational objects that have several parts, in order to model
-real-world objects that have several aspects. In that chapter we
-introduced the discipline of data abstraction, according to which data
-structures are specified in terms of constructors, which create data
-objects, and selectors, which access the parts of compound data
-objects. But we now know that there is another aspect of data that
-*Note Chapter 2:: did not address. The desire to model systems
-composed of objects that have changing state leads us to the need to
-modify compound data objects, as well as to construct and select from
-them. In order to model compound objects with changing state, we will
-design data abstractions to include, in addition to selectors and
-constructors, operations called "mutators", which modify data objects.
-For instance, modeling a banking system requires us to change account
-balances. Thus, a data structure for representing bank accounts might
-admit an operation
-
- (set-balance! <ACCOUNT> <NEW-VALUE>)
-
-that changes the balance of the designated account to the designated
-new value. Data objects for which mutators are defined are known as objects
-"mutable data objects".
-
- *Note Chapter 2:: introduced pairs as a general-purpose "glue" for
-synthesizing compound data. We begin this section by defining basic
-mutators for pairs, so that pairs can serve as building blocks for
-constructing mutable data objects. These mutators greatly enhance the
-representational power of pairs, enabling us to build data structures
-other than the sequences and trees that we worked with in section *Note
-2-2::. We also present some examples of simulations in which complex
-systems are modeled as collections of objects with local state.
-
-* Menu:
-
-* 3-3-1:: Mutable List Structure
-* 3-3-2:: Representing Queues
-* 3-3-3:: Representing Tables
-* 3-3-4:: A Simulator for Digital Circuits
-* 3-3-5:: Propagation of Constraints
-
-\1f
-File: sicp.info, Node: 3-3-1, Next: 3-3-2, Prev: 3-3, Up: 3-3
-
-3.3.1 Mutable List Structure
-----------------------------
-
-The basic operations on pairs--`cons', `car', and `cdr'--can be used to
-construct list structure and to select parts from list structure, but
-they are incapable of modifying list structure. The same is true of the
-list operations we have used so far, such as `append' and `list', since
-these can be defined in terms of `cons', `car', and `cdr'. To modify
-list structures we need new operations.
-
- *Figure 3.12:* Lists `x': `((a b) c d)' and `y': `(e f)'.
-
- +---+---+ +---+---+ +---+---+
- x -->| * | *-+---->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- | V V
- | +---+ +---+
- | | c | | d |
- | +---+ +---+
- | +---+---+ +---+---+
- +---------->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | a | | b |
- +---+ +---+
- +---+---+ +---+---+
- y -->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | e | | f |
- +---+ +---+
-
- *Figure 3.13:* Effect of `(set-car! x y)' on the lists in *Note
- Figure 3-12::.
-
- +---+---+ +---+---+ +---+---+
- x -->| * | *-+---->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- | V V
- | +---+ +---+
- | | c | | d |
- | +---+ +---+
- | +---+---+ +---+---+
- | | * | *-+---->| * | / |
- | +-|-+---+ +-|-+---+
- | V V
- | +---+ +---+
- | | a | | b |
- | +---+ +---+
- +---------->+---+---+ +---+---+
- | * | *-+---->| * | / |
- y -->+-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | e | | f |
- +---+ +---+
-
- *Figure 3.14:* Effect of `(define z (cons y (cdr x)))' on the
- lists in *Note Figure 3-12::.
-
- +---+---+ +---+---+ +---+---+
- x -->| * | *-+---->| * | *-+---->| * | / |
- +-|-+---+ +-->+-|-+---+ +-|-+---+
- | | V V
- | | +---+ +---+
- | | | c | | d |
- | | +---+ +---+
- | | +---+---+ +---+---+
- +-------+-->| * | *-+---->| * | / |
- | +-|-+---+ +-|-+---+
- +---+---+ | V V
- z -->| * | *-+-+ +---+ +---+
- +-|-+---+ | a | | b |
- | +---+ +---+
- +---------->+---+---+ +---+---+
- | * | *-+---->| * | / |
- y -->+-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | e | | f |
- +---+ +---+
-
- *Figure 3.15:* Effect of `(set-cdr! x y)' on the lists in *Note
- Figure 3-12::.
-
- +---+---+ +---+---+ +---+---+
- x -->| * | * | | * | *-+---->| * | / |
- +-|-+-|-+ +-|-+---+ +-|-+---+
- | | V V
- | | +---+ +---+
- | | | c | | d |
- | | +---+ +---+
- | | +---+---+ +---+---+
- +---+------>| * | *-+---->| * | / |
- | +-|-+---+ +-|-+---+
- | V V
- | +---+ +---+
- | | a | | b |
- | +---+ +---+
- +------>+---+---+ +---+---+
- | * | *-+---->| * | / |
- y -->+-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | e | | f |
- +---+ +---+
-
- The primitive mutators for pairs are `set-car!' and `set-cdr!'.
-`Set-car!' takes two arguments, the first of which must be a pair. It
-modifies this pair, replacing the `car' pointer by a pointer to the
-second argument of `set-car!'.(1)
-
- As an example, suppose that `x' is bound to the list `((a b) c d)'
-and `y' to the list `(e f)' as illustrated in *Note Figure 3-12::.
-Evaluating the expression ` (set-car! x y)' modifies the pair to which
-`x' is bound, replacing its `car' by the value of `y'. The result of
-the operation is shown in *Note Figure 3-13::. The structure `x' has
-been modified and would now be printed as `((e f) c d)'. The pairs
-representing the list `(a b)', identified by the pointer that was
-replaced, are now detached from the original structure.(2)
-
- Compare *Note Figure 3-13:: with *Note Figure 3-14::, which
-illustrates the result of executing `(define z (cons y (cdr x)))' with
-`x' and `y' bound to the original lists of *Note Figure 3-12::. The
-variable `z' is now bound to a new pair created by the `cons'
-operation; the list to which `x' is bound is unchanged.
-
- The `set-cdr!' operation is similar to `set-car!'. The only
-difference is that the `cdr' pointer of the pair, rather than the `car'
-pointer, is replaced. The effect of executing `(set-cdr! x y)' on the
-lists of *Note Figure 3-12:: is shown in *Note Figure 3-15::. Here the
-`cdr' pointer of `x' has been replaced by the pointer to `(e f)'.
-Also, the list `(c d)', which used to be the `cdr' of `x', is now
-detached from the structure.
-
- `Cons' builds new list structure by creating new pairs, while
-`set-car!' and `set-cdr!' modify existing pairs. Indeed, we could
-implement `cons' in terms of the two mutators, together with a procedure
-`get-new-pair', which returns a new pair that is not part of any
-existing list structure. We obtain the new pair, set its `car' and
-`cdr' pointers to the designated objects, and return the new pair as
-the result of the `cons'.(3)
-
- (define (cons x y)
- (let ((new (get-new-pair)))
- (set-car! new x)
- (set-cdr! new y)
- new))
-
- *Exercise 3.12:* The following procedure for appending lists was
- introduced in section *Note 2-2-1:::
-
- (define (append x y)
- (if (null? x)
- y
- (cons (car x) (append (cdr x) y))))
-
- `Append' forms a new list by successively `cons'ing the elements of
- `x' onto `y'. The procedure `append!' is similar to `append', but
- it is a mutator rather than a constructor. It appends the lists
- by splicing them together, modifying the final pair of `x' so that
- its `cdr' is now `y'. (It is an error to call `append!' with an
- empty `x'.)
-
- (define (append! x y)
- (set-cdr! (last-pair x) y)
- x)
-
- Here `last-pair' is a procedure that returns the last pair in its
- argument:
-
- (define (last-pair x)
- (if (null? (cdr x))
- x
- (last-pair (cdr x))))
-
- Consider the interaction
-
- (define x (list 'a 'b))
-
- (define y (list 'c 'd))
-
- (define z (append x y))
-
- z
- (a b c d)
-
- (cdr x)
- <RESPONSE>
-
- (define w (append! x y))
-
- w
- (a b c d)
-
- (cdr x)
- <RESPONSE>
-
- What are the missing <RESPONSE>s? Draw box-and-pointer diagrams to
- explain your answer.
-
- *Exercise 3.13:* Consider the following `make-cycle' procedure,
- which uses the `last-pair' procedure defined in *Note Exercise
- 3-12:::
-
- (define (make-cycle x)
- (set-cdr! (last-pair x) x)
- x)
-
- Draw a box-and-pointer diagram that shows the structure `z'
- created by
-
- (define z (make-cycle (list 'a 'b 'c)))
-
- What happens if we try to compute `(last-pair z)'?
-
- *Exercise 3.14:* The following procedure is quite useful, although
- obscure:
-
- (define (mystery x)
- (define (loop x y)
- (if (null? x)
- y
- (let ((temp (cdr x)))
- (set-cdr! x y)
- (loop temp x))))
- (loop x '()))
-
- `Loop' uses the "temporary" variable `temp' to hold the old value
- of the `cdr' of `x', since the `set-cdr!' on the next line
- destroys the `cdr'. Explain what `mystery' does in general.
- Suppose `v' is defined by `(define v (list 'a 'b 'c 'd))'. Draw the
- box-and-pointer diagram that represents the list to which `v' is
- bound. Suppose that we now evaluate `(define w (mystery v))'. Draw
- box-and-pointer diagrams that show the structures `v' and `w' after
- evaluating this expression. What would be printed as the values
- of `v' and `w'?
-
-Sharing and identity
-....................
-
-We mentioned in section *Note 3-1-3:: the theoretical issues of
-"sameness" and "change" raised by the introduction of assignment.
-These issues arise in practice when individual pairs are "shared" among
-different data objects. For example, consider the structure formed by
-
- (define x (list 'a 'b))
- (define z1 (cons x x))
-
- As shown in *Note Figure 3-16::, `z1' is a pair whose `car' and
-`cdr' both point to the same pair `x'. This sharing of `x' by the
-`car' and `cdr' of `z1' is a consequence of the straightforward way in
-which `cons' is implemented. In general, using `cons' to construct
-lists will result in an interlinked structure of pairs in which many
-individual pairs are shared by many different structures.
-
- *Figure 3.16:* The list `z1' formed by `(cons x x)'.
-
- +---+---+
- z1 -->| * | * |
- +-|-+-|-+
- V V
- +---+---+ +---+---+
- x -->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+
- V V
- +---+ +---+
- | a | | b |
- +---+ +---+
-
- *Figure 3.17:* The list `z2' formed by `(cons (list 'a 'b) (list
- 'a 'b))'.
-
- +---+---+ +---+---+ +---+---+
- z2 -->| * | *-+---->| * | *-+---->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- | V V
- | +---+ +---+
- | | a | | b |
- | +---+ +---+
- | ^ ^
- | | |
- | +-|-+---+ +-|-+---+
- +---------->| * | *-+---->| * | / |
- +---+---+ +---+---+
-
- In contrast to *Note Figure 3-16::, *Note Figure 3-17:: shows the
-structure created by
-
- (define z2 (cons (list 'a 'b) (list 'a 'b)))
-
- In this structure, the pairs in the two `(a b)' lists are distinct,
-although the actual symbols are shared.(4)
-
- When thought of as a list, `z1' and `z2' both represent "the same"
-list, `((a b) a b)'. In general, sharing is completely undetectable if
-we operate on lists using only `cons', `car', and `cdr'. However, if
-we allow mutators on list structure, sharing becomes significant. As an
-example of the difference that sharing can make, consider the following
-procedure, which modifies the `car' of the structure to which it is
-applied:
-
- (define (set-to-wow! x)
- (set-car! (car x) 'wow)
- x)
-
- Even though `z1' and `z2' are "the same" structure, applying
-`set-to-wow!' to them yields different results. With `z1', altering
-the `car' also changes the `cdr', because in `z1' the `car' and the
-`cdr' are the same pair. With `z2', the `car' and `cdr' are distinct,
-so `set-to-wow!' modifies only the `car':
-
- z1
- ((a b) a b)
-
- (set-to-wow! z1)
- ((wow b) wow b)
-
- z2
- ((a b) a b)
-
- (set-to-wow! z2)
- ((wow b) a b)
-
- One way to detect sharing in list structures is to use the predicate
-`eq?', which we introduced in section *Note 2-3-1:: as a way to test
-whether two symbols are equal. More generally, `(eq? x y)' tests
-whether `x' and `y' are the same object (that is, whether `x' and `y'
-are equal as pointers). Thus, with `z1' and `z2' as defined in figures
-*Note Figure 3-16:: and *Note Figure 3-17::, `(eq? (car z1) (cdr z1))'
-is true and `(eq? (car z2) (cdr z2))' is false.
-
- As will be seen in the following sections, we can exploit sharing to
-greatly extend the repertoire of data structures that can be
-represented by pairs. On the other hand, sharing can also be
-dangerous, since modifications made to structures will also affect
-other structures that happen to share the modified parts. The mutation
-operations `set-car!' and `set-cdr!' should be used with care; unless
-we have a good understanding of how our data objects are shared,
-mutation can have unanticipated results.(5)
-
- *Exercise 3.15:* Draw box-and-pointer diagrams to explain the
- effect of `set-to-wow!' on the structures `z1' and `z2' above.
-
- *Exercise 3.16:* Ben Bitdiddle decides to write a procedure to
- count the number of pairs in any list structure. "It's easy," he
- reasons. "The number of pairs in any structure is the number in
- the `car' plus the number in the `cdr' plus one more to count the
- current pair." So Ben writes the following procedure:
-
- (define (count-pairs x)
- (if (not (pair? x))
- 0
- (+ (count-pairs (car x))
- (count-pairs (cdr x))
- 1)))
-
- Show that this procedure is not correct. In particular, draw
- box-and-pointer diagrams representing list structures made up of
- exactly three pairs for which Ben's procedure would return 3;
- return 4; return 7; never return at all.
-
- *Exercise 3.17:* Devise a correct version of the `count-pairs'
- procedure of *Note Exercise 3-16:: that returns the number of
- distinct pairs in any structure. (Hint: Traverse the structure,
- maintaining an auxiliary data structure that is used to keep track
- of which pairs have already been counted.)
-
- *Exercise 3.18:* Write a procedure that examines a list and
- determines whether it contains a cycle, that is, whether a program
- that tried to find the end of the list by taking successive `cdr's
- would go into an infinite loop. *Note Exercise 3-13:: constructed
- such lists.
-
- *Exercise 3.19:* Redo *Note Exercise 3-18:: using an algorithm
- that takes only a constant amount of space. (This requires a very
- clever idea.)
-
-Mutation is just assignment
-...........................
-
-When we introduced compound data, we observed in section *Note 2-1-3::
-that pairs can be represented purely in terms of procedures:
-
- (define (cons x y)
- (define (dispatch m)
- (cond ((eq? m 'car) x)
- ((eq? m 'cdr) y)
- (else (error "Undefined operation -- CONS" m))))
- dispatch)
-
- (define (car z) (z 'car))
-
- (define (cdr z) (z 'cdr))
-
- The same observation is true for mutable data. We can implement
-mutable data objects as procedures using assignment and local state.
-For instance, we can extend the above pair implementation to handle
-`set-car!' and `set-cdr!' in a manner analogous to the way we
-implemented bank accounts using `make-account' in section *Note 3-1-1:::
-
- (define (cons x y)
- (define (set-x! v) (set! x v))
- (define (set-y! v) (set! y v))
- (define (dispatch m)
- (cond ((eq? m 'car) x)
- ((eq? m 'cdr) y)
- ((eq? m 'set-car!) set-x!)
- ((eq? m 'set-cdr!) set-y!)
- (else (error "Undefined operation -- CONS" m))))
- dispatch)
-
- (define (car z) (z 'car))
-
- (define (cdr z) (z 'cdr))
-
- (define (set-car! z new-value)
- ((z 'set-car!) new-value)
- z)
-
- (define (set-cdr! z new-value)
- ((z 'set-cdr!) new-value)
- z)
-
- Assignment is all that is needed, theoretically, to account for the
-behavior of mutable data. As soon as we admit `set!' to our language,
-we raise all the issues, not only of assignment, but of mutable data in
-general.(6)
-
- *Exercise 3.20:* Draw environment diagrams to illustrate the
- evaluation of the sequence of expressions
-
- (define x (cons 1 2))
- (define z (cons x x))
- (set-car! (cdr z) 17)
-
- (car x)
- 17
-
- using the procedural implementation of pairs given above. (Compare
- *Note Exercise 3-11::.)
-
- ---------- Footnotes ----------
-
- (1) `Set-car!' and `set-cdr!' return implementation-dependent
-values. Like `set!', they should be used only for their effect.
-
- (2) We see from this that mutation operations on lists can create
-"garbage" that is not part of any accessible structure. We will see in
-section *Note 5-3-2:: that Lisp memory-management systems include a "garbage
-collector", which identifies and recycles the memory space used by
-unneeded pairs.
-
- (3) `Get-new-pair' is one of the operations that must be implemented
-as part of the memory management required by a Lisp implementation. We
-will discuss this in section *Note 5-3-1::.
-
- (4) The two pairs are distinct because each call to `cons' returns a
-new pair. The symbols are shared; in Scheme there is a unique symbol
-with any given name. Since Scheme provides no way to mutate a symbol,
-this sharing is undetectable. Note also that the sharing is what
-enables us to compare symbols using `eq?', which simply checks equality
-of pointers.
-
- (5) The subtleties of dealing with sharing of mutable data objects
-reflect the underlying issues of "sameness" and "change" that were
-raised in section *Note 3-1-3::. We mentioned there that admitting
-change to our language requires that a compound object must have an
-"identity" that is something different from the pieces from which it is
-composed. In Lisp, we consider this "identity" to be the quality that
-is tested by `eq?', i.e., by equality of pointers. Since in most Lisp
-implementations a pointer is essentially a memory address, we are
-"solving the problem" of defining the identity of objects by
-stipulating that a data object "itself" is the information stored in
-some particular set of memory locations in the computer. This suffices
-for simple Lisp programs, but is hardly a general way to resolve the
-issue of "sameness" in computational models.
-
- (6) On the other hand, from the viewpoint of implementation,
-assignment requires us to modify the environment, which is itself a
-mutable data structure. Thus, assignment and mutation are equipotent:
-Each can be implemented in terms of the other.
-
-\1f
-File: sicp.info, Node: 3-3-2, Next: 3-3-3, Prev: 3-3-1, Up: 3-3
-
-3.3.2 Representing Queues
--------------------------
-
-The mutators `set-car!' and `set-cdr!' enable us to use pairs to
-construct data structures that cannot be built with `cons', `car', and
-`cdr' alone. This section shows how to use pairs to represent a data
-structure called a queue. Section *Note 3-3-3:: will show how to
-represent data structures called tables.
-
- A "queue" is a sequence in which items are inserted at one end
-(called the "rear" of the queue) and deleted from the other end (the "front").
-*Note Figure 3-18:: shows an initially empty queue in which the items
-`a' and `b' are inserted. Then `a' is removed, `c' and `d' are
-inserted, and `b' is removed. Because items are always removed in the
-order in which they are inserted, a queue is sometimes called a "FIFO"
-(first in, first out) buffer.
-
- *Figure 3.18:* Queue operations.
-
- Operation Resulting Queue
- (define q (make-queue))
- (insert-queue! q 'a) a
- (insert-queue! q 'b) a b
- (delete-queue! q) b
- (insert-queue! q 'c) b c
- (insert-queue! q 'd) b c d
- (delete-queue! q) c d
-
- In terms of data abstraction, we can regard a queue as defined by
-the following set of operations:
-
- * a constructor: `(make-queue)' returns an empty queue (a queue
- containing no items).
-
- * two selectors:
-
- (empty-queue? <QUEUE>)
-
- tests if the queue is empty.
-
- (front-queue <QUEUE>)
-
- returns the object at the front of the queue, signaling an error
- if the queue is empty; it does not modify the queue.
-
- * two mutators:
-
- (insert-queue! <QUEUE> <ITEM>)
-
- inserts the item at the rear of the queue and returns the modified
- queue as its value.
-
- (delete-queue! <QUEUE>)
-
- removes the item at the front of the queue and returns the
- modified queue as its value, signaling an error if the queue is
- empty before the deletion.
-
-
- Because a queue is a sequence of items, we could certainly represent
-it as an ordinary list; the front of the queue would be the `car' of
-the list, inserting an item in the queue would amount to appending a
-new element at the end of the list, and deleting an item from the queue
-would just be taking the `cdr' of the list. However, this
-representation is inefficient, because in order to insert an item we
-must scan the list until we reach the end. Since the only method we
-have for scanning a list is by successive `cdr' operations, this
-scanning requires [theta](n) steps for a list of n items. A simple
-modification to the list representation overcomes this disadvantage by
-allowing the queue operations to be implemented so that they require
-[theta](1) steps; that is, so that the number of steps needed is
-independent of the length of the queue.
-
- The difficulty with the list representation arises from the need to
-scan to find the end of the list. The reason we need to scan is that,
-although the standard way of representing a list as a chain of pairs
-readily provides us with a pointer to the beginning of the list, it
-gives us no easily accessible pointer to the end. The modification
-that avoids the drawback is to represent the queue as a list, together
-with an additional pointer that indicates the final pair in the list.
-That way, when we go to insert an item, we can consult the rear pointer
-and so avoid scanning the list.
-
- A queue is represented, then, as a pair of pointers, `front-ptr' and
-`rear-ptr', which indicate, respectively, the first and last pairs in an
-ordinary list. Since we would like the queue to be an identifiable
-object, we can use `cons' to combine the two pointers. Thus, the queue
-itself will be the `cons' of the two pointers. *Note Figure 3-19::
-illustrates this representation.
-
- *Figure 3.19:* Implementation of a queue as a list with front and
- rear pointers.
-
- +---+---+
- q -->| * | *-+-------------------+
- +-|-+---+ |
- | |
- | front-ptr | rear-ptr
- V V
- +---+---+ +---+---+ +---+---+
- | * | *-+--->| * | *-+--->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- V V V
- +---+ +---+ +---+
- | a | | b | | c |
- +---+ +---+ +---+
-
- To define the queue operations we use the following procedures,
-which enable us to select and to modify the front and rear pointers of
-a queue:
-
- (define (front-ptr queue) (car queue))
-
- (define (rear-ptr queue) (cdr queue))
-
- (define (set-front-ptr! queue item) (set-car! queue item))
-
- (define (set-rear-ptr! queue item) (set-cdr! queue item))
-
- Now we can implement the actual queue operations. We will consider
-a queue to be empty if its front pointer is the empty list:
-
- (define (empty-queue? queue) (null? (front-ptr queue)))
-
- The `make-queue' constructor returns, as an initially empty queue, a
-pair whose `car' and `cdr' are both the empty list:
-
- (define (make-queue) (cons '() '()))
-
- To select the item at the front of the queue, we return the `car' of
-the pair indicated by the front pointer:
-
- (define (front-queue queue)
- (if (empty-queue? queue)
- (error "FRONT called with an empty queue" queue)
- (car (front-ptr queue))))
-
- To insert an item in a queue, we follow the method whose result is
-indicated in *Note Figure 3-20::. We first create a new pair whose
-`car' is the item to be inserted and whose `cdr' is the empty list. If
-the queue was initially empty, we set the front and rear pointers of
-the queue to this new pair. Otherwise, we modify the final pair in the
-queue to point to the new pair, and also set the rear pointer to the
-new pair.
-
- *Figure 3.20:* Result of using `(insert-queue! q 'd)' on the
- queue of *Note Figure 3-19::.
-
- +---+---+
- q -->| * | *-+--------------------------------+
- +-|-+---+ |
- | |
- | front-ptr | rear-ptr
- V V
- +---+---+ +---+---+ +---+---+ +---+---+
- | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
- V V V V
- +---+ +---+ +---+ +---+
- | a | | b | | c | | d |
- +---+ +---+ +---+ +---+
-
- (define (insert-queue! queue item)
- (let ((new-pair (cons item '())))
- (cond ((empty-queue? queue)
- (set-front-ptr! queue new-pair)
- (set-rear-ptr! queue new-pair)
- queue)
- (else
- (set-cdr! (rear-ptr queue) new-pair)
- (set-rear-ptr! queue new-pair)
- queue))))
-
- To delete the item at the front of the queue, we merely modify the
-front pointer so that it now points at the second item in the queue,
-which can be found by following the `cdr' pointer of the first item
-(see *Note Figure 3-21::):(1)
-
- *Figure 3.21:* Result of using `(delete-queue! q)' on the queue
- of *Note Figure 3-20::.
-
- +---+---+
- q -->| * | *-+--------------------------------+
- +-|-+---+ |
- +------------+ |
- front-ptr | | rear-ptr
- V V
- +---+---+ +---+---+ +---+---+ +---+---+
- | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
- V V V V
- +---+ +---+ +---+ +---+
- | a | | b | | c | | d |
- +---+ +---+ +---+ +---+
-
- (define (delete-queue! queue)
- (cond ((empty-queue? queue)
- (error "DELETE! called with an empty queue" queue))
- (else
- (set-front-ptr! queue (cdr (front-ptr queue)))
- queue)))
-
- *Exercise 3.21:* Ben Bitdiddle decides to test the queue
- implementation described above. He types in the procedures to the
- Lisp interpreter and proceeds to try them out:
-
- (define q1 (make-queue))
-
- (insert-queue! q1 'a)
- ((a) a)
-
- (insert-queue! q1 'b)
- ((a b) b)
-
- (delete-queue! q1)
- ((b) b)
-
- (delete-queue! q1)
- (() b)
-
- "It's all wrong!" he complains. "The interpreter's response shows
- that the last item is inserted into the queue twice. And when I
- delete both items, the second `b' is still there, so the queue
- isn't empty, even though it's supposed to be." Eva Lu Ator
- suggests that Ben has misunderstood what is happening. "It's not
- that the items are going into the queue twice," she explains.
- "It's just that the standard Lisp printer doesn't know how to make
- sense of the queue representation. If you want to see the queue
- printed correctly, you'll have to define your own print procedure
- for queues." Explain what Eva Lu is talking about. In particular,
- show why Ben's examples produce the printed results that they do.
- Define a procedure `print-queue' that takes a queue as input and
- prints the sequence of items in the queue.
-
- *Exercise 3.22:* Instead of representing a queue as a pair of
- pointers, we can build a queue as a procedure with local state.
- The local state will consist of pointers to the beginning and the
- end of an ordinary list. Thus, the `make-queue' procedure will
- have the form
-
- (define (make-queue)
- (let ((front-ptr ... )
- (rear-ptr ... ))
- <DEFINITIONS OF INTERNAL PROCEDURES>
- (define (dispatch m) ...)
- dispatch))
-
- Complete the definition of `make-queue' and provide
- implementations of the queue operations using this representation.
-
- *Exercise 3.23:* A "deque" ("double-ended queue") is a sequence in
- which items can be inserted and deleted at either the front or the
- rear. Operations on deques are the constructor `make-deque', the
- predicate `empty-deque?', selectors `front-deque' and
- `rear-deque', and mutators `front-insert-deque!',
- `rear-insert-deque!', `front-delete-deque!', and
- `rear-delete-deque!'. Show how to represent deques using pairs,
- and give implementations of the operations.(2) All operations
- should be accomplished in [theta](1) steps.
-
- ---------- Footnotes ----------
-
- (1) If the first item is the final item in the queue, the front
-pointer will be the empty list after the deletion, which will mark the
-queue as empty; we needn't worry about updating the rear pointer, which
-will still point to the deleted item, because `empty-queue?' looks only
-at the front pointer.
-
- (2) Be careful not to make the interpreter try to print a structure
-that contains cycles. (See *Note Exercise 3-13::.)
-
-\1f
-File: sicp.info, Node: 3-3-3, Next: 3-3-4, Prev: 3-3-2, Up: 3-3
-
-3.3.3 Representing Tables
--------------------------
-
-When we studied various ways of representing sets in *Note Chapter 2::,
-we mentioned in section *Note 2-3-3:: the task of maintaining a table
-of records indexed by identifying keys. In the implementation of
-data-directed programming in section *Note 2-4-3::, we made extensive
-use of two-dimensional tables, in which information is stored and
-retrieved using two keys. Here we see how to build tables as mutable
-list structures.
-
- We first consider a one-dimensional table, in which each value is
-stored under a single key. We implement the table as a list of
-records, each of which is implemented as a pair consisting of a key and
-the associated value. The records are glued together to form a list by
-pairs whose `car's point to successive records. These gluing pairs are
-called the "backbone" of the table. In order to have a place that we
-can change when we add a new record to the table, we build the table as
-a "headed list". A headed list has a special backbone pair at the
-beginning, which holds a dummy "record"--in this case the arbitrarily
-chosen symbol `*table*'. *Note Figure 3-22:: shows the box-and-pointer
-diagram for the table
-
- a: 1
- b: 2
- c: 3
-
- *Figure 3.22:* A table represented as a headed list.
-
- +---+---+ +---+---+ +---+---+ +---+---+
- | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
- | | | |
- V V V V
- +---------+ +---+---+ +---+---+ +---+---+
- | *table* | | * | * | | * | * | | * | * |
- +---------+ +-|-+-|-+ +-|-+-|-+ +-|-+-|-+
- | | | | | |
- V V V V V V
- +---+ +---+ +---+ +---+ +---+ +---+
- | a | | 1 | | b | | 2 | | c | | 3 |
- +---+ +---+ +---+ +---+ +---+ +---+
-
- To extract information from a table we use the `lookup' procedure,
-which takes a key as argument and returns the associated value (or
-false if there is no value stored under that key). `Lookup' is defined
-in terms of the `assoc' operation, which expects a key and a list of
-records as arguments. Note that `assoc' never sees the dummy record.
-`Assoc' returns the record that has the given key as its `car'.(1)
-`Lookup' then checks to see that the resulting record returned by
-`assoc' is not false, and returns the value (the `cdr') of the record.
-
- (define (lookup key table)
- (let ((record (assoc key (cdr table))))
- (if record
- (cdr record)
- false)))
-
- (define (assoc key records)
- (cond ((null? records) false)
- ((equal? key (caar records)) (car records))
- (else (assoc key (cdr records)))))
-
- To insert a value in a table under a specified key, we first use
-`assoc' to see if there is already a record in the table with this key.
-If not, we form a new record by `cons'ing the key with the value, and
-insert this at the head of the table's list of records, after the dummy
-record. If there already is a record with this key, we set the `cdr'
-of this record to the designated new value. The header of the table
-provides us with a fixed location to modify in order to insert the new
-record.(2)
-
- (define (insert! key value table)
- (let ((record (assoc key (cdr table))))
- (if record
- (set-cdr! record value)
- (set-cdr! table
- (cons (cons key value) (cdr table)))))
- 'ok)
-
- To construct a new table, we simply create a list containing the
-symbol `*table*':
-
- (define (make-table)
- (list '*table*))
-
-Two-dimensional tables
-......................
-
-In a two-dimensional table, each value is indexed by two keys. We can
-construct such a table as a one-dimensional table in which each key
-identifies a subtable. *Note Figure 3-23:: shows the box-and-pointer
-diagram for the table
-
- math:
- +: 43
- -: 45
- *: 42
- letters:
- a: 97
- b: 98
-
-which has two subtables. (The subtables don't need a special header
-symbol, since the key that identifies the subtable serves this purpose.)
-
- *Figure 3.23:* A two-dimensional table.
-
- table
- |
- V
- +---+---+ +---+---+ +---+---+
- | * | *-+-->| * | *-+-->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- V | V
- +-------+ | +---+---+ +---+---+ +---+---+
- |*table*| | | * | *-+-->| * | *-+-->| * | / |
- +-------+ | +-|-+---+ +-|-+---+ +-|-+---+
- | V V V
- | +-------+ +---+---+ +---+---+
- | |letters| | * | * | | * | * |
- | +-------+ +-|-+-|-+ +-|-+-|-+
- | V V V V
- | +---+ +---+ +---+ +---+
- | | a | | 97| | b | | 98|
- | +---+ +---+ +---+ +---+
- V
- +---+---+ +---+---+ +---+---+ +---+---+
- | * | *-+-->| * | *-+-->| * | *-+-->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
- V V V V
- +------+ +---+---+ +---+---+ +---+---+
- | math | | * | * | | * | * | | * | * |
- +------+ +-|-+-|-+ +-|-+-|-+ +-|-+-|-+
- V V V V V V
- +---+ +---+ +---+ +---+ +---+ +---+
- | + | | 43| | - | | 45| | * | | 42|
- +---+ +---+ +---+ +---+ +---+ +---+
-
- When we look up an item, we use the first key to identify the
-correct subtable. Then we use the second key to identify the record
-within the subtable.
-
- (define (lookup key-1 key-2 table)
- (let ((subtable (assoc key-1 (cdr table))))
- (if subtable
- (let ((record (assoc key-2 (cdr subtable))))
- (if record
- (cdr record)
- false))
- false)))
-
- To insert a new item under a pair of keys, we use `assoc' to see if
-there is a subtable stored under the first key. If not, we build a new
-subtable containing the single record (`key-2', `value') and insert it
-into the table under the first key. If a subtable already exists for
-the first key, we insert the new record into this subtable, using the
-insertion method for one-dimensional tables described above:
-
- (define (insert! key-1 key-2 value table)
- (let ((subtable (assoc key-1 (cdr table))))
- (if subtable
- (let ((record (assoc key-2 (cdr subtable))))
- (if record
- (set-cdr! record value)
- (set-cdr! subtable
- (cons (cons key-2 value)
- (cdr subtable)))))
- (set-cdr! table
- (cons (list key-1
- (cons key-2 value))
- (cdr table)))))
- 'ok)
-
-Creating local tables
-.....................
-
-The `lookup' and `insert!' operations defined above take the table as
-an argument. This enables us to use programs that access more than one
-table. Another way to deal with multiple tables is to have separate
-`lookup' and `insert!' procedures for each table. We can do this by
-representing a table procedurally, as an object that maintains an
-internal table as part of its local state. When sent an appropriate
-message, this "table object" supplies the procedure with which to
-operate on the internal table. Here is a generator for two-dimensional
-tables represented in this fashion:
-
- (define (make-table)
- (let ((local-table (list '*table*)))
- (define (lookup key-1 key-2)
- (let ((subtable (assoc key-1 (cdr local-table))))
- (if subtable
- (let ((record (assoc key-2 (cdr subtable))))
- (if record
- (cdr record)
- false))
- false)))
- (define (insert! key-1 key-2 value)
- (let ((subtable (assoc key-1 (cdr local-table))))
- (if subtable
- (let ((record (assoc key-2 (cdr subtable))))
- (if record
- (set-cdr! record value)
- (set-cdr! subtable
- (cons (cons key-2 value)
- (cdr subtable)))))
- (set-cdr! local-table
- (cons (list key-1
- (cons key-2 value))
- (cdr local-table)))))
- 'ok)
- (define (dispatch m)
- (cond ((eq? m 'lookup-proc) lookup)
- ((eq? m 'insert-proc!) insert!)
- (else (error "Unknown operation -- TABLE" m))))
- dispatch))
-
- Using `make-table', we could implement the `get' and `put'
-operations used in section *Note 2-4-3:: for data-directed programming,
-as follows:
-
- (define operation-table (make-table))
- (define get (operation-table 'lookup-proc))
- (define put (operation-table 'insert-proc!))
-
- `Get' takes as arguments two keys, and `put' takes as arguments two
-keys and a value. Both operations access the same local table, which is
-encapsulated within the object created by the call to `make-table'.
-
- *Exercise 3.24:* In the table implementations above, the keys are
- tested for equality using `equal?' (called by `assoc'). This is
- not always the appropriate test. For instance, we might have a
- table with numeric keys in which we don't need an exact match to
- the number we're looking up, but only a number within some
- tolerance of it. Design a table constructor `make-table' that
- takes as an argument a `same-key?' procedure that will be used to
- test "equality" of keys. `Make-table' should return a `dispatch'
- procedure that can be used to access appropriate `lookup' and
- `insert!' procedures for a local table.
-
- *Exercise 3.25:* Generalizing one- and two-dimensional tables,
- show how to implement a table in which values are stored under an
- arbitrary number of keys and different values may be stored under
- different numbers of keys. The `lookup' and `insert!' procedures
- should take as input a list of keys used to access the table.
-
- *Exercise 3.26:* To search a table as implemented above, one needs
- to scan through the list of records. This is basically the
- unordered list representation of section *Note 2-3-3::. For large
- tables, it may be more efficient to structure the table in a
- different manner. Describe a table implementation where the (key,
- value) records are organized using a binary tree, assuming that
- keys can be ordered in some way (e.g., numerically or
- alphabetically). (Compare *Note Exercise 2-66:: of *Note Chapter
- 2::.)
-
- *Exercise 3.27:* "Memoization" (also called "tabulation") is a
- technique that enables a procedure to record, in a local table,
- values that have previously been computed. This technique can
- make a vast difference in the performance of a program. A memoized
- procedure maintains a table in which values of previous calls are
- stored using as keys the arguments that produced the values. When
- the memoized procedure is asked to compute a value, it first
- checks the table to see if the value is already there and, if so,
- just returns that value. Otherwise, it computes the new value in
- the ordinary way and stores this in the table. As an example of
- memoization, recall from section *Note 1-2-2:: the exponential
- process for computing Fibonacci numbers:
-
- (define (fib n)
- (cond ((= n 0) 0)
- ((= n 1) 1)
- (else (+ (fib (- n 1))
- (fib (- n 2))))))
-
- The memoized version of the same procedure is
-
- (define memo-fib
- (memoize (lambda (n)
- (cond ((= n 0) 0)
- ((= n 1) 1)
- (else (+ (memo-fib (- n 1))
- (memo-fib (- n 2))))))))
-
- where the memoizer is defined as
-
- (define (memoize f)
- (let ((table (make-table)))
- (lambda (x)
- (let ((previously-computed-result (lookup x table)))
- (or previously-computed-result
- (let ((result (f x)))
- (insert! x result table)
- result))))))
-
- Draw an environment diagram to analyze the computation of
- `(memo-fib 3)'. Explain why `memo-fib' computes the nth Fibonacci
- number in a number of steps proportional to n. Would the scheme
- still work if we had simply defined `memo-fib' to be `(memoize
- fib)'?
-
- ---------- Footnotes ----------
-
- (1) Because `assoc' uses `equal?', it can recognize keys that are
-symbols, numbers, or list structure.
-
- (2) Thus, the first backbone pair is the object that represents the
-table "itself"; that is, a pointer to the table is a pointer to this
-pair. This same backbone pair always starts the table. If we did not
-arrange things in this way, `insert!' would have to return a new value
-for the start of the table when it added a new record.
-
-\1f
-File: sicp.info, Node: 3-3-4, Next: 3-3-5, Prev: 3-3-3, Up: 3-3
-
-3.3.4 A Simulator for Digital Circuits
---------------------------------------
-
-Designing complex digital systems, such as computers, is an important
-engineering activity. Digital systems are constructed by
-interconnecting simple elements. Although the behavior of these
-individual elements is simple, networks of them can have very complex
-behavior. Computer simulation of proposed circuit designs is an
-important tool used by digital systems engineers. In this section we
-design a system for performing digital logic simulations. This system
-typifies a kind of program called an "event-driven simulation", in
-which actions ("events") trigger further events that happen at a later
-time, which in turn trigger more events, and so so.
-
- Our computational model of a circuit will be composed of objects that
-correspond to the elementary components from which the circuit is
-constructed. There are "wires", which carry "digital signals". A
-digital signal may at any moment have only one of two possible values,
-0 and 1. There are also various types of digital "function boxes",
-which connect wires carrying input signals to other output wires. Such
-boxes produce output signals computed from their input signals. The
-output signal is delayed by a time that depends on the type of the
-function box. For example, an "inverter" is a primitive function box
-that inverts its input. If the input signal to an inverter changes to
-0, then one inverter-delay later the inverter will change its output
-signal to 1. If the input signal to an inverter changes to 1, then one
-inverter-delay later the inverter will change its output signal to 0.
-We draw an inverter symbolically as in *Note Figure 3-24::. An "and-gate",
-also shown in *Note Figure 3-24::, is a primitive function box with two
-inputs and one output. It drives its output signal to a value that is
-the "logical and" of the inputs. That is, if both of its input signals
-become 1, then one and-gate-delay time later the and-gate will force
-its output signal to be 1; otherwise the output will be 0. An "or-gate"
-is a similar two-input primitive function box that drives its output
-signal to a value that is the "logical or" of the inputs. That is, the
-output will become 1 if at least one of the input signals is 1;
-otherwise the output will become 0.
-
- *Figure 3.24:* Primitive functions in the digital logic simulator.
-
- __ ___
- |\ --| \ --\ \
- --| >o-- | )-- ) >--
- |/ --|__/ --/__/
-
- Inverter And-gate Or-gate
-
- We can connect primitive functions together to construct more complex
-functions. To accomplish this we wire the outputs of some function
-boxes to the inputs of other function boxes. For example, the "half-adder"
-circuit shown in *Note Figure 3-25:: consists of an or-gate, two
-and-gates, and an inverter. It takes two input signals, A and B, and
-has two output signals, S and C. S will become 1 whenever precisely
-one of A and B is 1, and C will become 1 whenever A and B are both 1.
-We can see from the figure that, because of the delays involved, the
-outputs may be generated at different times. Many of the difficulties
-in the design of digital circuits arise from this fact.
-
- *Figure 3.25:* A half-adder circuit.
-
- +--------------------------------------+
- | ____ |
- A --------*---\ \ D ___ |
- | | > >---------------| \ |
- | +--|---/___/ | )----- S
- | | | |\ E +--|___/ |
- | | | +--| >o---+ |
- | | | ___ | |/ |
- | | +---| \ | |
- | | | )--*----------------------- C
- B -----*------|___/ |
- | |
- +--------------------------------------+
-
- We will now build a program for modeling the digital logic circuits
-we wish to study. The program will construct computational objects
-modeling the wires, which will "hold" the signals. Function boxes will
-be modeled by procedures that enforce the correct relationships among
-the signals.
-
- One basic element of our simulation will be a procedure `make-wire',
-which constructs wires. For example, we can construct six wires as
-follows:
-
- (define a (make-wire))
- (define b (make-wire))
- (define c (make-wire))
-
- (define d (make-wire))
- (define e (make-wire))
- (define s (make-wire))
-
- We attach a function box to a set of wires by calling a procedure
-that constructs that kind of box. The arguments to the constructor
-procedure are the wires to be attached to the box. For example, given
-that we can construct and-gates, or-gates, and inverters, we can wire
-together the half-adder shown in *Note Figure 3-25:::
-
- (or-gate a b d)
- ok
-
- (and-gate a b c)
- ok
-
- (inverter c e)
- ok
-
- (and-gate d e s)
- ok
-
- Better yet, we can explicitly name this operation by defining a
-procedure `half-adder' that constructs this circuit, given the four
-external wires to be attached to the half-adder:
-
- (define (half-adder a b s c)
- (let ((d (make-wire)) (e (make-wire)))
- (or-gate a b d)
- (and-gate a b c)
- (inverter c e)
- (and-gate d e s)
- 'ok))
-
- The advantage of making this definition is that we can use
-`half-adder' itself as a building block in creating more complex
-circuits. *Note Figure 3-26::, for example, shows a "full-adder"
-composed of two half-adders and an or-gate.(1) We can construct a
-full-adder as follows:
-
- (define (full-adder a b c-in sum c-out)
- (let ((s (make-wire))
- (c1 (make-wire))
- (c2 (make-wire)))
- (half-adder b c-in s c1)
- (half-adder a s sum c2)
- (or-gate c1 c2 c-out)
- 'ok))
-
- Having defined `full-adder' as a procedure, we can now use it as a
-building block for creating still more complex circuits. (For example,
-see *Note Exercise 3-30::.)
-
- *Figure 3.26:* A full-adder circuit.
-
- +----------------------------------+
- | +-------+ |
- A -----------------+ half +-------------- SUM
- | +-------+ | adder | ____ |
- B -----+ half +---+ +---\ \ |
- | | adder | +-------+ >or >----- Cout
- C -----+ +---------------/___/ |
- | +-------+ |
- +----------------------------------+
-
- In essence, our simulator provides us with the tools to construct a
-language of circuits. If we adopt the general perspective on languages
-with which we approached the study of Lisp in section *Note 1-1::, we
-can say that the primitive function boxes form the primitive elements
-of the language, that wiring boxes together provides a means of
-combination, and that specifying wiring patterns as procedures serves
-as a means of abstraction.
-
-Primitive function boxes
-........................
-
-The primitive function boxes implement the "forces" by which a change
-in the signal on one wire influences the signals on other wires. To
-build function boxes, we use the following operations on wires:
-
- * (get-signal <WIRE>)
-
- returns the current value of the signal on the wire.
-
- * (set-signal! <WIRE> <NEW VALUE>)
-
- changes the value of the signal on the wire to the new value.
-
- * (add-action! <WIRE> <PROCEDURE OF NO ARGUMENTS>)
-
- asserts that the designated procedure should be run whenever the
- signal on the wire changes value. Such procedures are the
- vehicles by which changes in the signal value on the wire are
- communicated to other wires.
-
-
- In addition, we will make use of a procedure `after-delay' that
-takes a time delay and a procedure to be run and executes the given
-procedure after the given delay.
-
- Using these procedures, we can define the primitive digital logic
-functions. To connect an input to an output through an inverter, we
-use `add-action!' to associate with the input wire a procedure that
-will be run whenever the signal on the input wire changes value. The
-procedure computes the `logical-not' of the input signal, and then,
-after one `inverter-delay', sets the output signal to be this new value:
-
- (define (inverter input output)
- (define (invert-input)
- (let ((new-value (logical-not (get-signal input))))
- (after-delay inverter-delay
- (lambda ()
- (set-signal! output new-value)))))
- (add-action! input invert-input)
- 'ok)
-
- (define (logical-not s)
- (cond ((= s 0) 1)
- ((= s 1) 0)
- (else (error "Invalid signal" s))))
-
- An and-gate is a little more complex. The action procedure must be
-run if either of the inputs to the gate changes. It computes the
-`logical-and' (using a procedure analogous to `logical-not') of the
-values of the signals on the input wires and sets up a change to the
-new value to occur on the output wire after one `and-gate-delay'.
-
- (define (and-gate a1 a2 output)
- (define (and-action-procedure)
- (let ((new-value
- (logical-and (get-signal a1) (get-signal a2))))
- (after-delay and-gate-delay
- (lambda ()
- (set-signal! output new-value)))))
- (add-action! a1 and-action-procedure)
- (add-action! a2 and-action-procedure)
- 'ok)
-
- *Exercise 3.28:* Define an or-gate as a primitive function box.
- Your `or-gate' constructor should be similar to `and-gate'.
-
- *Exercise 3.29:* Another way to construct an or-gate is as a
- compound digital logic device, built from and-gates and inverters.
- Define a procedure `or-gate' that accomplishes this. What is the
- delay time of the or-gate in terms of `and-gate-delay' and
- `inverter-delay'?
-
- *Exercise 3.30:* *Note Figure 3-27:: shows a "ripple-carry adder"
- formed by stringing together n full-adders. This is the simplest
- form of parallel adder for adding two n-bit binary numbers. The
- inputs A_1, A_2, A_3, ..., A_n and B_1, B_2, B_3, ..., B_n are the
- two binary numbers to be added (each A_k and B_k is a 0 or a 1).
- The circuit generates S_1, S_2, S_3, ..., S_n, the n bits of the
- sum, and C, the carry from the addition. Write a procedure
- `ripple-carry-adder' that generates this circuit. The procedure
- should take as arguments three lists of n wires each--the A_k, the
- B_k, and the S_k--and also another wire C. The major drawback of
- the ripple-carry adder is the need to wait for the carry signals
- to propagate. What is the delay needed to obtain the complete
- output from an n-bit ripple-carry adder, expressed in terms of the
- delays for and-gates, or-gates, and inverters?
-
- *Figure 3.27:* A ripple-carry adder for n-bit numbers.
-
- : : :
- : A_1 B_1 C_1 A_2 B_2 C_2 A_3 B_3 C_3: : A_n B_n C_n=0
- : | | +---+ | | +---+ | | +----- : | | +-
- | | | | | | | | | | | | : : | | |
- : ++---+---++ | ++---+---++ | ++---+---++ : : ++---+---++
- : | FA | | | FA | | | FA | : : | FA |
- : +--+---+--+ | +--+---+--+ | +--+---+--+ : : +--+---+--+
- : | | | | | | | | : : | |
- C ------+ | +-----+ | +-----+ | : ------+ |
- : | C_1 | C_2 | : :C_(n-1) |
- : | | | : : |
- S_1 S_2 S_3 S_n
-
-Representing wires
-..................
-
-A wire in our simulation will be a computational object with two local
-state variables: a `signal-value' (initially taken to be 0) and a
-collection of `action-procedures' to be run when the signal changes
-value. We implement the wire, using message-passing style, as a
-collection of local procedures together with a `dispatch' procedure
-that selects the appropriate local operation, just as we did with the
-simple bank-account object in section *Note 3-1-1:::
-
- (define (make-wire)
- (let ((signal-value 0) (action-procedures '()))
- (define (set-my-signal! new-value)
- (if (not (= signal-value new-value))
- (begin (set! signal-value new-value)
- (call-each action-procedures))
- 'done))
-
- (define (accept-action-procedure! proc)
- (set! action-procedures (cons proc action-procedures))
- (proc))
-
- (define (dispatch m)
- (cond ((eq? m 'get-signal) signal-value)
- ((eq? m 'set-signal!) set-my-signal!)
- ((eq? m 'add-action!) accept-action-procedure!)
- (else (error "Unknown operation -- WIRE" m))))
- dispatch))
-
- The local procedure `set-my-signal!' tests whether the new signal
-value changes the signal on the wire. If so, it runs each of the
-action procedures, using the following procedure `call-each', which
-calls each of the items in a list of no-argument procedures:
-
- (define (call-each procedures)
- (if (null? procedures)
- 'done
- (begin
- ((car procedures))
- (call-each (cdr procedures)))))
-
- The local procedure `accept-action-procedure!' adds the given
-procedure to the list of procedures to be run, and then runs the new
-procedure once. (See *Note Exercise 3-31::.)
-
- With the local `dispatch' procedure set up as specified, we can
-provide the following procedures to access the local operations on
-wires:(2)
-
- (define (get-signal wire)
- (wire 'get-signal))
-
- (define (set-signal! wire new-value)
- ((wire 'set-signal!) new-value))
-
- (define (add-action! wire action-procedure)
- ((wire 'add-action!) action-procedure))
-
- Wires, which have time-varying signals and may be incrementally
-attached to devices, are typical of mutable objects. We have modeled
-them as procedures with local state variables that are modified by
-assignment. When a new wire is created, a new set of state variables
-is allocated (by the `let' expression in `make-wire') and a new
-`dispatch' procedure is constructed and returned, capturing the
-environment with the new state variables.
-
- The wires are shared among the various devices that have been
-connected to them. Thus, a change made by an interaction with one
-device will affect all the other devices attached to the wire. The
-wire communicates the change to its neighbors by calling the action
-procedures provided to it when the connections were established.
-
-The agenda
-..........
-
-The only thing needed to complete the simulator is `after-delay'. The
-idea here is that we maintain a data structure, called an "agenda",
-that contains a schedule of things to do. The following operations are
-defined for agendas:
-
- * `(make-agenda)' returns a new empty agenda.
-
- * `(empty-agenda? <AGENDA>)' is true if the specified agenda is
- empty.
-
- * `(first-agenda-item <AGENDA>)' returns the first item on the
- agenda.
-
- * `(remove-first-agenda-item! <AGENDA>)' modifies the agenda by
- removing the first item.
-
- * `(add-to-agenda! <TIME> <ACTION> <AGENDA>)' modifies the agenda by
- adding the given action procedure to be run at the specified time.
-
- * `(current-time <AGENDA>)' returns the current simulation time.
-
-
- The particular agenda that we use is denoted by `the-agenda'. The
-procedure `after-delay' adds new elements to `the-agenda':
-
- (define (after-delay delay action)
- (add-to-agenda! (+ delay (current-time the-agenda))
- action
- the-agenda))
-
- The simulation is driven by the procedure `propagate', which
-operates on `the-agenda', executing each procedure on the agenda in
-sequence. In general, as the simulation runs, new items will be added
-to the agenda, and `propagate' will continue the simulation as long as
-there are items on the agenda:
-
- (define (propagate)
- (if (empty-agenda? the-agenda)
- 'done
- (let ((first-item (first-agenda-item the-agenda)))
- (first-item)
- (remove-first-agenda-item! the-agenda)
- (propagate))))
-
-A sample simulation
-...................
-
-The following procedure, which places a "probe" on a wire, shows the
-simulator in action. The probe tells the wire that, whenever its signal
-changes value, it should print the new signal value, together with the
-current time and a name that identifies the wire:
-
- (define (probe name wire)
- (add-action! wire
- (lambda ()
- (newline)
- (display name)
- (display " ")
- (display (current-time the-agenda))
- (display " New-value = ")
- (display (get-signal wire)))))
-
- We begin by initializing the agenda and specifying delays for the
-primitive function boxes:
-
- (define the-agenda (make-agenda))
- (define inverter-delay 2)
- (define and-gate-delay 3)
- (define or-gate-delay 5)
-
- Now we define four wires, placing probes on two of them:
-
- (define input-1 (make-wire))
- (define input-2 (make-wire))
- (define sum (make-wire))
- (define carry (make-wire))
-
- (probe 'sum sum)
- sum 0 New-value = 0
-
- (probe 'carry carry)
- carry 0 New-value = 0
-
- Next we connect the wires in a half-adder circuit (as in *Note
-Figure 3-25::), set the signal on `input-1' to 1, and run the
-simulation:
-
- (half-adder input-1 input-2 sum carry)
- ok
-
- (set-signal! input-1 1)
- done
-
- (propagate)
- sum 8 New-value = 1
- done
-
- The `sum' signal changes to 1 at time 8. We are now eight time
-units from the beginning of the simulation. At this point, we can set
-the signal on `input-2' to 1 and allow the values to propagate:
-
- (set-signal! input-2 1)
- done
-
- (propagate)
- carry 11 New-value = 1
- sum 16 New-value = 0
- done
-
- The `carry' changes to 1 at time 11 and the `sum' changes to 0 at
-time 16.
-
- *Exercise 3.31:* The internal procedure `accept-action-procedure!'
- defined in `make-wire' specifies that when a new action procedure
- is added to a wire, the procedure is immediately run. Explain why
- this initialization is necessary. In particular, trace through the
- half-adder example in the paragraphs above and say how the
- system's response would differ if we had defined
- `accept-action-procedure!' as
-
- (define (accept-action-procedure! proc)
- (set! action-procedures (cons proc action-procedures)))
-
-Implementing the agenda
-.......................
-
-Finally, we give details of the agenda data structure, which holds the
-procedures that are scheduled for future execution.
-
- The agenda is made up of "time segments". Each time segment is a
-pair consisting of a number (the time) and a queue (see *Note Exercise
-3-32::) that holds the procedures that are scheduled to be run during
-that time segment.
-
- (define (make-time-segment time queue)
- (cons time queue))
-
- (define (segment-time s) (car s))
-
- (define (segment-queue s) (cdr s))
-
- We will operate on the time-segment queues using the queue
-operations described in section *Note 3-3-2::.
-
- The agenda itself is a one-dimensional table of time segments. It
-differs from the tables described in section *Note 3-3-3:: in that the
-segments will be sorted in order of increasing time. In addition, we
-store the "current time" (i.e., the time of the last action that was
-processed) at the head of the agenda. A newly constructed agenda has
-no time segments and has a current time of 0:(3)
-
- (define (make-agenda) (list 0))
-
- (define (current-time agenda) (car agenda))
-
- (define (set-current-time! agenda time)
- (set-car! agenda time))
-
- (define (segments agenda) (cdr agenda))
-
- (define (set-segments! agenda segments)
- (set-cdr! agenda segments))
-
- (define (first-segment agenda) (car (segments agenda)))
-
- (define (rest-segments agenda) (cdr (segments agenda)))
-
- An agenda is empty if it has no time segments:
-
- (define (empty-agenda? agenda)
- (null? (segments agenda)))
-
- To add an action to an agenda, we first check if the agenda is
-empty. If so, we create a time segment for the action and install this
-in the agenda. Otherwise, we scan the agenda, examining the time of
-each segment. If we find a segment for our appointed time, we add the
-action to the associated queue. If we reach a time later than the one
-to which we are appointed, we insert a new time segment into the agenda
-just before it. If we reach the end of the agenda, we must create a
-new time segment at the end.
-
- (define (add-to-agenda! time action agenda)
- (define (belongs-before? segments)
- (or (null? segments)
- (< time (segment-time (car segments)))))
- (define (make-new-time-segment time action)
- (let ((q (make-queue)))
- (insert-queue! q action)
- (make-time-segment time q)))
- (define (add-to-segments! segments)
- (if (= (segment-time (car segments)) time)
- (insert-queue! (segment-queue (car segments))
- action)
- (let ((rest (cdr segments)))
- (if (belongs-before? rest)
- (set-cdr!
- segments
- (cons (make-new-time-segment time action)
- (cdr segments)))
- (add-to-segments! rest)))))
- (let ((segments (segments agenda)))
- (if (belongs-before? segments)
- (set-segments!
- agenda
- (cons (make-new-time-segment time action)
- segments))
- (add-to-segments! segments))))
-
- The procedure that removes the first item from the agenda deletes
-the item at the front of the queue in the first time segment. If this
-deletion makes the time segment empty, we remove it from the list of
-segments:(4)
-
- (define (remove-first-agenda-item! agenda)
- (let ((q (segment-queue (first-segment agenda))))
- (delete-queue! q)
- (if (empty-queue? q)
- (set-segments! agenda (rest-segments agenda)))))
-
- The first agenda item is found at the head of the queue in the first
-time segment. Whenever we extract an item, we also update the current
-time:(5)
-
- (define (first-agenda-item agenda)
- (if (empty-agenda? agenda)
- (error "Agenda is empty -- FIRST-AGENDA-ITEM")
- (let ((first-seg (first-segment agenda)))
- (set-current-time! agenda (segment-time first-seg))
- (front-queue (segment-queue first-seg)))))
-
- *Exercise 3.32:* The procedures to be run during each time segment
- of the agenda are kept in a queue. Thus, the procedures for each
- segment are called in the order in which they were added to the
- agenda (first in, first out). Explain why this order must be
- used. In particular, trace the behavior of an and-gate whose
- inputs change from 0,1 to 1,0 in the same segment and say how the
- behavior would differ if we stored a segment's procedures in an
- ordinary list, adding and removing procedures only at the front
- (last in, first out).
-
- ---------- Footnotes ----------
-
- (1) A full-adder is a basic circuit element used in adding two
-binary numbers. Here A and B are the bits at corresponding positions in
-the two numbers to be added, and C_(in) is the carry bit from the
-addition one place to the right. The circuit generates SUM, which is
-the sum bit in the corresponding position, and C_(out), which is the
-carry bit to be propagated to the left.
-
- (2) [Footnote 27] These procedures are simply syntactic sugar that
-allow us to use ordinary procedural syntax to access the local
-procedures of objects. It is striking that we can interchange the role
-of "procedures" and "data" in such a simple way. For example, if we
-write `(wire 'get-signal)' we think of `wire' as a procedure that is
-called with the message `get-signal' as input. Alternatively, writing
-`(get-signal wire)' encourages us to think of `wire' as a data object
-that is the input to a procedure `get-signal'. The truth of the matter
-is that, in a language in which we can deal with procedures as objects,
-there is no fundamental difference between "procedures" and "data," and
-we can choose our syntactic sugar to allow us to program in whatever
-style we choose.
-
- (3) The agenda is a headed list, like the tables in section *Note
-3-3-3::, but since the list is headed by the time, we do not need an
-additional dummy header (such as the `*table*' symbol used with tables).
-
- (4) Observe that the `if' expression in this procedure has no
-<ALTERNATIVE> expression. Such a "one-armed `if' statement" is used to
-decide whether to do something, rather than to select between two
-expressions. An `if' expression returns an unspecified value if the
-predicate is false and there is no <ALTERNATIVE>.
-
- (5) In this way, the current time will always be the time of the
-action most recently processed. Storing this time at the head of the
-agenda ensures that it will still be available even if the associated
-time segment has been deleted.
-
-\1f
-File: sicp.info, Node: 3-3-5, Prev: 3-3-4, Up: 3-3
-
-3.3.5 Propagation of Constraints
---------------------------------
-
-Computer programs are traditionally organized as one-directional
-computations, which perform operations on prespecified arguments to
-produce desired outputs. On the other hand, we often model systems in
-terms of relations among quantities. For example, a mathematical model
-of a mechanical structure might include the information that the
-deflection d of a metal rod is related to the force f on the rod, the
-length L of the rod, the cross-sectional area A, and the elastic
-modulus E via the equation
-
- dAE = FL
-
- Such an equation is not one-directional. Given any four of the
-quantities, we can use it to compute the fifth. Yet translating the
-equation into a traditional computer language would force us to choose
-one of the quantities to be computed in terms of the other four. Thus,
-a procedure for computing the area A could not be used to compute the
-deflection d, even though the computations of A and d arise from the
-same equation.(1)
-
- In this section, we sketch the design of a language that enables us
-to work in terms of relations themselves. The primitive elements of
-the language are "primitive constraints", which state that certain
-relations hold between quantities. For example, `(adder a b c)'
-specifies that the quantities a, b, and c must be related by the
-equation a + b = c, `(multiplier x y z)' expresses the constraint xy =
-z, and `(constant 3.14 x)' says that the value of x must be 3.14.
-
- Our language provides a means of combining primitive constraints in
-order to express more complex relations. We combine constraints by
-constructing "constraint networks", in which constraints are joined by "connectors".
-A connector is an object that "holds" a value that may participate in
-one or more constraints. For example, we know that the relationship
-between Fahrenheit and Celsius temperatures is
-
- 9C = 5(F - 32)
-
- Such a constraint can be thought of as a network consisting of
-primitive adder, multiplier, and constant constraints (*Note Figure
-3-28::). In the figure, we see on the left a multiplier box with three
-terminals, labeled m1, m2, and p. These connect the multiplier to the
-rest of the network as follows: The m1 terminal is linked to a
-connector C, which will hold the Celsius temperature. The m2 terminal
-is linked to a connector w, which is also linked to a constant box that
-holds 9. The p terminal, which the multiplier box constrains to be the
-product of m1 and m2, is linked to the p terminal of another multiplier
-box, whose m2 is connected to a constant 5 and whose m1 is connected to
-one of the terms in a sum.
-
- *Figure 3.28:* The relation 9C = 5(F - 32) expressed as a
- constraint network.
-
- +---------+ +---------+ v +---------+
- C -----+ m1 | u | m1 +-------+ a1 |
- | * p +-----+ p * | | + s +---- F
- +--+ m2 | | m2 +--+ +--+ a2 |
- | +---------+ +---------+ | | +---------+
- w | x| |y
- | +-----+ +-----+ | | +-----+
- +----+ 9 | | 5 +-----+ +-----+ 32 |
- +-----+ +-----+ +-----+
-
- Computation by such a network proceeds as follows: When a connector
-is given a value (by the user or by a constraint box to which it is
-linked), it awakens all of its associated constraints (except for the
-constraint that just awakened it) to inform them that it has a value.
-Each awakened constraint box then polls its connectors to see if there
-is enough information to determine a value for a connector. If so, the
-box sets that connector, which then awakens all of its associated
-constraints, and so on. For instance, in conversion between Celsius
-and Fahrenheit, w, x, and y are immediately set by the constant boxes
-to 9, 5, and 32, respectively. The connectors awaken the multipliers
-and the adder, which determine that there is not enough information to
-proceed. If the user (or some other part of the network) sets C to a
-value (say 25), the leftmost multiplier will be awakened, and it will
-set u to 25*9 = 225. Then u awakens the second multiplier, which sets
-v to 45, and v awakens the adder, which sets f to 77.
-
-Using the constraint system
-...........................
-
-To use the constraint system to carry out the temperature computation
-outlined above, we first create two connectors, `C' and `F', by calling
-the constructor `make-connector', and link `C' and `F' in an
-appropriate network:
-
- (define C (make-connector))
- (define F (make-connector))
- (celsius-fahrenheit-converter C F)
- ok
-
- The procedure that creates the network is defined as follows:
-
- (define (celsius-fahrenheit-converter c f)
- (let ((u (make-connector))
- (v (make-connector))
- (w (make-connector))
- (x (make-connector))
- (y (make-connector)))
- (multiplier c w u)
- (multiplier v x u)
- (adder v y f)
- (constant 9 w)
- (constant 5 x)
- (constant 32 y)
- 'ok))
-
- This procedure creates the internal connectors `u', `v', `w', `x',
-and `y', and links them as shown in *Note Figure 3-28:: using the
-primitive constraint constructors `adder', `multiplier', and
-`constant'. Just as with the digital-circuit simulator of section
-*Note 3-3-4::, expressing these combinations of primitive elements in
-terms of procedures automatically provides our language with a means of
-abstraction for compound objects.
-
- To watch the network in action, we can place probes on the
-connectors `C' and `F', using a `probe' procedure similar to the one we
-used to monitor wires in section *Note 3-3-4::. Placing a probe on a
-connector will cause a message to be printed whenever the connector is
-given a value:
-
- (probe "Celsius temp" C)
- (probe "Fahrenheit temp" F)
-
- Next we set the value of `C' to 25. (The third argument to
-`set-value!' tells `C' that this directive comes from the `user'.)
-
- (set-value! C 25 'user)
- Probe: Celsius temp = 25
- Probe: Fahrenheit temp = 77
- done
-
- The probe on `C' awakens and reports the value. `C' also propagates
-its value through the network as described above. This sets `F' to 77,
-which is reported by the probe on `F'.
-
- Now we can try to set `F' to a new value, say 212:
-
- (set-value! F 212 'user)
- Error! Contradiction (77 212)
-
- The connector complains that it has sensed a contradiction: Its
-value is 77, and someone is trying to set it to 212. If we really want
-to reuse the network with new values, we can tell `C' to forget its old
-value:
-
- (forget-value! C 'user)
- Probe: Celsius temp = ?
- Probe: Fahrenheit temp = ?
- done
-
- `C' finds that the `user', who set its value originally, is now
-retracting that value, so `C' agrees to lose its value, as shown by the
-probe, and informs the rest of the network of this fact. This
-information eventually propagates to `F', which now finds that it has
-no reason for continuing to believe that its own value is 77. Thus,
-`F' also gives up its value, as shown by the probe.
-
- Now that `F' has no value, we are free to set it to 212:
-
- (set-value! F 212 'user)
- Probe: Fahrenheit temp = 212
- Probe: Celsius temp = 100
- done
-
- This new value, when propagated through the network, forces `C' to
-have a value of 100, and this is registered by the probe on `C'.
-Notice that the very same network is being used to compute `C' given
-`F' and to compute `F' given `C'. This nondirectionality of
-computation is the distinguishing feature of constraint-based systems.
-
-Implementing the constraint system
-..................................
-
-The constraint system is implemented via procedural objects with local
-state, in a manner very similar to the digital-circuit simulator of
-section *Note 3-3-4::. Although the primitive objects of the
-constraint system are somewhat more complex, the overall system is
-simpler, since there is no concern about agendas and logic delays.
-
- The basic operations on connectors are the following:
-
- * `(has-value? <CONNECTOR>)' tells whether the connector has a value.
-
- * `(get-value <CONNECTOR>)' returns the connector's current value.
-
- * `(set-value! <CONNECTOR> <NEW-VALUE> <INFORMANT>)' indicates that
- the informant is requesting the connector to set its value to the
- new value.
-
- * `(forget-value! <CONNECTOR> <RETRACTOR>)' tells the connector that
- the retractor is requesting it to forget its value.
-
- * `(connect <CONNECTOR> <NEW-CONSTRAINT>)' tells the connector to
- participate in the new constraint.
-
-
- The connectors communicate with the constraints by means of the
-procedures `inform-about-value', which tells the given constraint that
-the connector has a value, and `inform-about-no-value', which tells the
-constraint that the connector has lost its value.
-
- `Adder' constructs an adder constraint among summand connectors `a1'
-and `a2' and a `sum' connector. An adder is implemented as a procedure
-with local state (the procedure `me' below):
-
- (define (adder a1 a2 sum)
- (define (process-new-value)
- (cond ((and (has-value? a1) (has-value? a2))
- (set-value! sum
- (+ (get-value a1) (get-value a2))
- me))
- ((and (has-value? a1) (has-value? sum))
- (set-value! a2
- (- (get-value sum) (get-value a1))
- me))
- ((and (has-value? a2) (has-value? sum))
- (set-value! a1
- (- (get-value sum) (get-value a2))
- me))))
- (define (process-forget-value)
- (forget-value! sum me)
- (forget-value! a1 me)
- (forget-value! a2 me)
- (process-new-value))
- (define (me request)
- (cond ((eq? request 'I-have-a-value)
- (process-new-value))
- ((eq? request 'I-lost-my-value)
- (process-forget-value))
- (else
- (error "Unknown request -- ADDER" request))))
- (connect a1 me)
- (connect a2 me)
- (connect sum me)
- me)
-
- `Adder' connects the new adder to the designated connectors and
-returns it as its value. The procedure `me', which represents the
-adder, acts as a dispatch to the local procedures. The following
-"syntax interfaces" (see footnote *Note Footnote 27:: in section *Note
-3-3-4::) are used in conjunction with the dispatch:
-
- (define (inform-about-value constraint)
- (constraint 'I-have-a-value))
-
- (define (inform-about-no-value constraint)
- (constraint 'I-lost-my-value))
-
- The adder's local procedure `process-new-value' is called when the
-adder is informed that one of its connectors has a value. The adder
-first checks to see if both `a1' and `a2' have values. If so, it tells
-`sum' to set its value to the sum of the two addends. The `informant'
-argument to `set-value!' is `me', which is the adder object itself. If
-`a1' and `a2' do not both have values, then the adder checks to see if
-perhaps `a1' and `sum' have values. If so, it sets `a2' to the
-difference of these two. Finally, if `a2' and `sum' have values, this
-gives the adder enough information to set `a1'. If the adder is told
-that one of its connectors has lost a value, it requests that all of its
-connectors now lose their values. (Only those values that were set by
-this adder are actually lost.) Then it runs `process-new-value'. The
-reason for this last step is that one or more connectors may still have
-a value (that is, a connector may have had a value that was not
-originally set by the adder), and these values may need to be
-propagated back through the adder.
-
- A multiplier is very similar to an adder. It will set its `product'
-to 0 if either of the factors is 0, even if the other factor is not
-known.
-
- (define (multiplier m1 m2 product)
- (define (process-new-value)
- (cond ((or (and (has-value? m1) (= (get-value m1) 0))
- (and (has-value? m2) (= (get-value m2) 0)))
- (set-value! product 0 me))
- ((and (has-value? m1) (has-value? m2))
- (set-value! product
- (* (get-value m1) (get-value m2))
- me))
- ((and (has-value? product) (has-value? m1))
- (set-value! m2
- (/ (get-value product) (get-value m1))
- me))
- ((and (has-value? product) (has-value? m2))
- (set-value! m1
- (/ (get-value product) (get-value m2))
- me))))
- (define (process-forget-value)
- (forget-value! product me)
- (forget-value! m1 me)
- (forget-value! m2 me)
- (process-new-value))
- (define (me request)
- (cond ((eq? request 'I-have-a-value)
- (process-new-value))
- ((eq? request 'I-lost-my-value)
- (process-forget-value))
- (else
- (error "Unknown request -- MULTIPLIER" request))))
- (connect m1 me)
- (connect m2 me)
- (connect product me)
- me)
-
- A `constant' constructor simply sets the value of the designated
-connector. Any `I-have-a-value' or `I-lost-my-value' message sent to
-the constant box will produce an error.
-
- (define (constant value connector)
- (define (me request)
- (error "Unknown request -- CONSTANT" request))
- (connect connector me)
- (set-value! connector value me)
- me)
-
- Finally, a probe prints a message about the setting or unsetting of
-the designated connector:
-
- (define (probe name connector)
- (define (print-probe value)
- (newline)
- (display "Probe: ")
- (display name)
- (display " = ")
- (display value))
- (define (process-new-value)
- (print-probe (get-value connector)))
- (define (process-forget-value)
- (print-probe "?"))
- (define (me request)
- (cond ((eq? request 'I-have-a-value)
- (process-new-value))
- ((eq? request 'I-lost-my-value)
- (process-forget-value))
- (else
- (error "Unknown request -- PROBE" request))))
- (connect connector me)
- me)
-
-Representing connectors
-.......................
-
-A connector is represented as a procedural object with local state
-variables `value', the current value of the connector; `informant', the
-object that set the connector's value; and `constraints', a list of the
-constraints in which the connector participates.
-
- (define (make-connector)
- (let ((value false) (informant false) (constraints '()))
- (define (set-my-value newval setter)
- (cond ((not (has-value? me))
- (set! value newval)
- (set! informant setter)
- (for-each-except setter
- inform-about-value
- constraints))
- ((not (= value newval))
- (error "Contradiction" (list value newval)))
- (else 'ignored)))
- (define (forget-my-value retractor)
- (if (eq? retractor informant)
- (begin (set! informant false)
- (for-each-except retractor
- inform-about-no-value
- constraints))
- 'ignored))
- (define (connect new-constraint)
- (if (not (memq new-constraint constraints))
- (set! constraints
- (cons new-constraint constraints)))
- (if (has-value? me)
- (inform-about-value new-constraint))
- 'done)
- (define (me request)
- (cond ((eq? request 'has-value?)
- (if informant true false))
- ((eq? request 'value) value)
- ((eq? request 'set-value!) set-my-value)
- ((eq? request 'forget) forget-my-value)
- ((eq? request 'connect) connect)
- (else (error "Unknown operation -- CONNECTOR"
- request))))
- me))
-
- The connector's local procedure `set-my-value' is called when there
-is a request to set the connector's value. If the connector does not
-currently have a value, it will set its value and remember as
-`informant' the constraint that requested the value to be set.(2) Then
-the connector will notify all of its participating constraints except
-the constraint that requested the value to be set. This is
-accomplished using the following iterator, which applies a designated
-procedure to all items in a list except a given one:
-
- (define (for-each-except exception procedure list)
- (define (loop items)
- (cond ((null? items) 'done)
- ((eq? (car items) exception) (loop (cdr items)))
- (else (procedure (car items))
- (loop (cdr items)))))
- (loop list))
-
- If a connector is asked to forget its value, it runs the local
-procedure `forget-my-value', which first checks to make sure that the
-request is coming from the same object that set the value originally.
-If so, the connector informs its associated constraints about the loss
-of the value.
-
- The local procedure `connect' adds the designated new constraint to
-the list of constraints if it is not already in that list. Then, if
-the connector has a value, it informs the new constraint of this fact.
-
- The connector's procedure `me' serves as a dispatch to the other
-internal procedures and also represents the connector as an object.
-The following procedures provide a syntax interface for the dispatch:
-
- (define (has-value? connector)
- (connector 'has-value?))
-
- (define (get-value connector)
- (connector 'value))
-
- (define (set-value! connector new-value informant)
- ((connector 'set-value!) new-value informant))
-
- (define (forget-value! connector retractor)
- ((connector 'forget) retractor))
-
- (define (connect connector new-constraint)
- ((connector 'connect) new-constraint))
-
- *Exercise 3.33:* Using primitive multiplier, adder, and constant
- constraints, define a procedure `averager' that takes three
- connectors `a', `b', and `c' as inputs and establishes the
- constraint that the value of `c' is the average of the values of
- `a' and `b'.
-
- *Exercise 3.34:* Louis Reasoner wants to build a squarer, a
- constraint device with two terminals such that the value of
- connector `b' on the second terminal will always be the square of
- the value `a' on the first terminal. He proposes the following
- simple device made from a multiplier:
-
- (define (squarer a b)
- (multiplier a a b))
-
- There is a serious flaw in this idea. Explain.
-
- *Exercise 3.35:* Ben Bitdiddle tells Louis that one way to avoid
- the trouble in *Note Exercise 3-34:: is to define a squarer as a
- new primitive constraint. Fill in the missing portions in Ben's
- outline for a procedure to implement such a constraint:
-
- (define (squarer a b)
- (define (process-new-value)
- (if (has-value? b)
- (if (< (get-value b) 0)
- (error "square less than 0 -- SQUARER" (get-value b))
- <ALTERNATIVE1>)
- <ALTERNATIVE2>))
- (define (process-forget-value) <BODY1>)
- (define (me request) <BODY2>)
- <REST OF DEFINITION>
- me)
-
- *Exercise 3.36:* Suppose we evaluate the following sequence of
- expressions in the global environment:
-
- (define a (make-connector))
- (define b (make-connector))
- (set-value! a 10 'user)
-
- At some time during evaluation of the `set-value!', the following
- expression from the connector's local procedure is evaluated:
-
- (for-each-except setter inform-about-value constraints)
-
- Draw an environment diagram showing the environment in which the
- above expression is evaluated.
-
- *Exercise 3.37:* The `celsius-fahrenheit-converter' procedure is
- cumbersome when compared with a more expression-oriented style of
- definition, such as
-
- (define (celsius-fahrenheit-converter x)
- (c+ (c* (c/ (cv 9) (cv 5))
- x)
- (cv 32)))
-
- (define C (make-connector))
- (define F (celsius-fahrenheit-converter C))
-
- Here `c+', `c*', etc. are the "constraint" versions of the
- arithmetic operations. For example, `c+' takes two connectors as
- arguments and returns a connector that is related to these by an
- adder constraint:
-
- (define (c+ x y)
- (let ((z (make-connector)))
- (adder x y z)
- z))
-
- Define analogous procedures `c-', `c*', `c/', and `cv' (constant
- value) that enable us to define compound constraints as in the
- converter example above.(3)
-
- ---------- Footnotes ----------
-
- (1) Constraint propagation first appeared in the incredibly
-forward-looking SKETCHPAD system of Ivan Sutherland (1963). A
-beautiful constraint-propagation system based on the Smalltalk language
-was developed by Alan Borning (1977) at Xerox Palo Alto Research
-Center. Sussman, Stallman, and Steele applied constraint propagation
-to electrical circuit analysis (Sussman and Stallman 1975; Sussman and
-Steele 1980). TK!Solver (Konopasek and Jayaraman 1984) is an extensive
-modeling environment based on constraints.
-
- (2) The `setter' might not be a constraint. In our temperature
-example, we used `user' as the `setter'.
-
- (3) The expression-oriented format is convenient because it avoids
-the need to name the intermediate expressions in a computation. Our
-original formulation of the constraint language is cumbersome in the
-same way that many languages are cumbersome when dealing with operations
-on compound data. For example, if we wanted to compute the product (a +
-b) * (c + d), where the variables represent vectors, we could work in
-"imperative style," using procedures that set the values of designated
-vector arguments but do not themselves return vectors as values:
-
- (v-sum a b temp1)
- (v-sum c d temp2)
- (v-prod temp1 temp2 answer)
-
- Alternatively, we could deal with expressions, using procedures that
-return vectors as values, and thus avoid explicitly mentioning `temp1'
-and `temp2':
-
- (define answer (v-prod (v-sum a b) (v-sum c d)))
-
- Since Lisp allows us to return compound objects as values of
-procedures, we can transform our imperative-style constraint language
-into an expression-oriented style as shown in this exercise. In
-languages that are impoverished in handling compound objects, such as
-Algol, Basic, and Pascal (unless one explicitly uses Pascal pointer
-variables), one is usually stuck with the imperative style when
-manipulating compound objects. Given the advantage of the
-expression-oriented format, one might ask if there is any reason to have
-implemented the system in imperative style, as we did in this section.
-One reason is that the non-expression-oriented constraint language
-provides a handle on constraint objects (e.g., the value of the `adder'
-procedure) as well as on connector objects. This is useful if we wish
-to extend the system with new operations that communicate with
-constraints directly rather than only indirectly via operations on
-connectors. Although it is easy to implement the expression-oriented
-style in terms of the imperative implementation, it is very difficult
-to do the converse.
-
-\1f
-File: sicp.info, Node: 3-4, Next: 3-5, Prev: 3-3, Up: Chapter 3
-
-3.4 Concurrency: Time Is of the Essence
-=======================================
-
-We've seen the power of computational objects with local state as tools
-for modeling. Yet, as section *Note 3-1-3:: warned, this power
-extracts a price: the loss of referential transparency, giving rise to
-a thicket of questions about sameness and change, and the need to
-abandon the substitution model of evaluation in favor of the more
-intricate environment model.
-
- The central issue lurking beneath the complexity of state, sameness,
-and change is that by introducing assignment we are forced to admit "time"
-into our computational models. Before we introduced assignment, all
-our programs were timeless, in the sense that any expression that has a
-value always has the same value. In contrast, recall the example of
-modeling withdrawals from a bank account and returning the resulting
-balance, introduced at the beginning of section *Note 3-1-1:::
-
- (withdraw 25)
- 75
-
- (withdraw 25)
- 50
-
- Here successive evaluations of the same expression yield different
-values. This behavior arises from the fact that the execution of
-assignment statements (in this case, assignments to the variable
-`balance') delineates "moments in time" when values change. The result
-of evaluating an expression depends not only on the expression itself,
-but also on whether the evaluation occurs before or after these
-moments. Building models in terms of computational objects with local
-state forces us to confront time as an essential concept in programming.
-
- We can go further in structuring computational models to match our
-perception of the physical world. Objects in the world do not change
-one at a time in sequence. Rather we perceive them as acting "concurrently"--all
-at once. So it is often natural to model systems as collections of
-computational processes that execute concurrently. Just as we can make
-our programs modular by organizing models in terms of objects with
-separate local state, it is often appropriate to divide computational
-models into parts that evolve separately and concurrently. Even if the
-programs are to be executed on a sequential computer, the practice of
-writing programs as if they were to be executed concurrently forces the
-programmer to avoid inessential timing constraints and thus makes
-programs more modular.
-
- In addition to making programs more modular, concurrent computation
-can provide a speed advantage over sequential computation. Sequential
-computers execute only one operation at a time, so the amount of time
-it takes to perform a task is proportional to the total number of
-operations performed.(1) However, if it is possible to decompose a
-problem into pieces that are relatively independent and need to
-communicate only rarely, it may be possible to allocate pieces to
-separate computing processors, producing a speed advantage proportional
-to the number of processors available.
-
- Unfortunately, the complexities introduced by assignment become even
-more problematic in the presence of concurrency. The fact of
-concurrent execution, either because the world operates in parallel or
-because our computers do, entails additional complexity in our
-understanding of time.
-
-* Menu:
-
-* 3-4-1:: The Nature of Time in Concurrent Systems
-* 3-4-2:: Mechanisms for Controlling Concurrency
-
- ---------- Footnotes ----------
-
- (1) Most real processors actually execute a few operations at a
-time, following a strategy called "pipelining". Although this
-technique greatly improves the effective utilization of the hardware,
-it is used only to speed up the execution of a sequential instruction
-stream, while retaining the behavior of the sequential program.
-
-\1f
-File: sicp.info, Node: 3-4-1, Next: 3-4-2, Prev: 3-4, Up: 3-4
-
-3.4.1 The Nature of Time in Concurrent Systems
-----------------------------------------------
-
-On the surface, time seems straightforward. It is an ordering imposed
-on events.(1) For any events A and B, either A occurs before B, A and
-B are simultaneous, or A occurs after B. For instance, returning to
-the bank account example, suppose that Peter withdraws $10 and Paul
-withdraws $25 from a joint account that initially contains $100, leaving
-$65 in the account. Depending on the order of the two withdrawals, the
-sequence of balances in the account is either $100 -> $90 -> $65 or
-$100 -> $75 -> $65. In a computer implementation of the banking
-system, this changing sequence of balances could be modeled by
-successive assignments to a variable `balance'.
-
- In complex situations, however, such a view can be problematic.
-Suppose that Peter and Paul, and other people besides, are accessing
-the same bank account through a network of banking machines distributed
-all over the world. The actual sequence of balances in the account
-will depend critically on the detailed timing of the accesses and the
-details of the communication among the machines.
-
- This indeterminacy in the order of events can pose serious problems
-in the design of concurrent systems. For instance, suppose that the
-withdrawals made by Peter and Paul are implemented as two separate
-processes sharing a common variable `balance', each process specified
-by the procedure given in section *Note 3-1-1:::
-
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
-
- If the two processes operate independently, then Peter might test the
-balance and attempt to withdraw a legitimate amount. However, Paul
-might withdraw some funds in between the time that Peter checks the
-balance and the time Peter completes the withdrawal, thus invalidating
-Peter's test.
-
- Things can be worse still. Consider the expression
-
- (set! balance (- balance amount))
-
-executed as part of each withdrawal process. This consists of three
-steps: (1) accessing the value of the `balance' variable; (2) computing
-the new balance; (3) setting `balance' to this new value. If Peter and
-Paul's withdrawals execute this statement concurrently, then the two
-withdrawals might interleave the order in which they access `balance'
-and set it to the new value.
-
- The timing diagram in *Note Figure 3-29:: depicts an order of events
-where `balance' starts at 100, Peter withdraws 10, Paul withdraws 25,
-and yet the final value of `balance' is 75. As shown in the diagram,
-the reason for this anomaly is that Paul's assignment of 75 to
-`balance' is made under the assumption that the value of `balance' to
-be decremented is 100. That assumption, however, became invalid when
-Peter changed `balance' to 90. This is a catastrophic failure for the
-banking system, because the total amount of money in the system is not
-conserved. Before the transactions, the total amount of money was
-$100. Afterwards, Peter has $10, Paul has $25, and the bank has $75.(2)
-
- The general phenomenon illustrated here is that several processes
-may share a common state variable. What makes this complicated is that
-more than one process may be trying to manipulate the shared state at
-the same time. For the bank account example, during each transaction,
-each customer should be able to act as if the other customers did not
-exist. When a customer changes the balance in a way that depends on
-the balance, he must be able to assume that, just before the moment of
-change, the balance is still what he thought it was.
-
-Correct behavior of concurrent programs
-.......................................
-
-The above example typifies the subtle bugs that can creep into
-concurrent programs. The root of this complexity lies in the
-assignments to variables that are shared among the different processes.
-We already know that we must be careful in writing programs that use
-`set!', because the results of a computation depend on the order in
-which the assignments occur.(3) With concurrent processes we must be
-especially careful about assignments, because we may not be able to
-control the order of the assignments made by the different processes.
-If several such changes might be made concurrently (as with two
-depositors accessing a joint account) we need some way to ensure that
-our system behaves correctly. For example, in the case of withdrawals
-from a joint bank account, we must ensure that money is conserved. To
-make concurrent programs behave correctly, we may have to place some
-restrictions on concurrent execution.
-
- *Figure 3.29:* Timing diagram showing how interleaving the order
- of events in two banking withdrawals can lead to an incorrect
- final balance.
-
- | Peter Bank Paul
- | ____
- | / \
- | .--------------| $100 |-------------.
- | | \____/ |
- | V V
- | .----------------------. .----------------------.
- | | Access balance: $100 | | Access balance: $100 |
- | `----------+-----------' `----------+-----------'
- | V V
- | .----------------------. .----------------------.
- | | new value: 100-10=90 | | new value: 100-25=75 |
- | `----------+-----------' `----------+-----------'
- | V |
- | .----------------------. |
- | | set! balance to $90 | |
- | `----------+-----------' ____ |
- | | / \ |
- | `------------->| $ 90 | V
- | \____/ .----------------------.
- | | new value: 100-25=75 |
- | ____ `----------+-----------'
- | / \ |
- | | $ 90 |<------------'
- V \____/
- time
-
- One possible restriction on concurrency would stipulate that no two
-operations that change any shared state variables can occur at the same
-time. This is an extremely stringent requirement. For distributed
-banking, it would require the system designer to ensure that only one
-transaction could proceed at a time. This would be both inefficient
-and overly conservative. *Note Figure 3-30:: shows Peter and Paul
-sharing a bank account, where Paul has a private account as well. The
-diagram illustrates two withdrawals from the shared account (one by
-Peter and one by Paul) and a deposit to Paul's private account.(4) The
-two withdrawals from the shared account must not be concurrent (since
-both access and update the same account), and Paul's deposit and
-withdrawal must not be concurrent (since both access and update the
-amount in Paul's wallet). But there should be no problem permitting
-Paul's deposit to his private account to proceed concurrently with
-Peter's withdrawal from the shared account.
-
- *Figure 3.30:* Concurrent deposits and withdrawals from a joint
- account in Bank1 and a private account in Bank2.
-
- | Peter Bank1 Paul Bank2
- | ____ ____ ____ ____
- | / \ / \ / \ / \
- | | $7 |--. .--| $100 | | $5 |--. .--| $300 |
- | \____/ V V \____/ \____/ V V \____/
- | +---+ +---+
- | | W | | D |
- | ____ ++-++ ____ ____ ++-++ ____
- | / \ | | / \ / \ | | / \
- | | $17 |<-' `->| $90 |--. .--| $0 |<-' `->| $305 |
- | \____/ \____/ V V \____/ \____/
- | +---+
- | | W |
- | ____ ____ ++-++ ____ ____
- | / \ / \ | | / \ / \
- | | $17 | | $65 |<-' `->| $25 | | $305 |
- | \____/ \____/ \____/ \____/
- V
- time
-
- A less stringent restriction on concurrency would ensure that a
-concurrent system produces the same result as if the processes had run
-sequentially in some order. There are two important aspects to this
-requirement. First, it does not require the processes to actually run
-sequentially, but only to produce results that are the same _as if_
-they had run sequentially. For the example in *Note Figure 3-30::, the
-designer of the bank account system can safely allow Paul's deposit and
-Peter's withdrawal to happen concurrently, because the net result will
-be the same as if the two operations had happened sequentially.
-Second, there may be more than one possible "correct" result produced
-by a concurrent program, because we require only that the result be the
-same as for _some_ sequential order. For example, suppose that Peter
-and Paul's joint account starts out with $100, and Peter deposits $40
-while Paul concurrently withdraws half the money in the account. Then
-sequential execution could result in the account balance being either
-$70 or $90 (see *Note Exercise 3-38::).(5)
-
- There are still weaker requirements for correct execution of
-concurrent programs. A program for simulating diffusion (say, the flow
-of heat in an object) might consist of a large number of processes,
-each one representing a small volume of space, that update their values
-concurrently. Each process repeatedly changes its value to the average
-of its own value and its neighbors' values. This algorithm converges
-to the right answer independent of the order in which the operations
-are done; there is no need for any restrictions on concurrent use of
-the shared values.
-
- *Exercise 3.38:* Suppose that Peter, Paul, and Mary share a joint
- bank account that initially contains $100. Concurrently, Peter
- deposits $10, Paul withdraws $20, and Mary withdraws half the
- money in the account, by executing the following commands:
-
- Peter: (set! balance (+ balance 10))
- Paul: (set! balance (- balance 20))
- Mary: (set! balance (- balance (/ balance 2)))
-
- a. List all the different possible values for `balance' after
- these three transactions have been completed, assuming that
- the banking system forces the three processes to run
- sequentially in some order.
-
- b. What are some other values that could be produced if the
- system allows the processes to be interleaved? Draw timing
- diagrams like the one in *Note Figure 3-29:: to explain how
- these values can occur.
-
-
- ---------- Footnotes ----------
-
- (1) To quote some graffiti seen on a Cambridge building wall: "Time
-is a device that was invented to keep everything from happening at
-once."
-
- (2) An even worse failure for this system could occur if the two
-`set!' operations attempt to change the balance simultaneously, in
-which case the actual data appearing in memory might end up being a
-random combination of the information being written by the two
-processes. Most computers have interlocks on the primitive
-memory-write operations, which protect against such simultaneous
-access. Even this seemingly simple kind of protection, however, raises
-implementation challenges in the design of multiprocessing computers,
-where elaborate "cache-coherence" protocols are required to ensure that
-the various processors will maintain a consistent view of memory
-contents, despite the fact that data may be replicated ("cached") among
-the different processors to increase the speed of memory access.
-
- (3) The factorial program in section *Note 3-1-3:: illustrates this
-for a single sequential process.
-
- (4) The columns show the contents of Peter's wallet, the joint
-account (in Bank1), Paul's wallet, and Paul's private account (in
-Bank2), before and after each withdrawal (W) and deposit (D). Peter
-withdraws $10 from Bank1; Paul deposits $5 in Bank2, then withdraws $25
-from Bank1.
-
- (5) [Footnote 39] A more formal way to express this idea is to say
-that concurrent programs are inherently "nondeterministic". That is,
-they are described not by single-valued functions, but by functions
-whose results are sets of possible values. In section *Note 4-3:: we
-will study a language for expressing nondeterministic computations.
-
-\1f
-File: sicp.info, Node: 3-4-2, Prev: 3-4-1, Up: 3-4
-
-3.4.2 Mechanisms for Controlling Concurrency
---------------------------------------------
-
-We've seen that the difficulty in dealing with concurrent processes is
-rooted in the need to consider the interleaving of the order of events
-in the different processes. For example, suppose we have two
-processes, one with three ordered events (a,b,c) and one with three
-ordered events (x,y,z). If the two processes run concurrently, with no
-constraints on how their execution is interleaved, then there are 20
-different possible orderings for the events that are consistent with
-the individual orderings for the two processes:
-
- (a,b,c,x,y,z) (a,x,b,y,c,z) (x,a,b,c,y,z) (x,a,y,z,b,c)
- (a,b,x,c,y,z) (a,x,b,y,z,c) (x,a,b,y,c,z) (x,y,a,b,c,z)
- (a,b,x,y,c,z) (a,x,y,b,c,z) (x,a,b,y,z,c) (x,y,a,b,z,c)
- (a,b,x,y,z,c) (a,x,y,b,z,c) (x,a,y,b,c,z) (x,y,a,z,b,c)
- (a,x,b,c,y,z) (a,x,y,z,b,c) (x,a,y,b,z,c) (x,y,z,a,b,c)
-
- As programmers designing this system, we would have to consider the
-effects of each of these 20 orderings and check that each behavior is
-acceptable. Such an approach rapidly becomes unwieldy as the numbers
-of processes and events increase.
-
- A more practical approach to the design of concurrent systems is to
-devise general mechanisms that allow us to constrain the interleaving
-of concurrent processes so that we can be sure that the program
-behavior is correct. Many mechanisms have been developed for this
-purpose. In this section, we describe one of them, the "serializer".
-
-Serializing access to shared state
-..................................
-
-Serialization implements the following idea: Processes will execute
-concurrently, but there will be certain collections of procedures that
-cannot be executed concurrently. More precisely, serialization creates
-distinguished sets of procedures such that only one execution of a
-procedure in each serialized set is permitted to happen at a time. If
-some procedure in the set is being executed, then a process that
-attempts to execute any procedure in the set will be forced to wait
-until the first execution has finished.
-
- We can use serialization to control access to shared variables. For
-example, if we want to update a shared variable based on the previous
-value of that variable, we put the access to the previous value of the
-variable and the assignment of the new value to the variable in the
-same procedure. We then ensure that no other procedure that assigns to
-the variable can run concurrently with this procedure by serializing
-all of these procedures with the same serializer. This guarantees that
-the value of the variable cannot be changed between an access and the
-corresponding assignment.
-
-Serializers in Scheme
-.....................
-
-To make the above mechanism more concrete, suppose that we have
-extended Scheme to include a procedure called `parallel-execute':
-
- (parallel-execute <P_1> <P_2> ... <P_K>)
-
- Each <P> must be a procedure of no arguments. `Parallel-execute'
-creates a separate process for each <P>, which applies <P> (to no
-arguments). These processes all run concurrently.(1)
-
- As an example of how this is used, consider
-
- (define x 10)
-
- (parallel-execute (lambda () (set! x (* x x)))
- (lambda () (set! x (+ x 1))))
-
- This creates two concurrent processes--P_1, which sets `x' to `x'
-times `x', and P_2, which increments `x'. After execution is complete,
-`x' will be left with one of five possible values, depending on the
-interleaving of the events of P_1 and P_2:
-
- 101: P_1 sets `x' to 100 and then P_2 increments
- `x' to 101.
- 121: P_2 increments `x' to 11 and then P_1 sets
- `x' to `x' times `x'.
- 110: P_2 changes `x' from 10 to 11 between the two
- times that P_1 accesses the value of `x' during
- the evaluation of `(* x x)'.
- 11: P_2 accesses `x', then P_1 sets `x' to
- 100, then P_2 sets `x'.
- 100: P_1 accesses `x' (twice), then P_2 sets
- `x' to 11, then P_1 sets `x'.
-
- We can constrain the concurrency by using serialized procedures,
-which are created by "serializers". Serializers are constructed by
-`make-serializer', whose implementation is given below. A serializer
-takes a procedure as argument and returns a serialized procedure that
-behaves like the original procedure. All calls to a given serializer
-return serialized procedures in the same set.
-
- Thus, in contrast to the example above, executing
-
- (define x 10)
-
- (define s (make-serializer))
-
- (parallel-execute (s (lambda () (set! x (* x x))))
- (s (lambda () (set! x (+ x 1)))))
-
-can produce only two possible values for `x', 101 or 121. The other
-possibilities are eliminated, because the execution of P_1 and P_2
-cannot be interleaved.
-
- Here is a version of the `make-account' procedure from section *Note
-3-1-1::, where the deposits and withdrawals have been serialized:
-
- (define (make-account balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (let ((protected (make-serializer)))
- (define (dispatch m)
- (cond ((eq? m 'withdraw) (protected withdraw))
- ((eq? m 'deposit) (protected deposit))
- ((eq? m 'balance) balance)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch))
-
- With this implementation, two processes cannot be withdrawing from or
-depositing into a single account concurrently. This eliminates the
-source of the error illustrated in *Note Figure 3-29::, where Peter
-changes the account balance between the times when Paul accesses the
-balance to compute the new value and when Paul actually performs the
-assignment. On the other hand, each account has its own serializer, so
-that deposits and withdrawals for different accounts can proceed
-concurrently.
-
- *Exercise 3.39:* Which of the five possibilities in the parallel
- execution shown above remain if we instead serialize execution as
- follows:
-
- (define x 10)
-
- (define s (make-serializer))
-
- (parallel-execute (lambda () (set! x ((s (lambda () (* x x))))))
- (s (lambda () (set! x (+ x 1)))))
-
- *Exercise 3.40:* Give all possible values of `x' that can result
- from executing
-
- (define x 10)
-
- (parallel-execute (lambda () (set! x (* x x)))
- (lambda () (set! x (* x x x))))
-
- Which of these possibilities remain if we instead use serialized
- procedures:
-
- (define x 10)
-
- (define s (make-serializer))
-
- (parallel-execute (s (lambda () (set! x (* x x))))
- (s (lambda () (set! x (* x x x)))))
-
- *Exercise 3.41:* Ben Bitdiddle worries that it would be better to
- implement the bank account as follows (where the commented line
- has been changed):
-
- (define (make-account balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- ;; continued on next page
-
- (let ((protected (make-serializer)))
- (define (dispatch m)
- (cond ((eq? m 'withdraw) (protected withdraw))
- ((eq? m 'deposit) (protected deposit))
- ((eq? m 'balance)
- ((protected (lambda () balance)))) ; serialized
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch))
-
- because allowing unserialized access to the bank balance can
- result in anomalous behavior. Do you agree? Is there any
- scenario that demonstrates Ben's concern?
-
- *Exercise 3.42:* Ben Bitdiddle suggests that it's a waste of time
- to create a new serialized procedure in response to every
- `withdraw' and `deposit' message. He says that `make-account'
- could be changed so that the calls to `protected' are done outside
- the `dispatch' procedure. That is, an account would return the
- same serialized procedure (which was created at the same time as
- the account) each time it is asked for a withdrawal procedure.
-
- (define (make-account balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (let ((protected (make-serializer)))
- (let ((protected-withdraw (protected withdraw))
- (protected-deposit (protected deposit)))
- (define (dispatch m)
- (cond ((eq? m 'withdraw) protected-withdraw)
- ((eq? m 'deposit) protected-deposit)
- ((eq? m 'balance) balance)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch)))
-
- Is this a safe change to make? In particular, is there any
- difference in what concurrency is allowed by these two versions of
- `make-account' ?
-
-Complexity of using multiple shared resources
-.............................................
-
-Serializers provide a powerful abstraction that helps isolate the
-complexities of concurrent programs so that they can be dealt with
-carefully and (hopefully) correctly. However, while using serializers
-is relatively straightforward when there is only a single shared
-resource (such as a single bank account), concurrent programming can be
-treacherously difficult when there are multiple shared resources.
-
- To illustrate one of the difficulties that can arise, suppose we
-wish to swap the balances in two bank accounts. We access each account
-to find the balance, compute the difference between the balances,
-withdraw this difference from one account, and deposit it in the other
-account. We could implement this as follows:(2)
-
- (define (exchange account1 account2)
- (let ((difference (- (account1 'balance)
- (account2 'balance))))
- ((account1 'withdraw) difference)
- ((account2 'deposit) difference)))
-
- This procedure works well when only a single process is trying to do
-the exchange. Suppose, however, that Peter and Paul both have access
-to accounts a1, a2, and a3, and that Peter exchanges a1 and a2 while
-Paul concurrently exchanges a1 and a3. Even with account deposits and
-withdrawals serialized for individual accounts (as in the `make-account'
-procedure shown above in this section), `exchange' can still produce
-incorrect results. For example, Peter might compute the difference in
-the balances for a1 and a2, but then Paul might change the balance in
-a1 before Peter is able to complete the exchange.(3) For correct
-behavior, we must arrange for the `exchange' procedure to lock out any
-other concurrent accesses to the accounts during the entire time of the
-exchange.
-
- One way we can accomplish this is by using both accounts'
-serializers to serialize the entire `exchange' procedure. To do this,
-we will arrange for access to an account's serializer. Note that we
-are deliberately breaking the modularity of the bank-account object by
-exposing the serializer. The following version of `make-account' is
-identical to the original version given in section *Note 3-1-1::,
-except that a serializer is provided to protect the balance variable,
-and the serializer is exported via message passing:
-
- (define (make-account-and-serializer balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (let ((balance-serializer (make-serializer)))
- (define (dispatch m)
- (cond ((eq? m 'withdraw) withdraw)
- ((eq? m 'deposit) deposit)
- ((eq? m 'balance) balance)
- ((eq? m 'serializer) balance-serializer)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch))
-
- We can use this to do serialized deposits and withdrawals. However,
-unlike our earlier serialized account, it is now the responsibility of
-each user of bank-account objects to explicitly manage the
-serialization, for example as follows:(4)
-
- (define (deposit account amount)
- (let ((s (account 'serializer))
- (d (account 'deposit)))
- ((s d) amount)))
-
- Exporting the serializer in this way gives us enough flexibility to
-implement a serialized exchange program. We simply serialize the
-original `exchange' procedure with the serializers for both accounts:
-
- (define (serialized-exchange account1 account2)
- (let ((serializer1 (account1 'serializer))
- (serializer2 (account2 'serializer)))
- ((serializer1 (serializer2 exchange))
- account1
- account2)))
-
- *Exercise 3.43:* Suppose that the balances in three accounts start
- out as $10, $20, and $30, and that multiple processes run,
- exchanging the balances in the accounts. Argue that if the
- processes are run sequentially, after any number of concurrent
- exchanges, the account balances should be $10, $20, and $30 in
- some order. Draw a timing diagram like the one in *Note Figure
- 3-29:: to show how this condition can be violated if the exchanges
- are implemented using the first version of the account-exchange
- program in this section. On the other hand, argue that even with
- this `exchange' program, the sum of the balances in the accounts
- will be preserved. Draw a timing diagram to show how even this
- condition would be violated if we did not serialize the
- transactions on individual accounts.
-
- *Exercise 3.44:* Consider the problem of transferring an amount
- from one account to another. Ben Bitdiddle claims that this can
- be accomplished with the following procedure, even if there are
- multiple people concurrently transferring money among multiple
- accounts, using any account mechanism that serializes deposit and
- withdrawal transactions, for example, the version of
- `make-account' in the text above.
-
- (define (transfer from-account to-account amount)
- ((from-account 'withdraw) amount)
- ((to-account 'deposit) amount))
-
- Louis Reasoner claims that there is a problem here, and that we
- need to use a more sophisticated method, such as the one required
- for dealing with the exchange problem. Is Louis right? If not,
- what is the essential difference between the transfer problem and
- the exchange problem? (You should assume that the balance in
- `from-account' is at least `amount'.)
-
- *Exercise 3.45:* Louis Reasoner thinks our bank-account system is
- unnecessarily complex and error-prone now that deposits and
- withdrawals aren't automatically serialized. He suggests that
- `make-account-and-serializer' should have exported the serializer
- (for use by such procedures as `serialized-exchange') in addition
- to (rather than instead of) using it to serialize accounts and
- deposits as `make-account' did. He proposes to redefine accounts
- as follows:
-
- (define (make-account-and-serializer balance)
- (define (withdraw amount)
- (if (>= balance amount)
- (begin (set! balance (- balance amount))
- balance)
- "Insufficient funds"))
- (define (deposit amount)
- (set! balance (+ balance amount))
- balance)
- (let ((balance-serializer (make-serializer)))
- (define (dispatch m)
- (cond ((eq? m 'withdraw) (balance-serializer withdraw))
- ((eq? m 'deposit) (balance-serializer deposit))
- ((eq? m 'balance) balance)
- ((eq? m 'serializer) balance-serializer)
- (else (error "Unknown request -- MAKE-ACCOUNT"
- m))))
- dispatch))
-
- Then deposits are handled as with the original `make-account':
-
- (define (deposit account amount)
- ((account 'deposit) amount))
-
- Explain what is wrong with Louis's reasoning. In particular,
- consider what happens when `serialized-exchange' is called.
-
-Implementing serializers
-........................
-
-We implement serializers in terms of a more primitive synchronization
-mechanism called a "mutex". A mutex is an object that supports two
-operations--the mutex can be "acquired", and the mutex can be "released".
-Once a mutex has been acquired, no other acquire operations on that
-mutex may proceed until the mutex is released.(5) In our
-implementation, each serializer has an associated mutex. Given a
-procedure `p', the serializer returns a procedure that acquires the
-mutex, runs `p', and then releases the mutex. This ensures that only
-one of the procedures produced by the serializer can be running at
-once, which is precisely the serialization property that we need to
-guarantee.
-
- (define (make-serializer)
- (let ((mutex (make-mutex)))
- (lambda (p)
- (define (serialized-p . args)
- (mutex 'acquire)
- (let ((val (apply p args)))
- (mutex 'release)
- val))
- serialized-p)))
-
- The mutex is a mutable object (here we'll use a one-element list,
-which we'll refer to as a "cell") that can hold the value true or
-false. When the value is false, the mutex is available to be acquired.
-When the value is true, the mutex is unavailable, and any process that
-attempts to acquire the mutex must wait.
-
- Our mutex constructor `make-mutex' begins by initializing the cell
-contents to false. To acquire the mutex, we test the cell. If the
-mutex is available, we set the cell contents to true and proceed.
-Otherwise, we wait in a loop, attempting to acquire over and over
-again, until we find that the mutex is available.(6) To release the
-mutex, we set the cell contents to false.
-
- (define (make-mutex)
- (let ((cell (list false)))
- (define (the-mutex m)
- (cond ((eq? m 'acquire)
- (if (test-and-set! cell)
- (the-mutex 'acquire))) ; retry
- ((eq? m 'release) (clear! cell))))
- the-mutex))
-
- (define (clear! cell)
- (set-car! cell false))
-
- `Test-and-set!' tests the cell and returns the result of the test.
-In addition, if the test was false, `test-and-set!' sets the cell
-contents to true before returning false. We can express this behavior
-as the following procedure:
-
- (define (test-and-set! cell)
- (if (car cell)
- true
- (begin (set-car! cell true)
- false)))
-
- However, this implementation of `test-and-set!' does not suffice as
-it stands. There is a crucial subtlety here, which is the essential
-place where concurrency control enters the system: The `test-and-set!'
-operation must be performed "atomically". That is, we must guarantee
-that, once a process has tested the cell and found it to be false, the
-cell contents will actually be set to true before any other process can
-test the cell. If we do not make this guarantee, then the mutex can
-fail in a way similar to the bank-account failure in *Note Figure
-3-29::. (See *Note Exercise 3-46::.)
-
- The actual implementation of `test-and-set!' depends on the details
-of how our system runs concurrent processes. For example, we might be
-executing concurrent processes on a sequential processor using a
-time-slicing mechanism that cycles through the processes, permitting
-each process to run for a short time before interrupting it and moving
-on to the next process. In that case, `test-and-set!' can work by
-disabling time slicing during the testing and setting.(7)
-Alternatively, multiprocessing computers provide instructions that
-support atomic operations directly in hardware.(8)
-
- *Exercise 3.46:* Suppose that we implement `test-and-set!' using
- an ordinary procedure as shown in the text, without attempting to
- make the operation atomic. Draw a timing diagram like the one in
- *Note Figure 3-29:: to demonstrate how the mutex implementation
- can fail by allowing two processes to acquire the mutex at the
- same time.
-
- *Exercise 3.47:* A semaphore (of size n) is a generalization of a
- mutex. Like a mutex, a semaphore supports acquire and release
- operations, but it is more general in that up to n processes can
- acquire it concurrently. Additional processes that attempt to
- acquire the semaphore must wait for release operations. Give
- implementations of semaphores
-
- a. in terms of mutexes
-
- b. in terms of atomic `test-and-set!' operations.
-
-
-Deadlock
-........
-
-Now that we have seen how to implement serializers, we can see that
-account exchanging still has a problem, even with the
-`serialized-exchange' procedure above. Imagine that Peter attempts to
-exchange a1 with a2 while Paul concurrently attempts to exchange a2
-with a1. Suppose that Peter's process reaches the point where it has
-entered a serialized procedure protecting a1 and, just after that,
-Paul's process enters a serialized procedure protecting a2. Now Peter
-cannot proceed (to enter a serialized procedure protecting a2) until
-Paul exits the serialized procedure protecting a2. Similarly, Paul
-cannot proceed until Peter exits the serialized procedure protecting
-a1. Each process is stalled forever, waiting for the other. This
-situation is called a "deadlock". Deadlock is always a danger in
-systems that provide concurrent access to multiple shared resources.
-
- One way to avoid the deadlock in this situation is to give each
-account a unique identification number and rewrite
-`serialized-exchange' so that a process will always attempt to enter a
-procedure protecting the lowest-numbered account first. Although this
-method works well for the exchange problem, there are other situations
-that require more sophisticated deadlock-avoidance techniques, or where
-deadlock cannot be avoided at all. (See *Note Exercise 3-48:: and
-*Note Exercise 3-49::.)(9)
-
- *Exercise 3.48:* Explain in detail why the deadlock-avoidance
- method described above, (i.e., the accounts are numbered, and each
- process attempts to acquire the smaller-numbered account first)
- avoids deadlock in the exchange problem. Rewrite
- `serialized-exchange' to incorporate this idea. (You will also
- need to modify `make-account' so that each account is created with
- a number, which can be accessed by sending an appropriate message.)
-
- *Exercise 3.49:* Give a scenario where the deadlock-avoidance
- mechanism described above does not work. (Hint: In the exchange
- problem, each process knows in advance which accounts it will need
- to get access to. Consider a situation where a process must get
- access to some shared resources before it can know which
- additional shared resources it will require.)
-
-Concurrency, time, and communication
-....................................
-
-We've seen how programming concurrent systems requires controlling the
-ordering of events when different processes access shared state, and
-we've seen how to achieve this control through judicious use of
-serializers. But the problems of concurrency lie deeper than this,
-because, from a fundamental point of view, it's not always clear what
-is meant by "shared state."
-
- Mechanisms such as `test-and-set!' require processes to examine a
-global shared flag at arbitrary times. This is problematic and
-inefficient to implement in modern high-speed processors, where due to
-optimization techniques such as pipelining and cached memory, the
-contents of memory may not be in a consistent state at every instant.
-In contemporary multiprocessing systems, therefore, the serializer
-paradigm is being supplanted by new approaches to concurrency
-control.(10)
-
- The problematic aspects of shared state also arise in large,
-distributed systems. For instance, imagine a distributed banking
-system where individual branch banks maintain local values for bank
-balances and periodically compare these with values maintained by other
-branches. In such a system the value of "the account balance" would be
-undetermined, except right after synchronization. If Peter deposits
-money in an account he holds jointly with Paul, when should we say that
-the account balance has changed--when the balance in the local branch
-changes, or not until after the synchronization? And if Paul accesses
-the account from a different branch, what are the reasonable
-constraints to place on the banking system such that the behavior is
-"correct"? The only thing that might matter for correctness is the
-behavior observed by Peter and Paul individually and the "state" of the
-account immediately after synchronization. Questions about the "real"
-account balance or the order of events between synchronizations may be
-irrelevant or meaningless.(11)
-
- The basic phenomenon here is that synchronizing different processes,
-establishing shared state, or imposing an order on events requires
-communication among the processes. In essence, any notion of time in
-concurrency control must be intimately tied to communication.(12) It
-is intriguing that a similar connection between time and communication
-also arises in the Theory of Relativity, where the speed of light (the
-fastest signal that can be used to synchronize events) is a fundamental
-constant relating time and space. The complexities we encounter in
-dealing with time and state in our computational models may in fact
-mirror a fundamental complexity of the physical universe.
-
- ---------- Footnotes ----------
-
- (1) `Parallel-execute' is not part of standard Scheme, but it can be
-implemented in MIT Scheme. In our implementation, the new concurrent
-processes also run concurrently with the original Scheme process.
-Also, in our implementation, the value returned by `parallel-execute'
-is a special control object that can be used to halt the newly created
-processes.
-
- (2) We have simplified `exchange' by exploiting the fact that our
-`deposit' message accepts negative amounts. (This is a serious bug in
-our banking system!)
-
- (3) If the account balances start out as $10, $20, and $30, then
-after any number of concurrent exchanges, the balances should still be
-$10, $20, and $30 in some order. Serializing the deposits to
-individual accounts is not sufficient to guarantee this. See *Note
-Exercise 3-43::.
-
- (4) *Note Exercise 3-45:: investigates why deposits and withdrawals
-are no longer automatically serialized by the account.
-
- (5) The term "mutex" is an abbreviation for "mutual exclusion". The
-general problem of arranging a mechanism that permits concurrent
-processes to safely share resources is called the mutual exclusion
-problem. Our mutex is a simple variant of the "semaphore" mechanism
-(see *Note Exercise 3-47::), which was introduced in the "THE"
-Multiprogramming System developed at the Technological University of
-Eindhoven and named for the university's initials in Dutch (Dijkstra
-1968a). The acquire and release operations were originally called P
-and V, from the Dutch words _passeren_ (to pass) and _vrijgeven_ (to
-release), in reference to the semaphores used on railroad systems.
-Dijkstra's classic exposition (1968b) was one of the first to clearly
-present the issues of concurrency control, and showed how to use
-semaphores to handle a variety of concurrency problems.
-
- (6) In most time-shared operating systems, processes that are
-blocked by a mutex do not waste time "busy-waiting" as above. Instead,
-the system schedules another process to run while the first is waiting,
-and the blocked process is awakened when the mutex becomes available.
-
- (7) In MIT Scheme for a single processor, which uses a time-slicing
-model, `test-and-set!' can be implemented as follows:
-
- (define (test-and-set! cell)
- (without-interrupts
- (lambda ()
- (if (car cell)
- true
- (begin (set-car! cell true)
- false)))))
-
- `Without-interrupts' disables time-slicing interrupts while its
-procedure argument is being executed.
-
- (8) There are many variants of such instructions--including
-test-and-set, test-and-clear, swap, compare-and-exchange, load-reserve,
-and store-conditional--whose design must be carefully matched to the
-machine's processor-memory interface. One issue that arises here is to
-determine what happens if two processes attempt to acquire the same
-resource at exactly the same time by using such an instruction. This
-requires some mechanism for making a decision about which process gets
-control. Such a mechanism is called an "arbiter". Arbiters usually
-boil down to some sort of hardware device. Unfortunately, it is
-possible to prove that one cannot physically construct a fair arbiter
-that works 100% of the time unless one allows the arbiter an
-arbitrarily long time to make its decision. The fundamental phenomenon
-here was originally observed by the fourteenth-century French
-philosopher Jean Buridan in his commentary on Aristotle's De caelo.
-Buridan argued that a perfectly rational dog placed between two equally
-attractive sources of food will starve to death, because it is
-incapable of deciding which to go to first.
-
- (9) The general technique for avoiding deadlock by numbering the
-shared resources and acquiring them in order is due to Havender (1968).
-Situations where deadlock cannot be avoided require "deadlock-recovery"
-methods, which entail having processes "back out" of the deadlocked
-state and try again. Deadlock-recovery mechanisms are widely used in
-database management systems, a topic that is treated in detail in Gray
-and Reuter 1993.
-
- (10) One such alternative to serialization is called "barrier
-synchronization". The programmer permits concurrent processes to
-execute as they please, but establishes certain synchronization points
-("barriers") through which no process can proceed until all the
-processes have reached the barrier. Modern processors provide machine
-instructions that permit programmers to establish synchronization
-points at places where consistency is required. The PowerPC^( TM), for
-example, includes for this purpose two instructions called SYNC and
-EIEIO (Enforced In-order Execution of Input/Output).
-
- (11) This may seem like a strange point of view, but there are
-systems that work this way. International charges to credit-card
-accounts, for example, are normally cleared on a per-country basis, and
-the charges made in different countries are periodically reconciled.
-Thus the account balance may be different in different countries.
-
- (12) For distributed systems, this perspective was pursued by
-Lamport (1978), who showed how to use communication to establish
-"global clocks" that can be used to establish orderings on events in
-distributed systems.
-
-\1f
-File: sicp.info, Node: 3-5, Prev: 3-4, Up: Chapter 3
-
-3.5 Streams
-===========
-
-We've gained a good understanding of assignment as a tool in modeling,
-as well as an appreciation of the complex problems that assignment
-raises. It is time to ask whether we could have gone about things in a
-different way, so as to avoid some of these problems. In this section,
-we explore an alternative approach to modeling state, based on data
-structures called "streams". As we shall see, streams can mitigate
-some of the complexity of modeling state.
-
- Let's step back and review where this complexity comes from. In an
-attempt to model real-world phenomena, we made some apparently
-reasonable decisions: We modeled real-world objects with local state by
-computational objects with local variables. We identified time
-variation in the real world with time variation in the computer. We
-implemented the time variation of the states of the model objects in
-the computer with assignments to the local variables of the model
-objects.
-
- Is there another approach? Can we avoid identifying time in the
-computer with time in the modeled world? Must we make the model change
-with time in order to model phenomena in a changing world? Think about
-the issue in terms of mathematical functions. We can describe the
-time-varying behavior of a quantity x as a function of time x(t). If
-we concentrate on x instant by instant, we think of it as a changing
-quantity. Yet if we concentrate on the entire time history of values,
-we do not emphasize change--the function itself does not change.(1)
-
- If time is measured in discrete steps, then we can model a time
-function as a (possibly infinite) sequence. In this section, we will
-see how to model change in terms of sequences that represent the time
-histories of the systems being modeled. To accomplish this, we
-introduce new data structures called "streams". From an abstract point
-of view, a stream is simply a sequence. However, we will find that the
-straightforward implementation of streams as lists (as in section *Note
-2-2-1::) doesn't fully reveal the power of stream processing. As an
-alternative, we introduce the technique of "delayed evaluation", which
-enables us to represent very large (even infinite) sequences as streams.
-
- Stream processing lets us model systems that have state without ever
-using assignment or mutable data. This has important implications,
-both theoretical and practical, because we can build models that avoid
-the drawbacks inherent in introducing assignment. On the other hand,
-the stream framework raises difficulties of its own, and the question
-of which modeling technique leads to more modular and more easily
-maintained systems remains open.
-
-* Menu:
-
-* 3-5-1:: Streams Are Delayed Lists
-* 3-5-2:: Infinite Streams
-* 3-5-3:: Exploiting the Stream Paradigm
-* 3-5-4:: Streams and Delayed Evaluation
-* 3-5-5:: Modularity of Functional Programs and Modularity of
- Objects
-
- ---------- Footnotes ----------
-
- (1) Physicists sometimes adopt this view by introducing the "world
-lines" of particles as a device for reasoning about motion. We've also
-already mentioned (section *Note 2-2-3::) that this is the natural way
-to think about signal-processing systems. We will explore applications
-of streams to signal processing in section *Note 3-5-3::.
-
-\1f
-File: sicp.info, Node: 3-5-1, Next: 3-5-2, Prev: 3-5, Up: 3-5
-
-3.5.1 Streams Are Delayed Lists
--------------------------------
-
-As we saw in section *Note 2-2-3::, sequences can serve as standard
-interfaces for combining program modules. We formulated powerful
-abstractions for manipulating sequences, such as `map', `filter', and
-`accumulate', that capture a wide variety of operations in a manner that
-is both succinct and elegant.
-
- Unfortunately, if we represent sequences as lists, this elegance is
-bought at the price of severe inefficiency with respect to both the
-time and space required by our computations. When we represent
-manipulations on sequences as transformations of lists, our programs
-must construct and copy data structures (which may be huge) at every
-step of a process.
-
- To see why this is true, let us compare two programs for computing
-the sum of all the prime numbers in an interval. The first program is
-written in standard iterative style:(1)
-
- (define (sum-primes a b)
- (define (iter count accum)
- (cond ((> count b) accum)
- ((prime? count) (iter (+ count 1) (+ count accum)))
- (else (iter (+ count 1) accum))))
- (iter a 0))
-
- The second program performs the same computation using the sequence
-operations of section *Note 2-2-3:::
-
- (define (sum-primes a b)
- (accumulate +
- 0
- (filter prime? (enumerate-interval a b))))
-
- In carrying out the computation, the first program needs to store
-only the sum being accumulated. In contrast, the filter in the second
-program cannot do any testing until `enumerate-interval' has
-constructed a complete list of the numbers in the interval. The filter
-generates another list, which in turn is passed to `accumulate' before
-being collapsed to form a sum. Such large intermediate storage is not
-needed by the first program, which we can think of as enumerating the
-interval incrementally, adding each prime to the sum as it is generated.
-
- The inefficiency in using lists becomes painfully apparent if we use
-the sequence paradigm to compute the second prime in the interval from
-10,000 to 1,000,000 by evaluating the expression
-
- (car (cdr (filter prime?
- (enumerate-interval 10000 1000000))))
-
- This expression does find the second prime, but the computational
-overhead is outrageous. We construct a list of almost a million
-integers, filter this list by testing each element for primality, and
-then ignore almost all of the result. In a more traditional
-programming style, we would interleave the enumeration and the
-filtering, and stop when we reached the second prime.
-
- Streams are a clever idea that allows one to use sequence
-manipulations without incurring the costs of manipulating sequences as
-lists. With streams we can achieve the best of both worlds: We can
-formulate programs elegantly as sequence manipulations, while attaining
-the efficiency of incremental computation. The basic idea is to
-arrange to construct a stream only partially, and to pass the partial
-construction to the program that consumes the stream. If the consumer
-attempts to access a part of the stream that has not yet been
-constructed, the stream will automatically construct just enough more
-of itself to produce the required part, thus preserving the illusion
-that the entire stream exists. In other words, although we will write
-programs as if we were processing complete sequences, we design our
-stream implementation to automatically and transparently interleave the
-construction of the stream with its use.
-
- On the surface, streams are just lists with different names for the
-procedures that manipulate them. There is a constructor,
-`cons-stream', and two selectors, `stream-car' and `stream-cdr', which
-satisfy the constraints
-
- (stream-car (cons-stream x y)) = x
- (stream-cdr (cons-stream x y)) = y
-
- There is a distinguishable object, `the-empty-stream', which cannot
-be the result of any `cons-stream' operation, and which can be
-identified with the predicate `stream-null?'.(2) Thus we can make and
-use streams, in just the same way as we can make and use lists, to
-represent aggregate data arranged in a sequence. In particular, we can
-build stream analogs of the list operations from *Note Chapter 2::,
-such as `list-ref', `map', and `for-each':(3)
-
- (define (stream-ref s n)
- (if (= n 0)
- (stream-car s)
- (stream-ref (stream-cdr s) (- n 1))))
-
- (define (stream-map proc s)
- (if (stream-null? s)
- the-empty-stream
- (cons-stream (proc (stream-car s))
- (stream-map proc (stream-cdr s)))))
-
- (define (stream-for-each proc s)
- (if (stream-null? s)
- 'done
- (begin (proc (stream-car s))
- (stream-for-each proc (stream-cdr s)))))
-
- `Stream-for-each' is useful for viewing streams:
-
- (define (display-stream s)
- (stream-for-each display-line s))
-
- (define (display-line x)
- (newline)
- (display x))
-
- To make the stream implementation automatically and transparently
-interleave the construction of a stream with its use, we will arrange
-for the `cdr' of a stream to be evaluated when it is accessed by the
-`stream-cdr' procedure rather than when the stream is constructed by
-`cons-stream'. This implementation choice is reminiscent of our
-discussion of rational numbers in section *Note 2-1-2::, where we saw
-that we can choose to implement rational numbers so that the reduction
-of numerator and denominator to lowest terms is performed either at
-construction time or at selection time. The two rational-number
-implementations produce the same data abstraction, but the choice has
-an effect on efficiency. There is a similar relationship between
-streams and ordinary lists. As a data abstraction, streams are the
-same as lists. The difference is the time at which the elements are
-evaluated. With ordinary lists, both the `car' and the `cdr' are
-evaluated at construction time. With streams, the `cdr' is evaluated
-at selection time.
-
- Our implementation of streams will be based on a special form called
-`delay'. Evaluating `(delay <EXP>)' does not evaluate the expression
-<EXP>, but rather returns a so-called object "delayed object", which we
-can think of as a "promise" to evaluate <EXP> at some future time. As
-a companion to `delay', there is a procedure called `force' that takes
-a delayed object as argument and performs the evaluation--in effect,
-forcing the `delay' to fulfill its promise. We will see below how
-`delay' and `force' can be implemented, but first let us use these to
-construct streams.
-
- `Cons-stream' is a special form defined so that
-
- (cons-stream <A> <B>)
-
-is equivalent to
-
- (cons <A> (delay <B>))
-
- What this means is that we will construct streams using pairs.
-However, rather than placing the value of the rest of the stream into
-the `cdr' of the pair we will put there a promise to compute the rest
-if it is ever requested. `Stream-car' and `stream-cdr' can now be
-defined as procedures:
-
- (define (stream-car stream) (car stream))
-
- (define (stream-cdr stream) (force (cdr stream)))
-
- `Stream-car' selects the `car' of the pair; `stream-cdr' selects the
-`cdr' of the pair and evaluates the delayed expression found there to
-obtain the rest of the stream.(4)
-
-The stream implementation in action
-...................................
-
-To see how this implementation behaves, let us analyze the "outrageous"
-prime computation we saw above, reformulated in terms of streams:
-
- (stream-car
- (stream-cdr
- (stream-filter prime?
- (stream-enumerate-interval 10000 1000000))))
-
- We will see that it does indeed work efficiently.
-
- We begin by calling `stream-enumerate-interval' with the arguments
-10,000 and 1,000,000. `Stream-enumerate-interval' is the stream analog
-of `enumerate-interval' (section *Note 2-2-3::):
-
- (define (stream-enumerate-interval low high)
- (if (> low high)
- the-empty-stream
- (cons-stream
- low
- (stream-enumerate-interval (+ low 1) high))))
-
-and thus the result returned by `stream-enumerate-interval', formed by
-the `cons-stream', is(5)
-
- (cons 10000
- (delay (stream-enumerate-interval 10001 1000000)))
-
- That is, `stream-enumerate-interval' returns a stream represented as
-a pair whose `car' is 10,000 and whose `cdr' is a promise to enumerate
-more of the interval if so requested. This stream is now filtered for
-primes, using the stream analog of the `filter' procedure (section
-*Note 2-2-3::):
-
- (define (stream-filter pred stream)
- (cond ((stream-null? stream) the-empty-stream)
- ((pred (stream-car stream))
- (cons-stream (stream-car stream)
- (stream-filter pred
- (stream-cdr stream))))
- (else (stream-filter pred (stream-cdr stream)))))
-
- `Stream-filter' tests the `stream-car' of the stream (the `car' of
-the pair, which is 10,000). Since this is not prime, `stream-filter'
-examines the `stream-cdr' of its input stream. The call to
-`stream-cdr' forces evaluation of the delayed
-`stream-enumerate-interval', which now returns
-
- (cons 10001
- (delay (stream-enumerate-interval 10002 1000000)))
-
- `Stream-filter' now looks at the `stream-car' of this stream, 10,001,
-sees that this is not prime either, forces another `stream-cdr', and so
-on, until `stream-enumerate-interval' yields the prime 10,007, whereupon
-`stream-filter', according to its definition, returns
-
- (cons-stream (stream-car stream)
- (stream-filter pred (stream-cdr stream)))
-
-which in this case is
-
- (cons 10007
- (delay
- (stream-filter
- prime?
- (cons 10008
- (delay
- (stream-enumerate-interval 10009
- 1000000))))))
-
- This result is now passed to `stream-cdr' in our original expression.
-This forces the delayed `stream-filter', which in turn keeps forcing the
-delayed `stream-enumerate-interval' until it finds the next prime, which
-is 10,009. Finally, the result passed to `stream-car' in our original
-expression is
-
- (cons 10009
- (delay
- (stream-filter
- prime?
- (cons 10010
- (delay
- (stream-enumerate-interval 10011
- 1000000))))))
-
- `Stream-car' returns 10,009, and the computation is complete. Only
-as many integers were tested for primality as were necessary to find
-the second prime, and the interval was enumerated only as far as was
-necessary to feed the prime filter.
-
- In general, we can think of delayed evaluation as "demand-driven"
-programming, whereby each stage in the stream process is activated only
-enough to satisfy the next stage. What we have done is to decouple the
-actual order of events in the computation from the apparent structure
-of our procedures. We write procedures as if the streams existed "all
-at once" when, in reality, the computation is performed incrementally,
-as in traditional programming styles.
-
-Implementing `delay' and `force'
-................................
-
-Although `delay' and `force' may seem like mysterious operations, their
-implementation is really quite straightforward. `Delay' must package
-an expression so that it can be evaluated later on demand, and we can
-accomplish this simply by treating the expression as the body of a
-procedure. `Delay' can be a special form such that
-
- (delay <EXP>)
-
-is syntactic sugar for
-
- (lambda () <EXP>)
-
- `Force' simply calls the procedure (of no arguments) produced by
-`delay', so we can implement `force' as a procedure:
-
- (define (force delayed-object)
- (delayed-object))
-
- This implementation suffices for `delay' and `force' to work as
-advertised, but there is an important optimization that we can include.
-In many applications, we end up forcing the same delayed object many
-times. This can lead to serious inefficiency in recursive programs
-involving streams. (See *Note Exercise 3-57::.) The solution is to
-build delayed objects so that the first time they are forced, they
-store the value that is computed. Subsequent forcings will simply
-return the stored value without repeating the computation. In other
-words, we implement `delay' as a special-purpose memoized procedure
-similar to the one described in *Note Exercise 3-27::. One way to
-accomplish this is to use the following procedure, which takes as
-argument a procedure (of no arguments) and returns a memoized version
-of the procedure. The first time the memoized procedure is run, it
-saves the computed result. On subsequent evaluations, it simply
-returns the result.
-
- (define (memo-proc proc)
- (let ((already-run? false) (result false))
- (lambda ()
- (if (not already-run?)
- (begin (set! result (proc))
- (set! already-run? true)
- result)
- result))))
-
- `Delay' is then defined so that `(delay <EXP>)' is equivalent to
-
- (memo-proc (lambda () <EXP>))
-
-and `force' is as defined previously.(6)
-
- *Exercise 3.50:* Complete the following definition, which
- generalizes `stream-map' to allow procedures that take multiple
- arguments, analogous to `map' in section *Note 2-2-3::, footnote
- *Note Footnote 12::.
-
- (define (stream-map proc . argstreams)
- (if (<??> (car argstreams))
- the-empty-stream
- (<??>
- (apply proc (map <??> argstreams))
- (apply stream-map
- (cons proc (map <??> argstreams))))))
-
- *Exercise 3.51:* In order to take a closer look at delayed
- evaluation, we will use the following procedure, which simply
- returns its argument after printing it:
-
- (define (show x)
- (display-line x)
- x)
-
- What does the interpreter print in response to evaluating each
- expression in the following sequence?(7)
-
- (define x (stream-map show (stream-enumerate-interval 0 10)))
-
- (stream-ref x 5)
-
- (stream-ref x 7)
-
- *Exercise 3.52:* Consider the sequence of expressions
-
- (define sum 0)
-
- (define (accum x)
- (set! sum (+ x sum))
- sum)
-
- (define seq (stream-map accum (stream-enumerate-interval 1 20)))
- (define y (stream-filter even? seq))
- (define z (stream-filter (lambda (x) (= (remainder x 5) 0))
- seq))
-
- (stream-ref y 7)
-
- (display-stream z)
-
- What is the value of `sum' after each of the above expressions is
- evaluated? What is the printed response to evaluating the
- `stream-ref' and `display-stream' expressions? Would these
- responses differ if we had implemented `(delay <EXP>)' simply as
- `(lambda () <EXP>)' without using the optimization provided by
- `memo-proc'? Explain
-
- ---------- Footnotes ----------
-
- (1) Assume that we have a predicate `prime?' (e.g., as in section
-*Note 1-2-6::) that tests for primality.
-
- (2) In the MIT implementation, `the-empty-stream' is the same as the
-empty list `'()', and `stream-null?' is the same as `null?'.
-
- (3) This should bother you. The fact that we are defining such
-similar procedures for streams and lists indicates that we are missing
-some underlying abstraction. Unfortunately, in order to exploit this
-abstraction, we will need to exert finer control over the process of
-evaluation than we can at present. We will discuss this point further
-at the end of section *Note 3-5-4::. In section *Note 4-2::, we'll
-develop a framework that unifies lists and streams.
-
- (4) Although `stream-car' and `stream-cdr' can be defined as
-procedures, `cons-stream' must be a special form. If `cons-stream'
-were a procedure, then, according to our model of evaluation,
-evaluating `(cons-stream <A> <B>)' would automatically cause <B> to be
-evaluated, which is precisely what we do not want to happen. For the
-same reason, `delay' must be a special form, though `force' can be an
-ordinary procedure.
-
- (5) The numbers shown here do not really appear in the delayed
-expression. What actually appears is the original expression, in an
-environment in which the variables are bound to the appropriate numbers.
-For example, `(+ low 1)' with `low' bound to 10,000 actually appears
-where `10001' is shown.
-
- (6) There are many possible implementations of streams other than
-the one described in this section. Delayed evaluation, which is the
-key to making streams practical, was inherent in Algol 60's "call-by-name"
-parameter-passing method. The use of this mechanism to implement
-streams was first described by Landin (1965). Delayed evaluation for
-streams was introduced into Lisp by Friedman and Wise (1976). In their
-implementation, `cons' always delays evaluating its arguments, so that
-lists automatically behave as streams. The memoizing optimization is
-also known as "call-by-need". The Algol community would refer to our
-original delayed objects as "call-by-name thunks" and to the optimized
-versions as "call-by-need thunks".
-
- (7) Exercises such as *Note Exercise 3-51:: and *Note Exercise
-3-52:: are valuable for testing our understanding of how `delay' works.
-On the other hand, intermixing delayed evaluation with printing--and,
-even worse, with assignment--is extremely confusing, and instructors of
-courses on computer languages have traditionally tormented their
-students with examination questions such as the ones in this section.
-Needless to say, writing programs that depend on such subtleties is
-odious programming style. Part of the power of stream processing is
-that it lets us ignore the order in which events actually happen in our
-programs. Unfortunately, this is precisely what we cannot afford to do
-in the presence of assignment, which forces us to be concerned with
-time and change.
-
-\1f
-File: sicp.info, Node: 3-5-2, Next: 3-5-3, Prev: 3-5-1, Up: 3-5
-
-3.5.2 Infinite Streams
-----------------------
-
-We have seen how to support the illusion of manipulating streams as
-complete entities even though, in actuality, we compute only as much of
-the stream as we need to access. We can exploit this technique to
-represent sequences efficiently as streams, even if the sequences are
-very long. What is more striking, we can use streams to represent
-sequences that are infinitely long. For instance, consider the
-following definition of the stream of positive integers:
-
- (define (integers-starting-from n)
- (cons-stream n (integers-starting-from (+ n 1))))
-
- (define integers (integers-starting-from 1))
-
- This makes sense because `integers' will be a pair whose `car' is 1
-and whose `cdr' is a promise to produce the integers beginning with 2.
-This is an infinitely long stream, but in any given time we can examine
-only a finite portion of it. Thus, our programs will never know that
-the entire infinite stream is not there.
-
- Using `integers' we can define other infinite streams, such as the
-stream of integers that are not divisible by 7:
-
- (define (divisible? x y) (= (remainder x y) 0))
-
- (define no-sevens
- (stream-filter (lambda (x) (not (divisible? x 7)))
- integers))
-
- Then we can find integers not divisible by 7 simply by accessing
-elements of this stream:
-
- (stream-ref no-sevens 100)
- 117
-
- In analogy with `integers', we can define the infinite stream of
-Fibonacci numbers:
-
- (define (fibgen a b)
- (cons-stream a (fibgen b (+ a b))))
-
- (define fibs (fibgen 0 1))
-
- `Fibs' is a pair whose `car' is 0 and whose `cdr' is a promise to
-evaluate `(fibgen 1 1)'. When we evaluate this delayed `(fibgen 1 1)',
-it will produce a pair whose `car' is 1 and whose `cdr' is a promise to
-evaluate `(fibgen 1 2)', and so on.
-
- For a look at a more exciting infinite stream, we can generalize the
-`no-sevens' example to construct the infinite stream of prime numbers,
-using a method known as the Eratosthenes "sieve of Eratosthenes".(1) We
-start with the integers beginning with 2, which is the first prime. To
-get the rest of the primes, we start by filtering the multiples of 2
-from the rest of the integers. This leaves a stream beginning with 3,
-which is the next prime. Now we filter the multiples of 3 from the
-rest of this stream. This leaves a stream beginning with 5, which is
-the next prime, and so on. In other words, we construct the primes by
-a sieving process, described as follows: To sieve a stream `S', form a
-stream whose first element is the first element of `S' and the rest of
-which is obtained by filtering all multiples of the first element of
-`S' out of the rest of `S' and sieving the result. This process is
-readily described in terms of stream operations:
-
- (define (sieve stream)
- (cons-stream
- (stream-car stream)
- (sieve (stream-filter
- (lambda (x)
- (not (divisible? x (stream-car stream))))
- (stream-cdr stream)))))
-
- (define primes (sieve (integers-starting-from 2)))
-
- Now to find a particular prime we need only ask for it:
-
- (stream-ref primes 50)
- 233
-
- It is interesting to contemplate the signal-processing system set up
-by `sieve', shown in the "Henderson diagram" in *Note Figure 3-31::.(2)
-The input stream feeds into an "un`cons'er" that separates the first
-element of the stream from the rest of the stream. The first element
-is used to construct a divisibility filter, through which the rest is
-passed, and the output of the filter is fed to another sieve box. Then
-the original first element is `cons'ed onto the output of the internal
-sieve to form the output stream. Thus, not only is the stream
-infinite, but the signal processor is also infinite, because the sieve
-contains a sieve within it.
-
- *Figure 3.31:* The prime sieve viewed as a signal-processing
- system.
-
- +---------------------------------------------------------------+
- | sieve |
- | |
- | __/| |\__ |
- | __/car|........................................| \__ |
- | _/ | : | \_ |
- ----><_ | V | cons _>---->
- | \__ | +------------+ +------------+ | __/ |
- | \cdr|--->| filter: | | sieve |--->| __/ |
- | \| | |--->| | |/ |
- | | not | | | |
- | | divisible? | | | |
- | +------------+ +------------+ |
- +---------------------------------------------------------------+
-
-Defining streams implicitly
-...........................
-
-The `integers' and `fibs' streams above were defined by specifying
-"generating" procedures that explicitly compute the stream elements one
-by one. An alternative way to specify streams is to take advantage of
-delayed evaluation to define streams implicitly. For example, the
-following expression defines the stream `ones' to be an infinite stream
-of ones:
-
- (define ones (cons-stream 1 ones))
-
- This works much like the definition of a recursive procedure: `ones'
-is a pair whose `car' is 1 and whose `cdr' is a promise to evaluate
-`ones'. Evaluating the `cdr' gives us again a 1 and a promise to
-evaluate `ones', and so on.
-
- We can do more interesting things by manipulating streams with
-operations such as `add-streams', which produces the elementwise sum of
-two given streams:(3)
-
- (define (add-streams s1 s2)
- (stream-map + s1 s2))
-
- Now we can define the integers as follows:
-
- (define integers (cons-stream 1 (add-streams ones integers)))
-
- This defines `integers' to be a stream whose first element is 1 and
-the rest of which is the sum of `ones' and `integers'. Thus, the second
-element of `integers' is 1 plus the first element of `integers', or 2;
-the third element of `integers' is 1 plus the second element of
-`integers', or 3; and so on. This definition works because, at any
-point, enough of the `integers' stream has been generated so that we
-can feed it back into the definition to produce the next integer.
-
- We can define the Fibonacci numbers in the same style:
-
- (define fibs
- (cons-stream 0
- (cons-stream 1
- (add-streams (stream-cdr fibs)
- fibs))))
-
- This definition says that `fibs' is a stream beginning with 0 and 1,
-such that the rest of the stream can be generated by adding `fibs' to
-itself shifted by one place:
-
- 1 1 2 3 5 8 13 21 ... = `(stream-cdr fibs)'
- 0 1 1 2 3 5 8 13 ... = `fibs'
- 0 1 1 2 3 5 8 13 21 34 ... = `fibs'
-
- `Scale-stream' is another useful procedure in formulating such stream
-definitions. This multiplies each item in a stream by a given constant:
-
- (define (scale-stream stream factor)
- (stream-map (lambda (x) (* x factor)) stream))
-
- For example,
-
- (define double (cons-stream 1 (scale-stream double 2)))
-
-produces the stream of powers of 2: 1, 2, 4, 8, 16, 32, ....
-
- An alternate definition of the stream of primes can be given by
-starting with the integers and filtering them by testing for primality.
-We will need the first prime, 2, to get started:
-
- (define primes
- (cons-stream
- 2
- (stream-filter prime? (integers-starting-from 3))))
-
- This definition is not so straightforward as it appears, because we
-will test whether a number n is prime by checking whether n is
-divisible by a prime (not by just any integer) less than or equal to
-_[sqrt]_(n):
-
- (define (prime? n)
- (define (iter ps)
- (cond ((> (square (stream-car ps)) n) true)
- ((divisible? n (stream-car ps)) false)
- (else (iter (stream-cdr ps)))))
- (iter primes))
-
- This is a recursive definition, since `primes' is defined in terms
-of the `prime?' predicate, which itself uses the `primes' stream. The
-reason this procedure works is that, at any point, enough of the
-`primes' stream has been generated to test the primality of the numbers
-we need to check next. That is, for every n we test for primality,
-either n is not prime (in which case there is a prime already generated
-that divides it) or n is prime (in which case there is a prime already
-generated--i.e., a prime less than n--that is greater than
-_[sqrt]_(n)).(4)
-
- *Exercise 3.53:* Without running the program, describe the
- elements of the stream defined by
-
- (define s (cons-stream 1 (add-streams s s)))
-
- *Exercise 3.54:* Define a procedure `mul-streams', analogous to
- `add-streams', that produces the elementwise product of its two
- input streams. Use this together with the stream of `integers' to
- complete the following definition of the stream whose nth element
- (counting from 0) is n + 1 factorial:
-
- (define factorials (cons-stream 1 (mul-streams <??> <??>)))
-
- *Exercise 3.55:* Define a procedure `partial-sums' that takes as
- argument a stream S and returns the stream whose elements are S_0,
- S_0 + S_1, S_0 + S_1 + S_2, .... For example, `(partial-sums
- integers)' should be the stream 1, 3, 6, 10, 15, ....
-
- *Exercise 3.56:* A famous problem, first raised by R. Hamming, is
- to enumerate, in ascending order with no repetitions, all positive
- integers with no prime factors other than 2, 3, or 5. One obvious
- way to do this is to simply test each integer in turn to see
- whether it has any factors other than 2, 3, and 5. But this is
- very inefficient, since, as the integers get larger, fewer and
- fewer of them fit the requirement. As an alternative, let us call
- the required stream of numbers `S' and notice the following facts
- about it.
-
- * `S' begins with 1.
-
- * The elements of `(scale-stream S 2)' are also elements of `S'.
-
- * The same is true for `(scale-stream S 3)' and `(scale-stream
- 5 S)'.
-
- * These are all the elements of `S'.
-
-
- Now all we have to do is combine elements from these sources. For
- this we define a procedure `merge' that combines two ordered
- streams into one ordered result stream, eliminating repetitions:
-
- (define (merge s1 s2)
- (cond ((stream-null? s1) s2)
- ((stream-null? s2) s1)
- (else
- (let ((s1car (stream-car s1))
- (s2car (stream-car s2)))
- (cond ((< s1car s2car)
- (cons-stream s1car (merge (stream-cdr s1) s2)))
- ((> s1car s2car)
- (cons-stream s2car (merge s1 (stream-cdr s2))))
- (else
- (cons-stream s1car
- (merge (stream-cdr s1)
- (stream-cdr s2)))))))))
-
- Then the required stream may be constructed with `merge', as
- follows:
-
- (define S (cons-stream 1 (merge <??> <??>)))
-
- Fill in the missing expressions in the places marked <??> above.
-
- *Exercise 3.57:* How many additions are performed when we compute
- the nth Fibonacci number using the definition of `fibs' based on
- the `add-streams' procedure? Show that the number of additions
- would be exponentially greater if we had implemented `(delay
- <EXP>)' simply as `(lambda () <EXP>)', without using the
- optimization provided by the `memo-proc' procedure described in
- section *Note 3-5-1::.(5)
-
- *Exercise 3.58:* Give an interpretation of the stream computed by
- the following procedure:
-
- (define (expand num den radix)
- (cons-stream
- (quotient (* num radix) den)
- (expand (remainder (* num radix) den) den radix)))
-
- (`Quotient' is a primitive that returns the integer quotient of two
- integers.) What are the successive elements produced by `(expand
- 1 7 10)'? What is produced by `(expand 3 8 10)'?
-
- *Exercise 3.59:* In section *Note 2-5-3:: we saw how to implement
- a polynomial arithmetic system representing polynomials as lists
- of terms. In a similar way, we can work with "power series", such
- as
-
- x^2 x^3 x^4
- e^x = 1 + x + ----- + ----- + --------- + ...
- 2 3 * 2 4 * 3 * 2
-
- x^2 x^4
- cos x = 1 - ----- + --------- - ...
- 2 4 * 3 * 2
-
- x^3 x^5
- sin x = x - ----- + ------------- - ...
- 3 * 2 5 * 4 * 3 * 2
-
- represented as infinite streams. We will represent the series a_0
- + a_1 x + a_2 x^2 + a_3 x^3 + ... as the stream whose elements are
- the coefficients a_0, a_1, a_2, a_3, ....
-
- a. The integral of the series a_0 + a_1 x + a_2 x^2 + a_3 x^3 +
- ... is the series
-
- 1 1 1
- c + a_0 x + --- x_1 r^2 + --- a_2 r^3 + --- a_3 r^4 + ...
- 2 3 4
-
- where c is any constant. Define a procedure
- `integrate-series' that takes as input a stream a_0, a_1,
- a_2, ... representing a power series and returns the stream
- a_0, (1/2)a_1, (1/3)a_2, ... of coefficients of the
- non-constant terms of the integral of the series. (Since the
- result has no constant term, it doesn't represent a power
- series; when we use `integrate-series', we will `cons' on the
- appropriate constant.)
-
- b. The function x |-> e^x is its own derivative. This implies
- that e^x and the integral of e^x are the same series, except
- for the constant term, which is e^0 = 1. Accordingly, we can
- generate the series for e^x as
-
- (define exp-series
- (cons-stream 1 (integrate-series exp-series)))
-
- Show how to generate the series for sine and cosine, starting
- from the facts that the derivative of sine is cosine and the
- derivative of cosine is the negative of sine:
-
- (define cosine-series
- (cons-stream 1 <??>))
-
- (define sine-series
- (cons-stream 0 <??>))
-
- *Exercise 3.60:* With power series represented as streams of
- coefficients as in *Note Exercise 3-59::, adding series is
- implemented by `add-streams'. Complete the definition of the
- following procedure for multiplying series:
-
- (define (mul-series s1 s2)
- (cons-stream <??> (add-streams <??> <??>)))
-
- You can test your procedure by verifying that sin^2 x + cos^2 x =
- 1, using the series from *Note Exercise 3-59::.
-
- *Exercise 3.61:* Let S be a power series (*Note Exercise 3-59::)
- whose constant term is 1. Suppose we want to find the power
- series 1/S, that is, the series X such that S * X = 1. Write S =
- 1 + S_R where S_R is the part of S after the constant term. Then
- we can solve for X as follows:
-
- S * X = 1
- (1 + S_R) * X = 1
- X + S_R * X = 1
- X = 1 - S_R * X
-
- In other words, X is the power series whose constant term is 1 and
- whose higher-order terms are given by the negative of S_R times X.
- Use this idea to write a procedure `invert-unit-series' that
- computes 1/S for a power series S with constant term 1. You will
- need to use `mul-series' from *Note Exercise 3-60::.
-
- *Exercise 3.62:* Use the results of *Note Exercise 3-60:: and
- *Note Exercise 3-61:: to define a procedure `div-series' that
- divides two power series. `Div-series' should work for any two
- series, provided that the denominator series begins with a nonzero
- constant term. (If the denominator has a zero constant term, then
- `div-series' should signal an error.) Show how to use
- `div-series' together with the result of *Note Exercise 3-59:: to
- generate the power series for tangent.
-
- ---------- Footnotes ----------
-
- (1) Eratosthenes, a third-century B.C. Alexandrian Greek
-philosopher, is famous for giving the first accurate estimate of the
-circumference of the Earth, which he computed by observing shadows cast
-at noon on the day of the summer solstice. Eratosthenes's sieve method,
-although ancient, has formed the basis for special-purpose hardware
-"sieves" that, until recently, were the most powerful tools in
-existence for locating large primes. Since the 70s, however, these
-methods have been superseded by outgrowths of the probabilistic
-techniques discussed in section *Note 1-2-6::.
-
- (2) We have named these figures after Peter Henderson, who was the
-first person to show us diagrams of this sort as a way of thinking
-about stream processing. Each solid line represents a stream of values
-being transmitted. The dashed line from the `car' to the `cons' and
-the `filter' indicates that this is a single value rather than a stream.
-
- (3) This uses the generalized version of `stream-map' from *Note
-Exercise 3-50::.
-
- (4) This last point is very subtle and relies on the fact that
-p_(n+1) <= p_n^2. (Here, p_k denotes the kth prime.) Estimates such
-as these are very difficult to establish. The ancient proof by Euclid
-that there are an infinite number of primes shows that p_(n+1)<= p_1
-p_2...p_n + 1, and no substantially better result was proved until
-1851, when the Russian mathematician P. L. Chebyshev established that
-p_(n+1)<= 2p_n for all n. This result, originally conjectured in 1845,
-is known as hypothesis "Bertrand's hypothesis". A proof can be found
-in section 22.3 of Hardy and Wright 1960.
-
- (5) This exercise shows how call-by-need is closely related to
-ordinary memoization as described in *Note Exercise 3-27::. In that
-exercise, we used assignment to explicitly construct a local table.
-Our call-by-need stream optimization effectively constructs such a
-table automatically, storing values in the previously forced parts of
-the stream.
-
-\1f
-File: sicp.info, Node: 3-5-3, Next: 3-5-4, Prev: 3-5-2, Up: 3-5
-
-3.5.3 Exploiting the Stream Paradigm
-------------------------------------
-
-Streams with delayed evaluation can be a powerful modeling tool,
-providing many of the benefits of local state and assignment.
-Moreover, they avoid some of the theoretical tangles that accompany the
-introduction of assignment into a programming language.
-
- The stream approach can be illuminating because it allows us to
-build systems with different module boundaries than systems organized
-around assignment to state variables. For example, we can think of an
-entire time series (or signal) as a focus of interest, rather than the
-values of the state variables at individual moments. This makes it
-convenient to combine and compare components of state from different
-moments.
-
-Formulating iterations as stream processes
-..........................................
-
-In section *Note 1-2-1::, we introduced iterative processes, which
-proceed by updating state variables. We know now that we can represent
-state as a "timeless" stream of values rather than as a set of
-variables to be updated. Let's adopt this perspective in revisiting
-the square-root procedure from section *Note 1-1-7::. Recall that the
-idea is to generate a sequence of better and better guesses for the
-square root of x by applying over and over again the procedure that
-improves guesses:
-
- (define (sqrt-improve guess x)
- (average guess (/ x guess)))
-
- In our original `sqrt' procedure, we made these guesses be the
-successive values of a state variable. Instead we can generate the
-infinite stream of guesses, starting with an initial guess of 1:(1)
-
- (define (sqrt-stream x)
- (define guesses
- (cons-stream 1.0
- (stream-map (lambda (guess)
- (sqrt-improve guess x))
- guesses)))
- guesses)
-
- (display-stream (sqrt-stream 2))
- 1.
- 1.5
- 1.4166666666666665
- 1.4142156862745097
- 1.4142135623746899
- ...
-
- We can generate more and more terms of the stream to get better and
-better guesses. If we like, we can write a procedure that keeps
-generating terms until the answer is good enough. (See *Note Exercise
-3-64::.)
-
- Another iteration that we can treat in the same way is to generate an
-approximation to [pi], based upon the alternating series that we saw in
-section *Note 1-3-1:::
-
- [pi] 1 1 1
- ---- = 1 - --- + --- - --- + ...
- 4 3 5 7
-
- We first generate the stream of summands of the series (the
-reciprocals of the odd integers, with alternating signs). Then we take
-the stream of sums of more and more terms (using the `partial-sums'
-procedure of *Note Exercise 3-55::) and scale the result by 4:
-
- (define (pi-summands n)
- (cons-stream (/ 1.0 n)
- (stream-map - (pi-summands (+ n 2)))))
-
- (define pi-stream
- (scale-stream (partial-sums (pi-summands 1)) 4))
-
- (display-stream pi-stream)
- 4.
- 2.666666666666667
- 3.466666666666667
- 2.8952380952380956
- 3.3396825396825403
- 2.9760461760461765
- 3.2837384837384844
- 3.017071817071818
- ...
-
- This gives us a stream of better and better approximations to [pi],
-although the approximations converge rather slowly. Eight terms of the
-sequence bound the value of [pi] between 3.284 and 3.017.
-
- So far, our use of the stream of states approach is not much
-different from updating state variables. But streams give us an
-opportunity to do some interesting tricks. For example, we can
-transform a stream with a "sequence accelerator" that converts a
-sequence of approximations to a new sequence that converges to the same
-value as the original, only faster.
-
- One such accelerator, due to the eighteenth-century Swiss
-mathematician Leonhard Euler, works well with sequences that are
-partial sums of alternating series (series of terms with alternating
-signs). In Euler's technique, if S_n is the nth term of the original
-sum sequence, then the accelerated sequence has terms
-
- (S_(n+1) - S_n)^2
- S_(n+1) - ------------------------
- S_(n-1) - 2S_n + S_(n+1)
-
- Thus, if the original sequence is represented as a stream of values,
-the transformed sequence is given by
-
- (define (euler-transform s)
- (let ((s0 (stream-ref s 0)) ; S_(n-1)
- (s1 (stream-ref s 1)) ; S_n
- (s2 (stream-ref s 2))) ; S_(n+1)
- (cons-stream (- s2 (/ (square (- s2 s1))
- (+ s0 (* -2 s1) s2)))
- (euler-transform (stream-cdr s)))))
-
- We can demonstrate Euler acceleration with our sequence of
-approximations to [pi]:
-
- (display-stream (euler-transform pi-stream))
- 3.166666666666667
- 3.1333333333333337
- 3.1452380952380956
- 3.13968253968254
- 3.1427128427128435
- 3.1408813408813416
- 3.142071817071818
- 3.1412548236077655
- ...
-
- Even better, we can accelerate the accelerated sequence, and
-recursively accelerate that, and so on. Namely, we create a stream of
-streams (a structure we'll call a "tableau") in which each stream is
-the transform of the preceding one:
-
- (define (make-tableau transform s)
- (cons-stream s
- (make-tableau transform
- (transform s))))
-
- The tableau has the form
-
- s_00 s_01 s_02 s_03 s_04 ...
- s_10 s_11 s_12 s_13 ...
- s_20 s_21 s_22 ...
- ...
-
- Finally, we form a sequence by taking the first term in each row of
-the tableau:
-
- (define (accelerated-sequence transform s)
- (stream-map stream-car
- (make-tableau transform s)))
-
- We can demonstrate this kind of "super-acceleration" of the [pi]
-sequence:
-
- (display-stream (accelerated-sequence euler-transform
- pi-stream))
- 4.
- 3.166666666666667
- 3.142105263157895
- 3.141599357319005
- 3.1415927140337785
- 3.1415926539752927
- 3.1415926535911765
- 3.141592653589778
- ...
-
- The result is impressive. Taking eight terms of the sequence yields
-the correct value of [pi] to 14 decimal places. If we had used only the
-original [pi] sequence, we would need to compute on the order of 10^13
-terms (i.e., expanding the series far enough so that the individual
-terms are less then 10^(-13)) to get that much accuracy!
-
- We could have implemented these acceleration techniques without
-using streams. But the stream formulation is particularly elegant and
-convenient because the entire sequence of states is available to us as
-a data structure that can be manipulated with a uniform set of
-operations.
-
- *Exercise 3.63:* Louis Reasoner asks why the `sqrt-stream'
- procedure was not written in the following more straightforward
- way, without the local variable `guesses':
-
- (define (sqrt-stream x)
- (cons-stream 1.0
- (stream-map (lambda (guess)
- (sqrt-improve guess x))
- (sqrt-stream x))))
-
- Alyssa P. Hacker replies that this version of the procedure is
- considerably less efficient because it performs redundant
- computation. Explain Alyssa's answer. Would the two versions
- still differ in efficiency if our implementation of `delay' used
- only `(lambda () <EXP>)' without using the optimization provided
- by `memo-proc' (section *Note 3-5-1::)?
-
- *Exercise 3.64:* Write a procedure `stream-limit' that takes as
- arguments a stream and a number (the tolerance). It should
- examine the stream until it finds two successive elements that
- differ in absolute value by less than the tolerance, and return
- the second of the two elements. Using this, we could compute
- square roots up to a given tolerance by
-
- (define (sqrt x tolerance)
- (stream-limit (sqrt-stream x) tolerance))
-
- *Exercise 3.65:* Use the series
-
- 1 1 1
- ln 2 = 1 - --- + --- - --- + ...
- 2 3 4
-
- to compute three sequences of approximations to the natural
- logarithm of 2, in the same way we did above for [pi]. How
- rapidly do these sequences converge?
-
-Infinite streams of pairs
-.........................
-
-In section *Note 2-2-3::, we saw how the sequence paradigm handles
-traditional nested loops as processes defined on sequences of pairs.
-If we generalize this technique to infinite streams, then we can write
-programs that are not easily represented as loops, because the
-"looping" must range over an infinite set.
-
- For example, suppose we want to generalize the `prime-sum-pairs'
-procedure of section *Note 2-2-3:: to produce the stream of pairs of
-_all_ integers (i,j) with i <= j such that i + j is prime. If
-`int-pairs' is the sequence of all pairs of integers (i,j) with i <= j,
-then our required stream is simply(2)
-
- (stream-filter (lambda (pair)
- (prime? (+ (car pair) (cadr pair))))
- int-pairs)
-
- Our problem, then, is to produce the stream `int-pairs'. More
-generally, suppose we have two streams S = (S_i) and T = (T_j), and
-imagine the infinite rectangular array
-
- (S_0, T_0) (S_0, T_1) (S_0, T_2) ...
- (S_1, T_0) (S_1, T_1) (S_1, T_2) ...
- (S_2, T_0) (S_2, T_1) (S_2, T_2) ...
- ...
-
- We wish to generate a stream that contains all the pairs in the
-array that lie on or above the diagonal, i.e., the pairs
-
- (S_0, T_0) (S_0, T_1) (S_0, T_2) ...
- (S_1, T_1) (S_1, T_2) ...
- (S_2, T_2) ...
- ...
-
-(If we take both S and T to be the stream of integers, then this will
-be our desired stream `int-pairs'.)
-
- Call the general stream of pairs `(pairs S T)', and consider it to be
-composed of three parts: the pair (S_0,T_0), the rest of the pairs in
-the first row, and the remaining pairs:(3)
-
- (S_0, T_0) | (S_0, T_1) (S_0, T_2) ...
- -----------+-----------------------------
- | (S_1, T_1) (S_1, T_2) ...
- | (S_2, T_2) ...
- | ...
-
- Observe that the third piece in this decomposition (pairs that are
-not in the first row) is (recursively) the pairs formed from
-`(stream-cdr S)' and `(stream-cdr T)'. Also note that the second piece
-(the rest of the first row) is
-
- (stream-map (lambda (x) (list (stream-car s) x))
- (stream-cdr t))
-
- Thus we can form our stream of pairs as follows:
-
- (define (pairs s t)
- (cons-stream
- (list (stream-car s) (stream-car t))
- (<COMBINE-IN-SOME-WAY>
- (stream-map (lambda (x) (list (stream-car s) x))
- (stream-cdr t))
- (pairs (stream-cdr s) (stream-cdr t)))))
-
- In order to complete the procedure, we must choose some way to
-combine the two inner streams. One idea is to use the stream analog of
-the `append' procedure from section *Note 2-2-1:::
-
- (define (stream-append s1 s2)
- (if (stream-null? s1)
- s2
- (cons-stream (stream-car s1)
- (stream-append (stream-cdr s1) s2))))
-
- This is unsuitable for infinite streams, however, because it takes
-all the elements from the first stream before incorporating the second
-stream. In particular, if we try to generate all pairs of positive
-integers using
-
- (pairs integers integers)
-
-our stream of results will first try to run through all pairs with the
-first integer equal to 1, and hence will never produce pairs with any
-other value of the first integer.
-
- To handle infinite streams, we need to devise an order of
-combination that ensures that every element will eventually be reached
-if we let our program run long enough. An elegant way to accomplish
-this is with the following `interleave' procedure:(4)
-
- (define (interleave s1 s2)
- (if (stream-null? s1)
- s2
- (cons-stream (stream-car s1)
- (interleave s2 (stream-cdr s1)))))
-
- Since `interleave' takes elements alternately from the two streams,
-every element of the second stream will eventually find its way into
-the interleaved stream, even if the first stream is infinite.
-
- We can thus generate the required stream of pairs as
-
- (define (pairs s t)
- (cons-stream
- (list (stream-car s) (stream-car t))
- (interleave
- (stream-map (lambda (x) (list (stream-car s) x))
- (stream-cdr t))
- (pairs (stream-cdr s) (stream-cdr t)))))
-
- *Exercise 3.66:* Examine the stream `(pairs integers integers)'.
- Can you make any general comments about the order in which the
- pairs are placed into the stream? For example, about how many
- pairs precede the pair (1,100)? the pair (99,100)? the pair
- (100,100)? (If you can make precise mathematical statements here,
- all the better. But feel free to give more qualitative answers if
- you find yourself getting bogged down.)
-
- *Exercise 3.67:* Modify the `pairs' procedure so that `(pairs
- integers integers)' will produce the stream of _all_ pairs of
- integers (i,j) (without the condition i <= j). Hint: You will
- need to mix in an additional stream.
-
- *Exercise 3.68:* Louis Reasoner thinks that building a stream of
- pairs from three parts is unnecessarily complicated. Instead of
- separating the pair (S_0,T_0) from the rest of the pairs in the
- first row, he proposes to work with the whole first row, as
- follows:
-
- (define (pairs s t)
- (interleave
- (stream-map (lambda (x) (list (stream-car s) x))
- t)
- (pairs (stream-cdr s) (stream-cdr t))))
-
- Does this work? Consider what happens if we evaluate `(pairs
- integers integers)' using Louis's definition of `pairs'.
-
- *Exercise 3.69:* Write a procedure `triples' that takes three
- infinite streams, S, T, and U, and produces the stream of triples
- (S_i,T_j,U_k) such that i <= j <= k. Use `triples' to generate
- the stream of all Pythagorean triples of positive integers, i.e.,
- the triples (i,j,k) such that i <= j and i^2 + j^2 = k^2.
-
- *Exercise 3.70:* It would be nice to be able to generate streams
- in which the pairs appear in some useful order, rather than in the
- order that results from an _ad hoc_ interleaving process. We can
- use a technique similar to the `merge' procedure of *Note Exercise
- 3-56::, if we define a way to say that one pair of integers is
- "less than" another. One way to do this is to define a "weighting
- function" W(i,j) and stipulate that (i_1,j_1) is less than
- (i_2,j_2) if W(i_1,j_1) < W(i_2,j_2). Write a procedure
- `merge-weighted' that is like `merge', except that
- `merge-weighted' takes an additional argument `weight', which is a
- procedure that computes the weight of a pair, and is used to
- determine the order in which elements should appear in the
- resulting merged stream.(5) Using this, generalize `pairs' to a
- procedure `weighted-pairs' that takes two streams, together with a
- procedure that computes a weighting function, and generates the
- stream of pairs, ordered according to weight. Use your procedure
- to generate
-
- a. the stream of all pairs of positive integers (i,j) with i <= j
- ordered according to the sum i + j
-
- b. the stream of all pairs of positive integers (i,j) with i <=
- j, where neither i nor j is divisible by 2, 3, or 5, and the
- pairs are ordered according to the sum 2 i + 3 j + 5 i j.
-
-
- *Exercise 3.71:* Numbers that can be expressed as the sum of two
- cubes in more than one way are sometimes called "Ramanujan
- numbers", in honor of the mathematician Srinivasa Ramanujan.(6)
- Ordered streams of pairs provide an elegant solution to the
- problem of computing these numbers. To find a number that can be
- written as the sum of two cubes in two different ways, we need
- only generate the stream of pairs of integers (i,j) weighted
- according to the sum i^3 + j^3 (see *Note Exercise 3-70::), then
- search the stream for two consecutive pairs with the same weight.
- Write a procedure to generate the Ramanujan numbers. The first
- such number is 1,729. What are the next five?
-
- *Exercise 3.72:* In a similar way to *Note Exercise 3-71::
- generate a stream of all numbers that can be written as the sum of
- two squares in three different ways (showing how they can be so
- written).
-
-Streams as signals
-..................
-
-We began our discussion of streams by describing them as computational
-analogs of the "signals" in signal-processing systems. In fact, we can
-use streams to model signal-processing systems in a very direct way,
-representing the values of a signal at successive time intervals as
-consecutive elements of a stream. For instance, we can implement an "integrator"
-or "summer" that, for an input stream x = (x_i), an initial value C,
-and a small increment dt, accumulates the sum
-
- i
- ---
- S_i = C + > x_j dt
- ---
- j=1
-
-and returns the stream of values S = (S_i). The following `integral'
-procedure is reminiscent of the "implicit style" definition of the
-stream of integers (section *Note 3-5-2::):
-
- (define (integral integrand initial-value dt)
- (define int
- (cons-stream initial-value
- (add-streams (scale-stream integrand dt)
- int)))
- int)
-
- *Note Figure 3-32:: is a picture of a signal-processing system that
-corresponds to the `integral' procedure. The input stream is scaled by
-dt and passed through an adder, whose output is passed back through the
-same adder. The self-reference in the definition of `int' is reflected
-in the figure by the feedback loop that connects the output of the
-adder to one of the inputs.
-
- *Figure 3.32:* The `integral' procedure viewed as a
- signal-processing system.
-
- initial-value
- |
- +-----------+ | |\__
- input | | |\__ +-->| \_ integral
- ------>| scale: dt +----->| \_ |cons_>--*------->
- | | | add_>---->| __/ |
- +-----------+ +-->| __/ |/ |
- | |/ |
- | |
- +------------------------+
-
- *Exercise 3.73:* We can model electrical circuits using streams to
- represent the values of currents or voltages at a sequence of
- times. For instance, suppose we have an "RC circuit" consisting
- of a resistor of resistance R and a capacitor of capacitance C in
- series. The voltage response v of the circuit to an injected
- current i is determined by the formula in *Note Figure 3-33::,
- whose structure is shown by the accompanying signal-flow diagram.
-
- Write a procedure `RC' that models this circuit. `RC' should take
- as inputs the values of R, C, and dt and should return a procedure
- that takes as inputs a stream representing the current i and an
- initial value for the capacitor voltage v_0 and produces as output
- the stream of voltages v. For example, you should be able to use
- `RC' to model an RC circuit with R = 5 ohms, C = 1 farad, and a
- 0.5-second time step by evaluating `(define RC1 (RC 5 1 0.5))'.
- This defines `RC1' as a procedure that takes a stream representing
- the time sequence of currents and an initial capacitor voltage and
- produces the output stream of voltages.
-
- *Figure 3.33:* An RC circuit and the associated signal-flow
- diagram.
-
- + -
- ->----'\/\/\,---| |---
- i C
-
- / t
- | i
- v = v + | dt + R i
- 0 |
- / 0
-
- +--------------+
- +-->| scale: R |---------------------+ |\_
- | +--------------+ | | \_
- | +-->| \ v
- i | +--------------+ +------------+ | add >--->
- ----+-->| scale: 1/C |---->| integral |----->| _/
- +--------------+ +------------+ | _/
- |/
-
- *Exercise 3.74:* Alyssa P. Hacker is designing a system to process
- signals coming from physical sensors. One important feature she
- wishes to produce is a signal that describes the "zero crossings"
- of the input signal. That is, the resulting signal should be + 1
- whenever the input signal changes from negative to positive, - 1
- whenever the input signal changes from positive to negative, and 0
- otherwise. (Assume that the sign of a 0 input is positive.) For
- example, a typical input signal with its associated zero-crossing
- signal would be
-
- ... 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 ...
- ... 0 0 0 0 0 -1 0 0 0 0 1 0 0 ...
-
- In Alyssa's system, the signal from the sensor is represented as a
- stream `sense-data' and the stream `zero-crossings' is the
- corresponding stream of zero crossings. Alyssa first writes a
- procedure `sign-change-detector' that takes two values as
- arguments and compares the signs of the values to produce an
- appropriate 0, 1, or - 1. She then constructs her zero-crossing
- stream as follows:
-
- (define (make-zero-crossings input-stream last-value)
- (cons-stream
- (sign-change-detector (stream-car input-stream) last-value)
- (make-zero-crossings (stream-cdr input-stream)
- (stream-car input-stream))))
-
- (define zero-crossings (make-zero-crossings sense-data 0))
-
- Alyssa's boss, Eva Lu Ator, walks by and suggests that this
- program is approximately equivalent to the following one, which
- uses the generalized version of `stream-map' from *Note Exercise
- 3-50:::
-
- (define zero-crossings
- (stream-map sign-change-detector sense-data <EXPRESSION>))
-
- Complete the program by supplying the indicated <EXPRESSION>.
-
- *Exercise 3.75:* Unfortunately, Alyssa's zero-crossing detector in
- *Note Exercise 3-74:: proves to be insufficient, because the noisy
- signal from the sensor leads to spurious zero crossings. Lem E.
- Tweakit, a hardware specialist, suggests that Alyssa smooth the
- signal to filter out the noise before extracting the zero
- crossings. Alyssa takes his advice and decides to extract the
- zero crossings from the signal constructed by averaging each value
- of the sense data with the previous value. She explains the
- problem to her assistant, Louis Reasoner, who attempts to
- implement the idea, altering Alyssa's program as follows:
-
- (define (make-zero-crossings input-stream last-value)
- (let ((avpt (/ (+ (stream-car input-stream) last-value) 2)))
- (cons-stream (sign-change-detector avpt last-value)
- (make-zero-crossings (stream-cdr input-stream)
- avpt))))
-
- This does not correctly implement Alyssa's plan. Find the bug
- that Louis has installed and fix it without changing the structure
- of the program. (Hint: You will need to increase the number of
- arguments to `make-zero-crossings'.)
-
- *Exercise 3.76:* Eva Lu Ator has a criticism of Louis's approach
- in *Note Exercise 3-75::. The program he wrote is not modular,
- because it intermixes the operation of smoothing with the
- zero-crossing extraction. For example, the extractor should not
- have to be changed if Alyssa finds a better way to condition her
- input signal. Help Louis by writing a procedure `smooth' that
- takes a stream as input and produces a stream in which each
- element is the average of two successive input stream elements.
- Then use `smooth' as a component to implement the zero-crossing
- detector in a more modular style.
-
- ---------- Footnotes ----------
-
- (1) We can't use `let' to bind the local variable `guesses', because
-the value of `guesses' depends on `guesses' itself. *Note Exercise
-3-63:: addresses why we want a local variable here.
-
- (2) As in section *Note 2-2-3::, we represent a pair of integers as
-a list rather than a Lisp pair.
-
- (3) See *Note Exercise 3-68:: for some insight into why we chose
-this decomposition.
-
- (4) The precise statement of the required property on the order of
-combination is as follows: There should be a function f of two
-arguments such that the pair corresponding to element i of the first
-stream and element j of the second stream will appear as element number
-f(i,j) of the output stream. The trick of using `interleave' to
-accomplish this was shown to us by David Turner, who employed it in the
-language KRC (Turner 1981).
-
- (5) We will require that the weighting function be such that the
-weight of a pair increases as we move out along a row or down along a
-column of the array of pairs.
-
- (6) To quote from G. H. Hardy's obituary of Ramanujan (Hardy 1921):
-"It was Mr. Littlewood (I believe) who remarked that `every positive
-integer was one of his friends.' I remember once going to see him when
-he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and
-remarked that the number seemed to me a rather dull one, and that I
-hoped it was not an unfavorable omen. `No,' he replied, `it is a very
-interesting number; it is the smallest number expressible as the sum of
-two cubes in two different ways.' " The trick of using weighted pairs
-to generate the Ramanujan numbers was shown to us by Charles Leiserson.
-
-\1f
-File: sicp.info, Node: 3-5-4, Next: 3-5-5, Prev: 3-5-3, Up: 3-5
-
-3.5.4 Streams and Delayed Evaluation
-------------------------------------
-
-The `integral' procedure at the end of the preceding section shows how
-we can use streams to model signal-processing systems that contain
-feedback loops. The feedback loop for the adder shown in *Note Figure
-3-32:: is modeled by the fact that `integral''s internal stream `int'
-is defined in terms of itself:
-
- (define int
- (cons-stream initial-value
- (add-streams (scale-stream integrand dt)
- int)))
-
- The interpreter's ability to deal with such an implicit definition
-depends on the `delay' that is incorporated into `cons-stream'.
-Without this `delay', the interpreter could not construct `int' before
-evaluating both arguments to `cons-stream', which would require that
-`int' already be defined. In general, `delay' is crucial for using
-streams to model signal-processing systems that contain loops. Without
-`delay', our models would have to be formulated so that the inputs to
-any signal-processing component would be fully evaluated before the
-output could be produced. This would outlaw loops.
-
- Unfortunately, stream models of systems with loops may require uses
-of `delay' beyond the "hidden" `delay' supplied by `cons-stream'. For
-instance, *Note Figure 3-34:: shows a signal-processing system for
-solving the differential equation dy/dt = f(y) where f is a given
-function. The figure shows a mapping component, which applies f to its
-input signal, linked in a feedback loop to an integrator in a manner
-very similar to that of the analog computer circuits that are actually
-used to solve such equations.
-
- *Figure 3.34:* An "analog computer circuit" that solves the
- equation dy/dt = f(y).
-
- y_0
- |
- V
- +----------+ dy +----------+ y
- +-->| map: f +------>| integral +--*----->
- | +----------+ +----------+ |
- | |
- +------------------------------------+
-
- Assuming we are given an initial value y_0 for y, we could try to
-model this system using the procedure
-
- (define (solve f y0 dt)
- (define y (integral dy y0 dt))
- (define dy (stream-map f y))
- y)
-
- This procedure does not work, because in the first line of `solve'
-the call to `integral' requires that the input `dy' be defined, which
-does not happen until the second line of `solve'.
-
- On the other hand, the intent of our definition does make sense,
-because we can, in principle, begin to generate the `y' stream without
-knowing `dy'. Indeed, `integral' and many other stream operations have
-properties similar to those of `cons-stream', in that we can generate
-part of the answer given only partial information about the arguments.
-For `integral', the first element of the output stream is the specified
-`initial-value'. Thus, we can generate the first element of the output
-stream without evaluating the integrand `dy'. Once we know the first
-element of `y', the `stream-map' in the second line of `solve' can
-begin working to generate the first element of `dy', which will produce
-the next element of `y', and so on.
-
- To take advantage of this idea, we will redefine `integral' to
-expect the integrand stream to be a "delayed argument". `Integral' will
-`force' the integrand to be evaluated only when it is required to
-generate more than the first element of the output stream:
-
- (define (integral delayed-integrand initial-value dt)
- (define int
- (cons-stream initial-value
- (let ((integrand (force delayed-integrand)))
- (add-streams (scale-stream integrand dt)
- int))))
- int)
-
- Now we can implement our `solve' procedure by delaying the
-evaluation of `dy' in the definition of `y':(1)
-
- (define (solve f y0 dt)
- (define y (integral (delay dy) y0 dt))
- (define dy (stream-map f y))
- y)
-
- In general, every caller of `integral' must now `delay' the integrand
-argument. We can demonstrate that the `solve' procedure works by
-approximating eapprox 2.718 by computing the value at y = 1 of the
-solution to the differential equation dy/dt = y with initial condition
-y(0) = 1:
-
- (stream-ref (solve (lambda (y) y) 1 0.001) 1000)
- 2.716924
-
- *Exercise 3.77:* The `integral' procedure used above was analogous
- to the "implicit" definition of the infinite stream of integers in
- section *Note 3-5-2::. Alternatively, we can give a definition of
- `integral' that is more like `integers-starting-from' (also in
- section *Note 3-5-2::):
-
- (define (integral integrand initial-value dt)
- (cons-stream initial-value
- (if (stream-null? integrand)
- the-empty-stream
- (integral (stream-cdr integrand)
- (+ (* dt (stream-car integrand))
- initial-value)
- dt))))
-
- When used in systems with loops, this procedure has the same
- problem as does our original version of `integral'. Modify the
- procedure so that it expects the `integrand' as a delayed argument
- and hence can be used in the `solve' procedure shown above.
-
- *Figure 3.35:* Signal-flow diagram for the solution to a
- second-order linear differential equation.
-
- dy_0 y_0
- | |
- V V
- ddy +----------+ dy +----------+ y
- +--------->| integral +-----*--+ integral +--*--->
- | +----------+ | +----------+ |
- | | |
- | +----------+ | |
- | __/|<--+ scale: a |<--+ |
- | _/ | +----------+ |
- +--<_add | |
- \__ | +----------+ |
- \|<--+ scale: b |<-------------------+
- +----------+
-
- *Exercise 3.78:* Consider the problem of designing a
- signal-processing system to study the homogeneous second-order
- linear differential equation
-
- d^2 y d y
- ----- - a ----- - by = 0
- d t^2 d t
-
- The output stream, modeling y, is generated by a network that
- contains a loop. This is because the value of d^2y/dt^2 depends
- upon the values of y and dy/dt and both of these are determined by
- integrating d^2y/dt^2. The diagram we would like to encode is
- shown in *Note Figure 3-35::. Write a procedure `solve-2nd' that
- takes as arguments the constants a, b, and dt and the initial
- values y_0 and dy_0 for y and dy/dt and generates the stream of
- successive values of y.
-
- *Exercise 3.79:* Generalize the `solve-2nd' procedure of *Note
- Exercise 3-78:: so that it can be used to solve general
- second-order differential equations d^2 y/dt^2 = f(dy/dt, y).
-
- *Exercise 3.80:* A "series RLC circuit" consists of a resistor, a
- capacitor, and an inductor connected in series, as shown in *Note
- Figure 3-36::. If R, L, and C are the resistance, inductance, and
- capacitance, then the relations between voltage (v) and current
- (i) for the three components are described by the equations
-
- v_R = i_R R
-
- d_(i L)
- v_L = L ---------
- d t
-
- d v_C
- i_C = C -------
- d t
-
- and the circuit connections dictate the relations
-
- i_R = i_L = -i_C
-
- v_C = v_L + v_R
-
- Combining these equations shows that the state of the circuit
- (summarized by v_C, the voltage across the capacitor, and i_L, the
- current in the inductor) is described by the pair of differential
- equations
-
- d v_C i_L
- ----- = - ---
- d t C
-
- d i_L 1 R
- ----- = --- v_C - --- i_L
- d t L L
-
- The signal-flow diagram representing this system of differential
- equations is shown in *Note Figure 3-37::.
-
- *Figure 3.36:* A series RLC circuit.
- + v_R -
- i_R
- +--->----'\/\/\,--------+
- | | i_L
- \|/ R \|/
- + | i_C |_ +
- -+- __)
- v_C -+- C (_) v_L
- | __)
- - | | -
- +-----------------------+
-
- *Figure 3.37:* A signal-flow diagram for the solution to a series
- RLC circuit.
-
- +-------------+
- +----------------+ scale: l/L |<--+
- | +-------------+ |
- | |
- | +-------------+ | v_C
- | dv_C +-->| integral +---*------>
- | | +-------------+
- | | ^
- | | | v_(C_0)
- | |
- | | +-------------+
- | +---+ scale: -l/C |<--+
- | +-------------+ |
- | |\__ |
- +->| \_ di_L +-------------+ | i_L
- | add_>------>| integral +---*------>
- +->| __/ +-------------+ |
- | |/ ^ |
- | | i_(L_0) |
- | |
- | +-------------+ |
- +----------------+ scale: -R/L |<--+
- +-------------+
-
- Write a procedure `RLC' that takes as arguments the parameters R, L,
-and C of the circuit and the time increment dt. In a manner similar to
-that of the `RC' procedure of *Note Exercise 3-73::, `RLC' should
-produce a procedure that takes the initial values of the state
-variables, v_(C_0) and i_(L_0), and produces a pair (using `cons') of
-the streams of states v_C and i_L. Using `RLC', generate the pair of
-streams that models the behavior of a series RLC circuit with R = 1
-ohm, C = 0.2 farad, L = 1 henry, dt = 0.1 second, and initial values
-i_(L_0) = 0 amps and v_(C_0) = 10 volts.
-
-Normal-order evaluation
-.......................
-
-The examples in this section illustrate how the explicit use of `delay'
-and `force' provides great programming flexibility, but the same
-examples also show how this can make our programs more complex. Our
-new `integral' procedure, for instance, gives us the power to model
-systems with loops, but we must now remember that `integral' should be
-called with a delayed integrand, and every procedure that uses
-`integral' must be aware of this. In effect, we have created two
-classes of procedures: ordinary procedures and procedures that take
-delayed arguments. In general, creating separate classes of procedures
-forces us to create separate classes of higher-order procedures as
-well.(2)
-
- One way to avoid the need for two different classes of procedures is
-to make all procedures take delayed arguments. We could adopt a model
-of evaluation in which all arguments to procedures are automatically
-delayed and arguments are forced only when they are actually needed
-(for example, when they are required by a primitive operation). This
-would transform our language to use normal-order evaluation, which we
-first described when we introduced the substitution model for
-evaluation in section *Note 1-1-5::. Converting to normal-order
-evaluation provides a uniform and elegant way to simplify the use of
-delayed evaluation, and this would be a natural strategy to adopt if we
-were concerned only with stream processing. In section *Note 4-2::,
-after we have studied the evaluator, we will see how to transform our
-language in just this way. Unfortunately, including delays in
-procedure calls wreaks havoc with our ability to design programs that
-depend on the order of events, such as programs that use assignment,
-mutate data, or perform input or output. Even the single `delay' in
-`cons-stream' can cause great confusion, as illustrated by *Note
-Exercise 3-51:: and *Note Exercise 3-52::. As far as anyone knows,
-mutability and delayed evaluation do not mix well in programming
-languages, and devising ways to deal with both of these at once is an
-active area of research.
-
- ---------- Footnotes ----------
-
- (1) This procedure is not guaranteed to work in all Scheme
-implementations, although for any implementation there is a simple
-variation that will work. The problem has to do with subtle
-differences in the ways that Scheme implementations handle internal
-definitions. (See section *Note 4-1-6::.)
-
- (2) This is a small reflection, in Lisp, of the difficulties that
-conventional strongly typed languages such as Pascal have in coping with
-higher-order procedures. In such languages, the programmer must
-specify the data types of the arguments and the result of each
-procedure: number, logical value, sequence, and so on. Consequently,
-we could not express an abstraction such as "map a given procedure
-`proc' over all the elements in a sequence" by a single higher-order
-procedure such as `stream-map'. Rather, we would need a different
-mapping procedure for each different combination of argument and result
-data types that might be specified for a `proc'. Maintaining a
-practical notion of "data type" in the presence of higher-order
-procedures raises many difficult issues. One way of dealing with this
-problem is illustrated by the language ML (Gordon, Milner, and
-Wadsworth 1979), whose "polymorphic data types" include templates for
-higher-order transformations between data types. Moreover, data types
-for most procedures in ML are never explicitly declared by the
-programmer. Instead, ML includes a "type-inferencing" mechanism that
-uses information in the environment to deduce the data types for newly
-defined procedures.
-
-\1f
-File: sicp.info, Node: 3-5-5, Prev: 3-5-4, Up: 3-5
-
-3.5.5 Modularity of Functional Programs and Modularity of Objects
------------------------------------------------------------------
-
-As we saw in section *Note 3-1-2::, one of the major benefits of
-introducing assignment is that we can increase the modularity of our
-systems by encapsulating, or "hiding," parts of the state of a large
-system within local variables. Stream models can provide an equivalent
-modularity without the use of assignment. As an illustration, we can
-reimplement the Monte Carlo estimation of [pi], which we examined in
-section *Note 3-1-2::, from a stream-processing point of view.
-
- The key modularity issue was that we wished to hide the internal
-state of a random-number generator from programs that used random
-numbers. We began with a procedure `rand-update', whose successive
-values furnished our supply of random numbers, and used this to produce
-a random-number generator:
-
- (define rand
- (let ((x random-init))
- (lambda ()
- (set! x (rand-update x))
- x)))
-
- In the stream formulation there is no random-number generator _per
-se_, just a stream of random numbers produced by successive calls to
-`rand-update':
-
- (define random-numbers
- (cons-stream random-init
- (stream-map rand-update random-numbers)))
-
- We use this to construct the stream of outcomes of the Cesa`ro
-experiment performed on consecutive pairs in the `random-numbers'
-stream:
-
- (define cesaro-stream
- (map-successive-pairs (lambda (r1 r2) (= (gcd r1 r2) 1))
- random-numbers))
-
- (define (map-successive-pairs f s)
- (cons-stream
- (f (stream-car s) (stream-car (stream-cdr s)))
- (map-successive-pairs f (stream-cdr (stream-cdr s)))))
-
- The `cesaro-stream' is now fed to a `monte-carlo' procedure, which
-produces a stream of estimates of probabilities. The results are then
-converted into a stream of estimates of [pi]. This version of the
-program doesn't need a parameter telling how many trials to perform.
-Better estimates of [pi] (from performing more experiments) are
-obtained by looking farther into the `pi' stream:
-
- (define (monte-carlo experiment-stream passed failed)
- (define (next passed failed)
- (cons-stream
- (/ passed (+ passed failed))
- (monte-carlo
- (stream-cdr experiment-stream) passed failed)))
- (if (stream-car experiment-stream)
- (next (+ passed 1) failed)
- (next passed (+ failed 1))))
-
- (define pi
- (stream-map (lambda (p) (sqrt (/ 6 p)))
- (monte-carlo cesaro-stream 0 0)))
-
- There is considerable modularity in this approach, because we still
-can formulate a general `monte-carlo' procedure that can deal with
-arbitrary experiments. Yet there is no assignment or local state.
-
- *Exercise 3.81:* *Note Exercise 3-6:: discussed generalizing the
- random-number generator to allow one to reset the random-number
- sequence so as to produce repeatable sequences of "random"
- numbers. Produce a stream formulation of this same generator that
- operates on an input stream of requests to `generate' a new random
- number or to `reset' the sequence to a specified value and that
- produces the desired stream of random numbers. Don't use
- assignment in your solution.
-
- *Exercise 3.82:* Redo *Note Exercise 3-5:: on Monte Carlo
- integration in terms of streams. The stream version of
- `estimate-integral' will not have an argument telling how many
- trials to perform. Instead, it will produce a stream of estimates
- based on successively more trials.
-
-A functional-programming view of time
-.....................................
-
-Let us now return to the issues of objects and state that were raised
-at the beginning of this chapter and examine them in a new light. We
-introduced assignment and mutable objects to provide a mechanism for
-modular construction of programs that model systems with state. We
-constructed computational objects with local state variables and used
-assignment to modify these variables. We modeled the temporal behavior
-of the objects in the world by the temporal behavior of the
-corresponding computational objects.
-
- Now we have seen that streams provide an alternative way to model
-objects with local state. We can model a changing quantity, such as
-the local state of some object, using a stream that represents the time
-history of successive states. In essence, we represent time
-explicitly, using streams, so that we decouple time in our simulated
-world from the sequence of events that take place during evaluation.
-Indeed, because of the presence of `delay' there may be little relation
-between simulated time in the model and the order of events during the
-evaluation.
-
- In order to contrast these two approaches to modeling, let us
-reconsider the implementation of a "withdrawal processor" that monitors
-the balance in a bank account. In section *Note 3-1-3:: we implemented
-a simplified version of such a processor:
-
- (define (make-simplified-withdraw balance)
- (lambda (amount)
- (set! balance (- balance amount))
- balance))
-
- Calls to `make-simplified-withdraw' produce computational objects,
-each with a local state variable `balance' that is decremented by
-successive calls to the object. The object takes an `amount' as an
-argument and returns the new balance. We can imagine the user of a
-bank account typing a sequence of inputs to such an object and
-observing the sequence of returned values shown on a display screen.
-
- Alternatively, we can model a withdrawal processor as a procedure
-that takes as input a balance and a stream of amounts to withdraw and
-produces the stream of successive balances in the account:
-
- (define (stream-withdraw balance amount-stream)
- (cons-stream
- balance
- (stream-withdraw (- balance (stream-car amount-stream))
- (stream-cdr amount-stream))))
-
- `Stream-withdraw' implements a well-defined mathematical function
-whose output is fully determined by its input. Suppose, however, that
-the input `amount-stream' is the stream of successive values typed by
-the user and that the resulting stream of balances is displayed. Then,
-from the perspective of the user who is typing values and watching
-results, the stream process has the same behavior as the object created
-by `make-simplified-withdraw'. However, with the stream version, there
-is no assignment, no local state variable, and consequently none of the
-theoretical difficulties that we encountered in section *Note 3-1-3::.
-Yet the system has state!
-
- This is really remarkable. Even though `stream-withdraw' implements
-a well-defined mathematical function whose behavior does not change,
-the user's perception here is one of interacting with a system that has
-a changing state. One way to resolve this paradox is to realize that
-it is the user's temporal existence that imposes state on the system.
-If the user could step back from the interaction and think in terms of
-streams of balances rather than individual transactions, the system
-would appear stateless.(1)
-
- From the point of view of one part of a complex process, the other
-parts appear to change with time. They have hidden time-varying local
-state. If we wish to write programs that model this kind of natural
-decomposition in our world (as we see it from our viewpoint as a part
-of that world) with structures in our computer, we make computational
-objects that are not functional--they must change with time. We model
-state with local state variables, and we model the changes of state
-with assignments to those variables. By doing this we make the time of
-execution of a computation model time in the world that we are part of,
-and thus we get "objects" in our computer.
-
- Modeling with objects is powerful and intuitive, largely because
-this matches the perception of interacting with a world of which we are
-part. However, as we've seen repeatedly throughout this chapter, these
-models raise thorny problems of constraining the order of events and of
-synchronizing multiple processes. The possibility of avoiding these
-problems has stimulated the development of "functional programming
-languages", which do not include any provision for assignment or
-mutable data. In such a language, all procedures implement
-well-defined mathematical functions of their arguments, whose behavior
-does not change. The functional approach is extremely attractive for
-dealing with concurrent systems.(2)
-
- On the other hand, if we look closely, we can see time-related
-problems creeping into functional models as well. One particularly
-troublesome area arises when we wish to design interactive systems,
-especially ones that model interactions between independent entities.
-For instance, consider once more the implementation a banking system
-that permits joint bank accounts. In a conventional system using
-assignment and objects, we would model the fact that Peter and Paul
-share an account by having both Peter and Paul send their transaction
-requests to the same bank-account object, as we saw in section *Note
-3-1-3::. From the stream point of view, where there are no "objects"
-_per se_, we have already indicated that a bank account can be modeled
-as a process that operates on a stream of transaction requests to
-produce a stream of responses. Accordingly, we could model the fact
-that Peter and Paul have a joint bank account by merging Peter's stream
-of transaction requests with Paul's stream of requests and feeding the
-result to the bank-account stream process, as shown in *Note Figure
-3-38::.
-
- *Figure 3.38:* A joint bank account, modeled by merging two
- streams of transaction requests.
-
- Peter's requests +---------+ +---------+
- ------------------>| | | |
- Paul's requests | merge |---->| bank |---->
- ------------------>| | | account |
- +---------+ +---------+
-
- The trouble with this formulation is in the notion of "merge". It
-will not do to merge the two streams by simply taking alternately one
-request from Peter and one request from Paul. Suppose Paul accesses the
-account only very rarely. We could hardly force Peter to wait for Paul
-to access the account before he could issue a second transaction.
-However such a merge is implemented, it must interleave the two
-transaction streams in some way that is constrained by "real time" as
-perceived by Peter and Paul, in the sense that, if Peter and Paul meet,
-they can agree that certain transactions were processed before the
-meeting, and other transactions were processed after the meeting.(3)
-This is precisely the same constraint that we had to deal with in
-section *Note 3-4-1::, where we found the need to introduce explicit
-synchronization to ensure a "correct" order of events in concurrent
-processing of objects with state. Thus, in an attempt to support the
-functional style, the need to merge inputs from different agents
-reintroduces the same problems that the functional style was meant to
-eliminate.
-
- We began this chapter with the goal of building computational models
-whose structure matches our perception of the real world we are trying
-to model. We can model the world as a collection of separate,
-time-bound, interacting objects with state, or we can model the world
-as a single, timeless, stateless unity. Each view has powerful
-advantages, but neither view alone is completely satisfactory. A grand
-unification has yet to emerge.(4)
-
- ---------- Footnotes ----------
-
- (1) Similarly in physics, when we observe a moving particle, we say
-that the position (state) of the particle is changing. However, from
-the perspective of the particle's world line in space-time there is no
-change involved.
-
- (2) John Backus, the inventor of Fortran, gave high visibility to
-functional programming when he was awarded the ACM Turing award in
-1978. His acceptance speech (Backus 1978) strongly advocated the
-functional approach. A good overview of functional programming is
-given in Henderson 1980 and in Darlington, Henderson, and Turner 1982.
-
- (3) Observe that, for any two streams, there is in general more than
-one acceptable order of interleaving. Thus, technically, "merge" is a
-relation rather than a function--the answer is not a deterministic
-function of the inputs. We already mentioned (*Note Footnote 39::)
-that nondeterminism is essential when dealing with concurrency. The
-merge relation illustrates the same essential nondeterminism, from the
-functional perspective. In section *Note 4-3::, we will look at
-nondeterminism from yet another point of view.
-
- (4) The object model approximates the world by dividing it into
-separate pieces. The functional model does not modularize along object
-boundaries. The object model is useful when the unshared state of the
-"objects" is much larger than the state that they share. An example of
-a place where the object viewpoint fails is quantum mechanics, where
-thinking of things as individual particles leads to paradoxes and
-confusions. Unifying the object view with the functional view may have
-little to do with programming, but rather with fundamental
-epistemological issues.
-
-\1f
-File: sicp.info, Node: Chapter 4, Next: Chapter 5, Prev: Chapter 3, Up: Top
-
-4 Metalinguistic Abstraction
-****************************
-
- ... It's in words that the magic is--Abracadabra, Open Sesame, and
- the rest--but the magic words in one story aren't magical in the
- next. The real magic is to understand which words work, and when,
- and for what; the trick is to learn the trick.
-
- ... And those words are made from the letters of our alphabet: a
- couple-dozen squiggles we can draw with the pen. This is the key!
- And the treasure, too, if we can only get our hands on it! It's
- as if--as if the key to the treasure _is_ the treasure!
-
- --John Barth, `Chimera'
-
- In our study of program design, we have seen that expert programmers
-control the complexity of their designs with the same general
-techniques used by designers of all complex systems. They combine
-primitive elements to form compound objects, they abstract compound
-objects to form higher-level building blocks, and they preserve
-modularity by adopting appropriate large-scale views of system
-structure. In illustrating these techniques, we have used Lisp as a
-language for describing processes and for constructing computational
-data objects and processes to model complex phenomena in the real
-world. However, as we confront increasingly complex problems, we will
-find that Lisp, or indeed any fixed programming language, is not
-sufficient for our needs. We must constantly turn to new languages in
-order to express our ideas more effectively. Establishing new
-languages is a powerful strategy for controlling complexity in
-engineering design; we can often enhance our ability to deal with a
-complex problem by adopting a new language that enables us to describe
-(and hence to think about) the problem in a different way, using
-primitives, means of combination, and means of abstraction that are
-particularly well suited to the problem at hand.(1)
-
- Programming is endowed with a multitude of languages. There are
-physical languages, such as the machine languages for particular
-computers. These languages are concerned with the representation of
-data and control in terms of individual bits of storage and primitive
-machine instructions. The machine-language programmer is concerned
-with using the given hardware to erect systems and utilities for the
-efficient implementation of resource-limited computations. High-level
-languages, erected on a machine-language substrate, hide concerns about
-the representation of data as collections of bits and the
-representation of programs as sequences of primitive instructions.
-These languages have means of combination and abstraction, such as
-procedure definition, that are appropriate to the larger-scale
-organization of systems.
-
- "Metalinguistic abstraction"--establishing new languages--plays an
-important role in all branches of engineering design. It is
-particularly important to computer programming, because in programming
-not only can we formulate new languages but we can also implement these
-languages by constructing evaluators. An "evaluator" (or "interpreter")
-for a programming language is a procedure that, when applied to an
-expression of the language, performs the actions required to evaluate
-that expression.
-
- It is no exaggeration to regard this as the most fundamental idea in
-programming:
-
- The evaluator, which determines the meaning of expressions in a
- programming language, is just another program.
-
- To appreciate this point is to change our images of ourselves as
-programmers. We come to see ourselves as designers of languages,
-rather than only users of languages designed by others.
-
- In fact, we can regard almost any program as the evaluator for some
-language. For instance, the polynomial manipulation system of section
-*Note 2-5-3:: embodies the rules of polynomial arithmetic and
-implements them in terms of operations on list-structured data. If we
-augment this system with procedures to read and print polynomial
-expressions, we have the core of a special-purpose language for dealing
-with problems in symbolic mathematics. The digital-logic simulator of
-section *Note 3-3-4:: and the constraint propagator of section *Note
-3-3-5:: are legitimate languages in their own right, each with its own
-primitives, means of combination, and means of abstraction. Seen from
-this perspective, the technology for coping with large-scale computer
-systems merges with the technology for building new computer languages,
-and computer science itself becomes no more (and no less) than the
-discipline of constructing appropriate descriptive languages.
-
- We now embark on a tour of the technology by which languages are
-established in terms of other languages. In this chapter we shall use
-Lisp as a base, implementing evaluators as Lisp procedures. Lisp is
-particularly well suited to this task, because of its ability to
-represent and manipulate symbolic expressions. We will take the first
-step in understanding how languages are implemented by building an
-evaluator for Lisp itself. The language implemented by our evaluator
-will be a subset of the Scheme dialect of Lisp that we use in this
-book. Although the evaluator described in this chapter is written for a
-particular dialect of Lisp, it contains the essential structure of an
-evaluator for any expression-oriented language designed for writing
-programs for a sequential machine. (In fact, most language processors
-contain, deep within them, a little "Lisp" evaluator.) The evaluator
-has been simplified for the purposes of illustration and discussion,
-and some features have been left out that would be important to include
-in a production-quality Lisp system. Nevertheless, this simple
-evaluator is adequate to execute most of the programs in this book.(2)
-
- An important advantage of making the evaluator accessible as a Lisp
-program is that we can implement alternative evaluation rules by
-describing these as modifications to the evaluator program. One place
-where we can use this power to good effect is to gain extra control
-over the ways in which computational models embody the notion of time,
-which was so central to the discussion in *Note Chapter 3::. There, we
-mitigated some of the complexities of state and assignment by using
-streams to decouple the representation of time in the world from time
-in the computer. Our stream programs, however, were sometimes
-cumbersome, because they were constrained by the applicative-order
-evaluation of Scheme. In section *Note 4-2::, we'll change the
-underlying language to provide for a more elegant approach, by
-modifying the evaluator to provide for "normal-order evaluation".
-
- Section *Note 4-3:: implements a more ambitious linguistic change,
-whereby expressions have many values, rather than just a single value.
-In this language of "nondeterministic computing", it is natural to
-express processes that generate all possible values for expressions and
-then search for those values that satisfy certain constraints. In
-terms of models of computation and time, this is like having time
-branch into a set of "possible futures" and then searching for
-appropriate time lines. With our nondeterministic evaluator, keeping
-track of multiple values and performing searches are handled
-automatically by the underlying mechanism of the language.
-
- In section *Note 4-4:: we implement a "logic-programming" language in
-which knowledge is expressed in terms of relations, rather than in
-terms of computations with inputs and outputs. Even though this makes
-the language drastically different from Lisp, or indeed from any
-conventional language, we will see that the logic-programming evaluator
-shares the essential structure of the Lisp evaluator.
-
-* Menu:
-
-* 4-1:: The Metacircular Evaluator
-* 4-2:: Variations on a Scheme -- Lazy Evaluation
-* 4-3:: Variations on a Scheme -- Nondeterministic Computing
-* 4-4:: Logic Programming
-
- ---------- Footnotes ----------
-
- (1) The same idea is pervasive throughout all of engineering. For
-example, electrical engineers use many different languages for
-describing circuits. Two of these are the language of electrical "networks"
-and the language of electrical "systems". The network language
-emphasizes the physical modeling of devices in terms of discrete
-electrical elements. The primitive objects of the network language are
-primitive electrical components such as resistors, capacitors,
-inductors, and transistors, which are characterized in terms of
-physical variables called voltage and current. When describing
-circuits in the network language, the engineer is concerned with the
-physical characteristics of a design. In contrast, the primitive
-objects of the system language are signal-processing modules such as
-filters and amplifiers. Only the functional behavior of the modules is
-relevant, and signals are manipulated without concern for their
-physical realization as voltages and currents. The system language is
-erected on the network language, in the sense that the elements of
-signal-processing systems are constructed from electrical networks.
-Here, however, the concerns are with the large-scale organization of
-electrical devices to solve a given application problem; the physical
-feasibility of the parts is assumed. This layered collection of
-languages is another example of the stratified design technique
-illustrated by the picture language of section *Note 2-2-4::.
-
- (2) The most important features that our evaluator leaves out are
-mechanisms for handling errors and supporting debugging. For a more
-extensive discussion of evaluators, see Friedman, Wand, and Haynes
-1992, which gives an exposition of programming languages that proceeds
-via a sequence of evaluators written in Scheme.
-
-\1f
-File: sicp.info, Node: 4-1, Next: 4-2, Prev: Chapter 4, Up: Chapter 4
-
-4.1 The Metacircular Evaluator
-==============================
-
-Our evaluator for Lisp will be implemented as a Lisp program. It may
-seem circular to think about evaluating Lisp programs using an
-evaluator that is itself implemented in Lisp. However, evaluation is a
-process, so it is appropriate to describe the evaluation process using
-Lisp, which, after all, is our tool for describing processes.(1) An
-evaluator that is written in the same language that it evaluates is
-said to be "metacircular".
-
- The metacircular evaluator is essentially a Scheme formulation of the
-environment model of evaluation described in section *Note 3-2::.
-Recall that the model has two basic parts:
-
- 1. To evaluate a combination (a compound expression other than a
- special form), evaluate the subexpressions and then apply the
- value of the operator subexpression to the values of the operand
- subexpressions.
-
- 2. To apply a compound procedure to a set of arguments, evaluate the
- body of the procedure in a new environment. To construct this
- environment, extend the environment part of the procedure object
- by a frame in which the formal parameters of the procedure are
- bound to the arguments to which the procedure is applied.
-
-
- These two rules describe the essence of the evaluation process, a
-basic cycle in which expressions to be evaluated in environments are
-reduced to procedures to be applied to arguments, which in turn are
-reduced to new expressions to be evaluated in new environments, and so
-on, until we get down to symbols, whose values are looked up in the
-environment, and to primitive procedures, which are applied directly
-(see *Note Figure 4-1::).(2) This evaluation cycle will be embodied by
-the interplay between the two critical procedures in the evaluator,
-`eval' and `apply', which are described in section *Note 4-1-1:: (see
-*Note Figure 4-1::).
-
- The implementation of the evaluator will depend upon procedures that
-define the "syntax" of the expressions to be evaluated. We will use
-data abstraction to make the evaluator independent of the
-representation of the language. For example, rather than committing to
-a choice that an assignment is to be represented by a list beginning
-with the symbol `set!' we use an abstract predicate `assignment?' to
-test for an assignment, and we use abstract selectors
-`assignment-variable' and `assignment-value' to access the parts of an
-assignment. Implementation of expressions will be described in detail
-in section *Note 4-1-2::. There are also operations, described in
-section *Note 4-1-3::, that specify the representation of procedures
-and environments. For example, `make-procedure' constructs compound
-procedures, `lookup-variable-value' accesses the values of variables,
-and `apply-primitive-procedure' applies a primitive procedure to a
-given list of arguments.
-
-* Menu:
-
-* 4-1-1:: The Core of the Evaluator
-* 4-1-2:: Representing Expressions
-* 4-1-3:: Evaluator Data Structures
-* 4-1-4:: Running the Evaluator as a Program
-* 4-1-5:: Data as Programs
-* 4-1-6:: Internal Definitions
-* 4-1-7:: Separating Syntactic Analysis from Execution
-
- ---------- Footnotes ----------
-
- (1) Even so, there will remain important aspects of the evaluation
-process that are not elucidated by our evaluator. The most important
-of these are the detailed mechanisms by which procedures call other
-procedures and return values to their callers. We will address these
-issues in *Note Chapter 5::, where we take a closer look at the
-evaluation process by implementing the evaluator as a simple register
-machine.
-
- (2) If we grant ourselves the ability to apply primitives, then what
-remains for us to implement in the evaluator? The job of the evaluator
-is not to specify the primitives of the language, but rather to provide
-the connective tissue--the means of combination and the means of
-abstraction--that binds a collection of primitives to form a language.
-Specifically:
-
- * The evaluator enables us to deal with nested expressions. For
- example, although simply applying primitives would suffice for
- evaluating the expression `(+ 1 6)', it is not adequate for
- handling `(+ 1 (* 2 3))'. As far as the primitive procedure `+'
- is concerned, its arguments must be numbers, and it would choke if
- we passed it the expression `(* 2 3)' as an argument. One
- important role of the evaluator is to choreograph procedure
- composition so that `(* 2 3)' is reduced to 6 before being passed
- as an argument to `+'.
-
- * The evaluator allows us to use variables. For example, the
- primitive procedure for addition has no way to deal with
- expressions such as `(+ x 1)'. We need an evaluator to keep track
- of variables and obtain their values before invoking the primitive
- procedures.
-
- * The evaluator allows us to define compound procedures. This
- involves keeping track of procedure definitions, knowing how to
- use these definitions in evaluating expressions, and providing a
- mechanism that enables procedures to accept arguments.
-
- * The evaluator provides the special forms, which must be evaluated
- differently from procedure calls.
-
-
-\1f
-File: sicp.info, Node: 4-1-1, Next: 4-1-2, Prev: 4-1, Up: 4-1
-
-4.1.1 The Core of the Evaluator
--------------------------------
-
- *Figure 4.1:* The `eval'-`apply' cycle exposes the essence of a
- computer language.
-
- .,ad88888888baa,
- _ ,d8P""" ""9888ba. _
- / .a8" ,ad88888888888a |\
- / aP' ,88888888888888888a \
- / ,8" ,88888888888888888888, \
- | ,8' (888888888888888888888, |
- / ,8' `8888888888888888888888 \
- | 8) `888888888888888888888, |
- Procedure, | 8 "88888 Apply 8888888) | Expression
- Arguments | 8 Eval `888888888888888888) | Environment
- | 8) "8888888888888888 |
- \ (b "88888888888888' /
- | `8, 8888888888888) |
- \ "8a ,888888888888) /
- \ V8, d88888888888" /
- _\| `8b, ,d8888888888P' _/
- `V8a, ,ad8888888888P'
- ""88888888888888888P"
- """"""""""""
-
- [graphic by Normand Veillux, modified]
-
-The evaluation process can be described as the interplay between two
-procedures: `eval' and `apply'.
-
-Eval
-....
-
-`Eval' takes as arguments an expression and an environment. It
-classifies the expression and directs its evaluation. `Eval' is
-structured as a case analysis of the syntactic type of the expression
-to be evaluated. In order to keep the procedure general, we express
-the determination of the type of an expression abstractly, making no
-commitment to any particular representation for the various types of
-expressions. Each type of expression has a predicate that tests for it
-and an abstract means for selecting its parts. This "abstract syntax"
-makes it easy to see how we can change the syntax of the language by
-using the same evaluator, but with a different collection of syntax
-procedures.
-
-Primitive expressions
-
- * For self-evaluating expressions, such as numbers, `eval' returns
- the expression itself.
-
- * `Eval' must look up variables in the environment to find their
- values.
-
-
-Special forms
-
- * For quoted expressions, `eval' returns the expression that was
- quoted.
-
- * An assignment to (or a definition of) a variable must recursively
- call `eval' to compute the new value to be associated with the
- variable. The environment must be modified to change (or create)
- the binding of the variable.
-
- * An `if' expression requires special processing of its parts, so as
- to evaluate the consequent if the predicate is true, and otherwise
- to evaluate the alternative.
-
- * A `lambda' expression must be transformed into an applicable
- procedure by packaging together the parameters and body specified
- by the `lambda' expression with the environment of the evaluation.
-
- * A `begin' expression requires evaluating its sequence of
- expressions in the order in which they appear.
-
- * A case analysis (`cond') is transformed into a nest of `if'
- expressions and then evaluated.
-
-
-Combinations
-
- * For a procedure application, `eval' must recursively evaluate the
- operator part and the operands of the combination. The resulting
- procedure and arguments are passed to `apply', which handles the
- actual procedure application.
-
-
- Here is the definition of `eval':
-
- (define (eval exp env)
- (cond ((self-evaluating? exp) exp)
- ((variable? exp) (lookup-variable-value exp env))
- ((quoted? exp) (text-of-quotation exp))
- ((assignment? exp) (eval-assignment exp env))
- ((definition? exp) (eval-definition exp env))
- ((if? exp) (eval-if exp env))
- ((lambda? exp)
- (make-procedure (lambda-parameters exp)
- (lambda-body exp)
- env))
- ((begin? exp)
- (eval-sequence (begin-actions exp) env))
- ((cond? exp) (eval (cond->if exp) env))
- ((application? exp)
- (apply (eval (operator exp) env)
- (list-of-values (operands exp) env)))
- (else
- (error "Unknown expression type -- EVAL" exp))))
-
- For clarity, `eval' has been implemented as a case analysis using
-`cond'. The disadvantage of this is that our procedure handles only a
-few distinguishable types of expressions, and no new ones can be
-defined without editing the definition of `eval'. In most Lisp
-implementations, dispatching on the type of an expression is done in a
-data-directed style. This allows a user to add new types of
-expressions that `eval' can distinguish, without modifying the
-definition of `eval' itself. (See *Note Exercise 4-3::.)
-
-Apply
-.....
-
-`Apply' takes two arguments, a procedure and a list of arguments to
-which the procedure should be applied. `Apply' classifies procedures
-into two kinds: It calls `apply-primitive-procedure' to apply
-primitives; it applies compound procedures by sequentially evaluating
-the expressions that make up the body of the procedure. The
-environment for the evaluation of the body of a compound procedure is
-constructed by extending the base environment carried by the procedure
-to include a frame that binds the parameters of the procedure to the
-arguments to which the procedure is to be applied. Here is the
-definition of `apply':
-
- (define (apply procedure arguments)
- (cond ((primitive-procedure? procedure)
- (apply-primitive-procedure procedure arguments))
- ((compound-procedure? procedure)
- (eval-sequence
- (procedure-body procedure)
- (extend-environment
- (procedure-parameters procedure)
- arguments
- (procedure-environment procedure))))
- (else
- (error
- "Unknown procedure type -- APPLY" procedure))))
-
-Procedure arguments
-...................
-
-When `eval' processes a procedure application, it uses `list-of-values'
-to produce the list of arguments to which the procedure is to be
-applied. `List-of-values' takes as an argument the operands of the
-combination. It evaluates each operand and returns a list of the
-corresponding values:(1)
-
- (define (list-of-values exps env)
- (if (no-operands? exps)
- '()
- (cons (eval (first-operand exps) env)
- (list-of-values (rest-operands exps) env))))
-
-Conditionals
-............
-
-`Eval-if' evaluates the predicate part of an `if' expression in the
-given environment. If the result is true, `eval-if' evaluates the
-consequent, otherwise it evaluates the alternative:
-
- (define (eval-if exp env)
- (if (true? (eval (if-predicate exp) env))
- (eval (if-consequent exp) env)
- (eval (if-alternative exp) env)))
-
- The use of `true?' in `eval-if' highlights the issue of the
-connection between an implemented language and an implementation
-language. The `if-predicate' is evaluated in the language being
-implemented and thus yields a value in that language. The interpreter
-predicate `true?' translates that value into a value that can be tested
-by the `if' in the implementation language: The metacircular
-representation of truth might not be the same as that of the underlying
-Scheme.(2)
-
-Sequences
-.........
-
-`Eval-sequence' is used by `apply' to evaluate the sequence of
-expressions in a procedure body and by `eval' to evaluate the sequence
-of expressions in a `begin' expression. It takes as arguments a
-sequence of expressions and an environment, and evaluates the
-expressions in the order in which they occur. The value returned is
-the value of the final expression.
-
- (define (eval-sequence exps env)
- (cond ((last-exp? exps) (eval (first-exp exps) env))
- (else (eval (first-exp exps) env)
- (eval-sequence (rest-exps exps) env))))
-
-Assignments and definitions
-...........................
-
-The following procedure handles assignments to variables. It calls
-`eval' to find the value to be assigned and transmits the variable and
-the resulting value to `set-variable-value!' to be installed in the
-designated environment.
-
- (define (eval-assignment exp env)
- (set-variable-value! (assignment-variable exp)
- (eval (assignment-value exp) env)
- env)
- 'ok)
-
-Definitions of variables are handled in a similar manner.(3)
-
- (define (eval-definition exp env)
- (define-variable! (definition-variable exp)
- (eval (definition-value exp) env)
- env)
- 'ok)
-
- We have chosen here to return the symbol `ok' as the value of an
-assignment or a definition.(4)
-
- *Exercise 4.1:* Notice that we cannot tell whether the
- metacircular evaluator evaluates operands from left to right or
- from right to left. Its evaluation order is inherited from the
- underlying Lisp: If the arguments to `cons' in `list-of-values'
- are evaluated from left to right, then `list-of-values' will
- evaluate operands from left to right; and if the arguments to
- `cons' are evaluated from right to left, then `list-of-values'
- will evaluate operands from right to left.
-
- Write a version of `list-of-values' that evaluates operands from
- left to right regardless of the order of evaluation in the
- underlying Lisp. Also write a version of `list-of-values' that
- evaluates operands from right to left.
-
- ---------- Footnotes ----------
-
- (1) We could have simplified the `application?' clause in `eval' by
-using `map' (and stipulating that `operands' returns a list) rather
-than writing an explicit `list-of-values' procedure. We chose not to
-use `map' here to emphasize the fact that the evaluator can be
-implemented without any use of higher-order procedures (and thus could
-be written in a language that doesn't have higher-order procedures),
-even though the language that it supports will include higher-order
-procedures.
-
- (2) In this case, the language being implemented and the
-implementation language are the same. Contemplation of the meaning of
-`true?' here yields expansion of consciousness without the abuse of
-substance.
-
- (3) This implementation of `define' ignores a subtle issue in the
-handling of internal definitions, although it works correctly in most
-cases. We will see what the problem is and how to solve it in section
-*Note 4-1-6::.
-
- (4) As we said when we introduced `define' and `set!', these values
-are implementation-dependent in Scheme--that is, the implementor can
-choose what value to return.
-
-\1f
-File: sicp.info, Node: 4-1-2, Next: 4-1-3, Prev: 4-1-1, Up: 4-1
-
-4.1.2 Representing Expressions
-------------------------------
-
-The evaluator is reminiscent of the symbolic differentiation program
-discussed in section *Note 2-3-2::. Both programs operate on symbolic
-expressions. In both programs, the result of operating on a compound
-expression is determined by operating recursively on the pieces of the
-expression and combining the results in a way that depends on the type
-of the expression. In both programs we used data abstraction to
-decouple the general rules of operation from the details of how
-expressions are represented. In the differentiation program this meant
-that the same differentiation procedure could deal with algebraic
-expressions in prefix form, in infix form, or in some other form. For
-the evaluator, this means that the syntax of the language being
-evaluated is determined solely by the procedures that classify and
-extract pieces of expressions.
-
- Here is the specification of the syntax of our language:
-
- * The only self-evaluating items are numbers and strings:
-
- (define (self-evaluating? exp)
- (cond ((number? exp) true)
- ((string? exp) true)
- (else false)))
-
- * Variables are represented by symbols:
-
- (define (variable? exp) (symbol? exp))
-
- * Quotations have the form `(quote <TEXT-OF-QUOTATION>)':(1)
-
- (define (quoted? exp)
- (tagged-list? exp 'quote))
-
- (define (text-of-quotation exp) (cadr exp))
-
- `Quoted?' is defined in terms of the procedure `tagged-list?',
- which identifies lists beginning with a designated symbol:
-
- (define (tagged-list? exp tag)
- (if (pair? exp)
- (eq? (car exp) tag)
- false))
-
- * Assignments have the form `(set! <VAR> <VALUE>)':
-
- (define (assignment? exp)
- (tagged-list? exp 'set!))
-
- (define (assignment-variable exp) (cadr exp))
-
- (define (assignment-value exp) (caddr exp))
-
- * Definitions have the form
-
- (define <VAR> <VALUE>)
-
- or the form
-
- (define (<VAR> <PARAMETER_1> ... <PARAMETER_N>)
- <BODY>)
-
- The latter form (standard procedure definition) is syntactic sugar
- for
-
- (define <VAR>
- (lambda (<PARAMETER_1> ... <PARAMETER_N>)
- <BODY>))
-
- The corresponding syntax procedures are the following:
-
- (define (definition? exp)
- (tagged-list? exp 'define))
-
- (define (definition-variable exp)
- (if (symbol? (cadr exp))
- (cadr exp)
- (caadr exp)))
-
- (define (definition-value exp)
- (if (symbol? (cadr exp))
- (caddr exp)
- (make-lambda (cdadr exp) ; formal parameters
- (cddr exp)))) ; body
-
- * `Lambda' expressions are lists that begin with the symbol `lambda':
-
- (define (lambda? exp) (tagged-list? exp 'lambda))
-
- (define (lambda-parameters exp) (cadr exp))
-
- (define (lambda-body exp) (cddr exp))
-
- We also provide a constructor for `lambda' expressions, which is
- used by `definition-value', above:
-
- (define (make-lambda parameters body)
- (cons 'lambda (cons parameters body)))
-
- * Conditionals begin with `if' and have a predicate, a consequent,
- and an (optional) alternative. If the expression has no
- alternative part, we provide `false' as the alternative.(2)
-
- (define (if? exp) (tagged-list? exp 'if))
-
- (define (if-predicate exp) (cadr exp))
-
- (define (if-consequent exp) (caddr exp))
-
- (define (if-alternative exp)
- (if (not (null? (cdddr exp)))
- (cadddr exp)
- 'false))
-
- We also provide a constructor for `if' expressions, to be used by
- `cond->if' to transform `cond' expressions into `if' expressions:
-
- (define (make-if predicate consequent alternative)
- (list 'if predicate consequent alternative))
-
- * `Begin' packages a sequence of expressions into a single
- expression. We include syntax operations on `begin' expressions
- to extract the actual sequence from the `begin' expression, as
- well as selectors that return the first expression and the rest of
- the expressions in the sequence.(3)
-
- (define (begin? exp) (tagged-list? exp 'begin))
-
- (define (begin-actions exp) (cdr exp))
-
- (define (last-exp? seq) (null? (cdr seq)))
-
- (define (first-exp seq) (car seq))
-
- (define (rest-exps seq) (cdr seq))
-
- We also include a constructor `sequence->exp' (for use by
- `cond->if') that transforms a sequence into a single expression,
- using `begin' if necessary:
-
- (define (sequence->exp seq)
- (cond ((null? seq) seq)
- ((last-exp? seq) (first-exp seq))
- (else (make-begin seq))))
-
- (define (make-begin seq) (cons 'begin seq))
-
- * A procedure application is any compound expression that is not one
- of the above expression types. The `car' of the expression is the
- operator, and the `cdr' is the list of operands:
-
- (define (application? exp) (pair? exp))
-
- (define (operator exp) (car exp))
-
- (define (operands exp) (cdr exp))
-
- (define (no-operands? ops) (null? ops))
-
- (define (first-operand ops) (car ops))
-
- (define (rest-operands ops) (cdr ops))
-
-
-Derived expressions
-...................
-
-Some special forms in our language can be defined in terms of
-expressions involving other special forms, rather than being
-implemented directly. One example is `cond', which can be implemented
-as a nest of `if' expressions. For example, we can reduce the problem
-of evaluating the expression
-
- (cond ((> x 0) x)
- ((= x 0) (display 'zero) 0)
- (else (- x)))
-
-to the problem of evaluating the following expression involving `if' and
-`begin' expressions:
-
- (if (> x 0)
- x
- (if (= x 0)
- (begin (display 'zero)
- 0)
- (- x)))
-
- Implementing the evaluation of `cond' in this way simplifies the
-evaluator because it reduces the number of special forms for which the
-evaluation process must be explicitly specified.
-
- We include syntax procedures that extract the parts of a `cond'
-expression, and a procedure `cond->if' that transforms `cond'
-expressions into `if' expressions. A case analysis begins with `cond'
-and has a list of predicate-action clauses. A clause is an `else'
-clause if its predicate is the symbol `else'.(4)
-
- (define (cond? exp) (tagged-list? exp 'cond))
-
- (define (cond-clauses exp) (cdr exp))
-
- (define (cond-else-clause? clause)
- (eq? (cond-predicate clause) 'else))
-
- (define (cond-predicate clause) (car clause))
-
- (define (cond-actions clause) (cdr clause))
-
- (define (cond->if exp)
- (expand-clauses (cond-clauses exp)))
-
- (define (expand-clauses clauses)
- (if (null? clauses)
- 'false ; no `else' clause
- (let ((first (car clauses))
- (rest (cdr clauses)))
- (if (cond-else-clause? first)
- (if (null? rest)
- (sequence->exp (cond-actions first))
- (error "ELSE clause isn't last -- COND->IF"
- clauses))
- (make-if (cond-predicate first)
- (sequence->exp (cond-actions first))
- (expand-clauses rest))))))
-
- Expressions (such as `cond') that we choose to implement as syntactic
-transformations are called "derived expressions". `Let' expressions
-are also derived expressions (see *Note Exercise 4-6::).(5)
-
- *Exercise 4.2:* Louis Reasoner plans to reorder the `cond' clauses
- in `eval' so that the clause for procedure applications appears
- before the clause for assignments. He argues that this will make
- the interpreter more efficient: Since programs usually contain more
- applications than assignments, definitions, and so on, his
- modified `eval' will usually check fewer clauses than the original
- `eval' before identifying the type of an expression.
-
- a. What is wrong with Louis's plan? (Hint: What will Louis's
- evaluator do with the expression `(define x 3)'?)
-
- b. Louis is upset that his plan didn't work. He is willing to
- go to any lengths to make his evaluator recognize procedure
- applications before it checks for most other kinds of
- expressions. Help him by changing the syntax of the
- evaluated language so that procedure applications start with
- `call'. For example, instead of `(factorial 3)' we will now
- have to write `(call factorial 3)' and instead of `(+ 1 2)'
- we will have to write `(call + 1 2)'.
-
-
- *Exercise 4.3:* Rewrite `eval' so that the dispatch is done in
- data-directed style. Compare this with the data-directed
- differentiation procedure of *Note Exercise 2-73::. (You may use
- the `car' of a compound expression as the type of the expression,
- as is appropriate for the syntax implemented in this section.)
-
- *Exercise 4.4:* Recall the definitions of the special forms `and'
- and `or' from *Note Chapter 1:::
-
- * `and': The expressions are evaluated from left to right. If
- any expression evaluates to false, false is returned; any
- remaining expressions are not evaluated. If all the
- expressions evaluate to true values, the value of the last
- expression is returned. If there are no expressions then
- true is returned.
-
- * `or': The expressions are evaluated from left to right. If
- any expression evaluates to a true value, that value is
- returned; any remaining expressions are not evaluated. If
- all expressions evaluate to false, or if there are no
- expressions, then false is returned.
-
-
- Install `and' and `or' as new special forms for the evaluator by
- defining appropriate syntax procedures and evaluation procedures
- `eval-and' and `eval-or'. Alternatively, show how to implement
- `and' and `or' as derived expressions.
-
- *Exercise 4.5:* Scheme allows an additional syntax for `cond'
- clauses, `(<TEST> => <RECIPIENT>)'. If <TEST> evaluates to a true
- value, then <RECIPIENT> is evaluated. Its value must be a
- procedure of one argument; this procedure is then invoked on the
- value of the <TEST>, and the result is returned as the value of
- the `cond' expression. For example
-
- (cond ((assoc 'b '((a 1) (b 2))) => cadr)
- (else false))
-
- returns 2. Modify the handling of `cond' so that it supports this
- extended syntax.
-
- *Exercise 4.6:* `Let' expressions are derived expressions, because
-
- (let ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
- <BODY>)
-
- is equivalent to
-
- ((lambda (<VAR_1> ... <VAR_N>)
- <BODY>)
- <EXP_1>
- ...
- <EXP_N>)
-
- Implement a syntactic transformation `let->combination' that
- reduces evaluating `let' expressions to evaluating combinations of
- the type shown above, and add the appropriate clause to `eval' to
- handle `let' expressions.
-
- *Exercise 4.7:* `Let*' is similar to `let', except that the
- bindings of the `let' variables are performed sequentially from
- left to right, and each binding is made in an environment in which
- all of the preceding bindings are visible. For example
-
- (let* ((x 3)
- (y (+ x 2))
- (z (+ x y 5)))
- (* x z))
-
- returns 39. Explain how a `let*' expression can be rewritten as a
- set of nested `let' expressions, and write a procedure
- `let*->nested-lets' that performs this transformation. If we have
- already implemented `let' (*Note Exercise 4-6::) and we want to
- extend the evaluator to handle `let*', is it sufficient to add a
- clause to `eval' whose action is
-
- (eval (let*->nested-lets exp) env)
-
- or must we explicitly expand `let*' in terms of non-derived
- expressions?
-
- *Exercise 4.8:* "Named `let'" is a variant of `let' that has the
- form
-
- (let <VAR> <BINDINGS> <BODY>)
-
- The <BINDINGS> and <BODY> are just as in ordinary `let', except
- that <VAR> is bound within <BODY> to a procedure whose body is
- <BODY> and whose parameters are the variables in the <BINDINGS>.
- Thus, one can repeatedly execute the <BODY> by invoking the
- procedure named <VAR>. For example, the iterative Fibonacci
- procedure (section *Note 1-2-2::) can be rewritten using named
- `let' as follows:
-
- (define (fib n)
- (let fib-iter ((a 1)
- (b 0)
- (count n))
- (if (= count 0)
- b
- (fib-iter (+ a b) a (- count 1)))))
-
- Modify `let->combination' of *Note Exercise 4-6:: to also support
- named `let'.
-
- *Exercise 4.9:* Many languages support a variety of iteration
- constructs, such as `do', `for', `while', and `until'. In Scheme,
- iterative processes can be expressed in terms of ordinary
- procedure calls, so special iteration constructs provide no
- essential gain in computational power. On the other hand, such
- constructs are often convenient. Design some iteration
- constructs, give examples of their use, and show how to implement
- them as derived expressions.
-
- *Exercise 4.10:* By using data abstraction, we were able to write
- an `eval' procedure that is independent of the particular syntax
- of the language to be evaluated. To illustrate this, design and
- implement a new syntax for Scheme by modifying the procedures in
- this section, without changing `eval' or `apply'.
-
- ---------- Footnotes ----------
-
- (1) As mentioned in section *Note 2-3-1::, the evaluator sees a
-quoted expression as a list beginning with `quote', even if the
-expression is typed with the quotation mark. For example, the
-expression `'a' would be seen by the evaluator as `(quote a)'. See
-*Note Exercise 2-55::.
-
- (2) The value of an `if' expression when the predicate is false and
-there is no alternative is unspecified in Scheme; we have chosen here
-to make it false. We will support the use of the variables `true' and
-`false' in expressions to be evaluated by binding them in the global
-environment. See section *Note 4-1-4::.
-
- (3) These selectors for a list of expressions--and the corresponding
-ones for a list of operands--are not intended as a data abstraction.
-They are introduced as mnemonic names for the basic list operations in
-order to make it easier to understand the explicit-control evaluator in
-section *Note 5-4::.
-
- (4) The value of a `cond' expression when all the predicates are
-false and there is no `else' clause is unspecified in Scheme; we have
-chosen here to make it false.
-
- (5) Practical Lisp systems provide a mechanism that allows a user to
-add new derived expressions and specify their implementation as
-syntactic transformations without modifying the evaluator. Such a
-user-defined transformation is called a "macro". Although it is easy
-to add an elementary mechanism for defining macros, the resulting
-language has subtle name-conflict problems. There has been much
-research on mechanisms for macro definition that do not cause these
-difficulties. See, for example, Kohlbecker 1986, Clinger and Rees
-1991, and Hanson 1991.
-
-\1f
-File: sicp.info, Node: 4-1-3, Next: 4-1-4, Prev: 4-1-2, Up: 4-1
-
-4.1.3 Evaluator Data Structures
--------------------------------
-
-In addition to defining the external syntax of expressions, the
-evaluator implementation must also define the data structures that the
-evaluator manipulates internally, as part of the execution of a
-program, such as the representation of procedures and environments and
-the representation of true and false.
-
-Testing of predicates
-.....................
-
-For conditionals, we accept anything to be true that is not the explicit
-`false' object.
-
- (define (true? x)
- (not (eq? x false)))
-
- (define (false? x)
- (eq? x false))
-
-Representing procedures
-.......................
-
-To handle primitives, we assume that we have available the following
-procedures:
-
- * `(apply-primitive-procedure <PROC> <ARGS>)'
-
- applies the given primitive procedure to the argument values in
- the list <ARGS> and returns the result of the application.
-
- * `(primitive-procedure? <PROC>)'
-
- tests whether <PROC> is a primitive procedure.
-
-
- These mechanisms for handling primitives are further described in
-section *Note 4-1-4::.
-
- Compound procedures are constructed from parameters, procedure
-bodies, and environments using the constructor `make-procedure':
-
- (define (make-procedure parameters body env)
- (list 'procedure parameters body env))
-
- (define (compound-procedure? p)
- (tagged-list? p 'procedure))
-
- (define (procedure-parameters p) (cadr p))
-
- (define (procedure-body p) (caddr p))
-
- (define (procedure-environment p) (cadddr p))
-
-Operations on Environments
-..........................
-
-The evaluator needs operations for manipulating environments. As
-explained in section *Note 3-2::, an environment is a sequence of
-frames, where each frame is a table of bindings that associate
-variables with their corresponding values. We use the following
-operations for manipulating environments:
-
- * `(lookup-variable-value <VAR> <ENV>)'
-
- returns the value that is bound to the symbol <VAR> in the
- environment <ENV>, or signals an error if the variable is unbound.
-
- * `(extend-environment <VARIABLES> <VALUES> <BASE-ENV>)'
-
- returns a new environment, consisting of a new frame in which the
- symbols in the list <VARIABLES> are bound to the corresponding
- elements in the list <VALUES>, where the enclosing environment is
- the environment <BASE-ENV>.
-
- * `(define-variable! <VAR> <VALUE> <ENV>)'
-
- adds to the first frame in the environment <ENV> a new binding that
- associates the variable <VAR> with the value <VALUE>.
-
- * `(set-variable-value! <VAR> <VALUE> <ENV>)'
-
- changes the binding of the variable <VAR> in the environment <ENV>
- so that the variable is now bound to the value <VALUE>, or signals
- an error if the variable is unbound.
-
-
- To implement these operations we represent an environment as a list
-of frames. The enclosing environment of an environment is the `cdr' of
-the list. The empty environment is simply the empty list.
-
- (define (enclosing-environment env) (cdr env))
-
- (define (first-frame env) (car env))
-
- (define the-empty-environment '())
-
- Each frame of an environment is represented as a pair of lists: a
-list of the variables bound in that frame and a list of the associated
-values.(1)
-
- (define (make-frame variables values)
- (cons variables values))
-
- (define (frame-variables frame) (car frame))
-
- (define (frame-values frame) (cdr frame))
-
- (define (add-binding-to-frame! var val frame)
- (set-car! frame (cons var (car frame)))
- (set-cdr! frame (cons val (cdr frame))))
-
- To extend an environment by a new frame that associates variables
-with values, we make a frame consisting of the list of variables and
-the list of values, and we adjoin this to the environment. We signal
-an error if the number of variables does not match the number of values.
-
- (define (extend-environment vars vals base-env)
- (if (= (length vars) (length vals))
- (cons (make-frame vars vals) base-env)
- (if (< (length vars) (length vals))
- (error "Too many arguments supplied" vars vals)
- (error "Too few arguments supplied" vars vals))))
-
- To look up a variable in an environment, we scan the list of
-variables in the first frame. If we find the desired variable, we
-return the corresponding element in the list of values. If we do not
-find the variable in the current frame, we search the enclosing
-environment, and so on. If we reach the empty environment, we signal
-an "unbound variable" error.
-
- (define (lookup-variable-value var env)
- (define (env-loop env)
- (define (scan vars vals)
- (cond ((null? vars)
- (env-loop (enclosing-environment env)))
- ((eq? var (car vars))
- (car vals))
- (else (scan (cdr vars) (cdr vals)))))
- (if (eq? env the-empty-environment)
- (error "Unbound variable" var)
- (let ((frame (first-frame env)))
- (scan (frame-variables frame)
- (frame-values frame)))))
- (env-loop env))
-
- To set a variable to a new value in a specified environment, we scan
-for the variable, just as in `lookup-variable-value', and change the
-corresponding value when we find it.
-
- (define (set-variable-value! var val env)
- (define (env-loop env)
- (define (scan vars vals)
- (cond ((null? vars)
- (env-loop (enclosing-environment env)))
- ((eq? var (car vars))
- (set-car! vals val))
- (else (scan (cdr vars) (cdr vals)))))
- (if (eq? env the-empty-environment)
- (error "Unbound variable -- SET!" var)
- (let ((frame (first-frame env)))
- (scan (frame-variables frame)
- (frame-values frame)))))
- (env-loop env))
-
- To define a variable, we search the first frame for a binding for
-the variable, and change the binding if it exists (just as in
-`set-variable-value!'). If no such binding exists, we adjoin one to
-the first frame.
-
- (define (define-variable! var val env)
- (let ((frame (first-frame env)))
- (define (scan vars vals)
- (cond ((null? vars)
- (add-binding-to-frame! var val frame))
- ((eq? var (car vars))
- (set-car! vals val))
- (else (scan (cdr vars) (cdr vals)))))
- (scan (frame-variables frame)
- (frame-values frame))))
-
- The method described here is only one of many plausible ways to
-represent environments. Since we used data abstraction to isolate the
-rest of the evaluator from the detailed choice of representation, we
-could change the environment representation if we wanted to. (See
-*Note Exercise 4-11::.) In a production-quality Lisp system, the speed
-of the evaluator's environment operations--especially that of variable
-lookup--has a major impact on the performance of the system. The
-representation described here, although conceptually simple, is not
-efficient and would not ordinarily be used in a production system.(2)
-
- *Exercise 4.11:* Instead of representing a frame as a pair of
- lists, we can represent a frame as a list of bindings, where each
- binding is a name-value pair. Rewrite the environment operations
- to use this alternative representation.
-
- *Exercise 4.12:* The procedures `set-variable-value!',
- `define-variable!', and `lookup-variable-value' can be expressed
- in terms of more abstract procedures for traversing the
- environment structure. Define abstractions that capture the
- common patterns and redefine the three procedures in terms of these
- abstractions.
-
- *Exercise 4.13:* Scheme allows us to create new bindings for
- variables by means of `define', but provides no way to get rid of
- bindings. Implement for the evaluator a special form
- `make-unbound!' that removes the binding of a given symbol from the
- environment in which the `make-unbound!' expression is evaluated.
- This problem is not completely specified. For example, should we
- remove only the binding in the first frame of the environment?
- Complete the specification and justify any choices you make.
-
- ---------- Footnotes ----------
-
- (1) Frames are not really a data abstraction in the following code:
-`Set-variable-value!' and `define-variable!' use `set-car!' to
-directly modify the values in a frame. The purpose of the frame
-procedures is to make the environment-manipulation procedures easy to
-read.
-
- (2) The drawback of this representation (as well as the variant in
-*Note Exercise 4-11::) is that the evaluator may have to search through
-many frames in order to find the binding for a given variable. (Such
-an approach is referred to as "deep binding".) One way to avoid this
-inefficiency is to make use of a strategy called "lexical addressing",
-which will be discussed in section *Note 5-5-6::.
-
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-
-4.1.4 Running the Evaluator as a Program
-----------------------------------------
-
-Given the evaluator, we have in our hands a description (expressed in
-Lisp) of the process by which Lisp expressions are evaluated. One
-advantage of expressing the evaluator as a program is that we can run
-the program. This gives us, running within Lisp, a working model of
-how Lisp itself evaluates expressions. This can serve as a framework
-for experimenting with evaluation rules, as we shall do later in this
-chapter.
-
- Our evaluator program reduces expressions ultimately to the
-application of primitive procedures. Therefore, all that we need to
-run the evaluator is to create a mechanism that calls on the underlying
-Lisp system to model the application of primitive procedures.
-
- There must be a binding for each primitive procedure name, so that
-when `eval' evaluates the operator of an application of a primitive, it
-will find an object to pass to `apply'. We thus set up a global
-environment that associates unique objects with the names of the
-primitive procedures that can appear in the expressions we will be
-evaluating. The global environment also includes bindings for the
-symbols `true' and `false', so that they can be used as variables in
-expressions to be evaluated.
-
- (define (setup-environment)
- (let ((initial-env
- (extend-environment (primitive-procedure-names)
- (primitive-procedure-objects)
- the-empty-environment)))
- (define-variable! 'true true initial-env)
- (define-variable! 'false false initial-env)
- initial-env))
-
- (define the-global-environment (setup-environment))
-
- It does not matter how we represent the primitive procedure objects,
-so long as `apply' can identify and apply them by using the procedures
-`primitive-procedure?' and `apply-primitive-procedure'. We have chosen
-to represent a primitive procedure as a list beginning with the symbol
-`primitive' and containing a procedure in the underlying Lisp that
-implements that primitive.
-
- (define (primitive-procedure? proc)
- (tagged-list? proc 'primitive))
-
- (define (primitive-implementation proc) (cadr proc))
-
- `Setup-environment' will get the primitive names and implementation
-procedures from a list:(1)
-
- (define primitive-procedures
- (list (list 'car car)
- (list 'cdr cdr)
- (list 'cons cons)
- (list 'null? null?)
- <MORE PRIMITIVES>
- ))
-
- (define (primitive-procedure-names)
- (map car
- primitive-procedures))
-
- (define (primitive-procedure-objects)
- (map (lambda (proc) (list 'primitive (cadr proc)))
- primitive-procedures))
-
- To apply a primitive procedure, we simply apply the implementation
-procedure to the arguments, using the underlying Lisp system:(2)
-
- (define (apply-primitive-procedure proc args)
- (apply-in-underlying-scheme
- (primitive-implementation proc) args))
-
- For convenience in running the metacircular evaluator, we provide a "driver
-loop" that models the read-eval-print loop of the underlying Lisp
-system. It prints a "prompt", reads an input expression, evaluates
-this expression in the global environment, and prints the result. We
-precede each printed result by an "output prompt" so as to distinguish
-the value of the expression from other output that may be printed.(3)
-
- (define input-prompt ";;; M-Eval input:")
- (define output-prompt ";;; M-Eval value:")
-
- (define (driver-loop)
- (prompt-for-input input-prompt)
- (let ((input (read)))
- (let ((output (eval input the-global-environment)))
- (announce-output output-prompt)
- (user-print output)))
- (driver-loop))
-
- (define (prompt-for-input string)
- (newline) (newline) (display string) (newline))
-
- (define (announce-output string)
- (newline) (display string) (newline))
-
- We use a special printing procedure, `user-print', to avoid printing
-the environment part of a compound procedure, which may be a very long
-list (or may even contain cycles).
-
- (define (user-print object)
- (if (compound-procedure? object)
- (display (list 'compound-procedure
- (procedure-parameters object)
- (procedure-body object)
- '<procedure-env>))
- (display object)))
-
- Now all we need to do to run the evaluator is to initialize the
-global environment and start the driver loop. Here is a sample
-interaction:
-
- (define the-global-environment (setup-environment))
-
- (driver-loop)
-
- ;;; M-Eval input:
- (define (append x y)
- (if (null? x)
- y
- (cons (car x)
- (append (cdr x) y))))
- ;;; M-Eval value:
- ok
-
- ;;; M-Eval input:
- (append '(a b c) '(d e f))
- ;;; M-Eval value:
- (a b c d e f)
-
- *Exercise 4.14:* Eva Lu Ator and Louis Reasoner are each
- experimenting with the metacircular evaluator. Eva types in the
- definition of `map', and runs some test programs that use it.
- They work fine. Louis, in contrast, has installed the system
- version of `map' as a primitive for the metacircular evaluator.
- When he tries it, things go terribly wrong. Explain why Louis's
- `map' fails even though Eva's works.
-
- ---------- Footnotes ----------
-
- (1) Any procedure defined in the underlying Lisp can be used as a
-primitive for the metacircular evaluator. The name of a primitive
-installed in the evaluator need not be the same as the name of its
-implementation in the underlying Lisp; the names are the same here
-because the metacircular evaluator implements Scheme itself. Thus, for
-example, we could put `(list 'first car)' or `(list 'square (lambda (x)
-(* x x)))' in the list of `primitive-procedures'.
-
- (2) `Apply-in-underlying-scheme' is the `apply' procedure we have
-used in earlier chapters. The metacircular evaluator's `apply'
-procedure (section *Note 4-1-1::) models the working of this primitive.
-Having two different things called `apply' leads to a technical
-problem in running the metacircular evaluator, because defining the
-metacircular evaluator's `apply' will mask the definition of the
-primitive. One way around this is to rename the metacircular `apply' to
-avoid conflict with the name of the primitive procedure. We have
-assumed instead that we have saved a reference to the underlying
-`apply' by doing
-
- (define apply-in-underlying-scheme apply)
-
-before defining the metacircular `apply'. This allows us to access the
-original version of `apply' under a different name.
-
- (3) The primitive procedure `read' waits for input from the user,
-and returns the next complete expression that is typed. For example,
-if the user types `(+ 23 x)', `read' returns a three-element list
-containing the symbol `+', the number 23, and the symbol `x'. If the
-user types `'x', `read' returns a two-element list containing the
-symbol `quote' and the symbol `x'.
-
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-
-4.1.5 Data as Programs
-----------------------
-
-In thinking about a Lisp program that evaluates Lisp expressions, an
-analogy might be helpful. One operational view of the meaning of a
-program is that a program is a description of an abstract (perhaps
-infinitely large) machine. For example, consider the familiar program
-to compute factorials:
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
-
- We may regard this program as the description of a machine
-containing parts that decrement, multiply, and test for equality,
-together with a two-position switch and another factorial machine. (The
-factorial machine is infinite because it contains another factorial
-machine within it.) *Note Figure 4-2:: is a flow diagram for the
-factorial machine, showing how the parts are wired together.
-
- *Figure 4.2:* The factorial program, viewed as an abstract machine.
-
- +-----------------------------------+
- | factorial |1 |
- | |1 V |
- | | +-----+ |
- | V | # | |
- | +-----+ | | |
- 6 --------*-----| = |------->| #-+-----> 720
- | | +-----+ | / | |
- | | | # | |
- | | +-----+ |
- | | ^ |
- | | | |
- | | +--+--+ |
- | *------------------->| * | |
- | | +-----+ |
- | V ^ |
- | +-----+ +-----------+ | |
- | | - +--->| factorial +---+ |
- | +-----+ +-----------+ |
- | ^ |
- | |1 |
- +-----------------------------------+
-
- In a similar way, we can regard the evaluator as a very special
-machine that takes as input a description of a machine. Given this
-input, the evaluator configures itself to emulate the machine
-described. For example, if we feed our evaluator the definition of
-`factorial', as shown in *Note Figure 4-3::, the evaluator will be able
-to compute factorials.
-
- *Figure 4.3:* The evaluator emulating a factorial machine.
-
- +--------+
- 6 ---->| eval |----> 720
- +--------+
- /
- . . . / . . .
- . . . ../. . .
- . ..
- . (define (factorial n) . . .
- . (if (= n 1) . .
- . 1 .
- . (* (factorial (- n 1)) n))) .
- . . . . .
- . . . . . . . . . . . .
- . ..
-
- From this perspective, our evaluator is seen to be a machine
-"universal machine". It mimics other machines when these are described
-as Lisp programs.(1) This is striking. Try to imagine an analogous
-evaluator for electrical circuits. This would be a circuit that takes
-as input a signal encoding the plans for some other circuit, such as a
-filter. Given this input, the circuit evaluator would then behave like
-a filter with the same description. Such a universal electrical
-circuit is almost unimaginably complex. It is remarkable that the
-program evaluator is a rather simple program.(2)
-
- Another striking aspect of the evaluator is that it acts as a bridge
-between the data objects that are manipulated by our programming
-language and the programming language itself. Imagine that the
-evaluator program (implemented in Lisp) is running, and that a user is
-typing expressions to the evaluator and observing the results. From
-the perspective of the user, an input expression such as `(* x x)' is
-an expression in the programming language, which the evaluator should
-execute. From the perspective of the evaluator, however, the
-expression is simply a list (in this case, a list of three symbols: `*',
-`x', and `x') that is to be manipulated according to a well-defined set
-of rules.
-
- That the user's programs are the evaluator's data need not be a
-source of confusion. In fact, it is sometimes convenient to ignore
-this distinction, and to give the user the ability to explicitly
-evaluate a data object as a Lisp expression, by making `eval' available
-for use in programs. Many Lisp dialects provide a primitive `eval'
-procedure that takes as arguments an expression and an environment and
-evaluates the expression relative to the environment.(3) Thus,
-
- (eval '(* 5 5) user-initial-environment)
-
-and
-
- (eval (cons '* (list 5 5)) user-initial-environment)
-
-will both return 25.(4)
-
- *Exercise 4.15:* Given a one-argument procedure `p' and an object
- `a', `p' is said to "halt" on `a' if evaluating the expression `(p
- a)' returns a value (as opposed to terminating with an error
- message or running forever). Show that it is impossible to write
- a procedure `halts?' that correctly determines whether `p' halts
- on `a' for any procedure `p' and object `a'. Use the following
- reasoning: If you had such a procedure `halts?', you could
- implement the following program:
-
- (define (run-forever) (run-forever))
-
- (define (try p)
- (if (halts? p p)
- (run-forever)
- 'halted))
-
- Now consider evaluating the expression `(try try)' and show that
- any possible outcome (either halting or running forever) violates
- the intended behavior of `halts?'.(5)
-
- ---------- Footnotes ----------
-
- (1) The fact that the machines are described in Lisp is inessential.
-If we give our evaluator a Lisp program that behaves as an evaluator
-for some other language, say C, the Lisp evaluator will emulate the C
-evaluator, which in turn can emulate any machine described as a C
-program. Similarly, writing a Lisp evaluator in C produces a C program
-that can execute any Lisp program. The deep idea here is that any
-evaluator can emulate any other. Thus, the notion of "what can in
-principle be computed" (ignoring practicalities of time and memory
-required) is independent of the language or the computer, and instead
-reflects an underlying notion of "computability". This was first
-demonstrated in a clear way by Alan M. Turing (1912-1954), whose 1936
-paper laid the foundations for theoretical computer science. In the
-paper, Turing presented a simple computational model--now known as a "Turing
-machine"--and argued that any "effective process" can be formulated as
-a program for such a machine. (This argument is known as the "Church-Turing
-thesis".) Turing then implemented a universal machine, i.e., a Turing
-machine that behaves as an evaluator for Turing-machine programs. He
-used this framework to demonstrate that there are well-posed problems
-that cannot be computed by Turing machines (see *Note Exercise 4-15::),
-and so by implication cannot be formulated as "effective processes."
-Turing went on to make fundamental contributions to practical computer
-science as well. For example, he invented the idea of structuring
-programs using general-purpose subroutines. See Hodges 1983 for a
-biography of Turing.
-
- (2) Some people find it counterintuitive that an evaluator, which is
-implemented by a relatively simple procedure, can emulate programs that
-are more complex than the evaluator itself. The existence of a
-universal evaluator machine is a deep and wonderful property of
-computation. theory "Recursion theory", a branch of mathematical
-logic, is concerned with logical limits of computation. Douglas
-Hofstadter's beautiful book `Go"del, Escher, Bach' (1979) explores some
-of these ideas.
-
- (3) Warning: This `eval' primitive is not identical to the `eval'
-procedure we implemented in section *Note 4-1-1::, because it uses
-_actual_ Scheme environments rather than the sample environment
-structures we built in section *Note 4-1-3::. These actual
-environments cannot be manipulated by the user as ordinary lists; they
-must be accessed via `eval' or other special operations. Similarly,
-the `apply' primitive we saw earlier is not identical to the
-metacircular `apply', because it uses actual Scheme procedures rather
-than the procedure objects we constructed in sections *Note 4-1-3:: and
-*Note 4-1-4::.
-
- (4) The MIT implementation of Scheme includes `eval', as well as a
-symbol `user-initial-environment' that is bound to the initial
-environment in which the user's input expressions are evaluated.
-
- (5) Although we stipulated that `halts?' is given a procedure
-object, notice that this reasoning still applies even if `halts?' can
-gain access to the procedure's text and its environment. This is
-Turing's celebrated "Halting Theorem", which gave the first clear
-example of a "non-computable" problem, i.e., a well-posed task that
-cannot be carried out as a computational procedure.
-
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-
-4.1.6 Internal Definitions
---------------------------
-
-Our environment model of evaluation and our metacircular evaluator
-execute definitions in sequence, extending the environment frame one
-definition at a time. This is particularly convenient for interactive
-program development, in which the programmer needs to freely mix the
-application of procedures with the definition of new procedures.
-However, if we think carefully about the internal definitions used to
-implement block structure (introduced in section *Note 1-1-8::), we
-will find that name-by-name extension of the environment may not be the
-best way to define local variables.
-
- Consider a procedure with internal definitions, such as
-
- (define (f x)
- (define (even? n)
- (if (= n 0)
- true
- (odd? (- n 1))))
- (define (odd? n)
- (if (= n 0)
- false
- (even? (- n 1))))
- <REST OF BODY OF `F'>)
-
- Our intention here is that the name `odd?' in the body of the
-procedure `even?' should refer to the procedure `odd?' that is defined
-after `even?'. The scope of the name `odd?' is the entire body of `f',
-not just the portion of the body of `f' starting at the point where the
-`define' for `odd?' occurs. Indeed, when we consider that `odd?' is
-itself defined in terms of `even?'--so that `even?' and `odd?' are
-mutually recursive procedures--we see that the only satisfactory
-interpretation of the two `define's is to regard them as if the names
-`even?' and `odd?' were being added to the environment simultaneously.
-More generally, in block structure, the scope of a local name is the
-entire procedure body in which the `define' is evaluated.
-
- As it happens, our interpreter will evaluate calls to `f' correctly,
-but for an "accidental" reason: Since the definitions of the internal
-procedures come first, no calls to these procedures will be evaluated
-until all of them have been defined. Hence, `odd?' will have been
-defined by the time `even?' is executed. In fact, our sequential
-evaluation mechanism will give the same result as a mechanism that
-directly implements simultaneous definition for any procedure in which
-the internal definitions come first in a body and evaluation of the
-value expressions for the defined variables doesn't actually use any of
-the defined variables. (For an example of a procedure that doesn't
-obey these restrictions, so that sequential definition isn't equivalent
-to simultaneous definition, see *Note Exercise 4-19::.)(1)
-
- There is, however, a simple way to treat definitions so that
-internally defined names have truly simultaneous scope--just create all
-local variables that will be in the current environment before
-evaluating any of the value expressions. One way to do this is by a
-syntax transformation on `lambda' expressions. Before evaluating the
-body of a `lambda' expression, we "scan out" and eliminate all the
-internal definitions in the body. The internally defined variables
-will be created with a `let' and then set to their values by
-assignment. For example, the procedure
-
- (lambda <VARS>
- (define u <E1>)
- (define v <E2>)
- <E3>)
-
-would be transformed into
-
- (lambda <VARS>
- (let ((u '*unassigned*)
- (v '*unassigned*))
- (set! u <E1>)
- (set! v <E2>)
- <E3>))
-
-where `*unassigned*' is a special symbol that causes looking up a
-variable to signal an error if an attempt is made to use the value of
-the not-yet-assigned variable.
-
- An alternative strategy for scanning out internal definitions is
-shown in *Note Exercise 4-18::. Unlike the transformation shown above,
-this enforces the restriction that the defined variables' values can be
-evaluated without using any of the variables' values.(2)
-
- *Exercise 4.16:* In this exercise we implement the method just
- described for interpreting internal definitions. We assume that
- the evaluator supports `let' (see *Note Exercise 4-6::).
-
- a. Change `lookup-variable-value' (section *Note 4-1-3::) to
- signal an error if the value it finds is the symbol
- `*unassigned*'.
-
- b. Write a procedure `scan-out-defines' that takes a procedure
- body and returns an equivalent one that has no internal
- definitions, by making the transformation described above.
-
- c. Install `scan-out-defines' in the interpreter, either in
- `make-procedure' or in `procedure-body' (see section *Note
- 4-1-3::). Which place is better? Why?
-
-
- *Exercise 4.17:* Draw diagrams of the environment in effect when
- evaluating the expression <E3> in the procedure in the text,
- comparing how this will be structured when definitions are
- interpreted sequentially with how it will be structured if
- definitions are scanned out as described. Why is there an extra
- frame in the transformed program? Explain why this difference in
- environment structure can never make a difference in the behavior
- of a correct program. Design a way to make the interpreter
- implement the "simultaneous" scope rule for internal definitions
- without constructing the extra frame.
-
- *Exercise 4.18:* Consider an alternative strategy for scanning out
- definitions that translates the example in the text to
-
- (lambda <VARS>
- (let ((u '*unassigned*)
- (v '*unassigned*))
- (let ((a <E1>)
- (b <E2>))
- (set! u a)
- (set! v b))
- <E3>))
-
- Here `a' and `b' are meant to represent new variable names, created
- by the interpreter, that do not appear in the user's program.
- Consider the `solve' procedure from section *Note 3-5-4:::
-
- (define (solve f y0 dt)
- (define y (integral (delay dy) y0 dt))
- (define dy (stream-map f y))
- y)
-
- Will this procedure work if internal definitions are scanned out
- as shown in this exercise? What if they are scanned out as shown
- in the text? Explain.
-
- *Exercise 4.19:* Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator
- are arguing about the desired result of evaluating the expression
-
- (let ((a 1))
- (define (f x)
- (define b (+ a x))
- (define a 5)
- (+ a b))
- (f 10))
-
- Ben asserts that the result should be obtained using the
- sequential rule for `define': `b' is defined to be 11, then `a' is
- defined to be 5, so the result is 16. Alyssa objects that mutual
- recursion requires the simultaneous scope rule for internal
- procedure definitions, and that it is unreasonable to treat
- procedure names differently from other names. Thus, she argues
- for the mechanism implemented in *Note Exercise 4-16::. This
- would lead to `a' being unassigned at the time that the value for
- `b' is to be computed. Hence, in Alyssa's view the procedure
- should produce an error. Eva has a third opinion. She says that
- if the definitions of `a' and `b' are truly meant to be
- simultaneous, then the value 5 for `a' should be used in
- evaluating `b'. Hence, in Eva's view `a' should be 5, `b' should
- be 15, and the result should be 20. Which (if any) of these
- viewpoints do you support? Can you devise a way to implement
- internal definitions so that they behave as Eva prefers?(3)
-
- *Exercise 4.20:* Because internal definitions look sequential but
- are actually simultaneous, some people prefer to avoid them
- entirely, and use the special form `letrec' instead. `Letrec'
- looks like `let', so it is not surprising that the variables it
- binds are bound simultaneously and have the same scope as each
- other. The sample procedure `f' above can be written without
- internal definitions, but with exactly the same meaning, as
-
- (define (f x)
- (letrec ((even?
- (lambda (n)
- (if (= n 0)
- true
- (odd? (- n 1)))))
- (odd?
- (lambda (n)
- (if (= n 0)
- false
- (even? (- n 1))))))
- <REST OF BODY OF `F'>))
-
- `Letrec' expressions, which have the form
-
- (letrec ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
- <BODY>)
-
- are a variation on `let' in which the expressions <EXP_K> that
- provide the initial values for the variables <VAR_K> are evaluated
- in an environment that includes all the `letrec' bindings. This
- permits recursion in the bindings, such as the mutual recursion of
- `even?' and `odd?' in the example above, or the evaluation of 10
- factorial with
-
- (letrec ((fact
- (lambda (n)
- (if (= n 1)
- 1
- (* n (fact (- n 1)))))))
- (fact 10))
-
- a. Implement `letrec' as a derived expression, by transforming a
- `letrec' expression into a `let' expression as shown in the
- text above or in *Note Exercise 4-18::. That is, the
- `letrec' variables should be created with a `let' and then be
- assigned their values with `set!'.
-
- b. Louis Reasoner is confused by all this fuss about internal
- definitions. The way he sees it, if you don't like to use
- `define' inside a procedure, you can just use `let'.
- Illustrate what is loose about his reasoning by drawing an
- environment diagram that shows the environment in which the
- <REST OF BODY OF `F'> is evaluated during evaluation of the
- expression `(f 5)', with `f' defined as in this exercise.
- Draw an environment diagram for the same evaluation, but with
- `let' in place of `letrec' in the definition of `f'.
-
-
- *Exercise 4.21:* Amazingly, Louis's intuition in *Note Exercise
- 4-20:: is correct. It is indeed possible to specify recursive
- procedures without using `letrec' (or even `define'), although the
- method for accomplishing this is much more subtle than Louis
- imagined. The following expression computes 10 factorial by
- applying a recursive factorial procedure:(4)
-
- ((lambda (n)
- ((lambda (fact)
- (fact fact n))
- (lambda (ft k)
- (if (= k 1)
- 1
- (* k (ft ft (- k 1)))))))
- 10)
-
- a. Check (by evaluating the expression) that this really does
- compute factorials. Devise an analogous expression for
- computing Fibonacci numbers.
-
- b. Consider the following procedure, which includes mutually
- recursive internal definitions:
-
- (define (f x)
- (define (even? n)
- (if (= n 0)
- true
- (odd? (- n 1))))
- (define (odd? n)
- (if (= n 0)
- false
- (even? (- n 1))))
- (even? x))
-
- Fill in the missing expressions to complete an alternative
- definition of `f', which uses neither internal definitions
- nor `letrec':
-
- (define (f x)
- ((lambda (even? odd?)
- (even? even? odd? x))
- (lambda (ev? od? n)
- (if (= n 0) true (od? <??> <??> <??>)))
- (lambda (ev? od? n)
- (if (= n 0) false (ev? <??> <??> <??>)))))
-
- ---------- Footnotes ----------
-
- (1) Wanting programs to not depend on this evaluation mechanism is
-the reason for the "management is not responsible" remark in *Note
-Footnote 28:: of *Note Chapter 1::. By insisting that internal
-definitions come first and do not use each other while the definitions
-are being evaluated, the IEEE standard for Scheme leaves implementors
-some choice in the mechanism used to evaluate these definitions. The
-choice of one evaluation rule rather than another here may seem like a
-small issue, affecting only the interpretation of "badly formed"
-programs. However, we will see in section *Note 5-5-6:: that moving to
-a model of simultaneous scoping for internal definitions avoids some
-nasty difficulties that would otherwise arise in implementing a
-compiler.
-
- (2) The IEEE standard for Scheme allows for different implementation
-strategies by specifying that it is up to the programmer to obey this
-restriction, not up to the implementation to enforce it. Some Scheme
-implementations, including MIT Scheme, use the transformation shown
-above. Thus, some programs that don't obey this restriction will in
-fact run in such implementations.
-
- (3) The MIT implementors of Scheme support Alyssa on the following
-grounds: Eva is in principle correct - the definitions should be
-regarded as simultaneous. But it seems difficult to implement a
-general, efficient mechanism that does what Eva requires. In the
-absence of such a mechanism, it is better to generate an error in the
-difficult cases of simultaneous definitions (Alyssa's notion) than to
-produce an incorrect answer (as Ben would have it).
-
- (4) This example illustrates a programming trick for formulating
-recursive procedures without using `define'. The most general trick of
-this sort is the Y "operator", which can be used to give a "pure
-[lambda]-calculus" implementation of recursion. (See Stoy 1977 for
-details on the [lambda] calculus, and Gabriel 1988 for an exposition of
-the Y operator in Scheme.)
-
-\1f
-File: sicp.info, Node: 4-1-7, Prev: 4-1-6, Up: 4-1
-
-4.1.7 Separating Syntactic Analysis from Execution
---------------------------------------------------
-
-The evaluator implemented above is simple, but it is very inefficient,
-because the syntactic analysis of expressions is interleaved with their
-execution. Thus if a program is executed many times, its syntax is
-analyzed many times. Consider, for example, evaluating `(factorial 4)'
-using the following definition of `factorial':
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
-
- Each time `factorial' is called, the evaluator must determine that
-the body is an `if' expression and extract the predicate. Only then
-can it evaluate the predicate and dispatch on its value. Each time it
-evaluates the expression `(* (factorial (- n 1)) n)', or the
-subexpressions `(factorial (- n 1))' and `(- n 1)', the evaluator must
-perform the case analysis in `eval' to determine that the expression is
-an application, and must extract its operator and operands. This
-analysis is expensive. Performing it repeatedly is wasteful.
-
- We can transform the evaluator to be significantly more efficient by
-arranging things so that syntactic analysis is performed only once.(1)
-We split `eval', which takes an expression and an environment, into two
-parts. The procedure `analyze' takes only the expression. It performs
-the syntactic analysis and returns a new procedure, the "execution
-procedure", that encapsulates the work to be done in executing the
-analyzed expression. The execution procedure takes an environment as
-its argument and completes the evaluation. This saves work because
-`analyze' will be called only once on an expression, while the
-execution procedure may be called many times.
-
- With the separation into analysis and execution, `eval' now becomes
-
- (define (eval exp env)
- ((analyze exp) env))
-
- The result of calling `analyze' is the execution procedure to be
-applied to the environment. The `analyze' procedure is the same case
-analysis as performed by the original `eval' of section *Note 4-1-1::,
-except that the procedures to which we dispatch perform only analysis,
-not full evaluation:
-
- (define (analyze exp)
- (cond ((self-evaluating? exp)
- (analyze-self-evaluating exp))
- ((quoted? exp) (analyze-quoted exp))
- ((variable? exp) (analyze-variable exp))
- ((assignment? exp) (analyze-assignment exp))
- ((definition? exp) (analyze-definition exp))
- ((if? exp) (analyze-if exp))
- ((lambda? exp) (analyze-lambda exp))
- ((begin? exp) (analyze-sequence (begin-actions exp)))
- ((cond? exp) (analyze (cond->if exp)))
- ((application? exp) (analyze-application exp))
- (else
- (error "Unknown expression type -- ANALYZE" exp))))
-
- Here is the simplest syntactic analysis procedure, which handles
-self-evaluating expressions. It returns an execution procedure that
-ignores its environment argument and just returns the expression:
-
- (define (analyze-self-evaluating exp)
- (lambda (env) exp))
-
- For a quoted expression, we can gain a little efficiency by
-extracting the text of the quotation only once, in the analysis phase,
-rather than in the execution phase.
-
- (define (analyze-quoted exp)
- (let ((qval (text-of-quotation exp)))
- (lambda (env) qval)))
-
- Looking up a variable value must still be done in the execution
-phase, since this depends upon knowing the environment.(2)
-
- (define (analyze-variable exp)
- (lambda (env) (lookup-variable-value exp env)))
-
- `Analyze-assignment' also must defer actually setting the variable
-until the execution, when the environment has been supplied. However,
-the fact that the `assignment-value' expression can be analyzed
-(recursively) during analysis is a major gain in efficiency, because
-the `assignment-value' expression will now be analyzed only once. The
-same holds true for definitions.
-
- (define (analyze-assignment exp)
- (let ((var (assignment-variable exp))
- (vproc (analyze (assignment-value exp))))
- (lambda (env)
- (set-variable-value! var (vproc env) env)
- 'ok)))
-
- (define (analyze-definition exp)
- (let ((var (definition-variable exp))
- (vproc (analyze (definition-value exp))))
- (lambda (env)
- (define-variable! var (vproc env) env)
- 'ok)))
-
- For `if' expressions, we extract and analyze the predicate,
-consequent, and alternative at analysis time.
-
- (define (analyze-if exp)
- (let ((pproc (analyze (if-predicate exp)))
- (cproc (analyze (if-consequent exp)))
- (aproc (analyze (if-alternative exp))))
- (lambda (env)
- (if (true? (pproc env))
- (cproc env)
- (aproc env)))))
-
- Analyzing a `lambda' expression also achieves a major gain in
-efficiency: We analyze the `lambda' body only once, even though
-procedures resulting from evaluation of the `lambda' may be applied
-many times.
-
- (define (analyze-lambda exp)
- (let ((vars (lambda-parameters exp))
- (bproc (analyze-sequence (lambda-body exp))))
- (lambda (env) (make-procedure vars bproc env))))
-
- Analysis of a sequence of expressions (as in a `begin' or the body
-of a `lambda' expression) is more involved.(3) Each expression in the
-sequence is analyzed, yielding an execution procedure. These execution
-procedures are combined to produce an execution procedure that takes an
-environment as argument and sequentially calls each individual execution
-procedure with the environment as argument.
-
- (define (analyze-sequence exps)
- (define (sequentially proc1 proc2)
- (lambda (env) (proc1 env) (proc2 env)))
- (define (loop first-proc rest-procs)
- (if (null? rest-procs)
- first-proc
- (loop (sequentially first-proc (car rest-procs))
- (cdr rest-procs))))
- (let ((procs (map analyze exps)))
- (if (null? procs)
- (error "Empty sequence -- ANALYZE"))
- (loop (car procs) (cdr procs))))
-
- To analyze an application, we analyze the operator and operands and
-construct an execution procedure that calls the operator execution
-procedure (to obtain the actual procedure to be applied) and the
-operand execution procedures (to obtain the actual arguments). We then
-pass these to `execute-application', which is the analog of `apply' in
-section *Note 4-1-1::. `Execute-application' differs from `apply' in
-that the procedure body for a compound procedure has already been
-analyzed, so there is no need to do further analysis. Instead, we just
-call the execution procedure for the body on the extended environment.
-
- (define (analyze-application exp)
- (let ((fproc (analyze (operator exp)))
- (aprocs (map analyze (operands exp))))
- (lambda (env)
- (execute-application (fproc env)
- (map (lambda (aproc) (aproc env))
- aprocs)))))
-
- (define (execute-application proc args)
- (cond ((primitive-procedure? proc)
- (apply-primitive-procedure proc args))
- ((compound-procedure? proc)
- ((procedure-body proc)
- (extend-environment (procedure-parameters proc)
- args
- (procedure-environment proc))))
- (else
- (error
- "Unknown procedure type -- EXECUTE-APPLICATION"
- proc))))
-
- Our new evaluator uses the same data structures, syntax procedures,
-and run-time support procedures as in sections *Note 4-1-2::, *Note
-4-1-3::, and *Note 4-1-4::.
-
- *Exercise 4.22:* Extend the evaluator in this section to support
- the special form `let'. (See *Note Exercise 4-6::.)
-
- *Exercise 4.23:* Alyssa P. Hacker doesn't understand why
- `analyze-sequence' needs to be so complicated. All the other
- analysis procedures are straightforward transformations of the
- corresponding evaluation procedures (or `eval' clauses) in section
- *Note 4-1-1::. She expected `analyze-sequence' to look like this:
-
- (define (analyze-sequence exps)
- (define (execute-sequence procs env)
- (cond ((null? (cdr procs)) ((car procs) env))
- (else ((car procs) env)
- (execute-sequence (cdr procs) env))))
- (let ((procs (map analyze exps)))
- (if (null? procs)
- (error "Empty sequence -- ANALYZE"))
- (lambda (env) (execute-sequence procs env))))
-
- Eva Lu Ator explains to Alyssa that the version in the text does
- more of the work of evaluating a sequence at analysis time.
- Alyssa's sequence-execution procedure, rather than having the
- calls to the individual execution procedures built in, loops
- through the procedures in order to call them: In effect, although
- the individual expressions in the sequence have been analyzed, the
- sequence itself has not been.
-
- Compare the two versions of `analyze-sequence'. For example,
- consider the common case (typical of procedure bodies) where the
- sequence has just one expression. What work will the execution
- procedure produced by Alyssa's program do? What about the
- execution procedure produced by the program in the text above?
- How do the two versions compare for a sequence with two
- expressions?
-
- *Exercise 4.24:* Design and carry out some experiments to compare
- the speed of the original metacircular evaluator with the version
- in this section. Use your results to estimate the fraction of time
- that is spent in analysis versus execution for various procedures.
-
- ---------- Footnotes ----------
-
- (1) This technique is an integral part of the compilation process,
-which we shall discuss in *Note Chapter 5::. Jonathan Rees wrote a
-Scheme interpreter like this in about 1982 for the T project (Rees and
-Adams 1982). Marc Feeley (1986) (see also Feeley and Lapalme 1987)
-independently invented this technique in his master's thesis.
-
- (2) There is, however, an important part of the variable search that
-_can_ be done as part of the syntactic analysis. As we will show in
-section *Note 5-5-6::, one can determine the position in the
-environment structure where the value of the variable will be found,
-thus obviating the need to scan the environment for the entry that
-matches the variable.
-
- (3) See *Note Exercise 4-23:: for some insight into the processing
-of sequences.
-
-\1f
-File: sicp.info, Node: 4-2, Next: 4-3, Prev: 4-1, Up: Chapter 4
-
-4.2 Variations on a Scheme - Lazy Evaluation
-============================================
-
-Now that we have an evaluator expressed as a Lisp program, we can
-experiment with alternative choices in language design simply by
-modifying the evaluator. Indeed, new languages are often invented by
-first writing an evaluator that embeds the new language within an
-existing high-level language. For example, if we wish to discuss some
-aspect of a proposed modification to Lisp with another member of the
-Lisp community, we can supply an evaluator that embodies the change.
-The recipient can then experiment with the new evaluator and send back
-comments as further modifications. Not only does the high-level
-implementation base make it easier to test and debug the evaluator; in
-addition, the embedding enables the designer to snarf(1) features from
-the underlying language, just as our embedded Lisp evaluator uses
-primitives and control structure from the underlying Lisp. Only later
-(if ever) need the designer go to the trouble of building a complete
-implementation in a low-level language or in hardware. In this section
-and the next we explore some variations on Scheme that provide
-significant additional expressive power.
-
-* Menu:
-
-* 4-2-1:: Normal Order and Applicative Order
-* 4-2-2:: An Interpreter with Lazy Evaluation
-* 4-2-3:: Streams as Lazy Lists
-
- ---------- Footnotes ----------
-
- (1) Snarf: "To grab, especially a large document or file for the
-purpose of using it either with or without the owner's permission."
-Snarf down: "To snarf, sometimes with the connotation of absorbing,
-processing, or understanding." (These definitions were snarfed from
-Steele et al. 1983. See also Raymond 1993.)
-
-\1f
-File: sicp.info, Node: 4-2-1, Next: 4-2-2, Prev: 4-2, Up: 4-2
-
-4.2.1 Normal Order and Applicative Order
-----------------------------------------
-
-In section *Note 1-1::, where we began our discussion of models of
-evaluation, we noted that Scheme is an "applicative-order" language,
-namely, that all the arguments to Scheme procedures are evaluated when
-the procedure is applied. In contrast, "normal-order" languages delay
-evaluation of procedure arguments until the actual argument values are
-needed. Delaying evaluation of procedure arguments until the last
-possible moment (e.g., until they are required by a primitive
-operation) is called evaluation "lazy evaluation".(1) Consider the
-procedure
-
- (define (try a b)
- (if (= a 0) 1 b))
-
- Evaluating `(try 0 (/ 1 0))' generates an error in Scheme. With lazy
-evaluation, there would be no error. Evaluating the expression would
-return 1, because the argument `(/ 1 0)' would never be evaluated.
-
- An example that exploits lazy evaluation is the definition of a
-procedure `unless'
-
- (define (unless condition usual-value exceptional-value)
- (if condition exceptional-value usual-value))
-
-that can be used in expressions such as
-
- (unless (= b 0)
- (/ a b)
- (begin (display "exception: returning 0")
- 0))
-
- This won't work in an applicative-order language because both the
-usual value and the exceptional value will be evaluated before `unless'
-is called (compare *Note Exercise 1-6::). An advantage of lazy
-evaluation is that some procedures, such as `unless', can do useful
-computation even if evaluation of some of their arguments would produce
-errors or would not terminate.
-
- If the body of a procedure is entered before an argument has been
-evaluated we say that the procedure is "non-strict" in that argument.
-If the argument is evaluated before the body of the procedure is
-entered we say that the procedure is "strict" in that argument.(2) In
-a purely applicative-order language, all procedures are strict in each
-argument. In a purely normal-order language, all compound procedures
-are non-strict in each argument, and primitive procedures may be either
-strict or non-strict. There are also languages (see *Note Exercise
-4-31::) that give programmers detailed control over the strictness of
-the procedures they define.
-
- A striking example of a procedure that can usefully be made
-non-strict is `cons' (or, in general, almost any constructor for data
-structures). One can do useful computation, combining elements to form
-data structures and operating on the resulting data structures, even if
-the values of the elements are not known. It makes perfect sense, for
-instance, to compute the length of a list without knowing the values of
-the individual elements in the list. We will exploit this idea in
-section *Note 4-2-3:: to implement the streams of *Note Chapter 3:: as
-lists formed of non-strict `cons' pairs.
-
- *Exercise 4.25:* Suppose that (in ordinary applicative-order
- Scheme) we define `unless' as shown above and then define
- `factorial' in terms of `unless' as
-
- (define (factorial n)
- (unless (= n 1)
- (* n (factorial (- n 1)))
- 1))
-
- What happens if we attempt to evaluate `(factorial 5)'? Will our
- definitions work in a normal-order language?
-
- *Exercise 4.26:* Ben Bitdiddle and Alyssa P. Hacker disagree over
- the importance of lazy evaluation for implementing things such as
- `unless'. Ben points out that it's possible to implement `unless'
- in applicative order as a special form. Alyssa counters that, if
- one did that, `unless' would be merely syntax, not a procedure
- that could be used in conjunction with higher-order procedures.
- Fill in the details on both sides of the argument. Show how to
- implement `unless' as a derived expression (like `cond' or `let'),
- and give an example of a situation where it might be useful to
- have `unless' available as a procedure, rather than as a special
- form.
-
- ---------- Footnotes ----------
-
- (1) The difference between the "lazy" terminology and the
-"normal-order" terminology is somewhat fuzzy. Generally, "lazy" refers
-to the mechanisms of particular evaluators, while "normal-order" refers
-to the semantics of languages, independent of any particular evaluation
-strategy. But this is not a hard-and-fast distinction, and the two
-terminologies are often used interchangeably.
-
- (2) The "strict" versus "non-strict" terminology means essentially
-the same thing as "applicative-order" versus "normal-order," except
-that it refers to individual procedures and arguments rather than to
-the language as a whole. At a conference on programming languages you
-might hear someone say, "The normal-order language Hassle has certain
-strict primitives. Other procedures take their arguments by lazy
-evaluation."
-
-\1f
-File: sicp.info, Node: 4-2-2, Next: 4-2-3, Prev: 4-2-1, Up: 4-2
-
-4.2.2 An Interpreter with Lazy Evaluation
------------------------------------------
-
-In this section we will implement a normal-order language that is the
-same as Scheme except that compound procedures are non-strict in each
-argument. Primitive procedures will still be strict. It is not
-difficult to modify the evaluator of section *Note 4-1-1:: so that the
-language it interprets behaves this way. Almost all the required
-changes center around procedure application.
-
- The basic idea is that, when applying a procedure, the interpreter
-must determine which arguments are to be evaluated and which are to be
-delayed. The delayed arguments are not evaluated; instead, they are
-transformed into objects called "thunks".(1) The thunk must contain the
-information required to produce the value of the argument when it is
-needed, as if it had been evaluated at the time of the application.
-Thus, the thunk must contain the argument expression and the
-environment in which the procedure application is being evaluated.
-
- The process of evaluating the expression in a thunk is called "forcing".(2)
-In general, a thunk will be forced only when its value is needed: when
-it is passed to a primitive procedure that will use the value of the
-thunk; when it is the value of a predicate of a conditional; and when
-it is the value of an operator that is about to be applied as a
-procedure. One design choice we have available is whether or not to "memoize"
-thunks, as we did with delayed objects in section *Note 3-5-1::. With
-memoization, the first time a thunk is forced, it stores the value that
-is computed. Subsequent forcings simply return the stored value
-without repeating the computation. We'll make our interpreter memoize,
-because this is more efficient for many applications. There are tricky
-considerations here, however.(3)
-
-Modifying the evaluator
-.......................
-
-The main difference between the lazy evaluator and the one in section
-*Note 4-1:: is in the handling of procedure applications in `eval' and
-`apply'.
-
- The `application?' clause of `eval' becomes
-
- ((application? exp)
- (apply (actual-value (operator exp) env)
- (operands exp)
- env))
-
- This is almost the same as the `application?' clause of `eval' in
-section *Note 4-1-1::. For lazy evaluation, however, we call `apply'
-with the operand expressions, rather than the arguments produced by
-evaluating them. Since we will need the environment to construct
-thunks if the arguments are to be delayed, we must pass this as well.
-We still evaluate the operator, because `apply' needs the actual
-procedure to be applied in order to dispatch on its type (primitive
-versus compound) and apply it.
-
- Whenever we need the actual value of an expression, we use
-
- (define (actual-value exp env)
- (force-it (eval exp env)))
-
-instead of just `eval', so that if the expression's value is a thunk, it
-will be forced.
-
- Our new version of `apply' is also almost the same as the version in
-section *Note 4-1-1::. The difference is that `eval' has passed in
-unevaluated operand expressions: For primitive procedures (which are
-strict), we evaluate all the arguments before applying the primitive;
-for compound procedures (which are non-strict) we delay all the
-arguments before applying the procedure.
-
- (define (apply procedure arguments env)
- (cond ((primitive-procedure? procedure)
- (apply-primitive-procedure
- procedure
- (list-of-arg-values arguments env))) ; changed
- ((compound-procedure? procedure)
- (eval-sequence
- (procedure-body procedure)
- (extend-environment
- (procedure-parameters procedure)
- (list-of-delayed-args arguments env) ; changed
- (procedure-environment procedure))))
- (else
- (error
- "Unknown procedure type -- APPLY" procedure))))
-
- The procedures that process the arguments are just like
-`list-of-values' from section *Note 4-1-1::, except that
-`list-of-delayed-args' delays the arguments instead of evaluating them,
-and `list-of-arg-values' uses `actual-value' instead of `eval':
-
- (define (list-of-arg-values exps env)
- (if (no-operands? exps)
- '()
- (cons (actual-value (first-operand exps) env)
- (list-of-arg-values (rest-operands exps)
- env))))
-
- (define (list-of-delayed-args exps env)
- (if (no-operands? exps)
- '()
- (cons (delay-it (first-operand exps) env)
- (list-of-delayed-args (rest-operands exps)
- env))))
-
- The other place we must change the evaluator is in the handling of
-`if', where we must use `actual-value' instead of `eval' to get the
-value of the predicate expression before testing whether it is true or
-false:
-
- (define (eval-if exp env)
- (if (true? (actual-value (if-predicate exp) env))
- (eval (if-consequent exp) env)
- (eval (if-alternative exp) env)))
-
- Finally, we must change the `driver-loop' procedure (section *Note
-4-1-4::) to use `actual-value' instead of `eval', so that if a delayed
-value is propagated back to the read-eval-print loop, it will be forced
-before being printed. We also change the prompts to indicate that this
-is the lazy evaluator:
-
- (define input-prompt ";;; L-Eval input:")
- (define output-prompt ";;; L-Eval value:")
-
- (define (driver-loop)
- (prompt-for-input input-prompt)
- (let ((input (read)))
- (let ((output
- (actual-value input the-global-environment)))
- (announce-output output-prompt)
- (user-print output)))
- (driver-loop))
-
- With these changes made, we can start the evaluator and test it. The
-successful evaluation of the `try' expression discussed in section
-*Note 4-2-1:: indicates that the interpreter is performing lazy
-evaluation:
-
- (define the-global-environment (setup-environment))
-
- (driver-loop)
-
- ;;; L-Eval input:
- (define (try a b)
- (if (= a 0) 1 b))
- ;;; L-Eval value:
- ok
-
- ;;; L-Eval input:
- (try 0 (/ 1 0))
- ;;; L-Eval value:
- 1
-
-Representing thunks
-...................
-
-Our evaluator must arrange to create thunks when procedures are applied
-to arguments and to force these thunks later. A thunk must package an
-expression together with the environment, so that the argument can be
-produced later. To force the thunk, we simply extract the expression
-and environment from the thunk and evaluate the expression in the
-environment. We use `actual-value' rather than `eval' so that in case
-the value of the expression is itself a thunk, we will force that, and
-so on, until we reach something that is not a thunk:
-
- (define (force-it obj)
- (if (thunk? obj)
- (actual-value (thunk-exp obj) (thunk-env obj))
- obj))
-
- One easy way to package an expression with an environment is to make
-a list containing the expression and the environment. Thus, we create
-a thunk as follows:
-
- (define (delay-it exp env)
- (list 'thunk exp env))
-
- (define (thunk? obj)
- (tagged-list? obj 'thunk))
-
- (define (thunk-exp thunk) (cadr thunk))
-
- (define (thunk-env thunk) (caddr thunk))
-
- Actually, what we want for our interpreter is not quite this, but
-rather thunks that have been memoized. When a thunk is forced, we will
-turn it into an evaluated thunk by replacing the stored expression with
-its value and changing the `thunk' tag so that it can be recognized as
-already evaluated.(4)
-
- (define (evaluated-thunk? obj)
- (tagged-list? obj 'evaluated-thunk))
-
- (define (thunk-value evaluated-thunk) (cadr evaluated-thunk))
-
- (define (force-it obj)
- (cond ((thunk? obj)
- (let ((result (actual-value
- (thunk-exp obj)
- (thunk-env obj))))
- (set-car! obj 'evaluated-thunk)
- (set-car! (cdr obj) result) ; replace `exp' with its value
- (set-cdr! (cdr obj) '()) ; forget unneeded `env'
- result))
- ((evaluated-thunk? obj)
- (thunk-value obj))
- (else obj)))
-
- Notice that the same `delay-it' procedure works both with and without
-memoization.
-
- *Exercise 4.27:* Suppose we type in the following definitions to
- the lazy evaluator:
-
- (define count 0)
-
- (define (id x)
- (set! count (+ count 1))
- x)
-
- Give the missing values in the following sequence of interactions,
- and explain your answers.(5)
-
- (define w (id (id 10)))
-
- ;;; L-Eval input:
- count
- ;;; L-Eval value:
- <RESPONSE>
-
- ;;; L-Eval input:
- w
- ;;; L-Eval value:
- <RESPONSE>
-
- ;;; L-Eval input:
- count
- ;;; L-Eval value:
- <RESPONSE>
-
- *Exercise 4.28:* `Eval' uses `actual-value' rather than `eval' to
- evaluate the operator before passing it to `apply', in order to
- force the value of the operator. Give an example that
- demonstrates the need for this forcing.
-
- *Exercise 4.29:* Exhibit a program that you would expect to run
- much more slowly without memoization than with memoization. Also,
- consider the following interaction, where the `id' procedure is
- defined as in *Note Exercise 4-27:: and `count' starts at 0:
-
- (define (square x)
- (* x x))
-
- ;;; L-Eval input:
- (square (id 10))
- ;;; L-Eval value:
- <RESPONSE>
-
- ;;; L-Eval input:
- count
- ;;; L-Eval value:
- <RESPONSE>
-
- Give the responses both when the evaluator memoizes and when it
- does not.
-
- *Exercise 4.30:* Cy D. Fect, a reformed C programmer, is worried
- that some side effects may never take place, because the lazy
- evaluator doesn't force the expressions in a sequence. Since the
- value of an expression in a sequence other than the last one is
- not used (the expression is there only for its effect, such as
- assigning to a variable or printing), there can be no subsequent
- use of this value (e.g., as an argument to a primitive procedure)
- that will cause it to be forced. Cy thus thinks that when
- evaluating sequences, we must force all expressions in the
- sequence except the final one. He proposes to modify
- `eval-sequence' from section *Note 4-1-1:: to use `actual-value'
- rather than `eval':
-
- (define (eval-sequence exps env)
- (cond ((last-exp? exps) (eval (first-exp exps) env))
- (else (actual-value (first-exp exps) env)
- (eval-sequence (rest-exps exps) env))))
-
- a. Ben Bitdiddle thinks Cy is wrong. He shows Cy the `for-each'
- procedure described in *Note Exercise 2-23::, which gives an
- important example of a sequence with side effects:
-
- (define (for-each proc items)
- (if (null? items)
- 'done
- (begin (proc (car items))
- (for-each proc (cdr items)))))
-
- He claims that the evaluator in the text (with the original
- `eval-sequence') handles this correctly:
-
- ;;; L-Eval input:
- (for-each (lambda (x) (newline) (display x))
- (list 57 321 88))
- 57
- 321
- 88
- ;;; L-Eval value:
- done
-
- Explain why Ben is right about the behavior of `for-each'.
-
- b. Cy agrees that Ben is right about the `for-each' example, but
- says that that's not the kind of program he was thinking
- about when he proposed his change to `eval-sequence'. He
- defines the following two procedures in the lazy evaluator:
-
- (define (p1 x)
- (set! x (cons x '(2)))
- x)
-
- (define (p2 x)
- (define (p e)
- e
- x)
- (p (set! x (cons x '(2)))))
-
- What are the values of `(p1 1)' and `(p2 1)' with the original
- `eval-sequence'? What would the values be with Cy's proposed
- change to `eval-sequence'?
-
- c. Cy also points out that changing `eval-sequence' as he
- proposes does not affect the behavior of the example in part
- a. Explain why this is true.
-
- d. How do you think sequences ought to be treated in the lazy
- evaluator? Do you like Cy's approach, the approach in the
- text, or some other approach?
-
-
- *Exercise 4.31:* The approach taken in this section is somewhat
- unpleasant, because it makes an incompatible change to Scheme. It
- might be nicer to implement lazy evaluation as an "upward-compatible
- extension", that is, so that ordinary Scheme programs will work as
- before. We can do this by extending the syntax of procedure
- declarations to let the user control whether or not arguments are
- to be delayed. While we're at it, we may as well also give the
- user the choice between delaying with and without memoization.
- For example, the definition
-
- (define (f a (b lazy) c (d lazy-memo))
- ...)
-
- would define `f' to be a procedure of four arguments, where the
- first and third arguments are evaluated when the procedure is
- called, the second argument is delayed, and the fourth argument is
- both delayed and memoized. Thus, ordinary procedure definitions
- will produce the same behavior as ordinary Scheme, while adding
- the `lazy-memo' declaration to each parameter of every compound
- procedure will produce the behavior of the lazy evaluator defined
- in this section. Design and implement the changes required to
- produce such an extension to Scheme. You will have to implement
- new syntax procedures to handle the new syntax for `define'. You
- must also arrange for `eval' or `apply' to determine when
- arguments are to be delayed, and to force or delay arguments
- accordingly, and you must arrange for forcing to memoize or not,
- as appropriate.
-
- ---------- Footnotes ----------
-
- (1) The word "thunk" was invented by an informal working group that
-was discussing the implementation of call-by-name in Algol 60. They
-observed that most of the analysis of ("thinking about") the expression
-could be done at compile time; thus, at run time, the expression would
-already have been "thunk" about (Ingerman et al. 1960).
-
- (2) This is analogous to the use of `force' on the delayed objects
-that were introduced in *Note Chapter 3:: to represent streams. The
-critical difference between what we are doing here and what we did in
-*Note Chapter 3:: is that we are building delaying and forcing into the
-evaluator, and thus making this uniform and automatic throughout the
-language.
-
- (3) Lazy evaluation combined with memoization is sometimes referred
-to as "call-by-need" argument passing, in contrast to "call-by-name"
-argument passing. (Call-by-name, introduced in Algol 60, is similar to
-non-memoized lazy evaluation.) As language designers, we can build our
-evaluator to memoize, not to memoize, or leave this an option for
-programmers (*Note Exercise 4-31::). As you might expect from *Note
-Chapter 3::, these choices raise issues that become both subtle and
-confusing in the presence of assignments. (See *Note Exercise 4-27::
-and *Note Exercise 4-29::.) An excellent article by Clinger (1982)
-attempts to clarify the multiple dimensions of confusion that arise
-here.
-
- (4) Notice that we also erase the `env' from the thunk once the
-expression's value has been computed. This makes no difference in the
-values returned by the interpreter. It does help save space, however,
-because removing the reference from the thunk to the `env' once it is
-no longer needed allows this structure to be "garbage-collected" and
-its space recycled, as we will discuss in section *Note 5-3::.
-
- Similarly, we could have allowed unneeded environments in the
-memoized delayed objects of section *Note 3-5-1:: to be
-garbage-collected, by having `memo-proc' do something like `(set! proc
-'())' to discard the procedure `proc' (which includes the environment
-in which the `delay' was evaluated) after storing its value.
-
- (5) This exercise demonstrates that the interaction between lazy
-evaluation and side effects can be very confusing. This is just what
-you might expect from the discussion in *Note Chapter 3::.
-
-\1f
-File: sicp.info, Node: 4-2-3, Prev: 4-2-2, Up: 4-2
-
-4.2.3 Streams as Lazy Lists
----------------------------
-
-In section *Note 3-5-1::, we showed how to implement streams as delayed
-lists. We introduced special forms `delay' and `cons-stream', which
-allowed us to construct a "promise" to compute the `cdr' of a stream,
-without actually fulfilling that promise until later. We could use
-this general technique of introducing special forms whenever we need
-more control over the evaluation process, but this is awkward. For one
-thing, a special form is not a first-class object like a procedure, so
-we cannot use it together with higher-order procedures.(1)
-Additionally, we were forced to create streams as a new kind of data
-object similar but not identical to lists, and this required us to
-reimplement many ordinary list operations (`map', `append', and so on)
-for use with streams.
-
- With lazy evaluation, streams and lists can be identical, so there
-is no need for special forms or for separate list and stream
-operations. All we need to do is to arrange matters so that `cons' is
-non-strict. One way to accomplish this is to extend the lazy evaluator
-to allow for non-strict primitives, and to implement `cons' as one of
-these. An easier way is to recall (section *Note 2-1-3::) that there
-is no fundamental need to implement `cons' as a primitive at all.
-Instead, we can represent pairs as procedures:(2)
-
- (define (cons x y)
- (lambda (m) (m x y)))
-
- (define (car z)
- (z (lambda (p q) p)))
-
- (define (cdr z)
- (z (lambda (p q) q)))
-
- In terms of these basic operations, the standard definitions of the
-list operations will work with infinite lists (streams) as well as
-finite ones, and the stream operations can be implemented as list
-operations. Here are some examples:
-
- (define (list-ref items n)
- (if (= n 0)
- (car items)
- (list-ref (cdr items) (- n 1))))
-
- (define (map proc items)
- (if (null? items)
- '()
- (cons (proc (car items))
- (map proc (cdr items)))))
-
- (define (scale-list items factor)
- (map (lambda (x) (* x factor))
- items))
-
- (define (add-lists list1 list2)
- (cond ((null? list1) list2)
- ((null? list2) list1)
- (else (cons (+ (car list1) (car list2))
- (add-lists (cdr list1) (cdr list2))))))
-
- (define ones (cons 1 ones))
-
- (define integers (cons 1 (add-lists ones integers)))
-
- ;;; L-Eval input:
- (list-ref integers 17)
- ;;; L-Eval value:
- 18
-
- Note that these lazy lists are even lazier than the streams of *Note
-Chapter 3::: The `car' of the list, as well as the `cdr', is
-delayed.(3) In fact, even accessing the `car' or `cdr' of a lazy pair
-need not force the value of a list element. The value will be forced
-only when it is really needed - e.g., for use as the argument of a
-primitive, or to be printed as an answer.
-
- Lazy pairs also help with the problem that arose with streams in
-section *Note 3-5-4::, where we found that formulating stream models of
-systems with loops may require us to sprinkle our programs with
-explicit `delay' operations, beyond the ones supplied by `cons-stream'.
-With lazy evaluation, all arguments to procedures are delayed
-uniformly. For instance, we can implement procedures to integrate
-lists and solve differential equations as we originally intended in
-section *Note 3-5-4:::
-
- (define (integral integrand initial-value dt)
- (define int
- (cons initial-value
- (add-lists (scale-list integrand dt)
- int)))
- int)
-
- (define (solve f y0 dt)
- (define y (integral dy y0 dt))
- (define dy (map f y))
- y)
-
- ;;; L-Eval input:
- (list-ref (solve (lambda (x) x) 1 0.001) 1000)
- ;;; L-Eval value:
- 2.716924
-
- *Exercise 4.32:* Give some examples that illustrate the difference
- between the streams of *Note Chapter 3:: and the "lazier" lazy
- lists described in this section. How can you take advantage of
- this extra laziness?
-
- *Exercise 4.33:* Ben Bitdiddle tests the lazy list implementation
- given above by evaluating the expression
-
- (car '(a b c))
-
- To his surprise, this produces an error. After some thought, he
- realizes that the "lists" obtained by reading in quoted
- expressions are different from the lists manipulated by the new
- definitions of `cons', `car', and `cdr'. Modify the evaluator's
- treatment of quoted expressions so that quoted lists typed at the
- driver loop will produce true lazy lists.
-
- *Exercise 4.34:* Modify the driver loop for the evaluator so that
- lazy pairs and lists will print in some reasonable way. (What are
- you going to do about infinite lists?) You may also need to modify
- the representation of lazy pairs so that the evaluator can
- identify them in order to print them.
-
- ---------- Footnotes ----------
-
- (1) This is precisely the issue with the `unless' procedure, as in
-*Note Exercise 4-26::.
-
- (2) This is the procedural representation described in *Note
-Exercise 2-4::. Essentially any procedural representation (e.g., a
-message-passing implementation) would do as well. Notice that we can
-install these definitions in the lazy evaluator simply by typing them
-at the driver loop. If we had originally included `cons', `car', and
-`cdr' as primitives in the global environment, they will be redefined.
-(Also see *Note Exercise 4-33:: and *Note Exercise 4-34::.)
-
- (3) This permits us to create delayed versions of more general kinds
-of list structures, not just sequences. Hughes 1990 discusses some
-applications of "lazy trees."
-
-\1f
-File: sicp.info, Node: 4-3, Next: 4-4, Prev: 4-2, Up: Chapter 4
-
-4.3 Variations on a Scheme - Nondeterministic Computing
-=======================================================
-
-In this section, we extend the Scheme evaluator to support a programming
-paradigm called "nondeterministic computing" by building into the
-evaluator a facility to support automatic search. This is a much more
-profound change to the language than the introduction of lazy
-evaluation in section *Note 4-2::.
-
- Nondeterministic computing, like stream processing, is useful for
-"generate and test" applications. Consider the task of starting with
-two lists of positive integers and finding a pair of integers--one from
-the first list and one from the second list--whose sum is prime. We
-saw how to handle this with finite sequence operations in section *Note
-2-2-3:: and with infinite streams in section *Note 3-5-3::. Our
-approach was to generate the sequence of all possible pairs and filter
-these to select the pairs whose sum is prime. Whether we actually
-generate the entire sequence of pairs first as in *Note Chapter 2::, or
-interleave the generating and filtering as in *Note Chapter 3::, is
-immaterial to the essential image of how the computation is organized.
-
- The nondeterministic approach evokes a different image. Imagine
-simply that we choose (in some way) a number from the first list and a
-number from the second list and require (using some mechanism) that
-their sum be prime. This is expressed by following procedure:
-
- (define (prime-sum-pair list1 list2)
- (let ((a (an-element-of list1))
- (b (an-element-of list2)))
- (require (prime? (+ a b)))
- (list a b)))
-
- It might seem as if this procedure merely restates the problem,
-rather than specifying a way to solve it. Nevertheless, this is a
-legitimate nondeterministic program.(1)
-
- The key idea here is that expressions in a nondeterministic language
-can have more than one possible value. For instance, `an-element-of'
-might return any element of the given list. Our nondeterministic
-program evaluator will work by automatically choosing a possible value
-and keeping track of the choice. If a subsequent requirement is not
-met, the evaluator will try a different choice, and it will keep trying
-new choices until the evaluation succeeds, or until we run out of
-choices. Just as the lazy evaluator freed the programmer from the
-details of how values are delayed and forced, the nondeterministic
-program evaluator will free the programmer from the details of how
-choices are made.
-
- It is instructive to contrast the different images of time evoked by
-nondeterministic evaluation and stream processing. Stream processing
-uses lazy evaluation to decouple the time when the stream of possible
-answers is assembled from the time when the actual stream elements are
-produced. The evaluator supports the illusion that all the possible
-answers are laid out before us in a timeless sequence. With
-nondeterministic evaluation, an expression represents the exploration
-of a set of possible worlds, each determined by a set of choices. Some
-of the possible worlds lead to dead ends, while others have useful
-values. The nondeterministic program evaluator supports the illusion
-that time branches, and that our programs have different possible
-execution histories. When we reach a dead end, we can revisit a
-previous choice point and proceed along a different branch.
-
- The nondeterministic program evaluator implemented below is called
-the `amb' evaluator because it is based on a new special form called
-`amb'. We can type the above definition of `prime-sum-pair' at the
-`amb' evaluator driver loop (along with definitions of `prime?',
-`an-element-of', and `require') and run the procedure as follows:
-
- ;;; Amb-Eval input:
- (prime-sum-pair '(1 3 5 8) '(20 35 110))
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- (3 20)
-
- The value returned was obtained after the evaluator repeatedly chose
-elements from each of the lists, until a successful choice was made.
-
- Section *Note 4-3-1:: introduces `amb' and explains how it supports
-nondeterminism through the evaluator's automatic search mechanism.
-Section *Note 4-3-2:: presents examples of nondeterministic programs,
-and section *Note 4-3-3:: gives the details of how to implement the
-`amb' evaluator by modifying the ordinary Scheme evaluator.
-
-* Menu:
-
-* 4-3-1:: Amb and Search
-* 4-3-2:: Examples of Nondeterministic Programs
-* 4-3-3:: Implementing the `Amb' Evaluator
-
- ---------- Footnotes ----------
-
- (1) We assume that we have previously defined a procedure `prime?'
-that tests whether numbers are prime. Even with `prime?' defined, the
-`prime-sum-pair' procedure may look suspiciously like the unhelpful
-"pseudo-Lisp" attempt to define the square-root function, which we
-described at the beginning of section *Note 1-1-7::. In fact, a
-square-root procedure along those lines can actually be formulated as a
-nondeterministic program. By incorporating a search mechanism into the
-evaluator, we are eroding the distinction between purely declarative
-descriptions and imperative specifications of how to compute answers.
-We'll go even farther in this direction in section *Note 4-4::.
-
-\1f
-File: sicp.info, Node: 4-3-1, Next: 4-3-2, Prev: 4-3, Up: 4-3
-
-4.3.1 Amb and Search
---------------------
-
-To extend Scheme to support nondeterminism, we introduce a new special
-form called `amb'.(1) The expression
-
- (amb <E_1> <E_2> ... <E_N>)
-
-returns the value of one of the n expressions <E_I> "ambiguously." For
-example, the expression
-
- (list (amb 1 2 3) (amb 'a 'b))
-
-can have six possible values:
-
- `(1 a)' `(1 b)' `(2 a)' `(2 b)' `(3 a)' `(3 b)'
-
- `Amb' with a single choice produces an ordinary (single) value.
-
- `Amb' with no choices--the expression `(amb)'--is an expression with
-no acceptable values. Operationally, we can think of `(amb)' as an
-expression that when evaluated causes the computation to "fail": The
-computation aborts and no value is produced. Using this idea, we can
-express the requirement that a particular predicate expression `p' must
-be true as follows:
-
- (define (require p)
- (if (not p) (amb)))
-
- With `amb' and `require', we can implement the `an-element-of'
-procedure used above:
-
- (define (an-element-of items)
- (require (not (null? items)))
- (amb (car items) (an-element-of (cdr items))))
-
- `An-element-of' fails if the list is empty. Otherwise it ambiguously
-returns either the first element of the list or an element chosen from
-the rest of the list.
-
- We can also express infinite ranges of choices. The following
-procedure potentially returns any integer greater than or equal to some
-given n:
-
- (define (an-integer-starting-from n)
- (amb n (an-integer-starting-from (+ n 1))))
-
- This is like the stream procedure `integers-starting-from' described
-in section *Note 3-5-2::, but with an important difference: The stream
-procedure returns an object that represents the sequence of all
-integers beginning with n, whereas the `amb' procedure returns a single
-integer.(2)
-
- Abstractly, we can imagine that evaluating an `amb' expression
-causes time to split into branches, where the computation continues on
-each branch with one of the possible values of the expression. We say
-that `amb' represents a "nondeterministic choice point". If we had a
-machine with a sufficient number of processors that could be
-dynamically allocated, we could implement the search in a
-straightforward way. Execution would proceed as in a sequential
-machine, until an `amb' expression is encountered. At this point, more
-processors would be allocated and initialized to continue all of the
-parallel executions implied by the choice. Each processor would proceed
-sequentially as if it were the only choice, until it either terminates
-by encountering a failure, or it further subdivides, or it finishes.(3)
-
- On the other hand, if we have a machine that can execute only one
-process (or a few concurrent processes), we must consider the
-alternatives sequentially. One could imagine modifying an evaluator to
-pick at random a branch to follow whenever it encounters a choice
-point. Random choice, however, can easily lead to failing values. We
-might try running the evaluator over and over, making random choices
-and hoping to find a non-failing value, but it is better to "systematically
-search" all possible execution paths. The `amb' evaluator that we will
-develop and work with in this section implements a systematic search as
-follows: When the evaluator encounters an application of `amb', it
-initially selects the first alternative. This selection may itself
-lead to a further choice. The evaluator will always initially choose
-the first alternative at each choice point. If a choice results in a
-failure, then the evaluator automagically(4) "backtracks" to the most
-recent choice point and tries the next alternative. If it runs out of
-alternatives at any choice point, the evaluator will back up to the
-previous choice point and resume from there. This process leads to a
-search strategy known as "depth-first search" or backtracking
-"chronological backtracking".(5)
-
-Driver loop
-...........
-
-The driver loop for the `amb' evaluator has some unusual properties. It
-reads an expression and prints the value of the first non-failing
-execution, as in the `prime-sum-pair' example shown above. If we want
-to see the value of the next successful execution, we can ask the
-interpreter to backtrack and attempt to generate a second non-failing
-execution. This is signaled by typing the symbol `try-again'. If any
-expression except `try-again' is given, the interpreter will start a
-new problem, discarding the unexplored alternatives in the previous
-problem. Here is a sample interaction:
-
- ;;; Amb-Eval input:
- (prime-sum-pair '(1 3 5 8) '(20 35 110))
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- (3 20)
-
- ;;; Amb-Eval input:
- try-again
- ;;; Amb-Eval value:
- (3 110)
-
- ;;; Amb-Eval input:
- try-again
- ;;; Amb-Eval value:
- (8 35)
-
- ;;; Amb-Eval input:
- try-again
- ;;; There are no more values of
- (prime-sum-pair (quote (1 3 5 8)) (quote (20 35 110)))
-
- ;;; Amb-Eval input:
- (prime-sum-pair '(19 27 30) '(11 36 58))
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- (30 11)
-
- *Exercise 4.35:* Write a procedure `an-integer-between' that
- returns an integer between two given bounds. This can be used to
- implement a procedure that finds Pythagorean triples, i.e.,
- triples of integers (i,j,k) between the given bounds such that i
- <= j and i^2 + j^2 = k^2, as follows:
-
- (define (a-pythagorean-triple-between low high)
- (let ((i (an-integer-between low high)))
- (let ((j (an-integer-between i high)))
- (let ((k (an-integer-between j high)))
- (require (= (+ (* i i) (* j j)) (* k k)))
- (list i j k)))))
-
- *Exercise 4.36:* *Note Exercise 3-69:: discussed how to generate
- the stream of _all_ Pythagorean triples, with no upper bound on
- the size of the integers to be searched. Explain why simply
- replacing `an-integer-between' by `an-integer-starting-from' in
- the procedure in *Note Exercise 4-35:: is not an adequate way to
- generate arbitrary Pythagorean triples. Write a procedure that
- actually will accomplish this. (That is, write a procedure for
- which repeatedly typing `try-again' would in principle eventually
- generate all Pythagorean triples.)
-
- *Exercise 4.37:* Ben Bitdiddle claims that the following method
- for generating Pythagorean triples is more efficient than the one
- in *Note Exercise 4-35::. Is he correct? (Hint: Consider the
- number of possibilities that must be explored.)
-
- (define (a-pythagorean-triple-between low high)
- (let ((i (an-integer-between low high))
- (hsq (* high high)))
- (let ((j (an-integer-between i high)))
- (let ((ksq (+ (* i i) (* j j))))
- (require (>= hsq ksq))
- (let ((k (sqrt ksq)))
- (require (integer? k))
- (list i j k))))))
-
- ---------- Footnotes ----------
-
- (1) The idea of `amb' for nondeterministic programming was first
-described in 1961 by John McCarthy (see McCarthy 1967).
-
- (2) In actuality, the distinction between nondeterministically
-returning a single choice and returning all choices depends somewhat on
-our point of view. From the perspective of the code that uses the
-value, the nondeterministic choice returns a single value. From the
-perspective of the programmer designing the code, the nondeterministic
-choice potentially returns all possible values, and the computation
-branches so that each value is investigated separately.
-
- (3) One might object that this is a hopelessly inefficient
-mechanism. It might require millions of processors to solve some
-easily stated problem this way, and most of the time most of those
-processors would be idle. This objection should be taken in the
-context of history. Memory used to be considered just such an
-expensive commodity. In 1964 a megabyte of RAM cost about $400,000.
-Now every personal computer has many megabytes of RAM, and most of the
-time most of that RAM is unused. It is hard to underestimate the cost
-of mass-produced electronics.
-
- (4) Automagically: "Automatically, but in a way which, for some
-reason (typically because it is too complicated, or too ugly, or
-perhaps even too trivial), the speaker doesn't feel like explaining."
-(Steele 1983, Raymond 1993)
-
- (5) [Footnote 4.47] The integration of automatic search strategies
-into programming languages has had a long and checkered history. The
-first suggestions that nondeterministic algorithms might be elegantly
-encoded in a programming language with search and automatic
-backtracking came from Robert Floyd (1967). Carl Hewitt (1969)
-invented a programming language called Planner that explicitly
-supported automatic chronological backtracking, providing for a
-built-in depth-first search strategy. Sussman, Winograd, and Charniak
-(1971) implemented a subset of this language, called MicroPlanner,
-which was used to support work in problem solving and robot planning.
-Similar ideas, arising from logic and theorem proving, led to the
-genesis in Edinburgh and Marseille of the elegant language Prolog
-(which we will discuss in section *Note 4-4::). After sufficient
-frustration with automatic search, McDermott and Sussman (1972)
-developed a language called Conniver, which included mechanisms for
-placing the search strategy under programmer control. This proved
-unwieldy, however, and Sussman and Stallman (1975) found a more
-tractable approach while investigating methods of symbolic analysis for
-electrical circuits. They developed a non-chronological backtracking
-scheme that was based on tracing out the logical dependencies
-connecting facts, a technique that has come to be known as "dependency-directed
-backtracking". Although their method was complex, it produced
-reasonably efficient programs because it did little redundant search.
-Doyle (1979) and McAllester (1978, 1980) generalized and clarified the
-methods of Stallman and Sussman, developing a new paradigm for
-formulating search that is now called "truth maintenance". Modern
-problem-solving systems all use some form of truth-maintenance system
-as a substrate. See Forbus and deKleer 1993 for a discussion of
-elegant ways to build truth-maintenance systems and applications using
-truth maintenance. Zabih, McAllester, and Chapman 1987 describes a
-nondeterministic extension to Scheme that is based on `amb'; it is
-similar to the interpreter described in this section, but more
-sophisticated, because it uses dependency-directed backtracking rather
-than chronological backtracking. Winston 1992 gives an introduction to
-both kinds of backtracking.
-
-\1f
-File: sicp.info, Node: 4-3-2, Next: 4-3-3, Prev: 4-3-1, Up: 4-3
-
-4.3.2 Examples of Nondeterministic Programs
--------------------------------------------
-
-Section *Note 4-3-3:: describes the implementation of the `amb'
-evaluator. First, however, we give some examples of how it can be
-used. The advantage of nondeterministic programming is that we can
-suppress the details of how search is carried out, thereby expressing
-our programs at a higher level of abstraction.
-
-Logic Puzzles
-.............
-
-The following puzzle (taken from Dinesman 1968) is typical of a large
-class of simple logic puzzles:
-
- Baker, Cooper, Fletcher, Miller, and Smith live on different
- floors of an apartment house that contains only five floors.
- Baker does not live on the top floor. Cooper does not live on the
- bottom floor. Fletcher does not live on either the top or the
- bottom floor. Miller lives on a higher floor than does Cooper.
- Smith does not live on a floor adjacent to Fletcher's. Fletcher
- does not live on a floor adjacent to Cooper's. Where does
- everyone live?
-
- We can determine who lives on each floor in a straightforward way by
-enumerating all the possibilities and imposing the given
-restrictions:(1)
-
- (define (multiple-dwelling)
- (let ((baker (amb 1 2 3 4 5))
- (cooper (amb 1 2 3 4 5))
- (fletcher (amb 1 2 3 4 5))
- (miller (amb 1 2 3 4 5))
- (smith (amb 1 2 3 4 5)))
- (require
- (distinct? (list baker cooper fletcher miller smith)))
- (require (not (= baker 5)))
- (require (not (= cooper 1)))
- (require (not (= fletcher 5)))
- (require (not (= fletcher 1)))
- (require (> miller cooper))
- (require (not (= (abs (- smith fletcher)) 1)))
- (require (not (= (abs (- fletcher cooper)) 1)))
- (list (list 'baker baker)
- (list 'cooper cooper)
- (list 'fletcher fletcher)
- (list 'miller miller)
- (list 'smith smith))))
-
- Evaluating the expression `(multiple-dwelling)' produces the result
-
- ((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1))
-
- Although this simple procedure works, it is very slow. *Note
-Exercise 4-39:: and *Note Exercise 4-40:: discuss some possible
-improvements.
-
- *Exercise 4.38:* Modify the multiple-dwelling procedure to omit
- the requirement that Smith and Fletcher do not live on adjacent
- floors. How many solutions are there to this modified puzzle?
-
- *Exercise 4.39:* Does the order of the restrictions in the
- multiple-dwelling procedure affect the answer? Does it affect the
- time to find an answer? If you think it matters, demonstrate a
- faster program obtained from the given one by reordering the
- restrictions. If you think it does not matter, argue your case.
-
- *Exercise 4.40:* In the multiple dwelling problem, how many sets
- of assignments are there of people to floors, both before and
- after the requirement that floor assignments be distinct? It is
- very inefficient to generate all possible assignments of people to
- floors and then leave it to backtracking to eliminate them. For
- example, most of the restrictions depend on only one or two of the
- person-floor variables, and can thus be imposed before floors have
- been selected for all the people. Write and demonstrate a much
- more efficient nondeterministic procedure that solves this problem
- based upon generating only those possibilities that are not already
- ruled out by previous restrictions. (Hint: This will require a
- nest of `let' expressions.)
-
- *Exercise 4.41:* Write an ordinary Scheme program to solve the
- multiple dwelling puzzle.
-
- *Exercise 4.42:* Solve the following "Liars" puzzle (from Phillips
- 1934):
-
- Five schoolgirls sat for an examination. Their parents--so they
- thought--showed an undue degree of interest in the result. They
- therefore agreed that, in writing home about the examination, each
- girl should make one true statement and one untrue one. The
- following are the relevant passages from their letters:
-
- * Betty: "Kitty was second in the examination. I was only
- third."
-
- * Ethel: "You'll be glad to hear that I was on top. Joan was
- second."
-
- * Joan: "I was third, and poor old Ethel was bottom."
-
- * Kitty: "I came out second. Mary was only fourth."
-
- * Mary: "I was fourth. Top place was taken by Betty."
-
-
- What in fact was the order in which the five girls were placed?
-
- *Exercise 4.43:* Use the `amb' evaluator to solve the following
- puzzle:(2)
-
- Mary Ann Moore's father has a yacht and so has each of his
- four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood,
- and Dr. Parker. Each of the five also has one daughter and
- each has named his yacht after a daughter of one of the
- others. Sir Barnacle's yacht is the Gabrielle, Mr. Moore
- owns the Lorna; Mr. Hall the Rosalind. The Melissa, owned by
- Colonel Downing, is named after Sir Barnacle's daughter.
- Gabrielle's father owns the yacht that is named after Dr.
- Parker's daughter. Who is Lorna's father?
-
- Try to write the program so that it runs efficiently (see *Note
- Exercise 4-40::). Also determine how many solutions there are if
- we are not told that Mary Ann's last name is Moore.
-
- *Exercise 4.44:* *Note Exercise 2-42:: described the "eight-queens
- puzzle" of placing queens on a chessboard so that no two attack
- each other. Write a nondeterministic program to solve this puzzle.
-
-Parsing natural language
-........................
-
-Programs designed to accept natural language as input usually start by
-attempting to "parse" the input, that is, to match the input against
-some grammatical structure. For example, we might try to recognize
-simple sentences consisting of an article followed by a noun followed
-by a verb, such as "The cat eats." To accomplish such an analysis, we
-must be able to identify the parts of speech of individual words. We
-could start with some lists that classify various words:(3)
-
- (define nouns '(noun student professor cat class))
-
- (define verbs '(verb studies lectures eats sleeps))
-
- (define articles '(article the a))
-
- We also need a "grammar", that is, a set of rules describing how
-grammatical elements are composed from simpler elements. A very simple
-grammar might stipulate that a sentence always consists of two
-pieces--a noun phrase followed by a verb--and that a noun phrase
-consists of an article followed by a noun. With this grammar, the
-sentence "The cat eats" is parsed as follows:
-
- (sentence (noun-phrase (article the) (noun cat))
- (verb eats))
-
- We can generate such a parse with a simple program that has separate
-procedures for each of the grammatical rules. To parse a sentence, we
-identify its two constituent pieces and return a list of these two
-elements, tagged with the symbol `sentence':
-
- (define (parse-sentence)
- (list 'sentence
- (parse-noun-phrase)
- (parse-word verbs)))
-
- A noun phrase, similarly, is parsed by finding an article followed
-by a noun:
-
- (define (parse-noun-phrase)
- (list 'noun-phrase
- (parse-word articles)
- (parse-word nouns)))
-
- At the lowest level, parsing boils down to repeatedly checking that
-the next unparsed word is a member of the list of words for the
-required part of speech. To implement this, we maintain a global
-variable `*unparsed*', which is the input that has not yet been parsed.
-Each time we check a word, we require that `*unparsed*' must be
-non-empty and that it should begin with a word from the designated
-list. If so, we remove that word from `*unparsed*' and return the word
-together with its part of speech (which is found at the head of the
-list):(4)
-
- (define (parse-word word-list)
- (require (not (null? *unparsed*)))
- (require (memq (car *unparsed*) (cdr word-list)))
- (let ((found-word (car *unparsed*)))
- (set! *unparsed* (cdr *unparsed*))
- (list (car word-list) found-word)))
-
- To start the parsing, all we need to do is set `*unparsed*' to be the
-entire input, try to parse a sentence, and check that nothing is left
-over:
-
- (define *unparsed* '())
-
- (define (parse input)
- (set! *unparsed* input)
- (let ((sent (parse-sentence)))
- (require (null? *unparsed*))
- sent))
-
- We can now try the parser and verify that it works for our simple
-test sentence:
-
- ;;; Amb-Eval input:
- (parse '(the cat eats))
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- (sentence (noun-phrase (article the) (noun cat)) (verb eats))
-
- The `amb' evaluator is useful here because it is convenient to
-express the parsing constraints with the aid of `require'. Automatic
-search and backtracking really pay off, however, when we consider more
-complex grammars where there are choices for how the units can be
-decomposed.
-
- Let's add to our grammar a list of prepositions:
-
- (define prepositions '(prep for to in by with))
-
-and define a prepositional phrase (e.g., "for the cat") to be a
-preposition followed by a noun phrase:
-
- (define (parse-prepositional-phrase)
- (list 'prep-phrase
- (parse-word prepositions)
- (parse-noun-phrase)))
-
- Now we can define a sentence to be a noun phrase followed by a verb
-phrase, where a verb phrase can be either a verb or a verb phrase
-extended by a prepositional phrase:(5)
-
- (define (parse-sentence)
- (list 'sentence
- (parse-noun-phrase)
- (parse-verb-phrase)))
-
- (define (parse-verb-phrase)
- (define (maybe-extend verb-phrase)
- (amb verb-phrase
- (maybe-extend (list 'verb-phrase
- verb-phrase
- (parse-prepositional-phrase)))))
- (maybe-extend (parse-word verbs)))
-
- While we're at it, we can also elaborate the definition of noun
-phrases to permit such things as "a cat in the class." What we used to
-call a noun phrase, we'll now call a simple noun phrase, and a noun
-phrase will now be either a simple noun phrase or a noun phrase
-extended by a prepositional phrase:
-
- (define (parse-simple-noun-phrase)
- (list 'simple-noun-phrase
- (parse-word articles)
- (parse-word nouns)))
-
- (define (parse-noun-phrase)
- (define (maybe-extend noun-phrase)
- (amb noun-phrase
- (maybe-extend (list 'noun-phrase
- noun-phrase
- (parse-prepositional-phrase)))))
- (maybe-extend (parse-simple-noun-phrase)))
-
- Our new grammar lets us parse more complex sentences. For example
-
- (parse '(the student with the cat sleeps in the class))
-
-produces
-
- (sentence
- (noun-phrase
- (simple-noun-phrase (article the) (noun student))
- (prep-phrase (prep with)
- (simple-noun-phrase
- (article the) (noun cat))))
- (verb-phrase
- (verb sleeps)
- (prep-phrase (prep in)
- (simple-noun-phrase
- (article the) (noun class)))))
-
- Observe that a given input may have more than one legal parse. In
-the sentence "The professor lectures to the student with the cat," it
-may be that the professor is lecturing with the cat, or that the
-student has the cat. Our nondeterministic program finds both
-possibilities:
-
- (parse '(the professor lectures to the student with the cat))
-
-produces
-
- (sentence
- (simple-noun-phrase (article the) (noun professor))
- (verb-phrase
- (verb-phrase
- (verb lectures)
- (prep-phrase (prep to)
- (simple-noun-phrase
- (article the) (noun student))))
- (prep-phrase (prep with)
- (simple-noun-phrase
- (article the) (noun cat)))))
-
- Asking the evaluator to try again yields
-
- (sentence
- (simple-noun-phrase (article the) (noun professor))
- (verb-phrase
- (verb lectures)
- (prep-phrase (prep to)
- (noun-phrase
- (simple-noun-phrase
- (article the) (noun student))
- (prep-phrase (prep with)
- (simple-noun-phrase
- (article the) (noun cat)))))))
-
- *Exercise 4.45:* With the grammar given above, the following
- sentence can be parsed in five different ways: "The professor
- lectures to the student in the class with the cat." Give the five
- parses and explain the differences in shades of meaning among them.
-
- *Exercise 4.46:* The evaluators in sections *Note 4-1:: and *Note
- 4-2:: do not determine what order operands are evaluated in. We
- will see that the `amb' evaluator evaluates them from left to
- right. Explain why our parsing program wouldn't work if the
- operands were evaluated in some other order.
-
- *Exercise 4.47:* Louis Reasoner suggests that, since a verb phrase
- is either a verb or a verb phrase followed by a prepositional
- phrase, it would be much more straightforward to define the
- procedure `parse-verb-phrase' as follows (and similarly for noun
- phrases):
-
- (define (parse-verb-phrase)
- (amb (parse-word verbs)
- (list 'verb-phrase
- (parse-verb-phrase)
- (parse-prepositional-phrase))))
-
- Does this work? Does the program's behavior change if we
- interchange the order of expressions in the `amb'
-
- *Exercise 4.48:* Extend the grammar given above to handle more
- complex sentences. For example, you could extend noun phrases and
- verb phrases to include adjectives and adverbs, or you could
- handle compound sentences.(6)
-
- *Exercise 4.49:* Alyssa P. Hacker is more interested in generating
- interesting sentences than in parsing them. She reasons that by
- simply changing the procedure `parse-word' so that it ignores the
- "input sentence" and instead always succeeds and generates an
- appropriate word, we can use the programs we had built for parsing
- to do generation instead. Implement Alyssa's idea, and show the
- first half-dozen or so sentences generated.(7)
-
- ---------- Footnotes ----------
-
- (1) Our program uses the following procedure to determine if the
-elements of a list are distinct:
-
- (define (distinct? items)
- (cond ((null? items) true)
- ((null? (cdr items)) true)
- ((member (car items) (cdr items)) false)
- (else (distinct? (cdr items)))))
-
- `Member' is like `memq' except that it uses `equal?' instead of
-`eq?' to test for equality.
-
- (2) This is taken from a booklet called "Problematical Recreations,"
-published in the 1960s by Litton Industries, where it is attributed to
-the `Kansas State Engineer'.
-
- (3) Here we use the convention that the first element of each list
-designates the part of speech for the rest of the words in the list.
-
- (4) Notice that `parse-word' uses `set!' to modify the unparsed
-input list. For this to work, our `amb' evaluator must undo the
-effects of `set!' operations when it backtracks.
-
- (5) Observe that this definition is recursive--a verb may be
-followed by any number of prepositional phrases.
-
- (6) This kind of grammar can become arbitrarily complex, but it is
-only a toy as far as real language understanding is concerned. Real
-natural-language understanding by computer requires an elaborate
-mixture of syntactic analysis and interpretation of meaning. On the
-other hand, even toy parsers can be useful in supporting flexible
-command languages for programs such as information-retrieval systems.
-Winston 1992 discusses computational approaches to real language
-understanding and also the applications of simple grammars to command
-languages.
-
- (7) Although Alyssa's idea works just fine (and is surprisingly
-simple), the sentences that it generates are a bit boring--they don't
-sample the possible sentences of this language in a very interesting
-way. In fact, the grammar is highly recursive in many places, and
-Alyssa's technique "falls into" one of these recursions and gets stuck.
-See *Note Exercise 4-50:: for a way to deal with this.
-
-\1f
-File: sicp.info, Node: 4-3-3, Prev: 4-3-2, Up: 4-3
-
-4.3.3 Implementing the `Amb' Evaluator
---------------------------------------
-
-The evaluation of an ordinary Scheme expression may return a value, may
-never terminate, or may signal an error. In nondeterministic Scheme
-the evaluation of an expression may in addition result in the discovery
-of a dead end, in which case evaluation must backtrack to a previous
-choice point. The interpretation of nondeterministic Scheme is
-complicated by this extra case.
-
- We will construct the `amb' evaluator for nondeterministic Scheme by
-modifying the analyzing evaluator of section *Note 4-1-7::.(1) As in
-the analyzing evaluator, evaluation of an expression is accomplished by
-calling an execution procedure produced by analysis of that expression.
-The difference between the interpretation of ordinary Scheme and the
-interpretation of nondeterministic Scheme will be entirely in the
-execution procedures.
-
-Execution procedures and continuations
-......................................
-
-Recall that the execution procedures for the ordinary evaluator take one
-argument: the environment of execution. In contrast, the execution
-procedures in the `amb' evaluator take three arguments: the
-environment, and two procedures called "continuation procedures". The
-evaluation of an expression will finish by calling one of these two
-continuations: If the evaluation results in a value, the "success
-continuation" is called with that value; if the evaluation results in
-the discovery of a dead end, the "failure continuation" is called.
-Constructing and calling appropriate continuations is the mechanism by
-which the nondeterministic evaluator implements backtracking.
-
- It is the job of the success continuation to receive a value and
-proceed with the computation. Along with that value, the success
-continuation is passed another failure continuation, which is to be
-called subsequently if the use of that value leads to a dead end.
-
- It is the job of the failure continuation to try another branch of
-the nondeterministic process. The essence of the nondeterministic
-language is in the fact that expressions may represent choices among
-alternatives. The evaluation of such an expression must proceed with
-one of the indicated alternative choices, even though it is not known
-in advance which choices will lead to acceptable results. To deal with
-this, the evaluator picks one of the alternatives and passes this value
-to the success continuation. Together with this value, the evaluator
-constructs and passes along a failure continuation that can be called
-later to choose a different alternative.
-
- A failure is triggered during evaluation (that is, a failure
-continuation is called) when a user program explicitly rejects the
-current line of attack (for example, a call to `require' may result in
-execution of `(amb)', an expression that always fails--see section
-*Note 4-3-1::). The failure continuation in hand at that point will
-cause the most recent choice point to choose another alternative. If
-there are no more alternatives to be considered at that choice point, a
-failure at an earlier choice point is triggered, and so on. Failure
-continuations are also invoked by the driver loop in response to a
-`try-again' request, to find another value of the expression.
-
- In addition, if a side-effect operation (such as assignment to a
-variable) occurs on a branch of the process resulting from a choice, it
-may be necessary, when the process finds a dead end, to undo the side
-effect before making a new choice. This is accomplished by having the
-side-effect operation produce a failure continuation that undoes the
-side effect and propagates the failure.
-
- In summary, failure continuations are constructed by
-
- * `amb' expressions--to provide a mechanism to make alternative
- choices if the current choice made by the `amb' expression leads
- to a dead end;
-
- * the top-level driver--to provide a mechanism to report failure
- when the choices are exhausted;
-
- * assignments--to intercept failures and undo assignments during
- backtracking.
-
-
- Failures are initiated only when a dead end is encountered. This
-occurs
-
- * if the user program executes `(amb)';
-
- * if the user types `try-again' at the top-level driver.
-
-
- Failure continuations are also called during processing of a failure:
-
- * When the failure continuation created by an assignment finishes
- undoing a side effect, it calls the failure continuation it
- intercepted, in order to propagate the failure back to the choice
- point that led to this assignment or to the top level.
-
- * When the failure continuation for an `amb' runs out of choices, it
- calls the failure continuation that was originally given to the
- `amb', in order to propagate the failure back to the previous
- choice point or to the top level.
-
-
-Structure of the evaluator
-..........................
-
-The syntax- and data-representation procedures for the `amb' evaluator,
-and also the basic `analyze' procedure, are identical to those in the
-evaluator of section *Note 4-1-7::, except for the fact that we need
-additional syntax procedures to recognize the `amb' special form:(2)
-
- (define (amb? exp) (tagged-list? exp 'amb))
-
- (define (amb-choices exp) (cdr exp))
-
- We must also add to the dispatch in `analyze' a clause that will
-recognize this special form and generate an appropriate execution
-procedure:
-
- ((amb? exp) (analyze-amb exp))
-
- The top-level procedure `ambeval' (similar to the version of `eval'
-given in section *Note 4-1-7::) analyzes the given expression and
-applies the resulting execution procedure to the given environment,
-together with two given continuations:
-
- (define (ambeval exp env succeed fail)
- ((analyze exp) env succeed fail))
-
- A success continuation is a procedure of two arguments: the value
-just obtained and another failure continuation to be used if that value
-leads to a subsequent failure. A failure continuation is a procedure of
-no arguments. So the general form of an execution procedure is
-
- (lambda (env succeed fail)
- ;; `succeed' is `(lambda (value fail) ...)'
- ;; `fail' is `(lambda () ...)'
- ...)
-
- For example, executing
-
- (ambeval <EXP>
- the-global-environment
- (lambda (value fail) value)
- (lambda () 'failed))
-
-will attempt to evaluate the given expression and will return either the
-expression's value (if the evaluation succeeds) or the symbol `failed'
-(if the evaluation fails). The call to `ambeval' in the driver loop
-shown below uses much more complicated continuation procedures, which
-continue the loop and support the `try-again' request.
-
- Most of the complexity of the `amb' evaluator results from the
-mechanics of passing the continuations around as the execution
-procedures call each other. In going through the following code, you
-should compare each of the execution procedures with the corresponding
-procedure for the ordinary evaluator given in section *Note 4-1-7::.
-
-Simple expressions
-..................
-
-The execution procedures for the simplest kinds of expressions are
-essentially the same as those for the ordinary evaluator, except for
-the need to manage the continuations. The execution procedures simply
-succeed with the value of the expression, passing along the failure
-continuation that was passed to them.
-
- (define (analyze-self-evaluating exp)
- (lambda (env succeed fail)
- (succeed exp fail)))
-
- (define (analyze-quoted exp)
- (let ((qval (text-of-quotation exp)))
- (lambda (env succeed fail)
- (succeed qval fail))))
-
- (define (analyze-variable exp)
- (lambda (env succeed fail)
- (succeed (lookup-variable-value exp env)
- fail)))
-
- (define (analyze-lambda exp)
- (let ((vars (lambda-parameters exp))
- (bproc (analyze-sequence (lambda-body exp))))
- (lambda (env succeed fail)
- (succeed (make-procedure vars bproc env)
- fail))))
-
- Notice that looking up a variable always "succeeds." If
-`lookup-variable-value' fails to find the variable, it signals an error,
-as usual. Such a "failure" indicates a program bug--a reference to an
-unbound variable; it is not an indication that we should try another
-nondeterministic choice instead of the one that is currently being
-tried.
-
-Conditionals and sequences
-..........................
-
-Conditionals are also handled in a similar way as in the ordinary
-evaluator. The execution procedure generated by `analyze-if' invokes
-the predicate execution procedure `pproc' with a success continuation
-that checks whether the predicate value is true and goes on to execute
-either the consequent or the alternative. If the execution of `pproc'
-fails, the original failure continuation for the `if' expression is
-called.
-
- (define (analyze-if exp)
- (let ((pproc (analyze (if-predicate exp)))
- (cproc (analyze (if-consequent exp)))
- (aproc (analyze (if-alternative exp))))
- (lambda (env succeed fail)
- (pproc env
- ;; success continuation for evaluating the predicate
- ;; to obtain `pred-value'
- (lambda (pred-value fail2)
- (if (true? pred-value)
- (cproc env succeed fail2)
- (aproc env succeed fail2)))
- ;; failure continuation for evaluating the predicate
- fail))))
-
- Sequences are also handled in the same way as in the previous
-evaluator, except for the machinations in the subprocedure
-`sequentially' that are required for passing the continuations.
-Namely, to sequentially execute `a' and then `b', we call `a' with a
-success continuation that calls `b'.
-
- (define (analyze-sequence exps)
- (define (sequentially a b)
- (lambda (env succeed fail)
- (a env
- ;; success continuation for calling `a'
- (lambda (a-value fail2)
- (b env succeed fail2))
- ;; failure continuation for calling `a'
- fail)))
- (define (loop first-proc rest-procs)
- (if (null? rest-procs)
- first-proc
- (loop (sequentially first-proc (car rest-procs))
- (cdr rest-procs))))
- (let ((procs (map analyze exps)))
- (if (null? procs)
- (error "Empty sequence -- ANALYZE"))
- (loop (car procs) (cdr procs))))
-
-Definitions and assignments
-...........................
-
-Definitions are another case where we must go to some trouble to manage
-the continuations, because it is necessary to evaluate the
-definition-value expression before actually defining the new variable.
-To accomplish this, the definition-value execution procedure `vproc' is
-called with the environment, a success continuation, and the failure
-continuation. If the execution of `vproc' succeeds, obtaining a value
-`val' for the defined variable, the variable is defined and the success
-is propagated:
-
- (define (analyze-definition exp)
- (let ((var (definition-variable exp))
- (vproc (analyze (definition-value exp))))
- (lambda (env succeed fail)
- (vproc env
- (lambda (val fail2)
- (define-variable! var val env)
- (succeed 'ok fail2))
- fail))))
-
- Assignments are more interesting. This is the first place where we
-really use the continuations, rather than just passing them around.
-The execution procedure for assignments starts out like the one for
-definitions. It first attempts to obtain the new value to be assigned
-to the variable. If this evaluation of `vproc' fails, the assignment
-fails.
-
- If `vproc' succeeds, however, and we go on to make the assignment,
-we must consider the possibility that this branch of the computation
-might later fail, which will require us to backtrack out of the
-assignment. Thus, we must arrange to undo the assignment as part of
-the backtracking process.(3)
-
- This is accomplished by giving `vproc' a success continuation
-(marked with the comment "*1*" below) that saves the old value of the
-variable before assigning the new value to the variable and proceeding
-from the assignment. The failure continuation that is passed along
-with the value of the assignment (marked with the comment "*2*" below)
-restores the old value of the variable before continuing the failure.
-That is, a successful assignment provides a failure continuation that
-will intercept a subsequent failure; whatever failure would otherwise
-have called `fail2' calls this procedure instead, to undo the
-assignment before actually calling `fail2'.
-
- (define (analyze-assignment exp)
- (let ((var (assignment-variable exp))
- (vproc (analyze (assignment-value exp))))
- (lambda (env succeed fail)
- (vproc env
- (lambda (val fail2) ; *1*
- (let ((old-value
- (lookup-variable-value var env)))
- (set-variable-value! var val env)
- (succeed 'ok
- (lambda () ; *2*
- (set-variable-value! var
- old-value
- env)
- (fail2)))))
- fail))))
-
-Procedure applications
-......................
-
-The execution procedure for applications contains no new ideas except
-for the technical complexity of managing the continuations. This
-complexity arises in `analyze-application', due to the need to keep
-track of the success and failure continuations as we evaluate the
-operands. We use a procedure `get-args' to evaluate the list of
-operands, rather than a simple `map' as in the ordinary evaluator.
-
- (define (analyze-application exp)
- (let ((fproc (analyze (operator exp)))
- (aprocs (map analyze (operands exp))))
- (lambda (env succeed fail)
- (fproc env
- (lambda (proc fail2)
- (get-args aprocs
- env
- (lambda (args fail3)
- (execute-application
- proc args succeed fail3))
- fail2))
- fail))))
-
- In `get-args', notice how `cdr'ing down the list of `aproc'
-execution procedures and `cons'ing up the resulting list of `args' is
-accomplished by calling each `aproc' in the list with a success
-continuation that recursively calls `get-args'. Each of these recursive
-calls to `get-args' has a success continuation whose value is the
-`cons' of the newly obtained argument onto the list of accumulated
-arguments:
-
- (define (get-args aprocs env succeed fail)
- (if (null? aprocs)
- (succeed '() fail)
- ((car aprocs) env
- ;; success continuation for this `aproc'
- (lambda (arg fail2)
- (get-args (cdr aprocs)
- env
- ;; success continuation for recursive
- ;; call to `get-args'
- (lambda (args fail3)
- (succeed (cons arg args)
- fail3))
- fail2))
- fail)))
-
- The actual procedure application, which is performed by
-`execute-application', is accomplished in the same way as for the
-ordinary evaluator, except for the need to manage the continuations.
-
- (define (execute-application proc args succeed fail)
- (cond ((primitive-procedure? proc)
- (succeed (apply-primitive-procedure proc args)
- fail))
- ((compound-procedure? proc)
- ((procedure-body proc)
- (extend-environment (procedure-parameters proc)
- args
- (procedure-environment proc))
- succeed
- fail))
- (else
- (error
- "Unknown procedure type -- EXECUTE-APPLICATION"
- proc))))
-
-Evaluating `amb' expressions
-............................
-
-The `amb' special form is the key element in the nondeterministic
-language. Here we see the essence of the interpretation process and
-the reason for keeping track of the continuations. The execution
-procedure for `amb' defines a loop `try-next' that cycles through the
-execution procedures for all the possible values of the `amb'
-expression. Each execution procedure is called with a failure
-continuation that will try the next one. When there are no more
-alternatives to try, the entire `amb' expression fails.
-
- (define (analyze-amb exp)
- (let ((cprocs (map analyze (amb-choices exp))))
- (lambda (env succeed fail)
- (define (try-next choices)
- (if (null? choices)
- (fail)
- ((car choices) env
- succeed
- (lambda ()
- (try-next (cdr choices))))))
- (try-next cprocs))))
-
-Driver loop
-...........
-
-The driver loop for the `amb' evaluator is complex, due to the mechanism
-that permits the user to try again in evaluating an expression. The
-driver uses a procedure called `internal-loop', which takes as argument
-a procedure `try-again'. The intent is that calling `try-again' should
-go on to the next untried alternative in the nondeterministic
-evaluation. `Internal-loop' either calls `try-again' in response to
-the user typing `try-again' at the driver loop, or else starts a new
-evaluation by calling `ambeval'.
-
- The failure continuation for this call to `ambeval' informs the user
-that there are no more values and re-invokes the driver loop.
-
- The success continuation for the call to `ambeval' is more subtle.
-We print the obtained value and then invoke the internal loop again
-with a `try-again' procedure that will be able to try the next
-alternative. This `next-alternative' procedure is the second argument
-that was passed to the success continuation. Ordinarily, we think of
-this second argument as a failure continuation to be used if the
-current evaluation branch later fails. In this case, however, we have
-completed a successful evaluation, so we can invoke the "failure"
-alternative branch in order to search for additional successful
-evaluations.
-
- (define input-prompt ";;; Amb-Eval input:")
- (define output-prompt ";;; Amb-Eval value:")
-
- (define (driver-loop)
- (define (internal-loop try-again)
- (prompt-for-input input-prompt)
- (let ((input (read)))
- (if (eq? input 'try-again)
- (try-again)
- (begin
- (newline)
- (display ";;; Starting a new problem ")
- (ambeval input
- the-global-environment
- ;; `ambeval' success
- (lambda (val next-alternative)
- (announce-output output-prompt)
- (user-print val)
- (internal-loop next-alternative))
- ;; `ambeval' failure
- (lambda ()
- (announce-output
- ";;; There are no more values of")
- (user-print input)
- (driver-loop)))))))
- (internal-loop
- (lambda ()
- (newline)
- (display ";;; There is no current problem")
- (driver-loop))))
-
- The initial call to `internal-loop' uses a `try-again' procedure that
-complains that there is no current problem and restarts the driver
-loop. This is the behavior that will happen if the user types
-`try-again' when there is no evaluation in progress.
-
- *Exercise 4.50:* Implement a new special form `ramb' that is like
- `amb' except that it searches alternatives in a random order,
- rather than from left to right. Show how this can help with
- Alyssa's problem in *Note Exercise 4-49::.
-
- *Exercise 4.51:* Implement a new kind of assignment called
- `permanent-set!' that is not undone upon failure. For example, we
- can choose two distinct elements from a list and count the number
- of trials required to make a successful choice as follows:
-
- (define count 0)
-
- (let ((x (an-element-of '(a b c)))
- (y (an-element-of '(a b c))))
- (permanent-set! count (+ count 1))
- (require (not (eq? x y)))
- (list x y count))
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- (a b 2)
-
- ;;; Amb-Eval input:
- try-again
- ;;; Amb-Eval value:
- (a c 3)
-
- What values would have been displayed if we had used `set!' here
- rather than `permanent-set!' ?
-
- *Exercise 4.52:* Implement a new construct called `if-fail' that
- permits the user to catch the failure of an expression. `If-fail'
- takes two expressions. It evaluates the first expression as usual
- and returns as usual if the evaluation succeeds. If the evaluation
- fails, however, the value of the second expression is returned, as
- in the following example:
-
- ;;; Amb-Eval input:
- (if-fail (let ((x (an-element-of '(1 3 5))))
- (require (even? x))
- x)
- 'all-odd)
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- all-odd
-
- ;;; Amb-Eval input:
- (if-fail (let ((x (an-element-of '(1 3 5 8))))
- (require (even? x))
- x)
- 'all-odd)
- ;;; Starting a new problem
- ;;; Amb-Eval value:
- 8
-
- *Exercise 4.53:* With `permanent-set!' as described in *Note
- Exercise 4-51:: and `if-fail' as in *Note Exercise 4-52::, what
- will be the result of evaluating
-
- (let ((pairs '()))
- (if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110))))
- (permanent-set! pairs (cons p pairs))
- (amb))
- pairs))
-
- *Exercise 4.54:* If we had not realized that `require' could be
- implemented as an ordinary procedure that uses `amb', to be
- defined by the user as part of a nondeterministic program, we
- would have had to implement it as a special form. This would
- require syntax procedures
-
- (define (require? exp) (tagged-list? exp 'require))
-
- (define (require-predicate exp) (cadr exp))
-
- and a new clause in the dispatch in `analyze'
-
- ((require? exp) (analyze-require exp))
-
- as well the procedure `analyze-require' that handles `require'
- expressions. Complete the following definition of
- `analyze-require'.
-
- (define (analyze-require exp)
- (let ((pproc (analyze (require-predicate exp))))
- (lambda (env succeed fail)
- (pproc env
- (lambda (pred-value fail2)
- (if <??>
- <??>
- (succeed 'ok fail2)))
- fail))))
-
- ---------- Footnotes ----------
-
- (1) We chose to implement the lazy evaluator in section *Note 4-2::
-as a modification of the ordinary metacircular evaluator of section
-*Note 4-1-1::. In contrast, we will base the `amb' evaluator on the
-analyzing evaluator of section *Note 4-1-7::, because the execution
-procedures in that evaluator provide a convenient framework for
-implementing backtracking.
-
- (2) We assume that the evaluator supports `let' (see *Note Exercise
-4-22::), which we have used in our nondeterministic programs.
-
- (3) We didn't worry about undoing definitions, since we can assume
-that internal definitions are scanned out (section *Note 4-1-6::).
-
-\1f
-File: sicp.info, Node: 4-4, Prev: 4-3, Up: Chapter 4
-
-4.4 Logic Programming
-=====================
-
-In *Note Chapter 1:: we stressed that computer science deals with
-imperative (how to) knowledge, whereas mathematics deals with
-declarative (what is) knowledge. Indeed, programming languages require
-that the programmer express knowledge in a form that indicates the
-step-by-step methods for solving particular problems. On the other
-hand, high-level languages provide, as part of the language
-implementation, a substantial amount of methodological knowledge that
-frees the user from concern with numerous details of how a specified
-computation will progress.
-
- Most programming languages, including Lisp, are organized around
-computing the values of mathematical functions. Expression-oriented
-languages (such as Lisp, Fortran, and Algol) capitalize on the "pun"
-that an expression that describes the value of a function may also be
-interpreted as a means of computing that value. Because of this, most
-programming languages are strongly biased toward unidirectional
-computations (computations with well-defined inputs and outputs).
-There are, however, radically different programming languages that
-relax this bias. We saw one such example in section *Note 3-3-5::,
-where the objects of computation were arithmetic constraints. In a
-constraint system the direction and the order of computation are not so
-well specified; in carrying out a computation the system must therefore
-provide more detailed "how to" knowledge than would be the case with an
-ordinary arithmetic computation. This does not mean, however, that the
-user is released altogether from the responsibility of providing
-imperative knowledge. There are many constraint networks that
-implement the same set of constraints, and the user must choose from
-the set of mathematically equivalent networks a suitable network to
-specify a particular computation.
-
- The nondeterministic program evaluator of section *Note 4-3:: also
-moves away from the view that programming is about constructing
-algorithms for computing unidirectional functions. In a
-nondeterministic language, expressions can have more than one value,
-and, as a result, the computation is dealing with relations rather than
-with single-valued functions. Logic programming extends this idea by
-combining a relational vision of programming with a powerful kind of
-symbolic pattern matching called "unification".(1)
-
- This approach, when it works, can be a very powerful way to write
-programs. Part of the power comes from the fact that a single "what
-is" fact can be used to solve a number of different problems that would
-have different "how to" components. As an example, consider the
-`append' operation, which takes two lists as arguments and combines
-their elements to form a single list. In a procedural language such as
-Lisp, we could define `append' in terms of the basic list constructor
-`cons', as we did in section *Note 2-2-1:::
-
- (define (append x y)
- (if (null? x)
- y
- (cons (car x) (append (cdr x) y))))
-
- This procedure can be regarded as a translation into Lisp of the
-following two rules, the first of which covers the case where the first
-list is empty and the second of which handles the case of a nonempty
-list, which is a `cons' of two parts:
-
- * For any list `y', the empty list and `y' `append' to form `y'.
-
- * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
- form `(cons u z)' if `v' and `y' `append' to form `z'.(2)
-
-
- Using the `append' procedure, we can answer questions such as
-
- Find the `append' of `(a b)' and `(c d)'.
-
- But the same two rules are also sufficient for answering the
-following sorts of questions, which the procedure can't answer:
-
- Find a list `y' that `append's with `(a b)' to produce `(a b c d)'.
-
- Find all `x' and `y' that `append' to form `(a b c d)'.
-
- In a logic programming language, the programmer writes an `append'
-"procedure" by stating the two rules about `append' given above. "How
-to" knowledge is provided automatically by the interpreter to allow this
-single pair of rules to be used to answer all three types of questions
-about `append'.(3)
-
- Contemporary logic programming languages (including the one we
-implement here) have substantial deficiencies, in that their general
-"how to" methods can lead them into spurious infinite loops or other
-undesirable behavior. Logic programming is an active field of research
-in computer science.(4)
-
- Earlier in this chapter we explored the technology of implementing
-interpreters and described the elements that are essential to an
-interpreter for a Lisp-like language (indeed, to an interpreter for any
-conventional language). Now we will apply these ideas to discuss an
-interpreter for a logic programming language. We call this language
-the "query language", because it is very useful for retrieving
-information from data bases by formulating "queries", or questions,
-expressed in the language. Even though the query language is very
-different from Lisp, we will find it convenient to describe the
-language in terms of the same general framework we have been using all
-along: as a collection of primitive elements, together with means of
-combination that enable us to combine simple elements to create more
-complex elements and means of abstraction that enable us to regard
-complex elements as single conceptual units. An interpreter for a
-logic programming language is considerably more complex than an
-interpreter for a language like Lisp. Nevertheless, we will see that
-our query-language interpreter contains many of the same elements found
-in the interpreter of section *Note 4-1::. In particular, there will
-be an "eval" part that classifies expressions according to type and an
-"apply" part that implements the language's abstraction mechanism
-(procedures in the case of Lisp, and "rules" in the case of logic
-programming). Also, a central role is played in the implementation by
-a frame data structure, which determines the correspondence between
-symbols and their associated values. One additional interesting aspect
-of our query-language implementation is that we make substantial use of
-streams, which were introduced in *Note Chapter 3::.
-
-* Menu:
-
-* 4-4-1:: Deductive Information Retrieval
-* 4-4-2:: How the Query System Works
-* 4-4-3:: Is Logic Programming Mathematical Logic?
-* 4-4-4:: Implementing the Query System
-
- ---------- Footnotes ----------
-
- (1) Logic programming has grown out of a long history of research in
-automatic theorem proving. Early theorem-proving programs could
-accomplish very little, because they exhaustively searched the space of
-possible proofs. The major breakthrough that made such a search
-plausible was the discovery in the early 1960s of the "unification
-algorithm" and the principle "resolution principle" (Robinson 1965).
-Resolution was used, for example, by Green and Raphael (1968) (see also
-Green 1969) as the basis for a deductive question-answering system.
-During most of this period, researchers concentrated on algorithms that
-are guaranteed to find a proof if one exists. Such algorithms were
-difficult to control and to direct toward a proof. Hewitt (1969)
-recognized the possibility of merging the control structure of a
-programming language with the operations of a logic-manipulation system,
-leading to the work in automatic search mentioned in section *Note
-4-3-1:: (footnote *Note Footnote 4-47::). At the same time that this
-was being done, Colmerauer, in Marseille, was developing rule-based
-systems for manipulating natural language (see Colmerauer et al. 1973).
-He invented a programming language called Prolog for representing
-those rules. Kowalski (1973; 1979), in Edinburgh, recognized that
-execution of a Prolog program could be interpreted as proving theorems
-(using a proof technique called linear Horn-clause resolution). The
-merging of the last two strands led to the logic-programming movement.
-Thus, in assigning credit for the development of logic programming, the
-French can point to Prolog's genesis at the University of Marseille,
-while the British can highlight the work at the University of
-Edinburgh. According to people at MIT, logic programming was developed
-by these groups in an attempt to figure out what Hewitt was talking
-about in his brilliant but impenetrable Ph.D. thesis. For a history of
-logic programming, see Robinson 1983.
-
- (2) To see the correspondence between the rules and the procedure,
-let `x' in the procedure (where `x' is nonempty) correspond to `(cons u
-v)' in the rule. Then `z' in the rule corresponds to the `append' of
-`(cdr x)' and `y'.
-
- (3) This certainly does not relieve the user of the entire problem
-of how to compute the answer. There are many different mathematically
-equivalent sets of rules for formulating the `append' relation, only
-some of which can be turned into effective devices for computing in any
-direction. In addition, sometimes "what is" information gives no clue
-"how to" compute an answer. For example, consider the problem of
-computing the y such that y^2 = x.
-
- (4) Interest in logic programming peaked during the early 80s when
-the Japanese government began an ambitious project aimed at building
-superfast computers optimized to run logic programming languages. The
-speed of such computers was to be measured in LIPS (Logical Inferences
-Per Second) rather than the usual FLOPS (FLoating-point Operations Per
-Second). Although the project succeeded in developing hardware and
-software as originally planned, the international computer industry
-moved in a different direction. See Feigenbaum and Shrobe 1993 for an
-overview evaluation of the Japanese project. The logic programming
-community has also moved on to consider relational programming based on
-techniques other than simple pattern matching, such as the ability to
-deal with numerical constraints such as the ones illustrated in the
-constraint-propagation system of section *Note 3-3-5::.
-
-\1f
-File: sicp.info, Node: 4-4-1, Next: 4-4-2, Prev: 4-4, Up: 4-4
-
-4.4.1 Deductive Information Retrieval
--------------------------------------
-
-Logic programming excels in providing interfaces to data bases for
-information retrieval. The query language we shall implement in this
-chapter is designed to be used in this way.
-
- In order to illustrate what the query system does, we will show how
-it can be used to manage the data base of personnel records for
-Microshaft, a thriving high-technology company in the Boston area. The
-language provides pattern-directed access to personnel information and
-can also take advantage of general rules in order to make logical
-deductions.
-
-A sample data base
-..................
-
-The personnel data base for Microshaft contains "assertions" about
-company personnel. Here is the information about Ben Bitdiddle, the
-resident computer wizard:
-
- (address (Bitdiddle Ben) (Slumerville (Ridge Road) 10))
- (job (Bitdiddle Ben) (computer wizard))
- (salary (Bitdiddle Ben) 60000)
-
- Each assertion is a list (in this case a triple) whose elements can
-themselves be lists.
-
- As resident wizard, Ben is in charge of the company's computer
-division, and he supervises two programmers and one technician. Here
-is the information about them:
-
- (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))
- (job (Hacker Alyssa P) (computer programmer))
- (salary (Hacker Alyssa P) 40000)
- (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
-
- (address (Fect Cy D) (Cambridge (Ames Street) 3))
- (job (Fect Cy D) (computer programmer))
- (salary (Fect Cy D) 35000)
- (supervisor (Fect Cy D) (Bitdiddle Ben))
-
- (address (Tweakit Lem E) (Boston (Bay State Road) 22))
- (job (Tweakit Lem E) (computer technician))
- (salary (Tweakit Lem E) 25000)
- (supervisor (Tweakit Lem E) (Bitdiddle Ben))
-
- There is also a programmer trainee, who is supervised by Alyssa:
-
- (address (Reasoner Louis) (Slumerville (Pine Tree Road) 80))
- (job (Reasoner Louis) (computer programmer trainee))
- (salary (Reasoner Louis) 30000)
- (supervisor (Reasoner Louis) (Hacker Alyssa P))
-
- All of these people are in the computer division, as indicated by
-the word `computer' as the first item in their job descriptions.
-
- Ben is a high-level employee. His supervisor is the company's big
-wheel himself:
-
- (supervisor (Bitdiddle Ben) (Warbucks Oliver))
-
- (address (Warbucks Oliver) (Swellesley (Top Heap Road)))
- (job (Warbucks Oliver) (administration big wheel))
- (salary (Warbucks Oliver) 150000)
-
- Besides the computer division supervised by Ben, the company has an
-accounting division, consisting of a chief accountant and his assistant:
-
- (address (Scrooge Eben) (Weston (Shady Lane) 10))
- (job (Scrooge Eben) (accounting chief accountant))
- (salary (Scrooge Eben) 75000)
- (supervisor (Scrooge Eben) (Warbucks Oliver))
-
- (address (Cratchet Robert) (Allston (N Harvard Street) 16))
- (job (Cratchet Robert) (accounting scrivener))
- (salary (Cratchet Robert) 18000)
- (supervisor (Cratchet Robert) (Scrooge Eben))
-
- There is also a secretary for the big wheel:
-
- (address (Aull DeWitt) (Slumerville (Onion Square) 5))
- (job (Aull DeWitt) (administration secretary))
- (salary (Aull DeWitt) 25000)
- (supervisor (Aull DeWitt) (Warbucks Oliver))
-
- The data base also contains assertions about which kinds of jobs can
-be done by people holding other kinds of jobs. For instance, a
-computer wizard can do the jobs of both a computer programmer and a
-computer technician:
-
- (can-do-job (computer wizard) (computer programmer))
- (can-do-job (computer wizard) (computer technician))
-
- A computer programmer could fill in for a trainee:
-
- (can-do-job (computer programmer)
- (computer programmer trainee))
-
- Also, as is well known,
-
- (can-do-job (administration secretary)
- (administration big wheel))
-
-Simple queries
-..............
-
-The query language allows users to retrieve information from the data
-base by posing queries in response to the system's prompt. For
-example, to find all computer programmers one can say
-
- ;;; Query input:
- (job ?x (computer programmer))
-
- The system will respond with the following items:
-
- ;;; Query results:
- (job (Hacker Alyssa P) (computer programmer))
- (job (Fect Cy D) (computer programmer))
-
- The input query specifies that we are looking for entries in the
-data base that match a certain "pattern". In this example, the pattern
-specifies entries consisting of three items, of which the first is the
-literal symbol `job', the second can be anything, and the third is the
-literal list `(computer programmer)'. The "anything" that can be the
-second item in the matching list is specified by a "pattern variable",
-`?x'. The general form of a pattern variable is a symbol, taken to be
-the name of the variable, preceded by a question mark. We will see
-below why it is useful to specify names for pattern variables rather
-than just putting `?' into patterns to represent "anything." The
-system responds to a simple query by showing all entries in the data
-base that match the specified pattern.
-
- A pattern can have more than one variable. For example, the query
-
- (address ?x ?y)
-
-will list all the employees' addresses.
-
- A pattern can have no variables, in which case the query simply
-determines whether that pattern is an entry in the data base. If so,
-there will be one match; if not, there will be no matches.
-
- The same pattern variable can appear more than once in a query,
-specifying that the same "anything" must appear in each position. This
-is why variables have names. For example,
-
- (supervisor ?x ?x)
-
-finds all people who supervise themselves (though there are no such
-assertions in our sample data base).
-
- The query
-
- (job ?x (computer ?type))
-
-matches all job entries whose third item is a two-element list whose
-first item is `computer':
-
- (job (Bitdiddle Ben) (computer wizard))
- (job (Hacker Alyssa P) (computer programmer))
- (job (Fect Cy D) (computer programmer))
- (job (Tweakit Lem E) (computer technician))
-
- This same pattern does _not_ match
-
- (job (Reasoner Louis) (computer programmer trainee))
-
-because the third item in the entry is a list of three elements, and the
-pattern's third item specifies that there should be two elements. If
-we wanted to change the pattern so that the third item could be any
-list beginning with `computer', we could specify(1)
-
- (job ?x (computer . ?type))
-
- For example,
-
- (computer . ?type)
-
-matches the data
-
- (computer programmer trainee)
-
-with `?type' as the list `(programmer trainee)'. It also matches the
-data
-
- (computer programmer)
-
-with `?type' as the list `(programmer)', and matches the data
-
- (computer)
-
-with `?type' as the empty list `()'.
-
- We can describe the query language's processing of simple queries as
-follows:
-
- * The system finds all assignments to variables in the query pattern
- that "satisfy" the pattern--that is, all sets of values for the
- variables such that if the pattern variables are "instantiated
- with" (replaced by) the values, the result is in the data base.
-
- * The system responds to the query by listing all instantiations of
- the query pattern with the variable assignments that satisfy it.
-
-
- Note that if the pattern has no variables, the query reduces to a
-determination of whether that pattern is in the data base. If so, the
-empty assignment, which assigns no values to variables, satisfies that
-pattern for that data base.
-
- *Exercise 4.55:* Give simple queries that retrieve the following
- information from the data base:
-
- 1. all people supervised by Ben Bitdiddle;
-
- 2. the names and jobs of all people in the accounting division;
-
- 3. the names and addresses of all people who live in Slumerville.
-
-
-Compound queries
-................
-
-Simple queries form the primitive operations of the query language. In
-order to form compound operations, the query language provides means of
-combination. One thing that makes the query language a logic
-programming language is that the means of combination mirror the means
-of combination used in forming logical expressions: `and', `or', and
-`not'. (Here `and', `or', and `not' are not the Lisp primitives, but
-rather operations built into the query language.)
-
- We can use `and' as follows to find the addresses of all the computer
-programmers:
-
- (and (job ?person (computer programmer))
- (address ?person ?where))
-
- The resulting output is
-
- (and (job (Hacker Alyssa P) (computer programmer))
- (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)))
-
- (and (job (Fect Cy D) (computer programmer))
- (address (Fect Cy D) (Cambridge (Ames Street) 3)))
-
- In general,
-
- (and <QUERY_1> <QUERY_2> ... <QUERY_N>)
-
-is satisfied by all sets of values for the pattern variables that
-simultaneously satisfy <QUERY_1> ... <QUERY_N>.
-
- As for simple queries, the system processes a compound query by
-finding all assignments to the pattern variables that satisfy the
-query, then displaying instantiations of the query with those values.
-
- Another means of constructing compound queries is through `or'. For
-example,
-
- (or (supervisor ?x (Bitdiddle Ben))
- (supervisor ?x (Hacker Alyssa P)))
-
-will find all employees supervised by Ben Bitdiddle or Alyssa P.
-Hacker:
-
- (or (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
- (supervisor (Hacker Alyssa P) (Hacker Alyssa P)))
-
- (or (supervisor (Fect Cy D) (Bitdiddle Ben))
- (supervisor (Fect Cy D) (Hacker Alyssa P)))
-
- (or (supervisor (Tweakit Lem E) (Bitdiddle Ben))
- (supervisor (Tweakit Lem E) (Hacker Alyssa P)))
-
- (or (supervisor (Reasoner Louis) (Bitdiddle Ben))
- (supervisor (Reasoner Louis) (Hacker Alyssa P)))
-
- In general,
-
- (or <QUERY_1> <QUERY_2> ... <QUERY_N>)
-
-is satisfied by all sets of values for the pattern variables that
-satisfy at least one of <QUERY_1> ... <QUERY_N>.
-
- Compound queries can also be formed with `not'. For example,
-
- (and (supervisor ?x (Bitdiddle Ben))
- (not (job ?x (computer programmer))))
-
-finds all people supervised by Ben Bitdiddle who are not computer
-programmers. In general,
-
- (not <QUERY_1>)
-
-is satisfied by all assignments to the pattern variables that do not
-satisfy <QUERY_1>.(2)
-
- The final combining form is called `lisp-value'. When `lisp-value'
-is the first element of a pattern, it specifies that the next element
-is a Lisp predicate to be applied to the rest of the (instantiated)
-elements as arguments. In general,
-
- (lisp-value <PREDICATE> <ARG_1> ... <ARG_N>)
-
-will be satisfied by assignments to the pattern variables for which the
-<PREDICATE> applied to the instantiated <ARG_1> ... <ARG_N> is true.
-For example, to find all people whose salary is greater than $30,000 we
-could write(3)
-
- (and (salary ?person ?amount)
- (lisp-value > ?amount 30000))
-
- *Exercise 4.56:* Formulate compound queries that retrieve the
- following information:
-
- a. the names of all people who are supervised by Ben Bitdiddle,
- together with their addresses;
-
- b. all people whose salary is less than Ben Bitdiddle's,
- together with their salary and Ben Bitdiddle's salary;
-
- c. all people who are supervised by someone who is not in the
- computer division, together with the supervisor's name and
- job.
-
-
-Rules
-.....
-
-In addition to primitive queries and compound queries, the query
-language provides means for abstracting queries. These are given by "rules".
-The rule
-
- (rule (lives-near ?person-1 ?person-2)
- (and (address ?person-1 (?town . ?rest-1))
- (address ?person-2 (?town . ?rest-2))
- (not (same ?person-1 ?person-2))))
-
-specifies that two people live near each other if they live in the same
-town. The final `not' clause prevents the rule from saying that all
-people live near themselves. The `same' relation is defined by a very
-simple rule:(4)
-
- (rule (same ?x ?x))
-
- The following rule declares that a person is a "wheel" in an
-organization if he supervises someone who is in turn a supervisor:
-
- (rule (wheel ?person)
- (and (supervisor ?middle-manager ?person)
- (supervisor ?x ?middle-manager)))
-
- The general form of a rule is
-
- (rule <CONCLUSION> <BODY>)
-
-where <CONCLUSION> is a pattern and <BODY> is any query.(5) We can
-think of a rule as representing a large (even infinite) set of
-assertions, namely all instantiations of the rule conclusion with
-variable assignments that satisfy the rule body. When we described
-simple queries (patterns), we said that an assignment to variables
-satisfies a pattern if the instantiated pattern is in the data base.
-But the pattern needn't be explicitly in the data base as an assertion.
-It can be an implicit assertion implied by a rule. For example, the
-query
-
- (lives-near ?x (Bitdiddle Ben))
-
-results in
-
- (lives-near (Reasoner Louis) (Bitdiddle Ben))
- (lives-near (Aull DeWitt) (Bitdiddle Ben))
-
- To find all computer programmers who live near Ben Bitdiddle, we can
-ask
-
- (and (job ?x (computer programmer))
- (lives-near ?x (Bitdiddle Ben)))
-
- As in the case of compound procedures, rules can be used as parts of
-other rules (as we saw with the `lives-near' rule above) or even be
-defined recursively. For instance, the rule
-
- (rule (outranked-by ?staff-person ?boss)
- (or (supervisor ?staff-person ?boss)
- (and (supervisor ?staff-person ?middle-manager)
- (outranked-by ?middle-manager ?boss))))
-
-says that a staff person is outranked by a boss in the organization if
-the boss is the person's supervisor or (recursively) if the person's
-supervisor is outranked by the boss.
-
- *Exercise 4.57:* Define a rule that says that person 1 can replace
- person 2 if either person 1 does the same job as person 2 or
- someone who does person 1's job can also do person 2's job, and if
- person 1 and person 2 are not the same person. Using your rule,
- give queries that find the following:
-
- a. all people who can replace Cy D. Fect;
-
- b. all people who can replace someone who is being paid more
- than they are, together with the two salaries.
-
-
- *Exercise 4.58:* Define a rule that says that a person is a "big
- shot" in a division if the person works in the division but does
- not have a supervisor who works in the division.
-
- *Exercise 4.59:* Ben Bitdiddle has missed one meeting too many.
- Fearing that his habit of forgetting meetings could cost him his
- job, Ben decides to do something about it. He adds all the weekly
- meetings of the firm to the Microshaft data base by asserting the
- following:
-
- (meeting accounting (Monday 9am))
- (meeting administration (Monday 10am))
- (meeting computer (Wednesday 3pm))
- (meeting administration (Friday 1pm))
-
- Each of the above assertions is for a meeting of an entire
- division. Ben also adds an entry for the company-wide meeting
- that spans all the divisions. All of the company's employees
- attend this meeting.
-
- (meeting whole-company (Wednesday 4pm))
-
- a. On Friday morning, Ben wants to query the data base for all
- the meetings that occur that day. What query should he use?
-
- b. Alyssa P. Hacker is unimpressed. She thinks it would be much
- more useful to be able to ask for her meetings by specifying
- her name. So she designs a rule that says that a person's
- meetings include all `whole-company' meetings plus all
- meetings of that person's division. Fill in the body of
- Alyssa's rule.
-
- (rule (meeting-time ?person ?day-and-time)
- <RULE-BODY>)
-
- c. Alyssa arrives at work on Wednesday morning and wonders what
- meetings she has to attend that day. Having defined the
- above rule, what query should she make to find this out?
-
-
- *Exercise 4.60:* By giving the query
-
- (lives-near ?person (Hacker Alyssa P))
-
- Alyssa P. Hacker is able to find people who live near her, with
- whom she can ride to work. On the other hand, when she tries to
- find all pairs of people who live near each other by querying
-
- (lives-near ?person-1 ?person-2)
-
- she notices that each pair of people who live near each other is
- listed twice; for example,
-
- (lives-near (Hacker Alyssa P) (Fect Cy D))
- (lives-near (Fect Cy D) (Hacker Alyssa P))
-
- Why does this happen? Is there a way to find a list of people who
- live near each other, in which each pair appears only once?
- Explain.
-
-Logic as programs
-.................
-
-We can regard a rule as a kind of logical implication: _If_ an
-assignment of values to pattern variables satisfies the body, _then_ it
-satisfies the conclusion. Consequently, we can regard the query
-language as having the ability to perform "logical deductions" based
-upon the rules. As an example, consider the `append' operation
-described at the beginning of section *Note 4-4::. As we said,
-`append' can be characterized by the following two rules:
-
- * For any list `y', the empty list and `y' `append' to form `y'.
-
- * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
- form `(cons u z)' if `v' and `y' `append' to form `z'.
-
-
- To express this in our query language, we define two rules for a
-relation
-
- (append-to-form x y z)
-
-which we can interpret to mean "`x' and `y' `append' to form `z'":
-
- (rule (append-to-form () ?y ?y))
-
- (rule (append-to-form (?u . ?v) ?y (?u . ?z))
- (append-to-form ?v ?y ?z))
-
- The first rule has no body, which means that the conclusion holds
-for any value of `?y'. Note how the second rule makes use of
-dotted-tail notation to name the `car' and `cdr' of a list.
-
- Given these two rules, we can formulate queries that compute the
-`append' of two lists:
-
- ;;; Query input:
- (append-to-form (a b) (c d) ?z)
- ;;; Query results:
- (append-to-form (a b) (c d) (a b c d))
-
- What is more striking, we can use the same rules to ask the question
-"Which list, when `append'ed to `(a b)', yields `(a b c d)'?" This is
-done as follows:
-
- ;;; Query input:
- (append-to-form (a b) ?y (a b c d))
- ;;; Query results:
- (append-to-form (a b) (c d) (a b c d))
-
- We can also ask for all pairs of lists that `append' to form `(a b c
-d)':
-
- ;;; Query input:
- (append-to-form ?x ?y (a b c d))
- ;;; Query results:
- (append-to-form () (a b c d) (a b c d))
- (append-to-form (a) (b c d) (a b c d))
- (append-to-form (a b) (c d) (a b c d))
- (append-to-form (a b c) (d) (a b c d))
- (append-to-form (a b c d) () (a b c d))
-
- The query system may seem to exhibit quite a bit of intelligence in
-using the rules to deduce the answers to the queries above. Actually,
-as we will see in the next section, the system is following a
-well-determined algorithm in unraveling the rules. Unfortunately,
-although the system works impressively in the `append' case, the
-general methods may break down in more complex cases, as we will see in
-section *Note 4-4-3::.
-
- *Exercise 4.61:* The following rules implement a `next-to'
- relation that finds adjacent elements of a list:
-
- (rule (?x next-to ?y in (?x ?y . ?u)))
-
- (rule (?x next-to ?y in (?v . ?z))
- (?x next-to ?y in ?z))
-
- What will the response be to the following queries?
-
- (?x next-to ?y in (1 (2 3) 4))
-
- (?x next-to 1 in (2 1 3 1))
-
- *Exercise 4.62:* Define rules to implement the `last-pair'
- operation of *Note Exercise 2-17::, which returns a list
- containing the last element of a nonempty list. Check your rules
- on queries such as `(last-pair (3) ?x)', `(last-pair (1 2 3) ?x)',
- and `(last-pair (2 ?x) (3))'. Do your rules work correctly on
- queries such as `(last-pair ?x (3))' ?
-
- *Exercise 4.63:* The following data base (see Genesis 4) traces
- the genealogy of the descendants of Ada back to Adam, by way of
- Cain:
-
- (son Adam Cain)
- (son Cain Enoch)
- (son Enoch Irad)
- (son Irad Mehujael)
- (son Mehujael Methushael)
- (son Methushael Lamech)
- (wife Lamech Ada)
- (son Ada Jabal)
- (son Ada Jubal)
-
- Formulate rules such as "If S is the son of f, and f is the son of
- G, then S is the grandson of G" and "If W is the wife of M, and S
- is the son of W, then S is the son of M" (which was supposedly
- more true in biblical times than today) that will enable the query
- system to find the grandson of Cain; the sons of Lamech; the
- grandsons of Methushael. (See *Note Exercise 4-69:: for some
- rules to deduce more complicated relationships.)
-
- ---------- Footnotes ----------
-
- (1) This uses the dotted-tail notation introduced in *Note Exercise
-2-20::.
-
- (2) Actually, this description of `not' is valid only for simple
-cases. The real behavior of `not' is more complex. We will examine
-`not''s peculiarities in sections *Note 4-4-2:: and *Note 4-4-3::.
-
- (3) `Lisp-value' should be used only to perform an operation not
-provided in the query language. In particular, it should not be used
-to test equality (since that is what the matching in the query language
-is designed to do) or inequality (since that can be done with the
-`same' rule shown below).
-
- (4) Notice that we do not need `same' in order to make two things be
-the same: We just use the same pattern variable for each--in effect, we
-have one thing instead of two things in the first place. For example,
-see `?town' in the `lives-near' rule and `?middle-manager' in the
-`wheel' rule below. `Same' is useful when we want to force two things
-to be different, such as `?person-1' and `?person-2' in the
-`lives-near' rule. Although using the same pattern variable in two
-parts of a query forces the same value to appear in both places, using
-different pattern variables does not force different values to appear.
-(The values assigned to different pattern variables may be the same or
-different.)
-
- (5) We will also allow rules without bodies, as in `same', and we
-will interpret such a rule to mean that the rule conclusion is
-satisfied by any values of the variables.
-
-\1f
-File: sicp.info, Node: 4-4-2, Next: 4-4-3, Prev: 4-4-1, Up: 4-4
-
-4.4.2 How the Query System Works
---------------------------------
-
-In section *Note 4-4-4:: we will present an implementation of the query
-interpreter as a collection of procedures. In this section we give an
-overview that explains the general structure of the system independent
-of low-level implementation details. After describing the
-implementation of the interpreter, we will be in a position to
-understand some of its limitations and some of the subtle ways in which
-the query language's logical operations differ from the operations of
-mathematical logic.
-
- It should be apparent that the query evaluator must perform some
-kind of search in order to match queries against facts and rules in the
-data base. One way to do this would be to implement the query system
-as a nondeterministic program, using the `amb' evaluator of section
-*Note 4-3:: (see *Note Exercise 4-78::). Another possibility is to
-manage the search with the aid of streams. Our implementation follows
-this second approach.
-
- The query system is organized around two central operations called "pattern
-matching" and "unification". We first describe pattern matching and
-explain how this operation, together with the organization of
-information in terms of streams of frames, enables us to implement both
-simple and compound queries. We next discuss unification, a
-generalization of pattern matching needed to implement rules. Finally,
-we show how the entire query interpreter fits together through a
-procedure that classifies expressions in a manner analogous to the way
-`eval' classifies expressions for the interpreter described in section
-*Note 4-1::.
-
-Pattern matching
-................
-
-A "pattern matcher" is a program that tests whether some datum fits a
-specified pattern. For example, the data list `((a b) c (a b))' matches
-the pattern `(?x c ?x)' with the pattern variable `?x' bound to `(a
-b)'. The same data list matches the pattern `(?x ?y ?z)' with `?x' and
-`?z' both bound to `(a b)' and `?y' bound to `c'. It also matches the
-pattern `((?x ?y) c (?x ?y))' with `?x' bound to `a' and `?y' bound to
-`b'. However, it does not match the pattern `(?x a ?y)', since that
-pattern specifies a list whose second element is the symbol `a'.
-
- The pattern matcher used by the query system takes as inputs a
-pattern, a datum, and a "frame" that specifies bindings for various
-pattern variables. It checks whether the datum matches the pattern in
-a way that is consistent with the bindings already in the frame. If
-so, it returns the given frame augmented by any bindings that may have
-been determined by the match. Otherwise, it indicates that the match
-has failed.
-
- For example, using the pattern `(?x ?y ?x)' to match `(a b a)' given
-an empty frame will return a frame specifying that `?x' is bound to `a'
-and `?y' is bound to `b'. Trying the match with the same pattern, the
-same datum, and a frame specifying that `?y' is bound to `a' will fail.
-Trying the match with the same pattern, the same datum, and a frame in
-which `?y' is bound to `b' and `?x' is unbound will return the given
-frame augmented by a binding of `?x' to `a'.
-
- The pattern matcher is all the mechanism that is needed to process
-simple queries that don't involve rules. For instance, to process the
-query
-
- (job ?x (computer programmer))
-
-we scan through all assertions in the data base and select those that
-match the pattern with respect to an initially empty frame. For each
-match we find, we use the frame returned by the match to instantiate
-the pattern with a value for `?x'.
-
-Streams of frames
-.................
-
-The testing of patterns against frames is organized through the use of
-streams. Given a single frame, the matching process runs through the
-data-base entries one by one. For each data-base entry, the matcher
-generates either a special symbol indicating that the match has failed
-or an extension to the frame. The results for all the data-base
-entries are collected into a stream, which is passed through a filter
-to weed out the failures. The result is a stream of all the frames
-that extend the given frame via a match to some assertion in the data
-base.(1)
-
- In our system, a query takes an input stream of frames and performs
-the above matching operation for every frame in the stream, as
-indicated in *Note Figure 4-4::. That is, for each frame in the input
-stream, the query generates a new stream consisting of all extensions
-to that frame by matches to assertions in the data base. All these
-streams are then combined to form one huge stream, which contains all
-possible extensions of every frame in the input stream. This stream is
-the output of the query.
-
- *Figure 4.4:* A query processes a stream of frames.
-
- output stream
- input stream +-------------+ of frames,
- of frames | query | filtered and extended
- ---------------->| +------------------------->
- | (job ?x ?y) |
- +-------------+
- ^
- |
- stream of assertions
- from data base
-
- To answer a simple query, we use the query with an input stream
-consisting of a single empty frame. The resulting output stream
-contains all extensions to the empty frame (that is, all answers to our
-query). This stream of frames is then used to generate a stream of
-copies of the original query pattern with the variables instantiated by
-the values in each frame, and this is the stream that is finally
-printed.
-
-Compound queries
-................
-
-The real elegance of the stream-of-frames implementation is evident
-when we deal with compound queries. The processing of compound queries
-makes use of the ability of our matcher to demand that a match be
-consistent with a specified frame. For example, to handle the `and' of
-two queries, such as
-
- (and (can-do-job ?x (computer programmer trainee))
- (job ?person ?x))
-
-(informally, "Find all people who can do the job of a computer
-programmer trainee"), we first find all entries that match the pattern
-
- (can-do-job ?x (computer programmer trainee))
-
- This produces a stream of frames, each of which contains a binding
-for `?x'. Then for each frame in the stream we find all entries that
-match
-
- (job ?person ?x)
-
-in a way that is consistent with the given binding for `?x'. Each such
-match will produce a frame containing bindings for `?x' and `?person'.
-The `and' of two queries can be viewed as a series combination of the
-two component queries, as shown in *Note Figure 4-5::. The frames that
-pass through the first query filter are filtered and further extended
-by the second query.
-
- *Figure 4.5:* The `and' combination of two queries is produced by
- operating on the stream of frames in series.
-
- +----------------------+
- | (and A B) |
- input stream | | output stream
- of frames | +---+ +---+ | of frames
- ------------------->| A +------>| B +-------------------->
- | +---+ +---+ |
- | ^ ^ |
- | | | |
- | +-----*-----+ |
- +-----------|----------+
- |
- data base
-
- *Note Figure 4-6:: shows the analogous method for computing the `or'
-of two queries as a parallel combination of the two component queries.
-The input stream of frames is extended separately by each query. The
-two resulting streams are then merged to produce the final output
-stream.
-
- *Figure 4.6:* The `or' combination of two queries is produced by
- operating on the stream of frames in parallel and merging the
- results.
-
- +---------------------------+
- | (or A B) |
- | +---+ |
- input | +->| A |------------+ | output
- stream of | | +---+ V | stream of
- frames | | ^ +-------+ | frames
- -------------* | | merge +--------------->
- | | | +-------+ |
- | | | ^ |
- | | | +---+ | |
- | +------->| B +------+ |
- | | +---+ |
- | | ^ |
- | | | |
- | +--*--+ |
- +---------|-----------------+
- |
- data base
-
- Even from this high-level description, it is apparent that the
-processing of compound queries can be slow. For example, since a query
-may produce more than one output frame for each input frame, and each
-query in an `and' gets its input frames from the previous query, an
-`and' query could, in the worst case, have to perform a number of
-matches that is exponential in the number of queries (see *Note
-Exercise 4-76::).(2) Though systems for handling only simple queries
-are quite practical, dealing with complex queries is extremely
-difficult.(3)
-
- From the stream-of-frames viewpoint, the `not' of some query acts as
-a filter that removes all frames for which the query can be satisfied.
-For instance, given the pattern
-
- (not (job ?x (computer programmer)))
-
-we attempt, for each frame in the input stream, to produce extension
-frames that satisfy `(job ?x (computer programmer))'. We remove from
-the input stream all frames for which such extensions exist. The
-result is a stream consisting of only those frames in which the binding
-for `?x' does not satisfy `(job ?x (computer programmer))'. For
-example, in processing the query
-
- (and (supervisor ?x ?y)
- (not (job ?x (computer programmer))))
-
-the first clause will generate frames with bindings for `?x' and `?y'.
-The `not' clause will then filter these by removing all frames in which
-the binding for `?x' satisfies the restriction that `?x' is a computer
-programmer.(4)
-
- The `lisp-value' special form is implemented as a similar filter on
-frame streams. We use each frame in the stream to instantiate any
-variables in the pattern, then apply the Lisp predicate. We remove
-from the input stream all frames for which the predicate fails.
-
-Unification
-...........
-
-In order to handle rules in the query language, we must be able to find
-the rules whose conclusions match a given query pattern. Rule
-conclusions are like assertions except that they can contain variables,
-so we will need a generalization of pattern matching--called "unification"--in
-which both the "pattern" and the "datum" may contain variables.
-
- A unifier takes two patterns, each containing constants and
-variables, and determines whether it is possible to assign values to
-the variables that will make the two patterns equal. If so, it returns
-a frame containing these bindings. For example, unifying `(?x a ?y)'
-and `(?y ?z a)' will specify a frame in which `?x', `?y', and `?z' must
-all be bound to `a'. On the other hand, unifying `(?x ?y a)' and `(?x
-b ?y)' will fail, because there is no value for `?y' that can make the
-two patterns equal. (For the second elements of the patterns to be
-equal, `?y' would have to be `b'; however, for the third elements to be
-equal, `?y' would have to be `a'.) The unifier used in the query
-system, like the pattern matcher, takes a frame as input and performs
-unifications that are consistent with this frame.
-
- The unification algorithm is the most technically difficult part of
-the query system. With complex patterns, performing unification may
-seem to require deduction. To unify `(?x ?x)' and `((a ?y c) (a b
-?z))', for example, the algorithm must infer that `?x' should be `(a b
-c)', `?y' should be `b', and `?z' should be `c'. We may think of this
-process as solving a set of equations among the pattern components. In
-general, these are simultaneous equations, which may require substantial
-manipulation to solve.(5) For example, unifying `(?x ?x)' and `((a ?y
-c) (a b ?z))' may be thought of as specifying the simultaneous equations
-
- ?x = (a ?y c)
- ?x = (a b ?z)
-
- These equations imply that
-
- (a ?y c) = (a b ?z)
-
-which in turn implies that
-
- a = a, ?y = b, c = ?z,
-
-and hence that
-
- ?x = (a b c)
-
- In a successful pattern match, all pattern variables become bound,
-and the values to which they are bound contain only constants. This is
-also true of all the examples of unification we have seen so far. In
-general, however, a successful unification may not completely determine
-the variable values; some variables may remain unbound and others may
-be bound to values that contain variables.
-
- Consider the unification of `(?x a)' and `((b ?y) ?z)'. We can
-deduce that `?x = (b ?y)' and `a = ?z', but we cannot further solve for
-`?x' or `?y'. The unification doesn't fail, since it is certainly
-possible to make the two patterns equal by assigning values to `?x' and
-`?y'. Since this match in no way restricts the values `?y' can take
-on, no binding for `?y' is put into the result frame. The match does,
-however, restrict the value of `?x'. Whatever value `?y' has, `?x'
-must be `(b ?y)'. A binding of `?x' to the pattern `(b ?y)' is thus
-put into the frame. If a value for `?y' is later determined and added
-to the frame (by a pattern match or unification that is required to be
-consistent with this frame), the previously bound `?x' will refer to
-this value.(6)
-
-Applying rules
-..............
-
-Unification is the key to the component of the query system that makes
-inferences from rules. To see how this is accomplished, consider
-processing a query that involves applying a rule, such as
-
- (lives-near ?x (Hacker Alyssa P))
-
- To process this query, we first use the ordinary pattern-match
-procedure described above to see if there are any assertions in the
-data base that match this pattern. (There will not be any in this
-case, since our data base includes no direct assertions about who lives
-near whom.) The next step is to attempt to unify the query pattern
-with the conclusion of each rule. We find that the pattern unifies
-with the conclusion of the rule
-
- (rule (lives-near ?person-1 ?person-2)
- (and (address ?person-1 (?town . ?rest-1))
- (address ?person-2 (?town . ?rest-2))
- (not (same ?person-1 ?person-2))))
-
-resulting in a frame specifying that `?person-2' is bound to `(Hacker
-Alyssa P)' and that `?x' should be bound to (have the same value as)
-`?person-1'. Now, relative to this frame, we evaluate the compound
-query given by the body of the rule. Successful matches will extend
-this frame by providing a binding for `?person-1', and consequently a
-value for `?x', which we can use to instantiate the original query
-pattern.
-
- In general, the query evaluator uses the following method to apply a
-rule when trying to establish a query pattern in a frame that specifies
-bindings for some of the pattern variables:
-
- * Unify the query with the conclusion of the rule to form, if
- successful, an extension of the original frame.
-
- * Relative to the extended frame, evaluate the query formed by the
- body of the rule.
-
-
- Notice how similar this is to the method for applying a procedure in
-the `eval'/`apply' evaluator for Lisp:
-
- * Bind the procedure's parameters to its arguments to form a frame
- that extends the original procedure environment.
-
- * Relative to the extended environment, evaluate the expression
- formed by the body of the procedure.
-
-
- The similarity between the two evaluators should come as no
-surprise. Just as procedure definitions are the means of abstraction
-in Lisp, rule definitions are the means of abstraction in the query
-language. In each case, we unwind the abstraction by creating
-appropriate bindings and evaluating the rule or procedure body relative
-to these.
-
-Simple queries
-..............
-
-We saw earlier in this section how to evaluate simple queries in the
-absence of rules. Now that we have seen how to apply rules, we can
-describe how to evaluate simple queries by using both rules and
-assertions.
-
- Given the query pattern and a stream of frames, we produce, for each
-frame in the input stream, two streams:
-
- * a stream of extended frames obtained by matching the pattern
- against all assertions in the data base (using the pattern
- matcher), and
-
- * a stream of extended frames obtained by applying all possible
- rules (using the unifier).(7)
-
-
- Appending these two streams produces a stream that consists of all
-the ways that the given pattern can be satisfied consistent with the
-original frame. These streams (one for each frame in the input stream)
-are now all combined to form one large stream, which therefore consists
-of all the ways that any of the frames in the original input stream can
-be extended to produce a match with the given pattern.
-
-The query evaluator and the driver loop
-.......................................
-
-Despite the complexity of the underlying matching operations, the
-system is organized much like an evaluator for any language. The
-procedure that coordinates the matching operations is called `qeval',
-and it plays a role analogous to that of the `eval' procedure for Lisp.
-`Qeval' takes as inputs a query and a stream of frames. Its output is
-a stream of frames, corresponding to successful matches to the query
-pattern, that extend some frame in the input stream, as indicated in
-*Note Figure 4-4::. Like `eval', `qeval' classifies the different
-types of expressions (queries) and dispatches to an appropriate
-procedure for each. There is a procedure for each special form (`and',
-`or', `not', and `lisp-value') and one for simple queries.
-
- The driver loop, which is analogous to the `driver-loop' procedure
-for the other evaluators in this chapter, reads queries from the
-terminal. For each query, it calls `qeval' with the query and a stream
-that consists of a single empty frame. This will produce the stream of
-all possible matches (all possible extensions to the empty frame). For
-each frame in the resulting stream, it instantiates the original query
-using the values of the variables found in the frame. This stream of
-instantiated queries is then printed.(8)
-
- The driver also checks for the special command `assert!', which
-signals that the input is not a query but rather an assertion or rule
-to be added to the data base. For instance,
-
- (assert! (job (Bitdiddle Ben) (computer wizard)))
-
- (assert! (rule (wheel ?person)
- (and (supervisor ?middle-manager ?person)
- (supervisor ?x ?middle-manager))))
-
- ---------- Footnotes ----------
-
- (1) Because matching is generally very expensive, we would like to
-avoid applying the full matcher to every element of the data base.
-This is usually arranged by breaking up the process into a fast, coarse
-match and the final match. The coarse match filters the data base to
-produce a small set of candidates for the final match. With care, we
-can arrange our data base so that some of the work of coarse matching
-can be done when the data base is constructed rather then when we want
-to select the candidates. This is called "indexing" the data base.
-There is a vast technology built around data-base-indexing schemes.
-Our implementation, described in section *Note 4-4-4::, contains a
-simple-minded form of such an optimization.
-
- (2) But this kind of exponential explosion is not common in `and'
-queries because the added conditions tend to reduce rather than expand
-the number of frames produced.
-
- (3) There is a large literature on data-base-management systems that
-is concerned with how to handle complex queries efficiently.
-
- (4) There is a subtle difference between this filter implementation
-of `not' and the usual meaning of `not' in mathematical logic. See
-section *Note 4-4-3::.
-
- (5) In one-sided pattern matching, all the equations that contain
-pattern variables are explicit and already solved for the unknown (the
-pattern variable).
-
- (6) Another way to think of unification is that it generates the
-most general pattern that is a specialization of the two input
-patterns. That is, the unification of `(?x a)' and `((b ?y) ?z)' is
-`((b ?y) a)', and the unification of `(?x a ?y)' and `(?y ?z a)',
-discussed above, is `(a a a)'. For our implementation, it is more
-convenient to think of the result of unification as a frame rather than
-a pattern.
-
- (7) Since unification is a generalization of matching, we could
-simplify the system by using the unifier to produce both streams.
-Treating the easy case with the simple matcher, however, illustrates
-how matching (as opposed to full-blown unification) can be useful in
-its own right.
-
- (8) The reason we use streams (rather than lists) of frames is that
-the recursive application of rules can generate infinite numbers of
-values that satisfy a query. The delayed evaluation embodied in
-streams is crucial here: The system will print responses one by one as
-they are generated, regardless of whether there are a finite or
-infinite number of responses.
-
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-
-4.4.3 Is Logic Programming Mathematical Logic?
-----------------------------------------------
-
-The means of combination used in the query language may at first seem
-identical to the operations `and', `or', and `not' of mathematical
-logic, and the application of query-language rules is in fact
-accomplished through a legitimate method of inference.(1) This
-identification of the query language with mathematical logic is not
-really valid, though, because the query language provides a structure
-"control structure" that interprets the logical statements
-procedurally. We can often take advantage of this control structure.
-For example, to find all of the supervisors of programmers we could
-formulate a query in either of two logically equivalent forms:
-
- (and (job ?x (computer programmer))
- (supervisor ?x ?y))
-
-or
-
- (and (supervisor ?x ?y)
- (job ?x (computer programmer)))
-
- If a company has many more supervisors than programmers (the usual
-case), it is better to use the first form rather than the second
-because the data base must be scanned for each intermediate result
-(frame) produced by the first clause of the `and'.
-
- The aim of logic programming is to provide the programmer with
-techniques for decomposing a computational problem into two separate
-problems: "what" is to be computed, and "how" this should be computed.
-This is accomplished by selecting a subset of the statements of
-mathematical logic that is powerful enough to be able to describe
-anything one might want to compute, yet weak enough to have a
-controllable procedural interpretation. The intention here is that, on
-the one hand, a program specified in a logic programming language
-should be an effective program that can be carried out by a computer.
-Control ("how" to compute) is effected by using the order of evaluation
-of the language. We should be able to arrange the order of clauses and
-the order of subgoals within each clause so that the computation is
-done in an order deemed to be effective and efficient. At the same
-time, we should be able to view the result of the computation ("what"
-to compute) as a simple consequence of the laws of logic.
-
- Our query language can be regarded as just such a procedurally
-interpretable subset of mathematical logic. An assertion represents a
-simple fact (an atomic proposition). A rule represents the implication
-that the rule conclusion holds for those cases where the rule body
-holds. A rule has a natural procedural interpretation: To establish
-the conclusion of the rule, establish the body of the rule. Rules,
-therefore, specify computations. However, because rules can also be
-regarded as statements of mathematical logic, we can justify any
-"inference" accomplished by a logic program by asserting that the same
-result could be obtained by working entirely within mathematical
-logic.(2)
-
-Infinite loops
-..............
-
-A consequence of the procedural interpretation of logic programs is
-that it is possible to construct hopelessly inefficient programs for
-solving certain problems. An extreme case of inefficiency occurs when
-the system falls into infinite loops in making deductions. As a simple
-example, suppose we are setting up a data base of famous marriages,
-including
-
- (assert! (married Minnie Mickey))
-
- If we now ask
-
- (married Mickey ?who)
-
-we will get no response, because the system doesn't know that if A is
-married to B, then B is married to A. So we assert the rule
-
- (assert! (rule (married ?x ?y)
- (married ?y ?x)))
-
-and again query
-
- (married Mickey ?who)
-
- Unfortunately, this will drive the system into an infinite loop, as
-follows:
-
- * The system finds that the `married' rule is applicable; that is,
- the rule conclusion `(married ?x ?y)' successfully unifies with
- the query pattern `(married Mickey ?who)' to produce a frame in
- which `?x' is bound to `Mickey' and `?y' is bound to `?who'. So
- the interpreter proceeds to evaluate the rule body `(married ?y
- ?x)' in this frame--in effect, to process the query `(married ?who
- Mickey)'.
-
- * One answer appears directly as an assertion in the data base:
- `(married Minnie Mickey)'.
-
- * The `married' rule is also applicable, so the interpreter again
- evaluates the rule body, which this time is equivalent to
- `(married Mickey ?who)'.
-
-
- The system is now in an infinite loop. Indeed, whether the system
-will find the simple answer `(married Minnie Mickey)' before it goes
-into the loop depends on implementation details concerning the order in
-which the system checks the items in the data base. This is a very
-simple example of the kinds of loops that can occur. Collections of
-interrelated rules can lead to loops that are much harder to
-anticipate, and the appearance of a loop can depend on the order of
-clauses in an `and' (see *Note Exercise 4-64::) or on low-level details
-concerning the order in which the system processes queries.(3)
-
-Problems with `not'
-...................
-
-Another quirk in the query system concerns `not'. Given the data base
-of section *Note 4-4-1::, consider the following two queries:
-
- (and (supervisor ?x ?y)
- (not (job ?x (computer programmer))))
-
- (and (not (job ?x (computer programmer)))
- (supervisor ?x ?y))
-
- These two queries do not produce the same result. The first query
-begins by finding all entries in the data base that match `(supervisor
-?x ?y)', and then filters the resulting frames by removing the ones in
-which the value of `?x' satisfies `(job ?x (computer programmer))'.
-The second query begins by filtering the incoming frames to remove
-those that can satisfy `(job ?x (computer programmer))'. Since the
-only incoming frame is empty, it checks the data base to see if there
-are any patterns that satisfy `(job ?x (computer programmer))'. Since
-there generally are entries of this form, the `not' clause filters out
-the empty frame and returns an empty stream of frames. Consequently,
-the entire compound query returns an empty stream.
-
- The trouble is that our implementation of `not' really is meant to
-serve as a filter on values for the variables. If a `not' clause is
-processed with a frame in which some of the variables remain unbound
-(as does `?x' in the example above), the system will produce unexpected
-results. Similar problems occur with the use of `lisp-value'--the Lisp
-predicate can't work if some of its arguments are unbound. See *Note
-Exercise 4-77::.
-
- There is also a much more serious way in which the `not' of the query
-language differs from the `not' of mathematical logic. In logic, we
-interpret the statement "not P" to mean that P is not true. In the
-query system, however, "not P" means that P is not deducible from the
-knowledge in the data base. For example, given the personnel data base
-of section *Note 4-4-1::, the system would happily deduce all sorts of
-`not' statements, such as that Ben Bitdiddle is not a baseball fan,
-that it is not raining outside, and that 2 + 2 is not 4.(4) In other
-words, the `not' of logic programming languages reflects the so-called "closed
-world assumption" that all relevant information has been included in
-the data base.(5)
-
- *Exercise 4.64:* Louis Reasoner mistakenly deletes the
- `outranked-by' rule (section *Note 4-4-1::) from the data base.
- When he realizes this, he quickly reinstalls it. Unfortunately,
- he makes a slight change in the rule, and types it in as
-
- (rule (outranked-by ?staff-person ?boss)
- (or (supervisor ?staff-person ?boss)
- (and (outranked-by ?middle-manager ?boss)
- (supervisor ?staff-person ?middle-manager))))
-
- Just after Louis types this information into the system, DeWitt
- Aull comes by to find out who outranks Ben Bitdiddle. He issues
- the query
-
- (outranked-by (Bitdiddle Ben) ?who)
-
- After answering, the system goes into an infinite loop. Explain
- why.
-
- *Exercise 4.65:* Cy D. Fect, looking forward to the day when he
- will rise in the organization, gives a query to find all the
- wheels (using the `wheel' rule of section *Note 4-4-1::):
-
- (wheel ?who)
-
- To his surprise, the system responds
-
- ;;; Query results:
- (wheel (Warbucks Oliver))
- (wheel (Bitdiddle Ben))
- (wheel (Warbucks Oliver))
- (wheel (Warbucks Oliver))
- (wheel (Warbucks Oliver))
-
- Why is Oliver Warbucks listed four times?
-
- *Exercise 4.66:* Ben has been generalizing the query system to
- provide statistics about the company. For example, to find the
- total salaries of all the computer programmers one will be able to
- say
-
- (sum ?amount
- (and (job ?x (computer programmer))
- (salary ?x ?amount)))
-
- In general, Ben's new system allows expressions of the form
-
- (accumulation-function <VARIABLE>
- <QUERY PATTERN>)
-
- where `accumulation-function' can be things like `sum', `average',
- or `maximum'. Ben reasons that it should be a cinch to implement
- this. He will simply feed the query pattern to `qeval'. This
- will produce a stream of frames. He will then pass this stream
- through a mapping function that extracts the value of the
- designated variable from each frame in the stream and feed the
- resulting stream of values to the accumulation function. Just as
- Ben completes the implementation and is about to try it out, Cy
- walks by, still puzzling over the `wheel' query result in exercise
- *Note Exercise 4-65::. When Cy shows Ben the system's response,
- Ben groans, "Oh, no, my simple accumulation scheme won't work!"
-
- What has Ben just realized? Outline a method he can use to
- salvage the situation.
-
- *Exercise 4.67:* Devise a way to install a loop detector in the
- query system so as to avoid the kinds of simple loops illustrated
- in the text and in *Note Exercise 4-64::. The general idea is that
- the system should maintain some sort of history of its current
- chain of deductions and should not begin processing a query that
- it is already working on. Describe what kind of information
- (patterns and frames) is included in this history, and how the
- check should be made. (After you study the details of the
- query-system implementation in section *Note 4-4-4::, you may want
- to modify the system to include your loop detector.)
-
- *Exercise 4.68:* Define rules to implement the `reverse' operation
- of *Note Exercise 2-18::, which returns a list containing the same
- elements as a given list in reverse order. (Hint: Use
- `append-to-form'.) Can your rules answer both `(reverse (1 2 3)
- ?x)' and `(reverse ?x (1 2 3))' ?
-
- *Exercise 4.69:* Beginning with the data base and the rules you
- formulated in *Note Exercise 4-63::, devise a rule for adding
- "greats" to a grandson relationship. This should enable the system
- to deduce that Irad is the great-grandson of Adam, or that Jabal
- and Jubal are the great-great-great-great-great-grandsons of Adam.
- (Hint: Represent the fact about Irad, for example, as `((great
- grandson) Adam Irad)'. Write rules that determine if a list ends
- in the word `grandson'. Use this to express a rule that allows
- one to derive the relationship `((great . ?rel) ?x ?y)', where
- `?rel' is a list ending in `grandson'.) Check your rules on
- queries such as `((great grandson) ?g ?ggs)' and `(?relationship
- Adam Irad)'.
-
- ---------- Footnotes ----------
-
- (1) That a particular method of inference is legitimate is not a
-trivial assertion. One must prove that if one starts with true
-premises, only true conclusions can be derived. The method of
-inference represented by rule applications is "modus ponens", the
-familiar method of inference that says that if A is true and _A implies
-B_ is true, then we may conclude that B is true.
-
- (2) We must qualify this statement by agreeing that, in speaking of
-the "inference" accomplished by a logic program, we assume that the
-computation terminates. Unfortunately, even this qualified statement
-is false for our implementation of the query language (and also false
-for programs in Prolog and most other current logic programming
-languages) because of our use of `not' and `lisp-value'. As we will
-describe below, the `not' implemented in the query language is not
-always consistent with the `not' of mathematical logic, and
-`lisp-value' introduces additional complications. We could implement a
-language consistent with mathematical logic by simply removing `not'
-and `lisp-value' from the language and agreeing to write programs using
-only simple queries, `and', and `or'. However, this would greatly
-restrict the expressive power of the language. One of the major
-concerns of research in logic programming is to find ways to achieve
-more consistency with mathematical logic without unduly sacrificing
-expressive power.
-
- (3) This is not a problem of the logic but one of the procedural
-interpretation of the logic provided by our interpreter. We could
-write an interpreter that would not fall into a loop here. For
-example, we could enumerate all the proofs derivable from our
-assertions and our rules in a breadth-first rather than a depth-first
-order. However, such a system makes it more difficult to take
-advantage of the order of deductions in our programs. One attempt to
-build sophisticated control into such a program is described in deKleer
-et al. 1977. Another technique, which does not lead to such serious
-control problems, is to put in special knowledge, such as detectors for
-particular kinds of loops (*Note Exercise 4-67::). However, there can
-be no general scheme for reliably preventing a system from going down
-infinite paths in performing deductions. Imagine a diabolical rule of
-the form "To show P(x) is true, show that P(f(x)) is true," for some
-suitably chosen function f.
-
- (4) Consider the query `(not (baseball-fan (Bitdiddle Ben)))'. The
-system finds that `(baseball-fan (Bitdiddle Ben))' is not in the data
-base, so the empty frame does not satisfy the pattern and is not
-filtered out of the initial stream of frames. The result of the query
-is thus the empty frame, which is used to instantiate the input query
-to produce `(not (baseball-fan (Bitdiddle Ben)))'.
-
- (5) A discussion and justification of this treatment of `not' can be
-found in the article by Clark (1978).
-
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-
-4.4.4 Implementing the Query System
------------------------------------
-
-Section *Note 4-4-2:: described how the query system works. Now we fill
-in the details by presenting a complete implementation of the system.
-
-* Menu:
-
-* 4-4-4-1:: The Driver Loop and Instantiation
-* 4-4-4-2:: The Evaluator
-* 4-4-4-3:: Finding Assertions by Pattern Matching
-* 4-4-4-4:: Rules and Unification
-* 4-4-4-5:: Maintaining the Data Base
-* 4-4-4-6:: Stream Operations
-* 4-4-4-7:: Query Syntax Procedures
-* 4-4-4-8:: Frames and Bindings
-
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-
-4.4.4.1 The Driver Loop and Instantiation
-.........................................
-
-The driver loop for the query system repeatedly reads input
-expressions. If the expression is a rule or assertion to be added to
-the data base, then the information is added. Otherwise the expression
-is assumed to be a query. The driver passes this query to the
-evaluator `qeval' together with an initial frame stream consisting of a
-single empty frame. The result of the evaluation is a stream of frames
-generated by satisfying the query with variable values found in the
-data base. These frames are used to form a new stream consisting of
-copies of the original query in which the variables are instantiated
-with values supplied by the stream of frames, and this final stream is
-printed at the terminal:
-
- (define input-prompt ";;; Query input:")
- (define output-prompt ";;; Query results:")
-
- (define (query-driver-loop)
- (prompt-for-input input-prompt)
- (let ((q (query-syntax-process (read))))
- (cond ((assertion-to-be-added? q)
- (add-rule-or-assertion! (add-assertion-body q))
- (newline)
- (display "Assertion added to data base.")
- (query-driver-loop))
- (else
- (newline)
- (display output-prompt)
- (display-stream
- (stream-map
- (lambda (frame)
- (instantiate q
- frame
- (lambda (v f)
- (contract-question-mark v))))
- (qeval q (singleton-stream '()))))
- (query-driver-loop)))))
-
- Here, as in the other evaluators in this chapter, we use an abstract
-syntax for the expressions of the query language. The implementation
-of the expression syntax, including the predicate
-`assertion-to-be-added?' and the selector `add-assertion-body', is
-given in section *Note 4-4-4-7::. `Add-rule-or-assertion!' is defined
-in section *Note 4-4-4-5::.
-
- Before doing any processing on an input expression, the driver loop
-transforms it syntactically into a form that makes the processing more
-efficient. This involves changing the representation of pattern
-variables. When the query is instantiated, any variables that remain
-unbound are transformed back to the input representation before being
-printed. These transformations are performed by the two procedures
-`query-syntax-process' and `contract-question-mark' (section *Note
-4-4-4-7::).
-
- To instantiate an expression, we copy it, replacing any variables in
-the expression by their values in a given frame. The values are
-themselves instantiated, since they could contain variables (for
-example, if `?x' in `exp' is bound to `?y' as the result of unification
-and `?y' is in turn bound to 5). The action to take if a variable
-cannot be instantiated is given by a procedural argument to
-`instantiate'.
-
- (define (instantiate exp frame unbound-var-handler)
- (define (copy exp)
- (cond ((var? exp)
- (let ((binding (binding-in-frame exp frame)))
- (if binding
- (copy (binding-value binding))
- (unbound-var-handler exp frame))))
- ((pair? exp)
- (cons (copy (car exp)) (copy (cdr exp))))
- (else exp)))
- (copy exp))
-
- The procedures that manipulate bindings are defined in section *Note
-4-4-4-8::.
-
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-File: sicp.info, Node: 4-4-4-2, Next: 4-4-4-3, Prev: 4-4-4-1, Up: 4-4-4
-
-4.4.4.2 The Evaluator
-.....................
-
-The `qeval' procedure, called by the `query-driver-loop', is the basic
-evaluator of the query system. It takes as inputs a query and a stream
-of frames, and it returns a stream of extended frames. It identifies
-special forms by a data-directed dispatch using `get' and `put', just
-as we did in implementing generic operations in *Note Chapter 2::. Any
-query that is not identified as a special form is assumed to be a
-simple query, to be processed by `simple-query'.
-
- (define (qeval query frame-stream)
- (let ((qproc (get (type query) 'qeval)))
- (if qproc
- (qproc (contents query) frame-stream)
- (simple-query query frame-stream))))
-
- `Type' and `contents', defined in section *Note 4-4-4-7::, implement
-the abstract syntax of the special forms.
-
-Simple queries
-..............
-
-The `simple-query' procedure handles simple queries. It takes as
-arguments a simple query (a pattern) together with a stream of frames,
-and it returns the stream formed by extending each frame by all
-data-base matches of the query.
-
- (define (simple-query query-pattern frame-stream)
- (stream-flatmap
- (lambda (frame)
- (stream-append-delayed
- (find-assertions query-pattern frame)
- (delay (apply-rules query-pattern frame))))
- frame-stream))
-
- For each frame in the input stream, we use `find-assertions' (section
-*Note 4-4-4-3::) to match the pattern against all assertions in the
-data base, producing a stream of extended frames, and we use
-`apply-rules' (section *Note 4-4-4-4::) to apply all possible rules,
-producing another stream of extended frames. These two streams are
-combined (using `stream-append-delayed', section *Note 4-4-4-6::) to
-make a stream of all the ways that the given pattern can be satisfied
-consistent with the original frame (see *Note Exercise 4-71::). The
-streams for the individual input frames are combined using
-`stream-flatmap' (section *Note 4-4-4-6::) to form one large stream of
-all the ways that any of the frames in the original input stream can be
-extended to produce a match with the given pattern.
-
-Compound queries
-................
-
-`And' queries are handled as illustrated in *Note Figure 4-5:: by the
-`conjoin' procedure. `Conjoin' takes as inputs the conjuncts and the
-frame stream and returns the stream of extended frames. First,
-`conjoin' processes the stream of frames to find the stream of all
-possible frame extensions that satisfy the first query in the
-conjunction. Then, using this as the new frame stream, it recursively
-applies `conjoin' to the rest of the queries.
-
- (define (conjoin conjuncts frame-stream)
- (if (empty-conjunction? conjuncts)
- frame-stream
- (conjoin (rest-conjuncts conjuncts)
- (qeval (first-conjunct conjuncts)
- frame-stream))))
-
- The expression
-
- (put 'and 'qeval conjoin)
-
-sets up `qeval' to dispatch to `conjoin' when an `and' form is
-encountered.
-
- `Or' queries are handled similarly, as shown in *Note Figure 4-6::.
-The output streams for the various disjuncts of the `or' are computed
-separately and merged using the `interleave-delayed' procedure from
-section *Note 4-4-4-6::. (See *Note Exercise 4-71:: and *Note Exercise
-4-72::.)
-
- (define (disjoin disjuncts frame-stream)
- (if (empty-disjunction? disjuncts)
- the-empty-stream
- (interleave-delayed
- (qeval (first-disjunct disjuncts) frame-stream)
- (delay (disjoin (rest-disjuncts disjuncts)
- frame-stream)))))
-
- (put 'or 'qeval disjoin)
-
- The predicates and selectors for the syntax of conjuncts and
-disjuncts are given in section *Note 4-4-4-7::.
-
-Filters
-.......
-
-`Not' is handled by the method outlined in section *Note 4-4-2::. We
-attempt to extend each frame in the input stream to satisfy the query
-being negated, and we include a given frame in the output stream only
-if it cannot be extended.
-
- (define (negate operands frame-stream)
- (stream-flatmap
- (lambda (frame)
- (if (stream-null? (qeval (negated-query operands)
- (singleton-stream frame)))
- (singleton-stream frame)
- the-empty-stream))
- frame-stream))
-
- (put 'not 'qeval negate)
-
- `Lisp-value' is a filter similar to `not'. Each frame in the stream
-is used to instantiate the variables in the pattern, the indicated
-predicate is applied, and the frames for which the predicate returns
-false are filtered out of the input stream. An error results if there
-are unbound pattern variables.
-
- (define (lisp-value call frame-stream)
- (stream-flatmap
- (lambda (frame)
- (if (execute
- (instantiate
- call
- frame
- (lambda (v f)
- (error "Unknown pat var -- LISP-VALUE" v))))
- (singleton-stream frame)
- the-empty-stream))
- frame-stream))
-
- (put 'lisp-value 'qeval lisp-value)
-
- `Execute', which applies the predicate to the arguments, must `eval'
-the predicate expression to get the procedure to apply. However, it
-must not evaluate the arguments, since they are already the actual
-arguments, not expressions whose evaluation (in Lisp) will produce the
-arguments. Note that `execute' is implemented using `eval' and `apply'
-from the underlying Lisp system.
-
- (define (execute exp)
- (apply (eval (predicate exp) user-initial-environment)
- (args exp)))
-
- The `always-true' special form provides for a query that is always
-satisfied. It ignores its contents (normally empty) and simply passes
-through all the frames in the input stream. `Always-true' is used by
-the `rule-body' selector (section *Note 4-4-4-7::) to provide bodies
-for rules that were defined without bodies (that is, rules whose
-conclusions are always satisfied).
-
- (define (always-true ignore frame-stream) frame-stream)
-
- (put 'always-true 'qeval always-true)
-
- The selectors that define the syntax of `not' and `lisp-value' are
-given in section *Note 4-4-4-7::.
-
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-File: sicp.info, Node: 4-4-4-3, Next: 4-4-4-4, Prev: 4-4-4-2, Up: 4-4-4
-
-4.4.4.3 Finding Assertions by Pattern Matching
-..............................................
-
-`Find-assertions', called by `simple-query' (section *Note 4-4-4-2::),
-takes as input a pattern and a frame. It returns a stream of frames,
-each extending the given one by a data-base match of the given pattern.
-It uses `fetch-assertions' (section *Note 4-4-4-5::) to get a stream
-of all the assertions in the data base that should be checked for a
-match against the pattern and the frame. The reason for
-`fetch-assertions' here is that we can often apply simple tests that
-will eliminate many of the entries in the data base from the pool of
-candidates for a successful match. The system would still work if we
-eliminated `fetch-assertions' and simply checked a stream of all
-assertions in the data base, but the computation would be less efficient
-because we would need to make many more calls to the matcher.
-
- (define (find-assertions pattern frame)
- (stream-flatmap (lambda (datum)
- (check-an-assertion datum pattern frame))
- (fetch-assertions pattern frame)))
-
- `Check-an-assertion' takes as arguments a pattern, a data object
-(assertion), and a frame and returns either a one-element stream
-containing the extended frame or `the-empty-stream' if the match fails.
-
- (define (check-an-assertion assertion query-pat query-frame)
- (let ((match-result
- (pattern-match query-pat assertion query-frame)))
- (if (eq? match-result 'failed)
- the-empty-stream
- (singleton-stream match-result))))
-
- The basic pattern matcher returns either the symbol `failed' or an
-extension of the given frame. The basic idea of the matcher is to
-check the pattern against the data, element by element, accumulating
-bindings for the pattern variables. If the pattern and the data object
-are the same, the match succeeds and we return the frame of bindings
-accumulated so far. Otherwise, if the pattern is a variable we extend
-the current frame by binding the variable to the data, so long as this
-is consistent with the bindings already in the frame. If the pattern
-and the data are both pairs, we (recursively) match the `car' of the
-pattern against the `car' of the data to produce a frame; in this frame
-we then match the `cdr' of the pattern against the `cdr' of the data.
-If none of these cases are applicable, the match fails and we return
-the symbol `failed'.
-
- (define (pattern-match pat dat frame)
- (cond ((eq? frame 'failed) 'failed)
- ((equal? pat dat) frame)
- ((var? pat) (extend-if-consistent pat dat frame))
- ((and (pair? pat) (pair? dat))
- (pattern-match (cdr pat)
- (cdr dat)
- (pattern-match (car pat)
- (car dat)
- frame)))
- (else 'failed)))
-
- Here is the procedure that extends a frame by adding a new binding,
-if this is consistent with the bindings already in the frame:
-
- (define (extend-if-consistent var dat frame)
- (let ((binding (binding-in-frame var frame)))
- (if binding
- (pattern-match (binding-value binding) dat frame)
- (extend var dat frame))))
-
- If there is no binding for the variable in the frame, we simply add
-the binding of the variable to the data. Otherwise we match, in the
-frame, the data against the value of the variable in the frame. If the
-stored value contains only constants, as it must if it was stored
-during pattern matching by `extend-if-consistent', then the match
-simply tests whether the stored and new values are the same. If so, it
-returns the unmodified frame; if not, it returns a failure indication.
-The stored value may, however, contain pattern variables if it was
-stored during unification (see section *Note 4-4-4-4::). The recursive
-match of the stored pattern against the new data will add or check
-bindings for the variables in this pattern. For example, suppose we
-have a frame in which `?x' is bound to `(f ?y)' and `?y' is unbound,
-and we wish to augment this frame by a binding of `?x' to `(f b)'. We
-look up `?x' and find that it is bound to `(f ?y)'. This leads us to
-match `(f ?y)' against the proposed new value `(f b)' in the same
-frame. Eventually this match extends the frame by adding a binding of
-`?y' to `b'. `?X' remains bound to `(f ?y)'. We never modify a stored
-binding and we never store more than one binding for a given variable.
-
- The procedures used by `extend-if-consistent' to manipulate bindings
-are defined in section *Note 4-4-4-8::.
-
-Patterns with dotted tails
-..........................
-
-If a pattern contains a dot followed by a pattern variable, the pattern
-variable matches the rest of the data list (rather than the next
-element of the data list), just as one would expect with the
-dotted-tail notation described in *Note Exercise 2-20::. Although the
-pattern matcher we have just implemented doesn't look for dots, it does
-behave as we want. This is because the Lisp `read' primitive, which is
-used by `query-driver-loop' to read the query and represent it as a
-list structure, treats dots in a special way.
-
- When `read' sees a dot, instead of making the next item be the next
-element of a list (the `car' of a `cons' whose `cdr' will be the rest
-of the list) it makes the next item be the `cdr' of the list structure.
-For example, the list structure produced by `read' for the pattern
-`(computer ?type)' could be constructed by evaluating the expression
-`(cons 'computer (cons '?type '()))', and that for `(computer . ?type)'
-could be constructed by evaluating the expression `(cons 'computer
-'?type)'.
-
- Thus, as `pattern-match' recursively compares `car's and `cdr's of a
-data list and a pattern that had a dot, it eventually matches the
-variable after the dot (which is a `cdr' of the pattern) against a
-sublist of the data list, binding the variable to that list. For
-example, matching the pattern `(computer . ?type)' against `(computer
-programmer trainee)' will match `?type' against the list `(programmer
-trainee)'.
-
-\1f
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-
-4.4.4.4 Rules and Unification
-.............................
-
-`Apply-rules' is the rule analog of `find-assertions' (section *Note
-4-4-4-3::). It takes as input a pattern and a frame, and it forms a
-stream of extension frames by applying rules from the data base.
-`Stream-flatmap' maps `apply-a-rule' down the stream of possibly
-applicable rules (selected by `fetch-rules', section *Note 4-4-4-5::)
-and combines the resulting streams of frames.
-
- (define (apply-rules pattern frame)
- (stream-flatmap (lambda (rule)
- (apply-a-rule rule pattern frame))
- (fetch-rules pattern frame)))
-
- `Apply-a-rule' applies rules using the method outlined in section
-*Note 4-4-2::. It first augments its argument frame by unifying the
-rule conclusion with the pattern in the given frame. If this succeeds,
-it evaluates the rule body in this new frame.
-
- Before any of this happens, however, the program renames all the
-variables in the rule with unique new names. The reason for this is to
-prevent the variables for different rule applications from becoming
-confused with each other. For instance, if two rules both use a
-variable named `?x', then each one may add a binding for `?x' to the
-frame when it is applied. These two `?x''s have nothing to do with
-each other, and we should not be fooled into thinking that the two
-bindings must be consistent. Rather than rename variables, we could
-devise a more clever environment structure; however, the renaming
-approach we have chosen here is the most straightforward, even if not
-the most efficient. (See *Note Exercise 4-79::.) Here is the
-`apply-a-rule' procedure:
-
- (define (apply-a-rule rule query-pattern query-frame)
- (let ((clean-rule (rename-variables-in rule)))
- (let ((unify-result
- (unify-match query-pattern
- (conclusion clean-rule)
- query-frame)))
- (if (eq? unify-result 'failed)
- the-empty-stream
- (qeval (rule-body clean-rule)
- (singleton-stream unify-result))))))
-
- The selectors `rule-body' and `conclusion' that extract parts of a
-rule are defined in section *Note 4-4-4-7::.
-
- We generate unique variable names by associating a unique identifier
-(such as a number) with each rule application and combining this
-identifier with the original variable names. For example, if the
-rule-application identifier is 7, we might change each `?x' in the rule
-to `?x-7' and each `?y' in the rule to `?y-7'. (`Make-new-variable' and
-`new-rule-application-id' are included with the syntax procedures in
-section *Note 4-4-4-7::.)
-
- (define (rename-variables-in rule)
- (let ((rule-application-id (new-rule-application-id)))
- (define (tree-walk exp)
- (cond ((var? exp)
- (make-new-variable exp rule-application-id))
- ((pair? exp)
- (cons (tree-walk (car exp))
- (tree-walk (cdr exp))))
- (else exp)))
- (tree-walk rule)))
-
- The unification algorithm is implemented as a procedure that takes
-as inputs two patterns and a frame and returns either the extended
-frame or the symbol `failed'. The unifier is like the pattern matcher
-except that it is symmetrical--variables are allowed on both sides of
-the match. `Unify-match' is basically the same as `pattern-match',
-except that there is extra code (marked "`***'" below) to handle the
-case where the object on the right side of the match is a variable.
-
- (define (unify-match p1 p2 frame)
- (cond ((eq? frame 'failed) 'failed)
- ((equal? p1 p2) frame)
- ((var? p1) (extend-if-possible p1 p2 frame))
- ((var? p2) (extend-if-possible p2 p1 frame)) ; ***
- ((and (pair? p1) (pair? p2))
- (unify-match (cdr p1)
- (cdr p2)
- (unify-match (car p1)
- (car p2)
- frame)))
- (else 'failed)))
-
- In unification, as in one-sided pattern matching, we want to accept
-a proposed extension of the frame only if it is consistent with
-existing bindings. The procedure `extend-if-possible' used in
-unification is the same as the `extend-if-consistent' used in pattern
-matching except for two special checks, marked "`***'" in the program
-below. In the first case, if the variable we are trying to match is
-not bound, but the value we are trying to match it with is itself a
-(different) variable, it is necessary to check to see if the value is
-bound, and if so, to match its value. If both parties to the match are
-unbound, we may bind either to the other.
-
- The second check deals with attempts to bind a variable to a pattern
-that includes that variable. Such a situation can occur whenever a
-variable is repeated in both patterns. Consider, for example, unifying
-the two patterns `(?x ?x)' and `(?y <EXPRESSION INVOLVING `?Y'>)' in a
-frame where both `?x' and `?y' are unbound. First `?x' is matched
-against `?y', making a binding of `?x' to `?y'. Next, the same `?x' is
-matched against the given expression involving `?y'. Since `?x' is
-already bound to `?y', this results in matching `?y' against the
-expression. If we think of the unifier as finding a set of values for
-the pattern variables that make the patterns the same, then these
-patterns imply instructions to find a `?y' such that `?y' is equal to
-the expression involving `?y'. There is no general method for solving
-such equations, so we reject such bindings; these cases are recognized
-by the predicate `depends-on?'.(1) On the other hand, we do not want
-to reject attempts to bind a variable to itself. For example, consider
-unifying `(?x ?x)' and `(?y ?y)'. The second attempt to bind `?x' to
-`?y' matches `?y' (the stored value of `?x') against `?y' (the new
-value of `?x'). This is taken care of by the `equal?' clause of
-`unify-match'.
-
- (define (extend-if-possible var val frame)
- (let ((binding (binding-in-frame var frame)))
- (cond (binding
- (unify-match
- (binding-value binding) val frame))
- ((var? val) ; ***
- (let ((binding (binding-in-frame val frame)))
- (if binding
- (unify-match
- var (binding-value binding) frame)
- (extend var val frame))))
- ((depends-on? val var frame) ; ***
- 'failed)
- (else (extend var val frame)))))
-
- `Depends-on?' is a predicate that tests whether an expression
-proposed to be the value of a pattern variable depends on the variable.
-This must be done relative to the current frame because the expression
-may contain occurrences of a variable that already has a value that
-depends on our test variable. The structure of `depends-on?' is a
-simple recursive tree walk in which we substitute for the values of
-variables whenever necessary.
-
- (define (depends-on? exp var frame)
- (define (tree-walk e)
- (cond ((var? e)
- (if (equal? var e)
- true
- (let ((b (binding-in-frame e frame)))
- (if b
- (tree-walk (binding-value b))
- false))))
- ((pair? e)
- (or (tree-walk (car e))
- (tree-walk (cdr e))))
- (else false)))
- (tree-walk exp))
-
- ---------- Footnotes ----------
-
- (1) In general, unifying `?y' with an expression involving `?y'
-would require our being able to find a fixed point of the equation `?y'
-= <EXPRESSION INVOLVING `?Y'>. It is sometimes possible to
-syntactically form an expression that appears to be the solution. For
-example, `?y' = `(f ?y)' seems to have the fixed point `(f (f (f ...
-)))', which we can produce by beginning with the expression `(f ?y)'
-and repeatedly substituting `(f ?y)' for `?y'. Unfortunately, not
-every such equation has a meaningful fixed point. The issues that
-arise here are similar to the issues of manipulating infinite series in
-mathematics. For example, we know that 2 is the solution to the
-equation y = 1 + y/2. Beginning with the expression 1 + y/2 and
-repeatedly substituting 1 + y/2 for y gives
-
- 2 = y = 1 + y/2 = 1 + (1 + y/2)/2 = 1 + 1/2 + y/4 = ...
-
-which leads to
-
- 2 = 1 + 1/2 + 1/4 + 1/8 + ...
-
-However, if we try the same manipulation beginning with the observation
-that - 1 is the solution to the equation y = 1 + 2y, we obtain
-
- -1 = y = 1 + 2y = 1 + 2(1 + 2y) = 1 + 2 + 4y = ...
-
-which leads to
-
- -1 = 1 + 2 + 4 + 8 + ...
-
-Although the formal manipulations used in deriving these two equations
-are identical, the first result is a valid assertion about infinite
-series but the second is not. Similarly, for our unification results,
-reasoning with an arbitrary syntactically constructed expression may
-lead to errors.
-
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-
-4.4.4.5 Maintaining the Data Base
-.................................
-
-One important problem in designing logic programming languages is that
-of arranging things so that as few irrelevant data-base entries as
-possible will be examined in checking a given pattern. In our system,
-in addition to storing all assertions in one big stream, we store all
-assertions whose `car's are constant symbols in separate streams, in a
-table indexed by the symbol. To fetch an assertion that may match a
-pattern, we first check to see if the `car' of the pattern is a
-constant symbol. If so, we return (to be tested using the matcher) all
-the stored assertions that have the same `car'. If the pattern's `car'
-is not a constant symbol, we return all the stored assertions.
-Cleverer methods could also take advantage of information in the frame,
-or try also to optimize the case where the `car' of the pattern is not
-a constant symbol. We avoid building our criteria for indexing (using
-the `car', handling only the case of constant symbols) into the program;
-instead we call on predicates and selectors that embody our criteria.
-
- (define THE-ASSERTIONS the-empty-stream)
-
- (define (fetch-assertions pattern frame)
- (if (use-index? pattern)
- (get-indexed-assertions pattern)
- (get-all-assertions)))
-
- (define (get-all-assertions) THE-ASSERTIONS)
-
- (define (get-indexed-assertions pattern)
- (get-stream (index-key-of pattern) 'assertion-stream))
-
- `Get-stream' looks up a stream in the table and returns an empty
-stream if nothing is stored there.
-
- (define (get-stream key1 key2)
- (let ((s (get key1 key2)))
- (if s s the-empty-stream)))
-
- Rules are stored similarly, using the `car' of the rule conclusion.
-Rule conclusions are arbitrary patterns, however, so they differ from
-assertions in that they can contain variables. A pattern whose `car'
-is a constant symbol can match rules whose conclusions start with a
-variable as well as rules whose conclusions have the same `car'. Thus,
-when fetching rules that might match a pattern whose `car' is a
-constant symbol we fetch all rules whose conclusions start with a
-variable as well as those whose conclusions have the same `car' as the
-pattern. For this purpose we store all rules whose conclusions start
-with a variable in a separate stream in our table, indexed by the
-symbol `?'.
-
- (define THE-RULES the-empty-stream)
-
- (define (fetch-rules pattern frame)
- (if (use-index? pattern)
- (get-indexed-rules pattern)
- (get-all-rules)))
-
- (define (get-all-rules) THE-RULES)
-
- (define (get-indexed-rules pattern)
- (stream-append
- (get-stream (index-key-of pattern) 'rule-stream)
- (get-stream '? 'rule-stream)))
-
- `Add-rule-or-assertion!' is used by `query-driver-loop' to add
-assertions and rules to the data base. Each item is stored in the
-index, if appropriate, and in a stream of all assertions or rules in
-the data base.
-
- (define (add-rule-or-assertion! assertion)
- (if (rule? assertion)
- (add-rule! assertion)
- (add-assertion! assertion)))
-
- (define (add-assertion! assertion)
- (store-assertion-in-index assertion)
- (let ((old-assertions THE-ASSERTIONS))
- (set! THE-ASSERTIONS
- (cons-stream assertion old-assertions))
- 'ok))
-
- (define (add-rule! rule)
- (store-rule-in-index rule)
- (let ((old-rules THE-RULES))
- (set! THE-RULES (cons-stream rule old-rules))
- 'ok))
-
- To actually store an assertion or a rule, we check to see if it can
-be indexed. If so, we store it in the appropriate stream.
-
- (define (store-assertion-in-index assertion)
- (if (indexable? assertion)
- (let ((key (index-key-of assertion)))
- (let ((current-assertion-stream
- (get-stream key 'assertion-stream)))
- (put key
- 'assertion-stream
- (cons-stream assertion
- current-assertion-stream))))))
-
- (define (store-rule-in-index rule)
- (let ((pattern (conclusion rule)))
- (if (indexable? pattern)
- (let ((key (index-key-of pattern)))
- (let ((current-rule-stream
- (get-stream key 'rule-stream)))
- (put key
- 'rule-stream
- (cons-stream rule
- current-rule-stream)))))))
-
- The following procedures define how the data-base index is used. A
-pattern (an assertion or a rule conclusion) will be stored in the table
-if it starts with a variable or a constant symbol.
-
- (define (indexable? pat)
- (or (constant-symbol? (car pat))
- (var? (car pat))))
-
- The key under which a pattern is stored in the table is either `?'
-(if it starts with a variable) or the constant symbol with which it
-starts.
-
- (define (index-key-of pat)
- (let ((key (car pat)))
- (if (var? key) '? key)))
-
- The index will be used to retrieve items that might match a pattern
-if the pattern starts with a constant symbol.
-
- (define (use-index? pat)
- (constant-symbol? (car pat)))
-
- *Exercise 4.70:* What is the purpose of the `let' bindings in the
- procedures `add-assertion!' and `add-rule!' ? What would be wrong
- with the following implementation of `add-assertion!' ? Hint:
- Recall the definition of the infinite stream of ones in section
- *Note 3-5-2::: `(define ones (cons-stream 1 ones))'.
-
- (define (add-assertion! assertion)
- (store-assertion-in-index assertion)
- (set! THE-ASSERTIONS
- (cons-stream assertion THE-ASSERTIONS))
- 'ok)
-
-\1f
-File: sicp.info, Node: 4-4-4-6, Next: 4-4-4-7, Prev: 4-4-4-5, Up: 4-4-4
-
-4.4.4.6 Stream Operations
-.........................
-
-The query system uses a few stream operations that were not presented in
-*Note Chapter 3::.
-
- `Stream-append-delayed' and `interleave-delayed' are just like
-`stream-append' and `interleave' (section *Note 3-5-3::), except that
-they take a delayed argument (like the `integral' procedure in section
-*Note 3-5-4::). This postpones looping in some cases (see *Note
-Exercise 4-71::).
-
- (define (stream-append-delayed s1 delayed-s2)
- (if (stream-null? s1)
- (force delayed-s2)
- (cons-stream
- (stream-car s1)
- (stream-append-delayed (stream-cdr s1) delayed-s2))))
-
- (define (interleave-delayed s1 delayed-s2)
- (if (stream-null? s1)
- (force delayed-s2)
- (cons-stream
- (stream-car s1)
- (interleave-delayed (force delayed-s2)
- (delay (stream-cdr s1))))))
-
- `Stream-flatmap', which is used throughout the query evaluator to
-map a procedure over a stream of frames and combine the resulting
-streams of frames, is the stream analog of the `flatmap' procedure
-introduced for ordinary lists in section *Note 2-2-3::. Unlike
-ordinary `flatmap', however, we accumulate the streams with an
-interleaving process, rather than simply appending them (see *Note
-Exercise 4-72:: and *Note Exercise 4-73::).
-
- (define (stream-flatmap proc s)
- (flatten-stream (stream-map proc s)))
-
- (define (flatten-stream stream)
- (if (stream-null? stream)
- the-empty-stream
- (interleave-delayed
- (stream-car stream)
- (delay (flatten-stream (stream-cdr stream))))))
-
- The evaluator also uses the following simple procedure to generate a
-stream consisting of a single element:
-
- (define (singleton-stream x)
- (cons-stream x the-empty-stream))
-
-\1f
-File: sicp.info, Node: 4-4-4-7, Next: 4-4-4-8, Prev: 4-4-4-6, Up: 4-4-4
-
-4.4.4.7 Query Syntax Procedures
-...............................
-
-`Type' and `contents', used by `qeval' (section *Note 4-4-4-2::),
-specify that a special form is identified by the symbol in its `car'.
-They are the same as the `type-tag' and `contents' procedures in
-section *Note 2-4-2::, except for the error message.
-
- (define (type exp)
- (if (pair? exp)
- (car exp)
- (error "Unknown expression TYPE" exp)))
-
- (define (contents exp)
- (if (pair? exp)
- (cdr exp)
- (error "Unknown expression CONTENTS" exp)))
-
- The following procedures, used by `query-driver-loop' (in section
-*Note 4-4-4-1::), specify that rules and assertions are added to the
-data base by expressions of the form `(assert! <RULE-OR-ASSERTION>)':
-
- (define (assertion-to-be-added? exp)
- (eq? (type exp) 'assert!))
-
- (define (add-assertion-body exp)
- (car (contents exp)))
-
- Here are the syntax definitions for the `and', `or', `not', and
-`lisp-value' special forms (section *Note 4-4-4-2::):
-
- (define (empty-conjunction? exps) (null? exps))
- (define (first-conjunct exps) (car exps))
- (define (rest-conjuncts exps) (cdr exps))
-
- (define (empty-disjunction? exps) (null? exps))
- (define (first-disjunct exps) (car exps))
- (define (rest-disjuncts exps) (cdr exps))
-
- (define (negated-query exps) (car exps))
-
- (define (predicate exps) (car exps))
- (define (args exps) (cdr exps))
-
- The following three procedures define the syntax of rules:
-
- (define (rule? statement)
- (tagged-list? statement 'rule))
-
- (define (conclusion rule) (cadr rule))
-
- (define (rule-body rule)
- (if (null? (cddr rule))
- '(always-true)
- (caddr rule)))
-
- `Query-driver-loop' (section *Note 4-4-4-1::) calls
-`query-syntax-process' to transform pattern variables in the expression,
-which have the form `?symbol', into the internal format `(? symbol)'.
-That is to say, a pattern such as `(job ?x ?y)' is actually represented
-internally by the system as `(job (? x) (? y))'. This increases the
-efficiency of query processing, since it means that the system can
-check to see if an expression is a pattern variable by checking whether
-the `car' of the expression is the symbol `?', rather than having to
-extract characters from the symbol. The syntax transformation is
-accomplished by the following procedure:(1)
-
- (define (query-syntax-process exp)
- (map-over-symbols expand-question-mark exp))
-
- (define (map-over-symbols proc exp)
- (cond ((pair? exp)
- (cons (map-over-symbols proc (car exp))
- (map-over-symbols proc (cdr exp))))
- ((symbol? exp) (proc exp))
- (else exp)))
-
- (define (expand-question-mark symbol)
- (let ((chars (symbol->string symbol)))
- (if (string=? (substring chars 0 1) "?")
- (list '?
- (string->symbol
- (substring chars 1 (string-length chars))))
- symbol)))
-
- Once the variables are transformed in this way, the variables in a
-pattern are lists starting with `?', and the constant symbols (which
-need to be recognized for data-base indexing, section *Note 4-4-4-5::)
-are just the symbols.
-
- (define (var? exp)
- (tagged-list? exp '?))
-
- (define (constant-symbol? exp) (symbol? exp))
-
- Unique variables are constructed during rule application (in section
-*Note 4-4-4-4::) by means of the following procedures. The unique
-identifier for a rule application is a number, which is incremented
-each time a rule is applied.
-
- (define rule-counter 0)
-
- (define (new-rule-application-id)
- (set! rule-counter (+ 1 rule-counter))
- rule-counter)
-
- (define (make-new-variable var rule-application-id)
- (cons '? (cons rule-application-id (cdr var))))
-
- When `query-driver-loop' instantiates the query to print the answer,
-it converts any unbound pattern variables back to the right form for
-printing, using
-
- (define (contract-question-mark variable)
- (string->symbol
- (string-append "?"
- (if (number? (cadr variable))
- (string-append (symbol->string (caddr variable))
- "-"
- (number->string (cadr variable)))
- (symbol->string (cadr variable))))))
-
- ---------- Footnotes ----------
-
- (1) Most Lisp systems give the user the ability to modify the
-ordinary `read' procedure to perform such transformations by defining "reader
-macro characters". Quoted expressions are already handled in this way:
-The reader automatically translates `'expression' into `(quote
-expression)' before the evaluator sees it. We could arrange for
-`?expression' to be transformed into `(? expression)' in the same way;
-however, for the sake of clarity we have included the transformation
-procedure here explicitly.
-
- `Expand-question-mark' and `contract-question-mark' use several
-procedures with `string' in their names. These are Scheme primitives.
-
-\1f
-File: sicp.info, Node: 4-4-4-8, Prev: 4-4-4-7, Up: 4-4-4
-
-4.4.4.8 Frames and Bindings
-...........................
-
-Frames are represented as lists of bindings, which are variable-value
-pairs:
-
- (define (make-binding variable value)
- (cons variable value))
-
- (define (binding-variable binding)
- (car binding))
-
- (define (binding-value binding)
- (cdr binding))
-
- (define (binding-in-frame variable frame)
- (assoc variable frame))
-
- (define (extend variable value frame)
- (cons (make-binding variable value) frame))
-
- *Exercise 4.71:* Louis Reasoner wonders why the `simple-query' and
- `disjoin' procedures (section *Note 4-4-4-2::) are implemented
- using explicit `delay' operations, rather than being defined as
- follows:
-
- (define (simple-query query-pattern frame-stream)
- (stream-flatmap
- (lambda (frame)
- (stream-append (find-assertions query-pattern frame)
- (apply-rules query-pattern frame)))
- frame-stream))
-
- (define (disjoin disjuncts frame-stream)
- (if (empty-disjunction? disjuncts)
- the-empty-stream
- (interleave
- (qeval (first-disjunct disjuncts) frame-stream)
- (disjoin (rest-disjuncts disjuncts) frame-stream))))
-
- Can you give examples of queries where these simpler definitions
- would lead to undesirable behavior?
-
- *Exercise 4.72:* Why do `disjoin' and `stream-flatmap' interleave
- the streams rather than simply append them? Give examples that
- illustrate why interleaving works better. (Hint: Why did we use
- `interleave' in section *Note 3-5-3::?)
-
- *Exercise 4.73:* Why does `flatten-stream' use `delay' explicitly?
- What would be wrong with defining it as follows:
-
- (define (flatten-stream stream)
- (if (stream-null? stream)
- the-empty-stream
- (interleave
- (stream-car stream)
- (flatten-stream (stream-cdr stream)))))
-
- *Exercise 4.74:* Alyssa P. Hacker proposes to use a simpler
- version of `stream-flatmap' in `negate', `lisp-value', and
- `find-assertions'. She observes that the procedure that is mapped
- over the frame stream in these cases always produces either the
- empty stream or a singleton stream, so no interleaving is needed
- when combining these streams.
-
- a. Fill in the missing expressions in Alyssa's program.
-
- (define (simple-stream-flatmap proc s)
- (simple-flatten (stream-map proc s)))
-
- (define (simple-flatten stream)
- (stream-map <??>
- (stream-filter <??> stream)))
-
- b. Does the query system's behavior change if we change it in
- this way?
-
-
- *Exercise 4.75:* Implement for the query language a new special
- form called `unique'. `Unique' should succeed if there is
- precisely one item in the data base satisfying a specified query.
- For example,
-
- (unique (job ?x (computer wizard)))
-
- should print the one-item stream
-
- (unique (job (Bitdiddle Ben) (computer wizard)))
-
- since Ben is the only computer wizard, and
-
- (unique (job ?x (computer programmer)))
-
- should print the empty stream, since there is more than one
- computer programmer. Moreover,
-
- (and (job ?x ?j) (unique (job ?anyone ?j)))
-
- should list all the jobs that are filled by only one person, and
- the people who fill them.
-
- There are two parts to implementing `unique'. The first is to
- write a procedure that handles this special form, and the second
- is to make `qeval' dispatch to that procedure. The second part is
- trivial, since `qeval' does its dispatching in a data-directed
- way. If your procedure is called `uniquely-asserted', all you
- need to do is
-
- (put 'unique 'qeval uniquely-asserted)
-
- and `qeval' will dispatch to this procedure for every query whose
- `type' (`car') is the symbol `unique'.
-
- The real problem is to write the procedure `uniquely-asserted'.
- This should take as input the `contents' (`cdr') of the `unique'
- query, together with a stream of frames. For each frame in the
- stream, it should use `qeval' to find the stream of all extensions
- to the frame that satisfy the given query. Any stream that does
- not have exactly one item in it should be eliminated. The
- remaining streams should be passed back to be accumulated into one
- big stream that is the result of the `unique' query. This is
- similar to the implementation of the `not' special form.
-
- Test your implementation by forming a query that lists all people
- who supervise precisely one person.
-
- *Exercise 4.76:* Our implementation of `and' as a series
- combination of queries (*Note Figure 4-5::) is elegant, but it is
- inefficient because in processing the second query of the `and' we
- must scan the data base for each frame produced by the first
- query. If the data base has n elements, and a typical query
- produces a number of output frames proportional to n (say n/k),
- then scanning the data base for each frame produced by the first
- query will require n^2/k calls to the pattern matcher. Another
- approach would be to process the two clauses of the `and'
- separately, then look for all pairs of output frames that are
- compatible. If each query produces n/k output frames, then this
- means that we must perform n^2/k^2 compatibility checks--a factor
- of k fewer than the number of matches required in our current
- method.
-
- Devise an implementation of `and' that uses this strategy. You
- must implement a procedure that takes two frames as inputs, checks
- whether the bindings in the frames are compatible, and, if so,
- produces a frame that merges the two sets of bindings. This
- operation is similar to unification.
-
- *Exercise 4.77:* In section *Note 4-4-3:: we saw that `not' and
- `lisp-value' can cause the query language to give "wrong" answers
- if these filtering operations are applied to frames in which
- variables are unbound. Devise a way to fix this shortcoming. One
- idea is to perform the filtering in a "delayed" manner by
- appending to the frame a "promise" to filter that is fulfilled
- only when enough variables have been bound to make the operation
- possible. We could wait to perform filtering until all other
- operations have been performed. However, for efficiency's sake, we
- would like to perform filtering as soon as possible so as to cut
- down on the number of intermediate frames generated.
-
- *Exercise 4.78:* Redesign the query language as a nondeterministic
- program to be implemented using the evaluator of section *Note
- 4-3::, rather than as a stream process. In this approach, each
- query will produce a single answer (rather than the stream of all
- answers) and the user can type `try-again' to see more answers.
- You should find that much of the mechanism we built in this
- section is subsumed by nondeterministic search and backtracking.
- You will probably also find, however, that your new query language
- has subtle differences in behavior from the one implemented here.
- Can you find examples that illustrate this difference?
-
- *Exercise 4.79:* When we implemented the Lisp evaluator in section
- *Note 4-1::, we saw how to use local environments to avoid name
- conflicts between the parameters of procedures. For example, in
- evaluating
-
- (define (square x)
- (* x x))
-
- (define (sum-of-squares x y)
- (+ (square x) (square y)))
-
- (sum-of-squares 3 4)
-
- there is no confusion between the `x' in `square' and the `x' in
- `sum-of-squares', because we evaluate the body of each procedure
- in an environment that is specially constructed to contain
- bindings for the local variables. In the query system, we used a
- different strategy to avoid name conflicts in applying rules.
- Each time we apply a rule we rename the variables with new names
- that are guaranteed to be unique. The analogous strategy for the
- Lisp evaluator would be to do away with local environments and
- simply rename the variables in the body of a procedure each time
- we apply the procedure.
-
- Implement for the query language a rule-application method that
- uses environments rather than renaming. See if you can build on
- your environment structure to create constructs in the query
- language for dealing with large systems, such as the rule analog
- of block-structured procedures. Can you relate any of this to the
- problem of making deductions in a context (e.g., "If I supposed
- that P were true, then I would be able to deduce A and B.") as a
- method of problem solving? (This problem is open-ended. A good
- answer is probably worth a Ph.D.)
-
-\1f
-File: sicp.info, Node: Chapter 5, Next: References, Prev: Chapter 4, Up: Top
-
-5 Computing with Register Machines
-**********************************
-
- My aim is to show that the heavenly machine is not a kind of
- divine, live being, but a kind of clockwork (and he who believes
- that a clock has soul attributes the maker's glory to the work),
- insofar as nearly all the manifold motions are caused by a most
- simple and material force, just as all motions of the clock are
- caused by a single weight.
-
- Johannes Kepler (letter to Herwart von Hohenburg, 1605)
-
- We began this book by studying processes and by describing processes
-in terms of procedures written in Lisp. To explain the meanings of
-these procedures, we used a succession of models of evaluation: the
-substitution model of *Note Chapter 1::, the environment model of *Note
-Chapter 3::, and the metacircular evaluator of *Note Chapter 4::. Our
-examination of the metacircular evaluator, in particular, dispelled
-much of the mystery of how Lisp-like languages are interpreted. But
-even the metacircular evaluator leaves important questions unanswered,
-because it fails to elucidate the mechanisms of control in a Lisp
-system. For instance, the evaluator does not explain how the
-evaluation of a subexpression manages to return a value to the
-expression that uses this value, nor does the evaluator explain how
-some recursive procedures generate iterative processes (that is, are
-evaluated using constant space) whereas other recursive procedures
-generate recursive processes. These questions remain unanswered
-because the metacircular evaluator is itself a Lisp program and hence
-inherits the control structure of the underlying Lisp system. In order
-to provide a more complete description of the control structure of the
-Lisp evaluator, we must work at a more primitive level than Lisp itself.
-
- In this chapter we will describe processes in terms of the
-step-by-step operation of a traditional computer. Such a computer, or machine
-"register machine", sequentially executes "instructions" that
-manipulate the contents of a fixed set of storage elements called "registers".
-A typical register-machine instruction applies a primitive operation
-to the contents of some registers and assigns the result to another
-register. Our descriptions of processes executed by register machines
-will look very much like "machine-language" programs for traditional
-computers. However, instead of focusing on the machine language of any
-particular computer, we will examine several Lisp procedures and design
-a specific register machine to execute each procedure. Thus, we will
-approach our task from the perspective of a hardware architect rather
-than that of a machine-language computer programmer. In designing
-register machines, we will develop mechanisms for implementing
-important programming constructs such as recursion. We will also
-present a language for describing designs for register machines. In
-section *Note 5-2:: we will implement a Lisp program that uses these
-descriptions to simulate the machines we design.
-
- Most of the primitive operations of our register machines are very
-simple. For example, an operation might add the numbers fetched from
-two registers, producing a result to be stored into a third register.
-Such an operation can be performed by easily described hardware. In
-order to deal with list structure, however, we will also use the memory
-operations `car', `cdr', and `cons', which require an elaborate
-storage-allocation mechanism. In section *Note 5-3:: we study their
-implementation in terms of more elementary operations.
-
- In section *Note 5-4::, after we have accumulated experience
-formulating simple procedures as register machines, we will design a
-machine that carries out the algorithm described by the metacircular
-evaluator of section *Note 4-1::. This will fill in the gap in our
-understanding of how Scheme expressions are interpreted, by providing
-an explicit model for the mechanisms of control in the evaluator. In
-section *Note 5-5:: we will study a simple compiler that translates
-Scheme programs into sequences of instructions that can be executed
-directly with the registers and operations of the evaluator register
-machine.
-
-* Menu:
-
-* 5-1:: Designing Register Machines
-* 5-2:: A Register-Machine Simulator
-* 5-3:: Storage Allocation and Garbage Collection
-* 5-4:: The Explicit-Control Evaluator
-* 5-5:: Compilation
-
-\1f
-File: sicp.info, Node: 5-1, Next: 5-2, Prev: Chapter 5, Up: Chapter 5
-
-5.1 Designing Register Machines
-===============================
-
-To design a register machine, we must design its "data paths"
-(registers and operations) and the "controller" that sequences these
-operations. To illustrate the design of a simple register machine, let
-us examine Euclid's Algorithm, which is used to compute the greatest
-common divisor (GCD) of two integers. As we saw in section *Note
-1-2-5::, Euclid's Algorithm can be carried out by an iterative process,
-as specified by the following procedure:
-
- (define (gcd a b)
- (if (= b 0)
- a
- (gcd b (remainder a b))))
-
- A machine to carry out this algorithm must keep track of two
-numbers, a and b, so let us assume that these numbers are stored in two
-registers with those names. The basic operations required are testing
-whether the contents of register `b' is zero and computing the
-remainder of the contents of register `a' divided by the contents of
-register `b'. The remainder operation is a complex process, but assume
-for the moment that we have a primitive device that computes
-remainders. On each cycle of the GCD algorithm, the contents of
-register `a' must be replaced by the contents of register `b', and the
-contents of `b' must be replaced by the remainder of the old contents
-of `a' divided by the old contents of `b'. It would be convenient if
-these replacements could be done simultaneously, but in our model of
-register machines we will assume that only one register can be assigned
-a new value at each step. To accomplish the replacements, our machine
-will use a third "temporary" register, which we call `t'. (First the
-remainder will be placed in `t', then the contents of `b' will be
-placed in `a', and finally the remainder stored in `t' will be placed
-in `b'.)
-
- We can illustrate the registers and operations required for this
-machine by using the data-path diagram shown in *Note Figure 5-1::. In
-this diagram, the registers (`a', `b', and `t') are represented by
-rectangles. Each way to assign a value to a register is indicated by
-an arrow with an `X' behind the head, pointing from the source of data
-to the register. We can think of the `X' as a button that, when
-pushed, allows the value at the source to "flow" into the designated
-register. The label next to each button is the name we will use to
-refer to the button. The names are arbitrary, and can be chosen to
-have mnemonic value (for example, `a<-b' denotes pushing the button
-that assigns the contents of register `b' to register `a'). The source
-of data for a register can be another register (as in the `a<-b'
-assignment), an operation result (as in the `t<-r' assignment), or a
-constant (a built-in value that cannot be changed, represented in a
-data-path diagram by a triangle containing the constant).
-
- An operation that computes a value from constants and the contents
-of registers is represented in a data-path diagram by a trapezoid
-containing a name for the operation. For example, the box marked `rem'
-in *Note Figure 5-1:: represents an operation that computes the
-remainder of the contents of the registers `a' and `b' to which it is
-attached. Arrows (without buttons) point from the input registers and
-constants to the box, and arrows connect the operation's output value
-to registers. A test is represented by a circle containing a name for
-the test. For example, our GCD machine has an operation that tests
-whether the contents of register `b' is zero. A test also has arrows
-from its input registers and constants, but it has no output arrows;
-its value is used by the controller rather than by the data paths.
-Overall, the data-path diagram shows the registers and operations that
-are required for the machine and how they must be connected. If we
-view the arrows as wires and the `X' buttons as switches, the data-path
-diagram is very like the wiring diagram for a machine that could be
-constructed from electrical components.
-
- *Figure 5.1:* Data paths for a GCD machine.
-
- ___
- +-----+ +-----+ / \
- | a |<--(X)----| b +--->| = |
- +--+--+ a<-b +-+---+ \___/
- | | ^ ^
- +------+ +----+ | |
- | | (X) b<-t |
- .--+---+--. | / \
- \ rem / | / O \
- \_____/ | +-----+
- | |
- (X) t<-r |
- | |
- V |
- +-----+ |
- | t +------+
- +-----+
-
- In order for the data paths to actually compute GCDs, the buttons
-must be pushed in the correct sequence. We will describe this sequence
-in terms of a controller diagram, as illustrated in *Note Figure 5-2::.
-The elements of the controller diagram indicate how the data-path
-components should be operated. The rectangular boxes in the controller
-diagram identify data-path buttons to be pushed, and the arrows
-describe the sequencing from one step to the next. The diamond in the
-diagram represents a decision. One of the two sequencing arrows will
-be followed, depending on the value of the data-path test identified in
-the diamond. We can interpret the controller in terms of a physical
-analogy: Think of the diagram as a maze in which a marble is rolling.
-When the marble rolls into a box, it pushes the data-path button that
-is named by the box. When the marble rolls into a decision node (such
-as the test for `b' = 0), it leaves the node on the path determined by
-the result of the indicated test. Taken together, the data paths and
-the controller completely describe a machine for computing GCDs. We
-start the controller (the rolling marble) at the place marked `start',
-after placing numbers in registers `a' and `b'. When the controller
-reaches `done', we will find the value of the GCD in register `a'.
-
- *Figure 5.2:* Controller for a GCD machine.
-
- start
- |
- V
- / \ yes
- +--->< = >-----> done
- | \ /
- | | no
- | V
- | +------+
- | | t<-r |
- | +---+--+
- | |
- | V
- | +------+
- | | a<-b |
- | +---+--+
- | |
- | V
- | +------+
- +--+ b<-t |
- +------+
-
- *Exercise 5.1:* Design a register machine to compute factorials
- using the iterative algorithm specified by the following
- procedure. Draw data-path and controller diagrams for this
- machine.
-
- (define (factorial n)
- (define (iter product counter)
- (if (> counter n)
- product
- (iter (* counter product)
- (+ counter 1))))
- (iter 1 1))
-
-* Menu:
-
-* 5-1-1:: A Language for Describing Register Machines
-* 5-1-2:: Abstraction in Machine Design
-* 5-1-3:: Subroutines
-* 5-1-4:: Using a Stack to Implement Recursion
-* 5-1-5:: Instruction Summary
-
-\1f
-File: sicp.info, Node: 5-1-1, Next: 5-1-2, Prev: 5-1, Up: 5-1
-
-5.1.1 A Language for Describing Register Machines
--------------------------------------------------
-
-Data-path and controller diagrams are adequate for representing simple
-machines such as GCD, but they are unwieldy for describing large
-machines such as a Lisp interpreter. To make it possible to deal with
-complex machines, we will create a language that presents, in textual
-form, all the information given by the data-path and controller
-diagrams. We will start with a notation that directly mirrors the
-diagrams.
-
- We define the data paths of a machine by describing the registers
-and the operations. To describe a register, we give it a name and
-specify the buttons that control assignment to it. We give each of
-these buttons a name and specify the source of the data that enters the
-register under the button's control. (The source is a register, a
-constant, or an operation.) To describe an operation, we give it a
-name and specify its inputs (registers or constants).
-
- We define the controller of a machine as a sequence of "instructions"
-together with "labels" that identify "entry points" in the sequence. An
-instruction is one of the following:
-
- * The name of a data-path button to push to assign a value to a
- register. (This corresponds to a box in the controller diagram.)
-
- * A `test' instruction, that performs a specified test.
-
- * A conditional branch (`branch' instruction) to a location
- indicated by a controller label, based on the result of the
- previous test. (The test and branch together correspond to a
- diamond in the controller diagram.) If the test is false, the
- controller should continue with the next instruction in the
- sequence. Otherwise, the controller should continue with the
- instruction after the label.
-
- * An unconditional branch (`goto' instruction) naming a controller
- label at which to continue execution.
-
-
- The machine starts at the beginning of the controller instruction
-sequence and stops when execution reaches the end of the sequence.
-Except when a branch changes the flow of control, instructions are
-executed in the order in which they are listed.
-
- *Figure 5.3:* A specification of the GCD machine.
-
- (data-paths
- (registers
- ((name a)
- (buttons ((name a<-b) (source (register b)))))
- ((name b)
- (buttons ((name b<-t) (source (register t)))))
- ((name t)
- (buttons ((name t<-r) (source (operation rem))))))
-
- (operations
- ((name rem)
- (inputs (register a) (register b)))
- ((name =)
- (inputs (register b) (constant 0)))))
-
- (controller
- test-b ; label
- (test =) ; test
- (branch (label gcd-done)) ; conditional branch
- (t<-r) ; button push
- (a<-b) ; button push
- (b<-t) ; button push
- (goto (label test-b)) ; unconditional branch
- gcd-done) ; label
-
-
- *Note Figure 5-3:: shows the GCD machine described in this way. This
-example only hints at the generality of these descriptions, since the
-GCD machine is a very simple case: Each register has only one button,
-and each button and test is used only once in the controller.
-
- Unfortunately, it is difficult to read such a description. In order
-to understand the controller instructions we must constantly refer back
-to the definitions of the button names and the operation names, and to
-understand what the buttons do we may have to refer to the definitions
-of the operation names. We will thus transform our notation to combine
-the information from the data-path and controller descriptions so that
-we see it all together.
-
- To obtain this form of description, we will replace the arbitrary
-button and operation names by the definitions of their behavior. That
-is, instead of saying (in the controller) "Push button `t<-r'" and
-separately saying (in the data paths) "Button `t<-r' assigns the value
-of the `rem' operation to register `t'" and "The `rem' operation's
-inputs are the contents of registers `a' and `b'," we will say (in the
-controller) "Push the button that assigns to register `t' the value of
-the `rem' operation on the contents of registers `a' and `b'."
-Similarly, instead of saying (in the controller) "Perform the `=' test"
-and separately saying (in the data paths) "The `=' test operates on the
-contents of register `b' and the constant 0," we will say "Perform the
-`=' test on the contents of register `b' and the constant 0." We will
-omit the data-path description, leaving only the controller sequence.
-Thus, the GCD machine is described as follows:
-
- (controller
- test-b
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label test-b))
- gcd-done)
-
- This form of description is easier to read than the kind illustrated
-in *Note Figure 5-3::, but it also has disadvantages:
-
- * It is more verbose for large machines, because complete
- descriptions of the data-path elements are repeated whenever the
- elements are mentioned in the controller instruction sequence.
- (This is not a problem in the GCD example, because each operation
- and button is used only once.) Moreover, repeating the data-path
- descriptions obscures the actual data-path structure of the
- machine; it is not obvious for a large machine how many registers,
- operations, and buttons there are and how they are interconnected.
-
- * Because the controller instructions in a machine definition look
- like Lisp expressions, it is easy to forget that they are not
- arbitrary Lisp expressions. They can notate only legal machine
- operations. For example, operations can operate directly only on
- constants and the contents of registers, not on the results of
- other operations.
-
-
- In spite of these disadvantages, we will use this register-machine
-language throughout this chapter, because we will be more concerned
-with understanding controllers than with understanding the elements and
-connections in data paths. We should keep in mind, however, that
-data-path design is crucial in designing real machines.
-
- *Exercise 5.2:* Use the register-machine language to describe the
- iterative factorial machine of *Note Exercise 5-1::.
-
-Actions
-.......
-
-Let us modify the GCD machine so that we can type in the numbers whose
-GCD we want and get the answer printed at our terminal. We will not
-discuss how to make a machine that can read and print, but will assume
-(as we do when we use `read' and `display' in Scheme) that they are
-available as primitive operations.(1)
-
- `Read' is like the operations we have been using in that it produces
-a value that can be stored in a register. But `read' does not take
-inputs from any registers; its value depends on something that happens
-outside the parts of the machine we are designing. We will allow our
-machine's operations to have such behavior, and thus will draw and
-notate the use of `read' just as we do any other operation that
-computes a value.
-
- `Print', on the other hand, differs from the operations we have been
-using in a fundamental way: It does not produce an output value to be
-stored in a register. Though it has an effect, this effect is not on a
-part of the machine we are designing. We will refer to this kind of
-operation as an "action". We will represent an action in a data-path
-diagram just as we represent an operation that computes a value--as a
-trapezoid that contains the name of the action. Arrows point to the
-action box from any inputs (registers or constants). We also associate
-a button with the action. Pushing the button makes the action happen.
-To make a controller push an action button we use a new kind of
-instruction called `perform'. Thus, the action of printing the
-contents of register `a' is represented in a controller sequence by the
-instruction
-
- (perform (op print) (reg a))
-
- *Note Figure 5-4:: shows the data paths and controller for the new
-GCD machine. Instead of having the machine stop after printing the
-answer, we have made it start over, so that it repeatedly reads a pair
-of numbers, computes their GCD, and prints the result. This structure
-is like the driver loops we used in the interpreters of *Note Chapter
-4::.
-
- *Figure 5.4:* A GCD machine that reads inputs and prints results.
-
- .--------.
- \ read /
- \____/
- |
- +-------*------+
- | |
- a<-rd (X) (X) b<-rd
- | |
- V V ___
- +-----+ +-----+ / \
- | a |<--(X)--+ b +----->| = |
- +-+-+-+ a<-b +-+---+ \___/
- | | | ^ ^
- +--+ +----+ +--+ | |
- | | | (X) b<-t / \
- V V V | / O \
- .---------. .---------. | /_____\
- --(X)->\ print / \ rem / |
- P \_____/ \_____/ |
- | |
- (X) t<-r |
- | |
- V |
- +-----+ |
- | t +----+
- +-----+
-
- (controller
- gcd-loop
- (assign a (op read))
- (assign b (op read))
- test-b
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label test-b))
- gcd-done
- (perform (op print) (reg a))
- (goto (label gcd-loop)))
-
-
- ---------- Footnotes ----------
-
- (1) This assumption glosses over a great deal of complexity.
-Usually a large portion of the implementation of a Lisp system is
-dedicated to making reading and printing work.
-
-\1f
-File: sicp.info, Node: 5-1-2, Next: 5-1-3, Prev: 5-1-1, Up: 5-1
-
-5.1.2 Abstraction in Machine Design
------------------------------------
-
-We will often define a machine to include "primitive" operations that
-are actually very complex. For example, in sections *Note 5-4:: and
-*Note 5-5:: we will treat Scheme's environment manipulations as
-primitive. Such abstraction is valuable because it allows us to ignore
-the details of parts of a machine so that we can concentrate on other
-aspects of the design. The fact that we have swept a lot of complexity
-under the rug, however, does not mean that a machine design is
-unrealistic. We can always replace the complex "primitives" by simpler
-primitive operations.
-
- Consider the GCD machine. The machine has an instruction that
-computes the remainder of the contents of registers `a' and `b' and
-assigns the result to register `t'. If we want to construct the GCD
-machine without using a primitive remainder operation, we must specify
-how to compute remainders in terms of simpler operations, such as
-subtraction. Indeed, we can write a Scheme procedure that finds
-remainders in this way:
-
- (define (remainder n d)
- (if (< n d)
- n
- (remainder (- n d) d)))
-
- We can thus replace the remainder operation in the GCD machine's data
-paths with a subtraction operation and a comparison test. *Note Figure
-5-5:: shows the data paths and controller for the elaborated machine.
-The instruction
-
- *Figure 5.5:* Data paths and controller for the elaborated GCD
- machine.
-
- ___
- +-----+ +-----+ / \
- | a |<--(X)---+ b +-------*-->| = |
- +--+--+ a<-b +-+---+ | \___/
- | | ^ |
- (X) t<-a | | |
- | | (X) b<-t |
- V | | _V_
- +-----+ | | / \
- | t +-------*---|--*-----| < |
- +-----+ | | \___/
- ^ V V
- | ---------
- (X) t<-d \ - /
- | --+--
- | |
- +------------+
-
-
- start
- |
- V
- / \ yes +-------+
- +->< = >----> done | t<-d |<--+
- | \ / +---+---+ |
- | | no | |
- | | V |
- | | +------+ / \ no |
- | +-->| t<-a +------->< < >-----+
- | +------+ \ /
- | | yes
- | +-------------------+
- | V
- | +-------+
- | | a<-b |
- | +---+---+
- | |
- | V
- | +-------+
- +--+ b<-t |
- +-------+
-
- (assign t (op rem) (reg a) (reg b))
-
-in the GCD controller definition is replaced by a sequence of
-instructions that contains a loop, as shown in *Note Figure 5-6::.
-
- *Figure 5.6:* Controller instruction sequence for the GCD machine
- in *Note Figure 5-5::.
-
- (controller
- test-b
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (reg a))
- rem-loop
- (test (op <) (reg t) (reg b))
- (branch (label rem-done))
- (assign t (op -) (reg t) (reg b))
- (goto (label rem-loop))
- rem-done
- (assign a (reg b))
- (assign b (reg t))
- (goto (label test-b))
- gcd-done)
-
-
- *Exercise 5.3:* Design a machine to compute square roots using
- Newton's method, as described in section *Note 1-1-7:::
-
- (define (sqrt x)
- (define (good-enough? guess)
- (< (abs (- (square guess) x)) 0.001))
- (define (improve guess)
- (average guess (/ x guess)))
- (define (sqrt-iter guess)
- (if (good-enough? guess)
- guess
- (sqrt-iter (improve guess))))
- (sqrt-iter 1.0))
-
- Begin by assuming that `good-enough?' and `improve' operations are
- available as primitives. Then show how to expand these in terms
- of arithmetic operations. Describe each version of the `sqrt'
- machine design by drawing a data-path diagram and writing a
- controller definition in the register-machine language.
-
-\1f
-File: sicp.info, Node: 5-1-3, Next: 5-1-4, Prev: 5-1-2, Up: 5-1
-
-5.1.3 Subroutines
------------------
-
-When designing a machine to perform a computation, we would often
-prefer to arrange for components to be shared by different parts of the
-computation rather than duplicate the components. Consider a machine
-that includes two GCD computations--one that finds the GCD of the
-contents of registers `a' and `b' and one that finds the GCD of the
-contents of registers `c' and `d'. We might start by assuming we have
-a primitive `gcd' operation, then expand the two instances of `gcd' in
-terms of more primitive operations. *Note Figure 5-7:: shows just the
-GCD portions of the resulting machine's data paths, without showing how
-they connect to the rest of the machine. The figure also shows the
-corresponding portions of the machine's controller sequence.
-
- *Figure 5.7:* Portions of the data paths and controller sequence
- for a machine with two GCD computations.
-
- ___ ___
- +-----+ +-----+ / \ +-----+ +-----+ / \
- | a |<-(X)---+ b |--->| = | | c |<-(X)---+ d |--->| = |
- +--+--+ a<-b ++----+ \___/ +--+--+ c<-d ++----+ \___/
- | | ^ ^ | | ^ ^
- `----. .---' | | `----. .---' | |
- V V (X) b<-t | V V (X) d<-t |
- ------- | / \ ------- | / \
- \ rem / | /_0_\ \ rem / | /_0_\
- --+-- | --+-- |
- | | | |
- (X) t<-r | (X) s<-r |
- | | | |
- V | V |
- +-----+ | +-----+ |
- | t +-----' | s +-----'
- +-----+ +-----+
-
- gcd-1
- (test (op =) (reg b) (const 0))
- (branch (label after-gcd-1))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label gcd-1))
- after-gcd-1
- ...
- gcd-2
- (test (op =) (reg d) (const 0))
- (branch (label after-gcd-2))
- (assign s (op rem) (reg c) (reg d))
- (assign c (reg d))
- (assign d (reg s))
- (goto (label gcd-2))
- after-gcd-2
-
-
- This machine has two remainder operation boxes and two boxes for
-testing equality. If the duplicated components are complicated, as is
-the remainder box, this will not be an economical way to build the
-machine. We can avoid duplicating the data-path components by using
-the same components for both GCD computations, provided that doing so
-will not affect the rest of the larger machine's computation. If the
-values in registers `a' and `b' are not needed by the time the
-controller gets to `gcd-2' (or if these values can be moved to other
-registers for safekeeping), we can change the machine so that it uses
-registers `a' and `b', rather than registers `c' and `d', in computing
-the second GCD as well as the first. If we do this, we obtain the
-controller sequence shown in *Note Figure 5-8::.
-
- We have removed the duplicate data-path components (so that the data
-paths are again as in *Note Figure 5-1::), but the controller now has
-two GCD sequences that differ only in their entry-point labels. It
-would be better to replace these two sequences by branches to a single
-sequence--a `gcd' "subroutine"--at the end of which we branch back to
-the correct place in the main instruction sequence. We can accomplish
-this as follows: Before branching to `gcd', we place a distinguishing
-value (such as 0 or 1) into a special register, `continue'. At the end
-of the `gcd' subroutine we return either to `after-gcd-1' or to
-`after-gcd-2', depending on the value of the `continue' register.
-*Note Figure 5-9:: shows the relevant portion of the resulting
-controller sequence, which includes only a single copy of the `gcd'
-instructions.
-
- *Figure 5.8:* Portions of the controller sequence for a machine
- that uses the same data-path components for two different GCD
- computations.
-
- gcd-1
- (test (op =) (reg b) (const 0))
- (branch (label after-gcd-1))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label gcd-1))
- after-gcd-1
- ...
- gcd-2
- (test (op =) (reg b) (const 0))
- (branch (label after-gcd-2))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label gcd-2))
- after-gcd-2
-
-
- *Figure 5.9:* Using a `continue' register to avoid the duplicate
- controller sequence in *Note Figure 5-8::.
-
- gcd
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label gcd))
- gcd-done
- (test (op =) (reg continue) (const 0))
- (branch (label after-gcd-1))
- (goto (label after-gcd-2))
- ...
- ;; Before branching to `gcd' from the first place where
- ;; it is needed, we place 0 in the `continue' register
- (assign continue (const 0))
- (goto (label gcd))
- after-gcd-1
- ...
- ;; Before the second use of `gcd', we place 1 in the `continue' register
- (assign continue (const 1))
- (goto (label gcd))
- after-gcd-2
-
-
- *Figure 5.10:* Assigning labels to the `continue' register
- simplifies and generalizes the strategy shown in *Note Figure
- 5-9::.
-
- gcd
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label gcd))
- gcd-done
- (goto (reg continue))
- ...
- ;; Before calling `gcd', we assign to `continue'
- ;; the label to which `gcd' should return.
- (assign continue (label after-gcd-1))
- (goto (label gcd))
- after-gcd-1
- ...
- ;; Here is the second call to `gcd', with a different continuation.
- (assign continue (label after-gcd-2))
- (goto (label gcd))
- after-gcd-2
-
-
- This is a reasonable approach for handling small problems, but it
-would be awkward if there were many instances of GCD computations in the
-controller sequence. To decide where to continue executing after the
-`gcd' subroutine, we would need tests in the data paths and branch
-instructions in the controller for all the places that use `gcd'. A
-more powerful method for implementing subroutines is to have the
-`continue' register hold the label of the entry point in the controller
-sequence at which execution should continue when the subroutine is
-finished. Implementing this strategy requires a new kind of connection
-between the data paths and the controller of a register machine: There
-must be a way to assign to a register a label in the controller
-sequence in such a way that this value can be fetched from the register
-and used to continue execution at the designated entry point.
-
- To reflect this ability, we will extend the `assign' instruction of
-the register-machine language to allow a register to be assigned as
-value a label from the controller sequence (as a special kind of
-constant). We will also extend the `goto' instruction to allow
-execution to continue at the entry point described by the contents of a
-register rather than only at an entry point described by a constant
-label. Using these new constructs we can terminate the `gcd'
-subroutine with a branch to the location stored in the `continue'
-register. This leads to the controller sequence shown in *Note Figure
-5-10::.
-
- A machine with more than one subroutine could use multiple
-continuation registers (e.g., `gcd-continue', `factorial-continue') or
-we could have all subroutines share a single `continue' register.
-Sharing is more economical, but we must be careful if we have a
-subroutine (`sub1') that calls another subroutine (`sub2'). Unless
-`sub1' saves the contents of `continue' in some other register before
-setting up `continue' for the call to `sub2', `sub1' will not know
-where to go when it is finished. The mechanism developed in the next
-section to handle recursion also provides a better solution to this
-problem of nested subroutine calls.
-
-\1f
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-
-5.1.4 Using a Stack to Implement Recursion
-------------------------------------------
-
-With the ideas illustrated so far, we can implement any iterative
-process by specifying a register machine that has a register
-corresponding to each state variable of the process. The machine
-repeatedly executes a controller loop, changing the contents of the
-registers, until some termination condition is satisfied. At each
-point in the controller sequence, the state of the machine
-(representing the state of the iterative process) is completely
-determined by the contents of the registers (the values of the state
-variables).
-
- Implementing recursive processes, however, requires an additional
-mechanism. Consider the following recursive method for computing
-factorials, which we first examined in section *Note 1-2-1:::
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
-
- As we see from the procedure, computing n! requires computing (n -
-1)!. Our GCD machine, modeled on the procedure
-
- (define (gcd a b)
- (if (= b 0)
- a
- (gcd b (remainder a b))))
-
-similarly had to compute another GCD. But there is an important
-difference between the `gcd' procedure, which reduces the original
-computation to a new GCD computation, and `factorial', which requires
-computing another factorial as a subproblem. In GCD, the answer to the
-new GCD computation is the answer to the original problem. To compute
-the next GCD, we simply place the new arguments in the input registers
-of the GCD machine and reuse the machine's data paths by executing the
-same controller sequence. When the machine is finished solving the
-final GCD problem, it has completed the entire computation.
-
- In the case of factorial (or any recursive process) the answer to
-the new factorial subproblem is not the answer to the original problem.
-The value obtained for (n - 1)! must be multiplied by n to get the
-final answer. If we try to imitate the GCD design, and solve the
-factorial subproblem by decrementing the `n' register and rerunning the
-factorial machine, we will no longer have available the old value of
-`n' by which to multiply the result. We thus need a second factorial
-machine to work on the subproblem. This second factorial computation
-itself has a factorial subproblem, which requires a third factorial
-machine, and so on. Since each factorial machine contains another
-factorial machine within it, the total machine contains an infinite
-nest of similar machines and hence cannot be constructed from a fixed,
-finite number of parts.
-
- Nevertheless, we can implement the factorial process as a register
-machine if we can arrange to use the same components for each nested
-instance of the machine. Specifically, the machine that computes n!
-should use the same components to work on the subproblem of computing
-(n - 1)!, on the subproblem for (n - 2)!, and so on. This is plausible
-because, although the factorial process dictates that an unbounded
-number of copies of the same machine are needed to perform a
-computation, only one of these copies needs to be active at any given
-time. When the machine encounters a recursive subproblem, it can
-suspend work on the main problem, reuse the same physical parts to work
-on the subproblem, then continue the suspended computation.
-
- In the subproblem, the contents of the registers will be different
-than they were in the main problem. (In this case the `n' register is
-decremented.) In order to be able to continue the suspended
-computation, the machine must save the contents of any registers that
-will be needed after the subproblem is solved so that these can be
-restored to continue the suspended computation. In the case of
-factorial, we will save the old value of `n', to be restored when we
-are finished computing the factorial of the decremented `n' register.(1)
-
- Since there is no _a priori_ limit on the depth of nested recursive
-calls, we may need to save an arbitrary number of register values.
-These values must be restored in the reverse of the order in which they
-were saved, since in a nest of recursions the last subproblem to be
-entered is the first to be finished. This dictates the use of a "stack",
-or "last in, first out" data structure, to save register values. We
-can extend the register-machine language to include a stack by adding
-two kinds of instructions: Values are placed on the stack using a
-`save' instruction and restored from the stack using a `restore'
-instruction. After a sequence of values has been `save'd on the stack,
-a sequence of `restore's will retrieve these values in reverse order.(2)
-
- With the aid of the stack, we can reuse a single copy of the
-factorial machine's data paths for each factorial subproblem. There is
-a similar design issue in reusing the controller sequence that operates
-the data paths. To reexecute the factorial computation, the controller
-cannot simply loop back to the beginning, as with an iterative process,
-because after solving the (n - 1)! subproblem the machine must still
-multiply the result by n. The controller must suspend its computation
-of n!, solve the (n - 1)! subproblem, then continue its computation of
-n!. This view of the factorial computation suggests the use of the
-subroutine mechanism described in section *Note 5-1-3::, which has the
-controller use a `continue' register to transfer to the part of the
-sequence that solves a subproblem and then continue where it left off
-on the main problem. We can thus make a factorial subroutine that
-returns to the entry point stored in the `continue' register. Around
-each subroutine call, we save and restore `continue' just as we do the
-`n' register, since each "level" of the factorial computation will use
-the same `continue' register. That is, the factorial subroutine must
-put a new value in `continue' when it calls itself for a subproblem,
-but it will need the old value in order to return to the place that
-called it to solve a subproblem.
-
- *Note Figure 5-11:: shows the data paths and controller for a
-machine that implements the recursive `factorial' procedure. The
-machine has a stack and three registers, called `n', `val', and
-`continue'. To simplify the data-path diagram, we have not named the
-register-assignment buttons, only the stack-operation buttons (`sc' and
-`sn' to save registers, `rc' and `rn' to restore registers). To
-operate the machine, we put in register `n' the number whose factorial
-we wish to compute and start the machine. When the machine reaches
-`fact-done', the computation is finished and the answer will be found
-in the `val' register. In the controller sequence, `n' and `continue'
-are saved before each recursive call and restored upon return from the
-call. Returning from a call is accomplished by branching to the
-location stored in `continue'. `Continue' is initialized when the
-machine starts so that the last return will go to `fact-done'. The
-`val' register, which holds the result of the factorial computation, is
-not saved before the recursive call, because the old contents of `val'
-is not useful after the subroutine returns. Only the new value, which
-is the value produced by the subcomputation, is needed.
-
- Although in principle the factorial computation requires an infinite
-machine, the machine in *Note Figure 5-11:: is actually finite except
-for the stack, which is potentially unbounded. Any particular physical
-implementation of a stack, however, will be of finite size, and this
-will limit the depth of recursive calls that can be handled by the
-machine. This implementation of factorial illustrates the general
-strategy for realizing recursive algorithms as ordinary register
-machines augmented by stacks. When a recursive subproblem is
-encountered, we save on the stack the registers whose current values
-will be required after the subproblem is solved, solve the recursive
-subproblem, then restore the saved registers and continue execution on
-the main problem. The `continue' register must always be saved.
-Whether there are other registers that need to be saved depends on the
-particular machine, since not all recursive computations need the
-original values of registers that are modified during solution of the
-subproblem (see *Note Exercise 5-4::).
-
-A double recursion
-..................
-
-Let us examine a more complex recursive process, the tree-recursive
-computation of the Fibonacci numbers, which we introduced in section
-*Note 1-2-2:::
-
- (define (fib n)
- (if (< n 2)
- n
- (+ (fib (- n 1)) (fib (- n 2)))))
-
- Just as with factorial, we can implement the recursive Fibonacci
-computation as a register machine with registers `n', `val', and
-`continue'. The machine is more complex than the one for factorial,
-because there are two places in the controller sequence where we need
-to perform recursive calls--once to compute Fib(n - 1) and once to
-compute Fib(n - 2). To set up for each of these calls, we save the
-registers whose values will be needed later, set the `n' register to
-the number whose Fib we need to compute recursively (n - 1 or n - 2),
-and assign to `continue' the entry point in the main sequence to which
-to return (`afterfib-n-1' or `afterfib-n-2', respectively). We then go
-to `fib-loop'. When we return from the recursive call, the answer is
-in `val'. *Note Figure 5-12:: shows the controller sequence for this
-machine.
-
- *Figure 5.11:* A recursive factorial machine.
-
- ___
- / \
- +----------*-----------| = |
- | | \___/
- (X) | ^
- | | |
- V | +---+---+ sn +-------+
- +-------+ | | +---(X)-->| |
- | val |<-(X)-|----------+ n | | stack |
- +-----+-+ | | |<--(X)---+ |
- ^ | | +-------+ rn +-+-----+
- | | | ^ | ^
- (X) | | | | |
- | | +----|--------* (X) | (X) sc
- | | | | | | rc (X) |
- | | | *----. | | | |
- | V V | V V | V |
- | ------- | ------- | +------+-+
- | \ * / | \ - / | |continue+--> controller
- | --+-- | --+-- | +--------+
- | | | | | ^ ^
- +-----+ | +-----+ | |
- | (X) (X)
- | | |
- / \ after- / \ / \ fact-
- /_1_\ fact /___\ /___\ done
-
- (controller
- (assign continue (label fact-done)) ; set up final return address
- fact-loop
- (test (op =) (reg n) (const 1))
- (branch (label base-case))
- ;; Set up for the recursive call by saving `n' and `continue'.
- ;; Set up `continue' so that the computation will continue
- ;; at `after-fact' when the subroutine returns.
- (save continue)
- (save n)
- (assign n (op -) (reg n) (const 1))
- (assign continue (label after-fact))
- (goto (label fact-loop))
- after-fact
- (restore n)
- (restore continue)
- (assign val (op *) (reg n) (reg val)) ; `val' now contains n(n - 1)!
- (goto (reg continue)) ; return to caller
- base-case
- (assign val (const 1)) ; base case: 1! = 1
- (goto (reg continue)) ; return to caller
- fact-done)
-
-
- *Figure 5.12:* Controller for a machine to compute Fibonacci
- numbers.
-
- (controller
- (assign continue (label fib-done))
- fib-loop
- (test (op <) (reg n) (const 2))
- (branch (label immediate-answer))
- ;; set up to compute _Fib_(n - 1)
- (save continue)
- (assign continue (label afterfib-n-1))
- (save n) ; save old value of `n'
- (assign n (op -) (reg n) (const 1)); clobber `n' to n - 1
- (goto (label fib-loop)) ; perform recursive call
- afterfib-n-1 ; upon return, `val' contains _Fib_(n - 1)
- (restore n)
- (restore continue)
- ;; set up to compute _Fib_(n - 2)
- (assign n (op -) (reg n) (const 2))
- (save continue)
- (assign continue (label afterfib-n-2))
- (save val) ; save _Fib_(n - 1)
- (goto (label fib-loop))
- afterfib-n-2 ; upon return, `val' contains _Fib_(n - 2)
- (assign n (reg val)) ; `n' now contains _Fib_(n - 2)
- (restore val) ; `val' now contains _Fib_(n - 1)
- (restore continue)
- (assign val ; _Fib_(n - 1) + _Fib_(n - 2)
- (op +) (reg val) (reg n))
- (goto (reg continue)) ; return to caller, answer is in `val'
- immediate-answer
- (assign val (reg n)) ; base case: _Fib_(n) = n
- (goto (reg continue))
- fib-done)
-
-
- *Exercise 5.4:* Specify register machines that implement each of
- the following procedures. For each machine, write a controller
- instruction sequence and draw a diagram showing the data paths.
-
- a. Recursive exponentiation:
-
- (define (expt b n)
- (if (= n 0)
- 1
- (* b (expt b (- n 1)))))
-
- b. Iterative exponentiation:
-
- (define (expt b n)
- (define (expt-iter counter product)
- (if (= counter 0)
- product
- (expt-iter (- counter 1) (* b product))))
- (expt-iter n 1))
-
-
- *Exercise 5.5:* Hand-simulate the factorial and Fibonacci
- machines, using some nontrivial input (requiring execution of at
- least one recursive call). Show the contents of the stack at each
- significant point in the execution.
-
- *Exercise 5.6:* Ben Bitdiddle observes that the Fibonacci
- machine's controller sequence has an extra `save' and an extra
- `restore', which can be removed to make a faster machine. Where
- are these instructions?
-
- ---------- Footnotes ----------
-
- (1) One might argue that we don't need to save the old `n'; after we
-decrement it and solve the subproblem, we could simply increment it to
-recover the old value. Although this strategy works for factorial, it
-cannot work in general, since the old value of a register cannot always
-be computed from the new one.
-
- (2) In section *Note 5-3:: we will see how to implement a stack in
-terms of more primitive operations.
-
-\1f
-File: sicp.info, Node: 5-1-5, Prev: 5-1-4, Up: 5-1
-
-5.1.5 Instruction Summary
--------------------------
-
-A controller instruction in our register-machine language has one of the
-following forms, where each <INPUT_I> is either `(reg <REGISTER-NAME>)'
-or `(const <CONSTANT-VALUE>)'. These instructions were introduced in
-section *Note 5-1-1:::
-
- (assign <REGISTER-NAME> (reg <REGISTER-NAME>))
-
- (assign <REGISTER-NAME> (const <CONSTANT-VALUE>))
-
- (assign <REGISTER-NAME> (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
-
- (perform (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
-
- (test (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
-
- (branch (label <LABEL-NAME>))
-
- (goto (label <LABEL-NAME>))
-
- The use of registers to hold labels was introduced in section *Note
-5-1-3:::
-
- (assign <REGISTER-NAME> (label <LABEL-NAME>))
-
- (goto (reg <REGISTER-NAME>))
-
- Instructions to use the stack were introduced in section *Note
-5-1-4:::
-
- (save <REGISTER-NAME>)
-
- (restore <REGISTER-NAME>)
-
- The only kind of <CONSTANT-VALUE> we have seen so far is a number,
-but later we will use strings, symbols, and lists. For example,
-`(const "abc")' is the string `"abc"',
-`(const abc)' is the symbol `abc',
-`(const (a b c))' is the list `(a b c)',
-and `(const ())' is the empty list.
-
-\1f
-File: sicp.info, Node: 5-2, Next: 5-3, Prev: 5-1, Up: Chapter 5
-
-5.2 A Register-Machine Simulator
-================================
-
-In order to gain a good understanding of the design of register
-machines, we must test the machines we design to see if they perform as
-expected. One way to test a design is to hand-simulate the operation
-of the controller, as in *Note Exercise 5-5::. But this is extremely
-tedious for all but the simplest machines. In this section we
-construct a simulator for machines described in the register-machine
-language. The simulator is a Scheme program with four interface
-procedures. The first uses a description of a register machine to
-construct a model of the machine (a data structure whose parts
-correspond to the parts of the machine to be simulated), and the other
-three allow us to simulate the machine by manipulating the model:
-
- (make-machine <REGISTER-NAMES> <OPERATIONS> <CONTROLLER>)
-
- constructs and returns a model of the machine with the given
- registers, operations, and controller.
-
- (set-register-contents! <MACHINE-MODEL> <REGISTER-NAME> <VALUE>)
-
- stores a value in a simulated register in the given machine.
-
- (get-register-contents <MACHINE-MODEL> <REGISTER-NAME>)
-
- returns the contents of a simulated register in the given machine.
-
- (start <MACHINE-MODEL>)
-
- simulates the execution of the given machine, starting from the
- beginning of the controller sequence and stopping when it reaches
- the end of the sequence.
-
- As an example of how these procedures are used, we can define
-`gcd-machine' to be a model of the GCD machine of section *Note 5-1-1::
-as follows:
-
- (define gcd-machine
- (make-machine
- '(a b t)
- (list (list 'rem remainder) (list '= =))
- '(test-b
- (test (op =) (reg b) (const 0))
- (branch (label gcd-done))
- (assign t (op rem) (reg a) (reg b))
- (assign a (reg b))
- (assign b (reg t))
- (goto (label test-b))
- gcd-done)))
-
- The first argument to `make-machine' is a list of register names.
-The next argument is a table (a list of two-element lists) that pairs
-each operation name with a Scheme procedure that implements the
-operation (that is, produces the same output value given the same input
-values). The last argument specifies the controller as a list of
-labels and machine instructions, as in section *Note 5-1::.
-
- To compute GCDs with this machine, we set the input registers, start
-the machine, and examine the result when the simulation terminates:
-
- (set-register-contents! gcd-machine 'a 206)
- done
-
- (set-register-contents! gcd-machine 'b 40)
- done
-
- (start gcd-machine)
- done
-
- (get-register-contents gcd-machine 'a)
- 2
-
- This computation will run much more slowly than a `gcd' procedure
-written in Scheme, because we will simulate low-level machine
-instructions, such as `assign', by much more complex operations.
-
- *Exercise 5.7:* Use the simulator to test the machines you
- designed in *Note Exercise 5-4::.
-
-* Menu:
-
-* 5-2-1:: The Machine Model
-* 5-2-2:: The Assembler
-* 5-2-3:: Generating Execution Procedures for Instructions
-* 5-2-4:: Monitoring Machine Performance
-
-\1f
-File: sicp.info, Node: 5-2-1, Next: 5-2-2, Prev: 5-2, Up: 5-2
-
-5.2.1 The Machine Model
------------------------
-
-The machine model generated by `make-machine' is represented as a
-procedure with local state using the message-passing techniques
-developed in *Note Chapter 3::. To build this model, `make-machine'
-begins by calling the procedure `make-new-machine' to construct the
-parts of the machine model that are common to all register machines.
-This basic machine model constructed by `make-new-machine' is
-essentially a container for some registers and a stack, together with
-an execution mechanism that processes the controller instructions one
-by one.
-
- `Make-machine' then extends this basic model (by sending it
-messages) to include the registers, operations, and controller of the
-particular machine being defined. First it allocates a register in the
-new machine for each of the supplied register names and installs the
-designated operations in the machine. Then it uses an "assembler"
-(described below in section *Note 5-2-2::) to transform the controller
-list into instructions for the new machine and installs these as the
-machine's instruction sequence. `Make-machine' returns as its value
-the modified machine model.
-
- (define (make-machine register-names ops controller-text)
- (let ((machine (make-new-machine)))
- (for-each (lambda (register-name)
- ((machine 'allocate-register) register-name))
- register-names)
- ((machine 'install-operations) ops)
- ((machine 'install-instruction-sequence)
- (assemble controller-text machine))
- machine))
-
-Registers
-.........
-
-We will represent a register as a procedure with local state, as in
-*Note Chapter 3::. The procedure `make-register' creates a register
-that holds a value that can be accessed or changed:
-
- (define (make-register name)
- (let ((contents '*unassigned*))
- (define (dispatch message)
- (cond ((eq? message 'get) contents)
- ((eq? message 'set)
- (lambda (value) (set! contents value)))
- (else
- (error "Unknown request -- REGISTER" message))))
- dispatch))
-
- The following procedures are used to access registers:
-
- (define (get-contents register)
- (register 'get))
-
- (define (set-contents! register value)
- ((register 'set) value))
-
-The stack
-.........
-
-We can also represent a stack as a procedure with local state. The
-procedure `make-stack' creates a stack whose local state consists of a
-list of the items on the stack. A stack accepts requests to `push' an
-item onto the stack, to `pop' the top item off the stack and return it,
-and to `initialize' the stack to empty.
-
- (define (make-stack)
- (let ((s '()))
- (define (push x)
- (set! s (cons x s)))
- (define (pop)
- (if (null? s)
- (error "Empty stack -- POP")
- (let ((top (car s)))
- (set! s (cdr s))
- top)))
- (define (initialize)
- (set! s '())
- 'done)
- (define (dispatch message)
- (cond ((eq? message 'push) push)
- ((eq? message 'pop) (pop))
- ((eq? message 'initialize) (initialize))
- (else (error "Unknown request -- STACK"
- message))))
- dispatch))
-
- The following procedures are used to access stacks:
-
- (define (pop stack)
- (stack 'pop))
-
- (define (push stack value)
- ((stack 'push) value))
-
-The basic machine
-.................
-
-The `make-new-machine' procedure, shown in *Note Figure 5-13::,
-constructs an object whose local state consists of a stack, an
-initially empty instruction sequence, a list of operations that
-initially contains an operation to initialize the stack, and a "register
-table" that initially contains two registers, named `flag' and `pc'
-(for "program counter"). The internal procedure `allocate-register'
-adds new entries to the register table, and the internal procedure
-`lookup-register' looks up registers in the table.
-
- The `flag' register is used to control branching in the simulated
-machine. `Test' instructions set the contents of `flag' to the result
-of the test (true or false). `Branch' instructions decide whether or
-not to branch by examining the contents of `flag'.
-
- The `pc' register determines the sequencing of instructions as the
-machine runs. This sequencing is implemented by the internal procedure
-`execute'. In the simulation model, each machine instruction is a data
-structure that includes a procedure of no arguments, called the procedure
-"instruction execution procedure", such that calling this procedure
-simulates executing the instruction. As the simulation runs, `pc'
-points to the place in the instruction sequence beginning with the next
-instruction to be executed. `Execute' gets that instruction, executes
-it by calling the instruction execution procedure, and repeats this
-cycle until there are no more instructions to execute (i.e., until `pc'
-points to the end of the instruction sequence).
-
- *Figure 5.13:* The `make-new-machine' procedure, which implements
- the basic machine model.
-
- (define (make-new-machine)
- (let ((pc (make-register 'pc))
- (flag (make-register 'flag))
- (stack (make-stack))
- (the-instruction-sequence '()))
- (let ((the-ops
- (list (list 'initialize-stack
- (lambda () (stack 'initialize)))))
- (register-table
- (list (list 'pc pc) (list 'flag flag))))
- (define (allocate-register name)
- (if (assoc name register-table)
- (error "Multiply defined register: " name)
- (set! register-table
- (cons (list name (make-register name))
- register-table)))
- 'register-allocated)
- (define (lookup-register name)
- (let ((val (assoc name register-table)))
- (if val
- (cadr val)
- (error "Unknown register:" name))))
- (define (execute)
- (let ((insts (get-contents pc)))
- (if (null? insts)
- 'done
- (begin
- ((instruction-execution-proc (car insts)))
- (execute)))))
- (define (dispatch message)
- (cond ((eq? message 'start)
- (set-contents! pc the-instruction-sequence)
- (execute))
- ((eq? message 'install-instruction-sequence)
- (lambda (seq) (set! the-instruction-sequence seq)))
- ((eq? message 'allocate-register) allocate-register)
- ((eq? message 'get-register) lookup-register)
- ((eq? message 'install-operations)
- (lambda (ops) (set! the-ops (append the-ops ops))))
- ((eq? message 'stack) stack)
- ((eq? message 'operations) the-ops)
- (else (error "Unknown request -- MACHINE" message))))
- dispatch)))
-
- As part of its operation, each instruction execution procedure
-modifies `pc' to indicate the next instruction to be executed.
-`Branch' and `goto' instructions change `pc' to point to the new
-destination. All other instructions simply advance `pc', making it
-point to the next instruction in the sequence. Observe that each call
-to `execute' calls `execute' again, but this does not produce an
-infinite loop because running the instruction execution procedure
-changes the contents of `pc'.
-
- `Make-new-machine' returns a `dispatch' procedure that implements
-message-passing access to the internal state. Notice that starting the
-machine is accomplished by setting `pc' to the beginning of the
-instruction sequence and calling `execute'.
-
- For convenience, we provide an alternate procedural interface to a
-machine's `start' operation, as well as procedures to set and examine
-register contents, as specified at the beginning of section *Note 5-2:::
-
- (define (start machine)
- (machine 'start))
-
- (define (get-register-contents machine register-name)
- (get-contents (get-register machine register-name)))
-
- (define (set-register-contents! machine register-name value)
- (set-contents! (get-register machine register-name) value)
- 'done)
-
- These procedures (and many procedures in sections *Note 5-2-2:: and
-*Note 5-2-3::) use the following to look up the register with a given
-name in a given machine:
-
- (define (get-register machine reg-name)
- ((machine 'get-register) reg-name))
-
-\1f
-File: sicp.info, Node: 5-2-2, Next: 5-2-3, Prev: 5-2-1, Up: 5-2
-
-5.2.2 The Assembler
--------------------
-
-The assembler transforms the sequence of controller expressions for a
-machine into a corresponding list of machine instructions, each with
-its execution procedure. Overall, the assembler is much like the
-evaluators we studied in *Note Chapter 4::--there is an input language
-(in this case, the register-machine language) and we must perform an
-appropriate action for each type of expression in the language.
-
- The technique of producing an execution procedure for each
-instruction is just what we used in section *Note 4-1-7:: to speed up
-the evaluator by separating analysis from runtime execution. As we saw
-in *Note Chapter 4::, much useful analysis of Scheme expressions could
-be performed without knowing the actual values of variables. Here,
-analogously, much useful analysis of register-machine-language
-expressions can be performed without knowing the actual contents of
-machine registers. For example, we can replace references to registers
-by pointers to the register objects, and we can replace references to
-labels by pointers to the place in the instruction sequence that the
-label designates.
-
- Before it can generate the instruction execution procedures, the
-assembler must know what all the labels refer to, so it begins by
-scanning the controller text to separate the labels from the
-instructions. As it scans the text, it constructs both a list of
-instructions and a table that associates each label with a pointer into
-that list. Then the assembler augments the instruction list by
-inserting the execution procedure for each instruction.
-
- The `assemble' procedure is the main entry to the assembler. It
-takes the controller text and the machine model as arguments and
-returns the instruction sequence to be stored in the model. `Assemble'
-calls `extract-labels' to build the initial instruction list and label
-table from the supplied controller text. The second argument to
-`extract-labels' is a procedure to be called to process these results:
-This procedure uses `update-insts!' to generate the instruction
-execution procedures and insert them into the instruction list, and
-returns the modified list.
-
- (define (assemble controller-text machine)
- (extract-labels controller-text
- (lambda (insts labels)
- (update-insts! insts labels machine)
- insts)))
-
- `Extract-labels' takes as arguments a list `text' (the sequence of
-controller instruction expressions) and a `receive' procedure.
-`Receive' will be called with two values: (1) a list `insts' of
-instruction data structures, each containing an instruction from `text';
-and (2) a table called `labels', which associates each label from
-`text' with the position in the list `insts' that the label designates.
-
- (define (extract-labels text receive)
- (if (null? text)
- (receive '() '())
- (extract-labels (cdr text)
- (lambda (insts labels)
- (let ((next-inst (car text)))
- (if (symbol? next-inst)
- (receive insts
- (cons (make-label-entry next-inst
- insts)
- labels))
- (receive (cons (make-instruction next-inst)
- insts)
- labels)))))))
-
- `Extract-labels' works by sequentially scanning the elements of the
-`text' and accumulating the `insts' and the `labels'. If an element is
-a symbol (and thus a label) an appropriate entry is added to the
-`labels' table. Otherwise the element is accumulated onto the `insts'
-list.(1)
-
- `Update-insts!' modifies the instruction list, which initially
-contains only the text of the instructions, to include the
-corresponding execution procedures:
-
- (define (update-insts! insts labels machine)
- (let ((pc (get-register machine 'pc))
- (flag (get-register machine 'flag))
- (stack (machine 'stack))
- (ops (machine 'operations)))
- (for-each
- (lambda (inst)
- (set-instruction-execution-proc!
- inst
- (make-execution-procedure
- (instruction-text inst) labels machine
- pc flag stack ops)))
- insts)))
-
- The machine instruction data structure simply pairs the instruction
-text with the corresponding execution procedure. The execution
-procedure is not yet available when `extract-labels' constructs the
-instruction, and is inserted later by `update-insts!'.
-
- (define (make-instruction text)
- (cons text '()))
-
- (define (instruction-text inst)
- (car inst))
-
- (define (instruction-execution-proc inst)
- (cdr inst))
-
- (define (set-instruction-execution-proc! inst proc)
- (set-cdr! inst proc))
-
- The instruction text is not used by our simulator, but it is handy
-to keep around for debugging (see *Note Exercise 5-16::).
-
- Elements of the label table are pairs:
-
- (define (make-label-entry label-name insts)
- (cons label-name insts))
-
- Entries will be looked up in the table with
-
- (define (lookup-label labels label-name)
- (let ((val (assoc label-name labels)))
- (if val
- (cdr val)
- (error "Undefined label -- ASSEMBLE" label-name))))
-
- *Exercise 5.8:* The following register-machine code is ambiguous,
- because the label `here' is defined more than once:
-
- start
- (goto (label here))
- here
- (assign a (const 3))
- (goto (label there))
- here
- (assign a (const 4))
- (goto (label there))
- there
-
- With the simulator as written, what will the contents of register
- `a' be when control reaches `there'? Modify the `extract-labels'
- procedure so that the assembler will signal an error if the same
- label name is used to indicate two different locations.
-
- ---------- Footnotes ----------
-
- (1) Using the `receive' procedure here is a way to get
-`extract-labels' to effectively return two values--`labels' and
-`insts'--without explicitly making a compound data structure to hold
-them. An alternative implementation, which returns an explicit pair of
-values, is
-
- (define (extract-labels text)
- (if (null? text)
- (cons '() '())
- (let ((result (extract-labels (cdr text))))
- (let ((insts (car result)) (labels (cdr result)))
- (let ((next-inst (car text)))
- (if (symbol? next-inst)
- (cons insts
- (cons (make-label-entry next-inst insts) labels))
- (cons (cons (make-instruction next-inst) insts)
- labels)))))))
-
-which would be called by `assemble' as follows:
-
- (define (assemble controller-text machine)
- (let ((result (extract-labels controller-text)))
- (let ((insts (car result)) (labels (cdr result)))
- (update-insts! insts labels machine)
- insts)))
-
- You can consider our use of `receive' as demonstrating an elegant
-way to return multiple values, or simply an excuse to show off a
-programming trick. An argument like `receive' that is the next
-procedure to be invoked is called a "continuation." Recall that we
-also used continuations to implement the backtracking control structure
-in the `amb' evaluator in section *Note 4-3-3::.
-
-\1f
-File: sicp.info, Node: 5-2-3, Next: 5-2-4, Prev: 5-2-2, Up: 5-2
-
-5.2.3 Generating Execution Procedures for Instructions
-------------------------------------------------------
-
-The assembler calls `make-execution-procedure' to generate the execution
-procedure for an instruction. Like the `analyze' procedure in the
-evaluator of section *Note 4-1-7::, this dispatches on the type of
-instruction to generate the appropriate execution procedure.
-
- (define (make-execution-procedure inst labels machine
- pc flag stack ops)
- (cond ((eq? (car inst) 'assign)
- (make-assign inst machine labels ops pc))
- ((eq? (car inst) 'test)
- (make-test inst machine labels ops flag pc))
- ((eq? (car inst) 'branch)
- (make-branch inst machine labels flag pc))
- ((eq? (car inst) 'goto)
- (make-goto inst machine labels pc))
- ((eq? (car inst) 'save)
- (make-save inst machine stack pc))
- ((eq? (car inst) 'restore)
- (make-restore inst machine stack pc))
- ((eq? (car inst) 'perform)
- (make-perform inst machine labels ops pc))
- (else (error "Unknown instruction type -- ASSEMBLE"
- inst))))
-
- For each type of instruction in the register-machine language, there
-is a generator that builds an appropriate execution procedure. The
-details of these procedures determine both the syntax and meaning of
-the individual instructions in the register-machine language. We use
-data abstraction to isolate the detailed syntax of register-machine
-expressions from the general execution mechanism, as we did for
-evaluators in section *Note 4-1-2::, by using syntax procedures to
-extract and classify the parts of an instruction.
-
-`Assign' instructions
-.....................
-
-The `make-assign' procedure handles `assign' instructions:
-
- (define (make-assign inst machine labels operations pc)
- (let ((target
- (get-register machine (assign-reg-name inst)))
- (value-exp (assign-value-exp inst)))
- (let ((value-proc
- (if (operation-exp? value-exp)
- (make-operation-exp
- value-exp machine labels operations)
- (make-primitive-exp
- (car value-exp) machine labels))))
- (lambda () ; execution procedure for `assign'
- (set-contents! target (value-proc))
- (advance-pc pc)))))
-
- `Make-assign' extracts the target register name (the second element
-of the instruction) and the value expression (the rest of the list that
-forms the instruction) from the `assign' instruction using the selectors
-
- (define (assign-reg-name assign-instruction)
- (cadr assign-instruction))
-
- (define (assign-value-exp assign-instruction)
- (cddr assign-instruction))
-
- The register name is looked up with `get-register' to produce the
-target register object. The value expression is passed to
-`make-operation-exp' if the value is the result of an operation, and to
-`make-primitive-exp' otherwise. These procedures (shown below) parse
-the value expression and produce an execution procedure for the value.
-This is a procedure of no arguments, called `value-proc', which will be
-evaluated during the simulation to produce the actual value to be
-assigned to the register. Notice that the work of looking up the
-register name and parsing the value expression is performed just once,
-at assembly time, not every time the instruction is simulated. This
-saving of work is the reason we use execution procedures, and
-corresponds directly to the saving in work we obtained by separating
-program analysis from execution in the evaluator of section *Note
-4-1-7::.
-
- The result returned by `make-assign' is the execution procedure for
-the `assign' instruction. When this procedure is called (by the machine
-model's `execute' procedure), it sets the contents of the target
-register to the result obtained by executing `value-proc'. Then it
-advances the `pc' to the next instruction by running the procedure
-
- (define (advance-pc pc)
- (set-contents! pc (cdr (get-contents pc))))
-
- `Advance-pc' is the normal termination for all instructions except
-`branch' and `goto'.
-
-`Test', `branch', and `goto' instructions
-.........................................
-
-`Make-test' handles `test' instructions in a similar way. It extracts
-the expression that specifies the condition to be tested and generates
-an execution procedure for it. At simulation time, the procedure for
-the condition is called, the result is assigned to the `flag' register,
-and the `pc' is advanced:
-
- (define (make-test inst machine labels operations flag pc)
- (let ((condition (test-condition inst)))
- (if (operation-exp? condition)
- (let ((condition-proc
- (make-operation-exp
- condition machine labels operations)))
- (lambda ()
- (set-contents! flag (condition-proc))
- (advance-pc pc)))
- (error "Bad TEST instruction -- ASSEMBLE" inst))))
-
- (define (test-condition test-instruction)
- (cdr test-instruction))
-
- The execution procedure for a `branch' instruction checks the
-contents of the `flag' register and either sets the contents of the
-`pc' to the branch destination (if the branch is taken) or else just
-advances the `pc' (if the branch is not taken). Notice that the
-indicated destination in a `branch' instruction must be a label, and
-the `make-branch' procedure enforces this. Notice also that the label
-is looked up at assembly time, not each time the `branch' instruction
-is simulated.
-
- (define (make-branch inst machine labels flag pc)
- (let ((dest (branch-dest inst)))
- (if (label-exp? dest)
- (let ((insts
- (lookup-label labels (label-exp-label dest))))
- (lambda ()
- (if (get-contents flag)
- (set-contents! pc insts)
- (advance-pc pc))))
- (error "Bad BRANCH instruction -- ASSEMBLE" inst))))
-
- (define (branch-dest branch-instruction)
- (cadr branch-instruction))
-
- A `goto' instruction is similar to a branch, except that the
-destination may be specified either as a label or as a register, and
-there is no condition to check--the `pc' is always set to the new
-destination.
-
- (define (make-goto inst machine labels pc)
- (let ((dest (goto-dest inst)))
- (cond ((label-exp? dest)
- (let ((insts
- (lookup-label labels
- (label-exp-label dest))))
- (lambda () (set-contents! pc insts))))
- ((register-exp? dest)
- (let ((reg
- (get-register machine
- (register-exp-reg dest))))
- (lambda ()
- (set-contents! pc (get-contents reg)))))
- (else (error "Bad GOTO instruction -- ASSEMBLE"
- inst)))))
-
- (define (goto-dest goto-instruction)
- (cadr goto-instruction))
-
-Other instructions
-..................
-
-The stack instructions `save' and `restore' simply use the stack with
-the designated register and advance the `pc':
-
- (define (make-save inst machine stack pc)
- (let ((reg (get-register machine
- (stack-inst-reg-name inst))))
- (lambda ()
- (push stack (get-contents reg))
- (advance-pc pc))))
-
- (define (make-restore inst machine stack pc)
- (let ((reg (get-register machine
- (stack-inst-reg-name inst))))
- (lambda ()
- (set-contents! reg (pop stack))
- (advance-pc pc))))
-
- (define (stack-inst-reg-name stack-instruction)
- (cadr stack-instruction))
-
- The final instruction type, handled by `make-perform', generates an
-execution procedure for the action to be performed. At simulation
-time, the action procedure is executed and the `pc' advanced.
-
- (define (make-perform inst machine labels operations pc)
- (let ((action (perform-action inst)))
- (if (operation-exp? action)
- (let ((action-proc
- (make-operation-exp
- action machine labels operations)))
- (lambda ()
- (action-proc)
- (advance-pc pc)))
- (error "Bad PERFORM instruction -- ASSEMBLE" inst))))
-
- (define (perform-action inst) (cdr inst))
-
-Execution procedures for subexpressions
-.......................................
-
-The value of a `reg', `label', or `const' expression may be needed for
-assignment to a register (`make-assign') or for input to an operation
-(`make-operation-exp', below). The following procedure generates
-execution procedures to produce values for these expressions during the
-simulation:
-
- (define (make-primitive-exp exp machine labels)
- (cond ((constant-exp? exp)
- (let ((c (constant-exp-value exp)))
- (lambda () c)))
- ((label-exp? exp)
- (let ((insts
- (lookup-label labels
- (label-exp-label exp))))
- (lambda () insts)))
- ((register-exp? exp)
- (let ((r (get-register machine
- (register-exp-reg exp))))
- (lambda () (get-contents r))))
- (else
- (error "Unknown expression type -- ASSEMBLE" exp))))
-
- The syntax of `reg', `label', and `const' expressions is determined
-by
-
- (define (register-exp? exp) (tagged-list? exp 'reg))
-
- (define (register-exp-reg exp) (cadr exp))
-
- (define (constant-exp? exp) (tagged-list? exp 'const))
-
- (define (constant-exp-value exp) (cadr exp))
-
- (define (label-exp? exp) (tagged-list? exp 'label))
-
- (define (label-exp-label exp) (cadr exp))
-
- `Assign', `perform', and `test' instructions may include the
-application of a machine operation (specified by an `op' expression) to
-some operands (specified by `reg' and `const' expressions). The
-following procedure produces an execution procedure for an "operation
-expression"--a list containing the operation and operand expressions
-from the instruction:
-
- (define (make-operation-exp exp machine labels operations)
- (let ((op (lookup-prim (operation-exp-op exp) operations))
- (aprocs
- (map (lambda (e)
- (make-primitive-exp e machine labels))
- (operation-exp-operands exp))))
- (lambda ()
- (apply op (map (lambda (p) (p)) aprocs)))))
-
- The syntax of operation expressions is determined by
-
- (define (operation-exp? exp)
- (and (pair? exp) (tagged-list? (car exp) 'op)))
-
- (define (operation-exp-op operation-exp)
- (cadr (car operation-exp)))
-
- (define (operation-exp-operands operation-exp)
- (cdr operation-exp))
-
- Observe that the treatment of operation expressions is very much
-like the treatment of procedure applications by the
-`analyze-application' procedure in the evaluator of section *Note
-4-1-7:: in that we generate an execution procedure for each operand.
-At simulation time, we call the operand procedures and apply the Scheme
-procedure that simulates the operation to the resulting values. The
-simulation procedure is found by looking up the operation name in the
-operation table for the machine:
-
- (define (lookup-prim symbol operations)
- (let ((val (assoc symbol operations)))
- (if val
- (cadr val)
- (error "Unknown operation -- ASSEMBLE" symbol))))
-
- *Exercise 5.9:* The treatment of machine operations above permits
- them to operate on labels as well as on constants and the contents
- of registers. Modify the expression-processing procedures to
- enforce the condition that operations can be used only with
- registers and constants.
-
- *Exercise 5.10:* Design a new syntax for register-machine
- instructions and modify the simulator to use your new syntax. Can
- you implement your new syntax without changing any part of the
- simulator except the syntax procedures in this section?
-
- *Exercise 5.11:* When we introduced `save' and `restore' in
- section *Note 5-1-4::, we didn't specify what would happen if you
- tried to restore a register that was not the last one saved, as in
- the sequence
-
- (save y)
- (save x)
- (restore y)
-
- There are several reasonable possibilities for the meaning of
- `restore':
-
- a. `(restore y)' puts into `y' the last value saved on the stack,
- regardless of what register that value came from. This is
- the way our simulator behaves. Show how to take advantage of
- this behavior to eliminate one instruction from the Fibonacci
- machine of section *Note 5-1-4:: (*Note Figure 5-12::).
-
- b. `(restore y)' puts into `y' the last value saved on the
- stack, but only if that value was saved from `y'; otherwise,
- it signals an error. Modify the simulator to behave this
- way. You will have to change `save' to put the register name
- on the stack along with the value.
-
- c. `(restore y)' puts into `y' the last value saved from `y'
- regardless of what other registers were saved after `y' and
- not restored. Modify the simulator to behave this way. You
- will have to associate a separate stack with each register.
- You should make the `initialize-stack' operation initialize
- all the register stacks.
-
-
- *Exercise 5.12:* The simulator can be used to help determine the
- data paths required for implementing a machine with a given
- controller. Extend the assembler to store the following
- information in the machine model:
-
- * a list of all instructions, with duplicates removed, sorted
- by instruction type (`assign', `goto', and so on);
-
- * a list (without duplicates) of the registers used to hold
- entry points (these are the registers referenced by `goto'
- instructions);
-
- * a list (without duplicates) of the registers that are `save'd
- or `restore'd;
-
- * for each register, a list (without duplicates) of the sources
- from which it is assigned (for example, the sources for
- register `val' in the factorial machine of *Note Figure
- 5-11:: are `(const 1)' and `((op *) (reg n) (reg val))').
-
-
- Extend the message-passing interface to the machine to provide
- access to this new information. To test your analyzer, define the
- Fibonacci machine from *Note Figure 5-12:: and examine the lists
- you constructed.
-
- *Exercise 5.13:* Modify the simulator so that it uses the
- controller sequence to determine what registers the machine has
- rather than requiring a list of registers as an argument to
- `make-machine'. Instead of pre-allocating the registers in
- `make-machine', you can allocate them one at a time when they are
- first seen during assembly of the instructions.
-
-\1f
-File: sicp.info, Node: 5-2-4, Prev: 5-2-3, Up: 5-2
-
-5.2.4 Monitoring Machine Performance
-------------------------------------
-
-Simulation is useful not only for verifying the correctness of a
-proposed machine design but also for measuring the machine's
-performance. For example, we can install in our simulation program a
-"meter" that measures the number of stack operations used in a
-computation. To do this, we modify our simulated stack to keep track
-of the number of times registers are saved on the stack and the maximum
-depth reached by the stack, and add a message to the stack's interface
-that prints the statistics, as shown below. We also add an operation
-to the basic machine model to print the stack statistics, by
-initializing `the-ops' in `make-new-machine' to
-
- (list (list 'initialize-stack
- (lambda () (stack 'initialize)))
- (list 'print-stack-statistics
- (lambda () (stack 'print-statistics))))
-
- Here is the new version of `make-stack':
-
- (define (make-stack)
- (let ((s '())
- (number-pushes 0)
- (max-depth 0)
- (current-depth 0))
- (define (push x)
- (set! s (cons x s))
- (set! number-pushes (+ 1 number-pushes))
- (set! current-depth (+ 1 current-depth))
- (set! max-depth (max current-depth max-depth)))
- (define (pop)
- (if (null? s)
- (error "Empty stack -- POP")
- (let ((top (car s)))
- (set! s (cdr s))
- (set! current-depth (- current-depth 1))
- top)))
- (define (initialize)
- (set! s '())
- (set! number-pushes 0)
- (set! max-depth 0)
- (set! current-depth 0)
- 'done)
- (define (print-statistics)
- (newline)
- (display (list 'total-pushes '= number-pushes
- 'maximum-depth '= max-depth)))
- (define (dispatch message)
- (cond ((eq? message 'push) push)
- ((eq? message 'pop) (pop))
- ((eq? message 'initialize) (initialize))
- ((eq? message 'print-statistics)
- (print-statistics))
- (else
- (error "Unknown request -- STACK" message))))
- dispatch))
-
- *Note Exercise 5-15:: through *Note Exercise 5-19:: describe other
-useful monitoring and debugging features that can be added to the
-register-machine simulator.
-
- *Exercise 5.14:* Measure the number of pushes and the maximum
- stack depth required to compute n! for various small values of n
- using the factorial machine shown in *Note Figure 5-11::. From
- your data determine formulas in terms of n for the total number of
- push operations and the maximum stack depth used in computing n!
- for any n > 1. Note that each of these is a linear function of n
- and is thus determined by two constants. In order to get the
- statistics printed, you will have to augment the factorial machine
- with instructions to initialize the stack and print the
- statistics. You may want to also modify the machine so that it
- repeatedly reads a value for n, computes the factorial, and prints
- the result (as we did for the GCD machine in *Note Figure 5-4::),
- so that you will not have to repeatedly invoke
- `get-register-contents', `set-register-contents!', and `start'.
-
- *Exercise 5.15:* Add counting "instruction counting" to the
- register machine simulation. That is, have the machine model keep
- track of the number of instructions executed. Extend the machine
- model's interface to accept a new message that prints the value of
- the instruction count and resets the count to zero.
-
- *Exercise 5.16:* Augment the simulator to provide for "instruction
- tracing". That is, before each instruction is executed, the
- simulator should print the text of the instruction. Make the
- machine model accept `trace-on' and `trace-off' messages to turn
- tracing on and off.
-
- *Exercise 5.17:* Extend the instruction tracing of *Note Exercise
- 5-16:: so that before printing an instruction, the simulator
- prints any labels that immediately precede that instruction in the
- controller sequence. Be careful to do this in a way that does not
- interfere with instruction counting (*Note Exercise 5-15::). You
- will have to make the simulator retain the necessary label
- information.
-
- *Exercise 5.18:* Modify the `make-register' procedure of section
- *Note 5-2-1:: so that registers can be traced. Registers should
- accept messages that turn tracing on and off. When a register is
- traced, assigning a value to the register should print the name of
- the register, the old contents of the register, and the new
- contents being assigned. Extend the interface to the machine
- model to permit you to turn tracing on and off for designated
- machine registers.
-
- *Exercise 5.19:* Alyssa P. Hacker wants a "breakpoint" feature in
- the simulator to help her debug her machine designs. You have
- been hired to install this feature for her. She wants to be able
- to specify a place in the controller sequence where the simulator
- will stop and allow her to examine the state of the machine. You
- are to implement a procedure
-
- (set-breakpoint <MACHINE> <LABEL> <N>)
-
- that sets a breakpoint just before the nth instruction after the
- given label. For example,
-
- (set-breakpoint gcd-machine 'test-b 4)
-
- installs a breakpoint in `gcd-machine' just before the assignment
- to register `a'. When the simulator reaches the breakpoint it
- should print the label and the offset of the breakpoint and stop
- executing instructions. Alyssa can then use
- `get-register-contents' and `set-register-contents!' to manipulate
- the state of the simulated machine. She should then be able to
- continue execution by saying
-
- (proceed-machine <MACHINE>)
-
- She should also be able to remove a specific breakpoint by means of
-
- (cancel-breakpoint <MACHINE> <LABEL> <N>)
-
- or to remove all breakpoints by means of
-
- (cancel-all-breakpoints <MACHINE>)
-
-\1f
-File: sicp.info, Node: 5-3, Next: 5-4, Prev: 5-2, Up: Chapter 5
-
-5.3 Storage Allocation and Garbage Collection
-=============================================
-
-In section *Note 5-4::, we will show how to implement a Scheme
-evaluator as a register machine. In order to simplify the discussion,
-we will assume that our register machines can be equipped with a "list-structured
-memory", in which the basic operations for manipulating list-structured
-data are primitive. Postulating the existence of such a memory is a
-useful abstraction when one is focusing on the mechanisms of control in
-a Scheme interpreter, but this does not reflect a realistic view of the
-actual primitive data operations of contemporary computers. To obtain
-a more complete picture of how a Lisp system operates, we must
-investigate how list structure can be represented in a way that is
-compatible with conventional computer memories.
-
- There are two considerations in implementing list structure. The
-first is purely an issue of representation: how to represent the
-"box-and-pointer" structure of Lisp pairs, using only the storage and
-addressing capabilities of typical computer memories. The second issue
-concerns the management of memory as a computation proceeds. The
-operation of a Lisp system depends crucially on the ability to
-continually create new data objects. These include objects that are
-explicitly created by the Lisp procedures being interpreted as well as
-structures created by the interpreter itself, such as environments and
-argument lists. Although the constant creation of new data objects
-would pose no problem on a computer with an infinite amount of rapidly
-addressable memory, computer memories are available only in finite
-sizes (more's the pity). Lisp systems thus provide an "automatic
-storage allocation" facility to support the illusion of an infinite
-memory. When a data object is no longer needed, the memory allocated
-to it is automatically recycled and used to construct new data objects.
-There are various techniques for providing such automatic storage
-allocation. The method we shall discuss in this section is called "garbage
-collection".
-
-* Menu:
-
-* 5-3-1:: Memory as Vectors
-* 5-3-2:: Maintaining the Illusion of Infinite Memory
-
-\1f
-File: sicp.info, Node: 5-3-1, Next: 5-3-2, Prev: 5-3, Up: 5-3
-
-5.3.1 Memory as Vectors
------------------------
-
-A conventional computer memory can be thought of as an array of
-cubbyholes, each of which can contain a piece of information. Each
-cubbyhole has a unique name, called its "address" or "location".
-Typical memory systems provide two primitive operations: one that
-fetches the data stored in a specified location and one that assigns
-new data to a specified location. Memory addresses can be incremented
-to support sequential access to some set of the cubbyholes. More
-generally, many important data operations require that memory addresses
-be treated as data, which can be stored in memory locations and
-manipulated in machine registers. The representation of list structure
-is one application of such "address arithmetic".
-
- To model computer memory, we use a new kind of data structure called
-a "vector". Abstractly, a vector is a compound data object whose
-individual elements can be accessed by means of an integer index in an
-amount of time that is independent of the index.(1) In order to
-describe memory operations, we use two primitive Scheme procedures for
-manipulating vectors:
-
- * `(vector-ref <VECTOR> <N>)' returns the nth element of the vector.
-
- * `(vector-set! <VECTOR> <N> <VALUE>)' sets the nth element of the
- vector to the designated value.
-
-
- For example, if `v' is a vector, then `(vector-ref v 5)' gets the
-fifth entry in the vector `v' and `(vector-set! v 5 7)' changes the
-value of the fifth entry of the vector `v' to 7.(2) For computer
-memory, this access can be implemented through the use of address
-arithmetic to combine a address "base address" that specifies the
-beginning location of a vector in memory with an "index" that specifies
-the offset of a particular element of the vector.
-
-Representing Lisp data
-......................
-
-We can use vectors to implement the basic pair structures required for a
-list-structured memory. Let us imagine that computer memory is divided
-into two vectors: `the-cars' and `the-cdrs'. We will represent list
-structure as follows: A pointer to a pair is an index into the two
-vectors. The `car' of the pair is the entry in `the-cars' with the
-designated index, and the `cdr' of the pair is the entry in `the-cdrs'
-with the designated index. We also need a representation for objects
-other than pairs (such as numbers and symbols) and a way to distinguish
-one kind of data from another. There are many methods of accomplishing
-this, but they all reduce to using "typed pointers", that is, to
-extending the notion of "pointer" to include information on data
-type.(3) The data type enables the system to distinguish a pointer to a
-pair (which consists of the "pair" data type and an index into the
-memory vectors) from pointers to other kinds of data (which consist of
-some other data type and whatever is being used to represent data of
-that type). Two data objects are considered to be the same (`eq?') if
-their pointers are identical.(4) *Note Figure 5-14:: illustrates the
-use of this method to represent the list `((1 2) 3 4)', whose
-box-and-pointer diagram is also shown. We use letter prefixes to
-denote the data-type information. Thus, a pointer to the pair with
-index 5 is denoted `p5', the empty list is denoted by the pointer `e0',
-and a pointer to the number 4 is denoted `n4'. In the box-and-pointer
-diagram, we have indicated at the lower left of each pair the vector
-index that specifies where the `car' and `cdr' of the pair are stored.
-The blank locations in `the-cars' and `the-cdrs' may contain parts of
-other list structures (not of interest here).
-
- *Figure 5.14:* Box-and-pointer and memory-vector representations
- of the list `((1 2) 3 4)'.
-
- +---+---+ +---+---+ +---+---+
- ((1 2) 3 4) -->| * | *-+-------------->| * | *-+--->| * | / |
- +-|-+---+ +-|-+---+ +-|-+---+
- 1 | 2 | 4 |
- V V V
- +---+---+ +---+---+ +---+ +---+
- | * | *-+--->| * | / | | 3 | | 4 |
- +-|-+---+ +-|-+---+ +---+ +---+
- 5 | 7 |
- V V
- +---+ +---+
- | 1 | | 2 |
- +---+ +---+
-
- Index 0 1 2 3 4 5 6 7 8 ...
- +----+----+----+----+----+----+----+----+----+----
- the-cars | | p5 | n3 | | n4 | n1 | | n2 | | ...
- +----+----+----+----+----+----+----+----+----+----
- the-cdrs | | p2 | p4 | | e0 | p7 | | e0 | | ...
- +----+----+----+----+----+----+----+----+----+----
-
- A pointer to a number, such as `n4', might consist of a type
-indicating numeric data together with the actual representation of the
-number 4.(5) To deal with numbers that are too large to be represented
-in the fixed amount of space allocated for a single pointer, we could
-use a distinct "bignum" data type, for which the pointer designates a
-list in which the parts of the number are stored.(6)
-
- A symbol might be represented as a typed pointer that designates a
-sequence of the characters that form the symbol's printed
-representation. This sequence is constructed by the Lisp reader when
-the character string is initially encountered in input. Since we want
-two instances of a symbol to be recognized as the "same" symbol by
-`eq?' and we want `eq?' to be a simple test for equality of pointers,
-we must ensure that if the reader sees the same character string twice,
-it will use the same pointer (to the same sequence of characters) to
-represent both occurrences. To accomplish this, the reader maintains a
-table, traditionally called the "obarray", of all the symbols it has
-ever encountered. When the reader encounters a character string and is
-about to construct a symbol, it checks the obarray to see if it has ever
-before seen the same character string. If it has not, it uses the
-characters to construct a new symbol (a typed pointer to a new
-character sequence) and enters this pointer in the obarray. If the
-reader has seen the string before, it returns the symbol pointer stored
-in the obarray. This process of replacing character strings by unique
-pointers is called "interning" symbols.
-
-Implementing the primitive list operations
-..........................................
-
-Given the above representation scheme, we can replace each "primitive"
-list operation of a register machine with one or more primitive vector
-operations. We will use two registers, `the-cars' and `the-cdrs', to
-identify the memory vectors, and will assume that `vector-ref' and
-`vector-set!' are available as primitive operations. We also assume
-that numeric operations on pointers (such as incrementing a pointer,
-using a pair pointer to index a vector, or adding two numbers) use only
-the index portion of the typed pointer.
-
- For example, we can make a register machine support the instructions
-
- (assign <REG_1> (op car) (reg <REG_2>))
-
- (assign <REG_1> (op cdr) (reg <REG_2>))
-
-if we implement these, respectively, as
-
- (assign <REG_1> (op vector-ref) (reg the-cars) (reg <REG_2>))
-
- (assign <REG_1> (op vector-ref) (reg the-cdrs) (reg <REG_2>))
-
- The instructions
-
- (perform (op set-car!) (reg <REG_1>) (reg <REG_2>))
-
- (perform (op set-cdr!) (reg <REG_1>) (reg <REG_2>))
-
-are implemented as
-
- (perform
- (op vector-set!) (reg the-cars) (reg <REG_1>) (reg <REG_2>))
-
- (perform
- (op vector-set!) (reg the-cdrs) (reg <REG_1>) (reg <REG_2>))
-
- `Cons' is performed by allocating an unused index and storing the
-arguments to `cons' in `the-cars' and `the-cdrs' at that indexed vector
-position. We presume that there is a special register, `free', that
-always holds a pair pointer containing the next available index, and
-that we can increment the index part of that pointer to find the next
-free location.(7) For example, the instruction
-
- (assign <REG_1> (op cons) (reg <REG_2>) (reg <REG_3>))
-
-is implemented as the following sequence of vector operations:(8)
-
- (perform
- (op vector-set!) (reg the-cars) (reg free) (reg <REG_2>))
- (perform
- (op vector-set!) (reg the-cdrs) (reg free) (reg <REG_3>))
- (assign <REG_1> (reg free))
- (assign free (op +) (reg free) (const 1))
-
- The `eq?' operation
-
- (op eq?) (reg <REG_1>) (reg <REG_2>)
-
-simply tests the equality of all fields in the registers, and
-predicates such as `pair?', `null?', `symbol?', and `number?' need only
-check the type field.
-
-Implementing stacks
-...................
-
-Although our register machines use stacks, we need do nothing special
-here, since stacks can be modeled in terms of lists. The stack can be
-a list of the saved values, pointed to by a special register
-`the-stack'. Thus, ` (save <REG>)' can be implemented as
-
- (assign the-stack (op cons) (reg <REG>) (reg the-stack))
-
-Similarly, `(restore <REG>)' can be implemented as
-
- (assign <REG> (op car) (reg the-stack))
- (assign the-stack (op cdr) (reg the-stack))
-
-and `(perform (op initialize-stack))' can be implemented as
-
- (assign the-stack (const ()))
-
- These operations can be further expanded in terms of the vector
-operations given above. In conventional computer architectures,
-however, it is usually advantageous to allocate the stack as a separate
-vector. Then pushing and popping the stack can be accomplished by
-incrementing or decrementing an index into that vector.
-
- *Exercise 5.20:* Draw the box-and-pointer representation and the
- memory-vector representation (as in *Note Figure 5-14::) of the
- list structure produced by
-
- (define x (cons 1 2))
- (define y (list x x))
-
- with the `free' pointer initially `p1'. What is the final value of
- `free' ? What pointers represent the values of `x' and `y' ?
-
- *Exercise 5.21:* Implement register machines for the following
- procedures. Assume that the list-structure memory operations are
- available as machine primitives.
-
- a. Recursive `count-leaves':
-
- (define (count-leaves tree)
- (cond ((null? tree) 0)
- ((not (pair? tree)) 1)
- (else (+ (count-leaves (car tree))
- (count-leaves (cdr tree))))))
-
- b. Recursive `count-leaves' with explicit counter:
-
- (define (count-leaves tree)
- (define (count-iter tree n)
- (cond ((null? tree) n)
- ((not (pair? tree)) (+ n 1))
- (else (count-iter (cdr tree)
- (count-iter (car tree) n)))))
- (count-iter tree 0))
-
- *Exercise 5.22:* *Note Exercise 3-12:: of section *Note 3-3-1::
- presented an `append' procedure that appends two lists to form a
- new list and an `append!' procedure that splices two lists
- together. Design a register machine to implement each of these
- procedures. Assume that the list-structure memory operations are
- available as primitive operations.
-
- ---------- Footnotes ----------
-
- (1) We could represent memory as lists of items. However, the
-access time would then not be independent of the index, since accessing
-the nth element of a list requires n - 1 `cdr' operations.
-
- (2) For completeness, we should specify a `make-vector' operation
-that constructs vectors. However, in the present application we will
-use vectors only to model fixed divisions of the computer memory.
-
- (3) This is precisely the same "tagged data" idea we introduced in
-*Note Chapter 2:: for dealing with generic operations. Here, however,
-the data types are included at the primitive machine level rather than
-constructed through the use of lists.
-
- (4) Type information may be encoded in a variety of ways, depending
-on the details of the machine on which the Lisp system is to be
-implemented. The execution efficiency of Lisp programs will be
-strongly dependent on how cleverly this choice is made, but it is
-difficult to formulate general design rules for good choices. The most
-straightforward way to implement typed pointers is to allocate a fixed
-set of bits in each pointer to be a "type field" that encodes the data
-type. Important questions to be addressed in designing such a
-representation include the following: How many type bits are required?
-How large must the vector indices be? How efficiently can the
-primitive machine instructions be used to manipulate the type fields of
-pointers? Machines that include special hardware for the efficient
-handling of type fields are said to have architectures "tagged
-architectures".
-
- (5) This decision on the representation of numbers determines whether
-`eq?', which tests equality of pointers, can be used to test for
-equality of numbers. If the pointer contains the number itself, then
-equal numbers will have the same pointer. But if the pointer contains
-the index of a location where the number is stored, equal numbers will
-be guaranteed to have equal pointers only if we are careful never to
-store the same number in more than one location.
-
- (6) This is just like writing a number as a sequence of digits,
-except that each "digit" is a number between 0 and the largest number
-that can be stored in a single pointer.
-
- (7) There are other ways of finding free storage. For example, we
-could link together all the unused pairs into a "free list". Our free
-locations are consecutive (and hence can be accessed by incrementing a
-pointer) because we are using a compacting garbage collector, as we
-will see in section *Note 5-3-2::.
-
- (8) This is essentially the implementation of `cons' in terms of
-`set-car!' and `set-cdr!', as described in section *Note 3-3-1::. The
-operation `get-new-pair' used in that implementation is realized here
-by the `free' pointer.
-
-\1f
-File: sicp.info, Node: 5-3-2, Prev: 5-3-1, Up: 5-3
-
-5.3.2 Maintaining the Illusion of Infinite Memory
--------------------------------------------------
-
-The representation method outlined in section *Note 5-3-1:: solves the
-problem of implementing list structure, provided that we have an
-infinite amount of memory. With a real computer we will eventually run
-out of free space in which to construct new pairs.(1) However, most of
-the pairs generated in a typical computation are used only to hold
-intermediate results. After these results are accessed, the pairs are
-no longer needed--they are "garbage". For instance, the computation
-
- (accumulate + 0 (filter odd? (enumerate-interval 0 n)))
-
-constructs two lists: the enumeration and the result of filtering the
-enumeration. When the accumulation is complete, these lists are no
-longer needed, and the allocated memory can be reclaimed. If we can
-arrange to collect all the garbage periodically, and if this turns out
-to recycle memory at about the same rate at which we construct new
-pairs, we will have preserved the illusion that there is an infinite
-amount of memory.
-
- In order to recycle pairs, we must have a way to determine which
-allocated pairs are not needed (in the sense that their contents can no
-longer influence the future of the computation). The method we shall
-examine for accomplishing this is known as "garbage collection".
-Garbage collection is based on the observation that, at any moment in a
-Lisp interpretation, the only objects that can affect the future of the
-computation are those that can be reached by some succession of `car'
-and `cdr' operations starting from the pointers that are currently in
-the machine registers.(2) Any memory cell that is not so accessible may
-be recycled.
-
- There are many ways to perform garbage collection. The method we
-shall examine here is called "stop-and-copy". The basic idea is to
-divide memory into two halves: "working memory" and "free memory."
-When `cons' constructs pairs, it allocates these in working memory.
-When working memory is full, we perform garbage collection by locating
-all the useful pairs in working memory and copying these into
-consecutive locations in free memory. (The useful pairs are located by
-tracing all the `car' and `cdr' pointers, starting with the machine
-registers.) Since we do not copy the garbage, there will presumably be
-additional free memory that we can use to allocate new pairs. In
-addition, nothing in the working memory is needed, since all the useful
-pairs in it have been copied. Thus, if we interchange the roles of
-working memory and free memory, we can continue processing; new pairs
-will be allocated in the new working memory (which was the old free
-memory). When this is full, we can copy the useful pairs into the new
-free memory (which was the old working memory).(3)
-
-Implementation of a stop-and-copy garbage collector
-...................................................
-
-We now use our register-machine language to describe the stop-and-copy
-algorithm in more detail. We will assume that there is a register
-called `root' that contains a pointer to a structure that eventually
-points at all accessible data. This can be arranged by storing the
-contents of all the machine registers in a pre-allocated list pointed
-at by `root' just before starting garbage collection.(4) We also assume
-that, in addition to the current working memory, there is free memory
-available into which we can copy the useful data. The current working
-memory consists of vectors whose base addresses are in registers called
-`the-cars' and `the-cdrs', and the free memory is in registers called
-`new-cars' and `new-cdrs'.
-
- Garbage collection is triggered when we exhaust the free cells in
-the current working memory, that is, when a `cons' operation attempts
-to increment the `free' pointer beyond the end of the memory vector.
-When the garbage-collection process is complete, the `root' pointer
-will point into the new memory, all objects accessible from the `root'
-will have been moved to the new memory, and the `free' pointer will
-indicate the next place in the new memory where a new pair can be
-allocated. In addition, the roles of working memory and new memory
-will have been interchanged--new pairs will be constructed in the new
-memory, beginning at the place indicated by `free', and the (previous)
-working memory will be available as the new memory for the next garbage
-collection. *Note Figure 5-15:: shows the arrangement of memory just
-before and just after garbage collection.
-
- *Figure 5.15:* Reconfiguration of memory by the garbage-collection
- process.
-
- Just before garbage collection
-
- +------------------------------------+
- the-cars | | working
- | mixture of useful data and garbage | memory
- the-cdrs | |
- +------------------------------------+
- ^
- | free
-
- +------------------------------------+
- new-cars | | free
- | free memory | memory
- new-cdrs | |
- +------------------------------------+
-
- Just after garbage collection
-
- +------------------------------------+
- new-cars | | new
- | discarded memory | free
- new-cdrs | | memory
- +------------------------------------+
-
- +------------------+-----------------+
- the-cars | | | new
- | useful data | free area | working
- the-cdrs | | | memory
- +------------------+-----------------+
- ^
- | free
-
- The state of the garbage-collection process is controlled by
-maintaining two pointers: `free' and `scan'. These are initialized to
-point to the beginning of the new memory. The algorithm begins by
-relocating the pair pointed at by `root' to the beginning of the new
-memory. The pair is copied, the `root' pointer is adjusted to point to
-the new location, and the `free' pointer is incremented. In addition,
-the old location of the pair is marked to show that its contents have
-been moved. This marking is done as follows: In the `car' position, we
-place a special tag that signals that this is an already-moved object.
-(Such an object is traditionally called a "broken heart".)(5) In the
-`cdr' position we place a "forwarding address" that points at the
-location to which the object has been moved.
-
- After relocating the root, the garbage collector enters its basic
-cycle. At each step in the algorithm, the `scan' pointer (initially
-pointing at the relocated root) points at a pair that has been moved to
-the new memory but whose `car' and `cdr' pointers still refer to
-objects in the old memory. These objects are each relocated, and the
-`scan' pointer is incremented. To relocate an object (for example, the
-object indicated by the `car' pointer of the pair we are scanning) we
-check to see if the object has already been moved (as indicated by the
-presence of a broken-heart tag in the `car' position of the object).
-If the object has not already been moved, we copy it to the place
-indicated by `free', update `free', set up a broken heart at the
-object's old location, and update the pointer to the object (in this
-example, the `car' pointer of the pair we are scanning) to point to the
-new location. If the object has already been moved, its forwarding
-address (found in the `cdr' position of the broken heart) is
-substituted for the pointer in the pair being scanned. Eventually, all
-accessible objects will have been moved and scanned, at which point the
-`scan' pointer will overtake the `free' pointer and the process will
-terminate.
-
- We can specify the stop-and-copy algorithm as a sequence of
-instructions for a register machine. The basic step of relocating an
-object is accomplished by a subroutine called
-`relocate-old-result-in-new'. This subroutine gets its argument, a
-pointer to the object to be relocated, from a register named `old'. It
-relocates the designated object (incrementing `free' in the process),
-puts a pointer to the relocated object into a register called `new',
-and returns by branching to the entry point stored in the register
-`relocate-continue'. To begin garbage collection, we invoke this
-subroutine to relocate the `root' pointer, after initializing `free'
-and `scan'. When the relocation of `root' has been accomplished, we
-install the new pointer as the new `root' and enter the main loop of the
-garbage collector.
-
- begin-garbage-collection
- (assign free (const 0))
- (assign scan (const 0))
- (assign old (reg root))
- (assign relocate-continue (label reassign-root))
- (goto (label relocate-old-result-in-new))
- reassign-root
- (assign root (reg new))
- (goto (label gc-loop))
-
- In the main loop of the garbage collector we must determine whether
-there are any more objects to be scanned. We do this by testing
-whether the `scan' pointer is coincident with the `free' pointer. If
-the pointers are equal, then all accessible objects have been
-relocated, and we branch to `gc-flip', which cleans things up so that
-we can continue the interrupted computation. If there are still pairs
-to be scanned, we call the relocate subroutine to relocate the `car' of
-the next pair (by placing the `car' pointer in `old'). The
-`relocate-continue' register is set up so that the subroutine will
-return to update the `car' pointer.
-
- gc-loop
- (test (op =) (reg scan) (reg free))
- (branch (label gc-flip))
- (assign old (op vector-ref) (reg new-cars) (reg scan))
- (assign relocate-continue (label update-car))
- (goto (label relocate-old-result-in-new))
-
- At `update-car', we modify the `car' pointer of the pair being
-scanned, then proceed to relocate the `cdr' of the pair. We return to
-`update-cdr' when that relocation has been accomplished. After
-relocating and updating the `cdr', we are finished scanning that pair,
-so we continue with the main loop.
-
- update-car
- (perform
- (op vector-set!) (reg new-cars) (reg scan) (reg new))
- (assign old (op vector-ref) (reg new-cdrs) (reg scan))
- (assign relocate-continue (label update-cdr))
- (goto (label relocate-old-result-in-new))
-
- update-cdr
- (perform
- (op vector-set!) (reg new-cdrs) (reg scan) (reg new))
- (assign scan (op +) (reg scan) (const 1))
- (goto (label gc-loop))
-
- The subroutine `relocate-old-result-in-new' relocates objects as
-follows: If the object to be relocated (pointed at by `old') is not a
-pair, then we return the same pointer to the object unchanged (in
-`new'). (For example, we may be scanning a pair whose `car' is the
-number 4. If we represent the `car' by `n4', as described in section
-*Note 5-3-1::, then we want the "relocated" `car' pointer to still be
-`n4'.) Otherwise, we must perform the relocation. If the `car'
-position of the pair to be relocated contains a broken-heart tag, then
-the pair has in fact already been moved, so we retrieve the forwarding
-address (from the `cdr' position of the broken heart) and return this
-in `new'. If the pointer in `old' points at a yet-unmoved pair, then
-we move the pair to the first free cell in new memory (pointed at by
-`free') and set up the broken heart by storing a broken-heart tag and
-forwarding address at the old location. `Relocate-old-result-in-new'
-uses a register `oldcr' to hold the `car' or the `cdr' of the object
-pointed at by `old'.(6)
-
- relocate-old-result-in-new
- (test (op pointer-to-pair?) (reg old))
- (branch (label pair))
- (assign new (reg old))
- (goto (reg relocate-continue))
- pair
- (assign oldcr (op vector-ref) (reg the-cars) (reg old))
- (test (op broken-heart?) (reg oldcr))
- (branch (label already-moved))
- (assign new (reg free)) ; new location for pair
- ;; Update `free' pointer.
- (assign free (op +) (reg free) (const 1))
- ;; Copy the `car' and `cdr' to new memory.
- (perform (op vector-set!)
- (reg new-cars) (reg new) (reg oldcr))
- (assign oldcr (op vector-ref) (reg the-cdrs) (reg old))
- (perform (op vector-set!)
- (reg new-cdrs) (reg new) (reg oldcr))
- ;; Construct the broken heart.
- (perform (op vector-set!)
- (reg the-cars) (reg old) (const broken-heart))
- (perform
- (op vector-set!) (reg the-cdrs) (reg old) (reg new))
- (goto (reg relocate-continue))
- already-moved
- (assign new (op vector-ref) (reg the-cdrs) (reg old))
- (goto (reg relocate-continue))
-
- At the very end of the garbage-collection process, we interchange
-the role of old and new memories by interchanging pointers:
-interchanging `the-cars' with `new-cars', and `the-cdrs' with
-`new-cdrs'. We will then be ready to perform another garbage
-collection the next time memory runs out.
-
- gc-flip
- (assign temp (reg the-cdrs))
- (assign the-cdrs (reg new-cdrs))
- (assign new-cdrs (reg temp))
- (assign temp (reg the-cars))
- (assign the-cars (reg new-cars))
- (assign new-cars (reg temp))
-
- ---------- Footnotes ----------
-
- (1) This may not be true eventually, because memories may get large
-enough so that it would be impossible to run out of free memory in the
-lifetime of the computer. For example, there are about 3*(10^13),
-microseconds in a year, so if we were to `cons' once per microsecond we
-would need about 10^15 cells of memory to build a machine that could
-operate for 30 years without running out of memory. That much memory
-seems absurdly large by today's standards, but it is not physically
-impossible. On the other hand, processors are getting faster and a
-future computer may have large numbers of processors operating in
-parallel on a single memory, so it may be possible to use up memory
-much faster than we have postulated.
-
- (2) We assume here that the stack is represented as a list as
-described in section *Note 5-3-1::, so that items on the stack are
-accessible via the pointer in the stack register.
-
- (3) This idea was invented and first implemented by Minsky, as part
-of the implementation of Lisp for the PDP-1 at the MIT Research
-Laboratory of Electronics. It was further developed by Fenichel and
-Yochelson (1969) for use in the Lisp implementation for the Multics
-time-sharing system. Later, Baker (1978) developed a "real-time"
-version of the method, which does not require the computation to stop
-during garbage collection. Baker's idea was extended by Hewitt,
-Lieberman, and Moon (see Lieberman and Hewitt 1983) to take advantage
-of the fact that some structure is more volatile and other structure is
-more permanent.
-
- An alternative commonly used garbage-collection technique is the "mark-sweep"
-method. This consists of tracing all the structure accessible from the
-machine registers and marking each pair we reach. We then scan all of
-memory, and any location that is unmarked is "swept up" as garbage and
-made available for reuse. A full discussion of the mark-sweep method
-can be found in Allen 1978.
-
- The Minsky-Fenichel-Yochelson algorithm is the dominant algorithm in
-use for large-memory systems because it examines only the useful part
-of memory. This is in contrast to mark-sweep, in which the sweep phase
-must check all of memory. A second advantage of stop-and-copy is that
-it is a "compacting" garbage collector. That is, at the end of the
-garbage-collection phase the useful data will have been moved to
-consecutive memory locations, with all garbage pairs compressed out.
-This can be an extremely important performance consideration in
-machines with virtual memory, in which accesses to widely separated
-memory addresses may require extra paging operations.
-
- (4) This list of registers does not include the registers used by
-the storage-allocation system--`root', `the-cars', `the-cdrs', and the
-other registers that will be introduced in this section.
-
- (5) The term _broken heart_ was coined by David Cressey, who wrote a
-garbage collector for MDL, a dialect of Lisp developed at MIT during
-the early 1970s.
-
- (6) The garbage collector uses the low-level predicate
-`pointer-to-pair?' instead of the list-structure `pair?' operation
-because in a real system there might be various things that are treated
-as pairs for garbage-collection purposes. For example, in a Scheme
-system that conforms to the IEEE standard a procedure object may be
-implemented as a special kind of "pair" that doesn't satisfy the
-`pair?' predicate. For simulation purposes, `pointer-to-pair?' can be
-implemented as `pair?'.
-
-\1f
-File: sicp.info, Node: 5-4, Next: 5-5, Prev: 5-3, Up: Chapter 5
-
-5.4 The Explicit-Control Evaluator
-==================================
-
-In section *Note 5-1:: we saw how to transform simple Scheme programs
-into descriptions of register machines. We will now perform this
-transformation on a more complex program, the metacircular evaluator of
-sections *Note 4-1-1::-*Note 4-1-4::, which shows how the behavior of a
-Scheme interpreter can be described in terms of the procedures `eval'
-and `apply'. The "explicit-control evaluator" that we develop in this
-section shows how the underlying procedure-calling and argument-passing
-mechanisms used in the evaluation process can be described in terms of
-operations on registers and stacks. In addition, the explicit-control
-evaluator can serve as an implementation of a Scheme interpreter,
-written in a language that is very similar to the native machine
-language of conventional computers. The evaluator can be executed by
-the register-machine simulator of section *Note 5-2::. Alternatively,
-it can be used as a starting point for building a machine-language
-implementation of a Scheme evaluator, or even a special-purpose machine
-for evaluating Scheme expressions. *Note Figure 5-16:: shows such a
-hardware implementation: a silicon chip that acts as an evaluator for
-Scheme. The chip designers started with the data-path and controller
-specifications for a register machine similar to the evaluator
-described in this section and used design automation programs to
-construct the integrated-circuit layout.(1)
-
-Registers and operations
-........................
-
-In designing the explicit-control evaluator, we must specify the
-operations to be used in our register machine. We described the
-metacircular evaluator in terms of abstract syntax, using procedures
-such as `quoted?' and `make-procedure'. In implementing the register
-machine, we could expand these procedures into sequences of elementary
-list-structure memory operations, and implement these operations on our
-register machine. However, this would make our evaluator very long,
-obscuring the basic structure with details. To clarify the
-presentation, we will include as primitive operations of the register
-machine the syntax procedures given in section *Note 4-1-2:: and the
-procedures for representing environments and other run-time data given
-in sections *Note 4-1-3:: and *Note 4-1-4::. In order to completely
-specify an evaluator that could be programmed in a low-level machine
-language or implemented in hardware, we would replace these operations
-by more elementary operations, using the list-structure implementation
-we described in section *Note 5-3::.
-
- *Figure 5.16:* A silicon-chip implementation of an evaluator for
- Scheme.
-
- [This figure is missing.]
-
-
- Our Scheme evaluator register machine includes a stack and seven
-registers: `exp', `env', `val', `continue', `proc', `argl', and `unev'.
-`Exp' is used to hold the expression to be evaluated, and `env'
-contains the environment in which the evaluation is to be performed.
-At the end of an evaluation, `val' contains the value obtained by
-evaluating the expression in the designated environment. The
-`continue' register is used to implement recursion, as explained in
-section *Note 5-1-4::. (The evaluator needs to call itself
-recursively, since evaluating an expression requires evaluating its
-subexpressions.) The registers `proc', `argl', and `unev' are used in
-evaluating combinations.
-
- We will not provide a data-path diagram to show how the registers and
-operations of the evaluator are connected, nor will we give the
-complete list of machine operations. These are implicit in the
-evaluator's controller, which will be presented in detail.
-
-* Menu:
-
-* 5-4-1:: The Core of the Explicit-Control Evaluator
-* 5-4-2:: Sequence Evaluation and Tail Recursion
-* 5-4-3:: Conditionals, Assignments, and Definitions
-* 5-4-4:: Running the Evaluator
-
- ---------- Footnotes ----------
-
- (1) See Batali et al. 1982 for more information on the chip and the
-method by which it was designed.
-
-\1f
-File: sicp.info, Node: 5-4-1, Next: 5-4-2, Prev: 5-4, Up: 5-4
-
-5.4.1 The Core of the Explicit-Control Evaluator
-------------------------------------------------
-
-The central element in the evaluator is the sequence of instructions
-beginning at `eval-dispatch'. This corresponds to the `eval' procedure
-of the metacircular evaluator described in section *Note 4-1-1::. When
-the controller starts at `eval-dispatch', it evaluates the expression
-specified by `exp' in the environment specified by `env'. When
-evaluation is complete, the controller will go to the entry point
-stored in `continue', and the `val' register will hold the value of the
-expression. As with the metacircular `eval', the structure of
-`eval-dispatch' is a case analysis on the syntactic type of the
-expression to be evaluated.(1)
-
- eval-dispatch
- (test (op self-evaluating?) (reg exp))
- (branch (label ev-self-eval))
- (test (op variable?) (reg exp))
- (branch (label ev-variable))
- (test (op quoted?) (reg exp))
- (branch (label ev-quoted))
- (test (op assignment?) (reg exp))
- (branch (label ev-assignment))
- (test (op definition?) (reg exp))
- (branch (label ev-definition))
- (test (op if?) (reg exp))
- (branch (label ev-if))
- (test (op lambda?) (reg exp))
- (branch (label ev-lambda))
- (test (op begin?) (reg exp))
- (branch (label ev-begin))
- (test (op application?) (reg exp))
- (branch (label ev-application))
- (goto (label unknown-expression-type))
-
-Evaluating simple expressions
-.............................
-
-Numbers and strings (which are self-evaluating), variables, quotations,
-and `lambda' expressions have no subexpressions to be evaluated. For
-these, the evaluator simply places the correct value in the `val'
-register and continues execution at the entry point specified by
-`continue'. Evaluation of simple expressions is performed by the
-following controller code:
-
- ev-self-eval
- (assign val (reg exp))
- (goto (reg continue))
- ev-variable
- (assign val (op lookup-variable-value) (reg exp) (reg env))
- (goto (reg continue))
- ev-quoted
- (assign val (op text-of-quotation) (reg exp))
- (goto (reg continue))
- ev-lambda
- (assign unev (op lambda-parameters) (reg exp))
- (assign exp (op lambda-body) (reg exp))
- (assign val (op make-procedure)
- (reg unev) (reg exp) (reg env))
- (goto (reg continue))
-
- Observe how `ev-lambda' uses the `unev' and `exp' registers to hold
-the parameters and body of the lambda expression so that they can be
-passed to the `make-procedure' operation, along with the environment in
-`env'.
-
-Evaluating procedure applications
-.................................
-
-A procedure application is specified by a combination containing an
-operator and operands. The operator is a subexpression whose value is
-a procedure, and the operands are subexpressions whose values are the
-arguments to which the procedure should be applied. The metacircular
-`eval' handles applications by calling itself recursively to evaluate
-each element of the combination, and then passing the results to
-`apply', which performs the actual procedure application. The
-explicit-control evaluator does the same thing; these recursive calls
-are implemented by `goto' instructions, together with use of the stack
-to save registers that will be restored after the recursive call
-returns. Before each call we will be careful to identify which
-registers must be saved (because their values will be needed later).(2)
-
- We begin the evaluation of an application by evaluating the operator
-to produce a procedure, which will later be applied to the evaluated
-operands. To evaluate the operator, we move it to the `exp' register
-and go to `eval-dispatch'. The environment in the `env' register is
-already the correct one in which to evaluate the operator. However, we
-save `env' because we will need it later to evaluate the operands. We
-also extract the operands into `unev' and save this on the stack. We
-set up `continue' so that `eval-dispatch' will resume at
-`ev-appl-did-operator' after the operator has been evaluated. First,
-however, we save the old value of `continue', which tells the controller
-where to continue after the application.
-
- ev-application
- (save continue)
- (save env)
- (assign unev (op operands) (reg exp))
- (save unev)
- (assign exp (op operator) (reg exp))
- (assign continue (label ev-appl-did-operator))
- (goto (label eval-dispatch))
-
- Upon returning from evaluating the operator subexpression, we
-proceed to evaluate the operands of the combination and to accumulate
-the resulting arguments in a list, held in `argl'. First we restore
-the unevaluated operands and the environment. We initialize `argl' to
-an empty list. Then we assign to the `proc' register the procedure
-that was produced by evaluating the operator. If there are no
-operands, we go directly to `apply-dispatch'. Otherwise we save `proc'
-on the stack and start the argument-evaluation loop:(3)
-
- ev-appl-did-operator
- (restore unev) ; the operands
- (restore env)
- (assign argl (op empty-arglist))
- (assign proc (reg val)) ; the operator
- (test (op no-operands?) (reg unev))
- (branch (label apply-dispatch))
- (save proc)
-
- Each cycle of the argument-evaluation loop evaluates an operand from
-the list in `unev' and accumulates the result into `argl'. To evaluate
-an operand, we place it in the `exp' register and go to `eval-dispatch',
-after setting `continue' so that execution will resume with the
-argument-accumulation phase. But first we save the arguments
-accumulated so far (held in `argl'), the environment (held in `env'),
-and the remaining operands to be evaluated (held in `unev'). A special
-case is made for the evaluation of the last operand, which is handled at
-`ev-appl-last-arg'.
-
- ev-appl-operand-loop
- (save argl)
- (assign exp (op first-operand) (reg unev))
- (test (op last-operand?) (reg unev))
- (branch (label ev-appl-last-arg))
- (save env)
- (save unev)
- (assign continue (label ev-appl-accumulate-arg))
- (goto (label eval-dispatch))
-
- When an operand has been evaluated, the value is accumulated into
-the list held in `argl'. The operand is then removed from the list of
-unevaluated operands in `unev', and the argument-evaluation continues.
-
- ev-appl-accumulate-arg
- (restore unev)
- (restore env)
- (restore argl)
- (assign argl (op adjoin-arg) (reg val) (reg argl))
- (assign unev (op rest-operands) (reg unev))
- (goto (label ev-appl-operand-loop))
-
- Evaluation of the last argument is handled differently. There is no
-need to save the environment or the list of unevaluated operands before
-going to `eval-dispatch', since they will not be required after the
-last operand is evaluated. Thus, we return from the evaluation to a
-special entry point `ev-appl-accum-last-arg', which restores the
-argument list, accumulates the new argument, restores the saved
-procedure, and goes off to perform the application.(4)
-
- ev-appl-last-arg
- (assign continue (label ev-appl-accum-last-arg))
- (goto (label eval-dispatch))
- ev-appl-accum-last-arg
- (restore argl)
- (assign argl (op adjoin-arg) (reg val) (reg argl))
- (restore proc)
- (goto (label apply-dispatch))
-
- The details of the argument-evaluation loop determine the order in
-which the interpreter evaluates the operands of a combination (e.g.,
-left to right or right to left--see *Note Exercise 3-8::). This order
-is not determined by the metacircular evaluator, which inherits its
-control structure from the underlying Scheme in which it is
-implemented.(5) Because the `first-operand' selector (used in
-`ev-appl-operand-loop' to extract successive operands from `unev') is
-implemented as `car' and the `rest-operands' selector is implemented as
-`cdr', the explicit-control evaluator will evaluate the operands of a
-combination in left-to-right order.
-
-Procedure application
-.....................
-
-The entry point `apply-dispatch' corresponds to the `apply' procedure
-of the metacircular evaluator. By the time we get to `apply-dispatch',
-the `proc' register contains the procedure to apply and `argl' contains
-the list of evaluated arguments to which it must be applied. The saved
-value of `continue' (originally passed to `eval-dispatch' and saved at
-`ev-application'), which tells where to return with the result of the
-procedure application, is on the stack. When the application is
-complete, the controller transfers to the entry point specified by the
-saved `continue', with the result of the application in `val'. As with
-the metacircular `apply', there are two cases to consider. Either the
-procedure to be applied is a primitive or it is a compound procedure.
-
- apply-dispatch
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-apply))
- (test (op compound-procedure?) (reg proc))
- (branch (label compound-apply))
- (goto (label unknown-procedure-type))
-
- We assume that each primitive is implemented so as to obtain its
-arguments from `argl' and place its result in `val'. To specify how
-the machine handles primitives, we would have to provide a sequence of
-controller instructions to implement each primitive and arrange for
-`primitive-apply' to dispatch to the instructions for the primitive
-identified by the contents of `proc'. Since we are interested in the
-structure of the evaluation process rather than the details of the
-primitives, we will instead just use an `apply-primitive-procedure'
-operation that applies the procedure in `proc' to the arguments in
-`argl'. For the purpose of simulating the evaluator with the simulator
-of section *Note 5-2:: we use the procedure
-`apply-primitive-procedure', which calls on the underlying Scheme system
-to perform the application, just as we did for the metacircular
-evaluator in section *Note 4-1-4::. After computing the value of the
-primitive application, we restore `continue' and go to the designated
-entry point.
-
- primitive-apply
- (assign val (op apply-primitive-procedure)
- (reg proc)
- (reg argl))
- (restore continue)
- (goto (reg continue))
-
- To apply a compound procedure, we proceed just as with the
-metacircular evaluator. We construct a frame that binds the
-procedure's parameters to the arguments, use this frame to extend the
-environment carried by the procedure, and evaluate in this extended
-environment the sequence of expressions that forms the body of the
-procedure. `Ev-sequence', described below in section *Note 5-4-2::,
-handles the evaluation of the sequence.
-
- compound-apply
- (assign unev (op procedure-parameters) (reg proc))
- (assign env (op procedure-environment) (reg proc))
- (assign env (op extend-environment)
- (reg unev) (reg argl) (reg env))
- (assign unev (op procedure-body) (reg proc))
- (goto (label ev-sequence))
-
- `Compound-apply' is the only place in the interpreter where the `env'
-register is ever assigned a new value. Just as in the metacircular
-evaluator, the new environment is constructed from the environment
-carried by the procedure, together with the argument list and the
-corresponding list of variables to be bound.
-
- ---------- Footnotes ----------
-
- (1) In our controller, the dispatch is written as a sequence of
-`test' and `branch' instructions. Alternatively, it could have been
-written in a data-directed style (and in a real system it probably
-would have been) to avoid the need to perform sequential tests and to
-facilitate the definition of new expression types. A machine designed
-to run Lisp would probably include a `dispatch-on-type' instruction
-that would efficiently execute such data-directed dispatches.
-
- (2) This is an important but subtle point in translating algorithms
-from a procedural language, such as Lisp, to a register-machine
-language. As an alternative to saving only what is needed, we could
-save all the registers (except `val') before each recursive call. This
-is called a "framed-stack" discipline. This would work but might save
-more registers than necessary; this could be an important consideration
-in a system where stack operations are expensive. Saving registers
-whose contents will not be needed later may also hold onto useless data
-that could otherwise be garbage-collected, freeing space to be reused.
-
- (3) We add to the evaluator data-structure procedures in section
-*Note 4-1-3:: the following two procedures for manipulating argument
-lists:
-
- (define (empty-arglist) '())
-
- (define (adjoin-arg arg arglist)
- (append arglist (list arg)))
-
- We also use an additional syntax procedure to test for the last
-operand in a combination:
-
- (define (last-operand? ops)
- (null? (cdr ops)))
-
- (4) The optimization of treating the last operand specially is known
-as "evlis tail recursion" (see Wand 1980). We could be somewhat more
-efficient in the argument evaluation loop if we made evaluation of the
-first operand a special case too. This would permit us to postpone
-initializing `argl' until after evaluating the first operand, so as to
-avoid saving `argl' in this case. The compiler in section *Note 5-5::
-performs this optimization. (Compare the `construct-arglist' procedure
-of section *Note 5-5-3::.)
-
- (5) The order of operand evaluation in the metacircular evaluator is
-determined by the order of evaluation of the arguments to `cons' in the
-procedure `list-of-values' of section *Note 4-1-1:: (see *Note Exercise
-4-1::).
-
-\1f
-File: sicp.info, Node: 5-4-2, Next: 5-4-3, Prev: 5-4-1, Up: 5-4
-
-5.4.2 Sequence Evaluation and Tail Recursion
---------------------------------------------
-
-The portion of the explicit-control evaluator at `ev-sequence' is
-analogous to the metacircular evaluator's `eval-sequence' procedure. It
-handles sequences of expressions in procedure bodies or in explicit
-`begin' expressions.
-
- Explicit `begin' expressions are evaluated by placing the sequence of
-expressions to be evaluated in `unev', saving `continue' on the stack,
-and jumping to `ev-sequence'.
-
- ev-begin
- (assign unev (op begin-actions) (reg exp))
- (save continue)
- (goto (label ev-sequence))
-
- The implicit sequences in procedure bodies are handled by jumping to
-`ev-sequence' from `compound-apply', at which point `continue' is
-already on the stack, having been saved at `ev-application'.
-
- The entries at `ev-sequence' and `ev-sequence-continue' form a loop
-that successively evaluates each expression in a sequence. The list of
-unevaluated expressions is kept in `unev'. Before evaluating each
-expression, we check to see if there are additional expressions to be
-evaluated in the sequence. If so, we save the rest of the unevaluated
-expressions (held in `unev') and the environment in which these must be
-evaluated (held in `env') and call `eval-dispatch' to evaluate the
-expression. The two saved registers are restored upon the return from
-this evaluation, at `ev-sequence-continue'.
-
- The final expression in the sequence is handled differently, at the
-entry point `ev-sequence-last-exp'. Since there are no more
-expressions to be evaluated after this one, we need not save `unev' or
-`env' before going to `eval-dispatch'. The value of the whole sequence
-is the value of the last expression, so after the evaluation of the
-last expression there is nothing left to do except continue at the
-entry point currently held on the stack (which was saved by
-`ev-application' or `ev-begin'.) Rather than setting up `continue' to
-arrange for `eval-dispatch' to return here and then restoring
-`continue' from the stack and continuing at that entry point, we
-restore `continue' from the stack before going to `eval-dispatch', so
-that `eval-dispatch' will continue at that entry point after evaluating
-the expression.
-
- ev-sequence
- (assign exp (op first-exp) (reg unev))
- (test (op last-exp?) (reg unev))
- (branch (label ev-sequence-last-exp))
- (save unev)
- (save env)
- (assign continue (label ev-sequence-continue))
- (goto (label eval-dispatch))
- ev-sequence-continue
- (restore env)
- (restore unev)
- (assign unev (op rest-exps) (reg unev))
- (goto (label ev-sequence))
- ev-sequence-last-exp
- (restore continue)
- (goto (label eval-dispatch))
-
-Tail recursion
-..............
-
-In *Note Chapter 1:: we said that the process described by a procedure
-such as
-
- (define (sqrt-iter guess x)
- (if (good-enough? guess x)
- guess
- (sqrt-iter (improve guess x)
- x)))
-
-is an iterative process. Even though the procedure is syntactically
-recursive (defined in terms of itself), it is not logically necessary
-for an evaluator to save information in passing from one call to
-`sqrt-iter' to the next.(1) An evaluator that can execute a procedure
-such as `sqrt-iter' without requiring increasing storage as the
-procedure continues to call itself is called a "tail-recursive"
-evaluator. The metacircular implementation of the evaluator in *Note
-Chapter 4:: does not specify whether the evaluator is tail-recursive,
-because that evaluator inherits its mechanism for saving state from the
-underlying Scheme. With the explicit-control evaluator, however, we
-can trace through the evaluation process to see when procedure calls
-cause a net accumulation of information on the stack.
-
- Our evaluator is tail-recursive, because in order to evaluate the
-final expression of a sequence we transfer directly to `eval-dispatch'
-without saving any information on the stack. Hence, evaluating the
-final expression in a sequence--even if it is a procedure call (as in
-`sqrt-iter', where the `if' expression, which is the last expression in
-the procedure body, reduces to a call to `sqrt-iter')--will not cause
-any information to be accumulated on the stack.(2)
-
- If we did not think to take advantage of the fact that it was
-unnecessary to save information in this case, we might have implemented
-`eval-sequence' by treating all the expressions in a sequence in the
-same way--saving the registers, evaluating the expression, returning to
-restore the registers, and repeating this until all the expressions
-have been evaluated:(3)
-
- ev-sequence
- (test (op no-more-exps?) (reg unev))
- (branch (label ev-sequence-end))
- (assign exp (op first-exp) (reg unev))
- (save unev)
- (save env)
- (assign continue (label ev-sequence-continue))
- (goto (label eval-dispatch))
- ev-sequence-continue
- (restore env)
- (restore unev)
- (assign unev (op rest-exps) (reg unev))
- (goto (label ev-sequence))
- ev-sequence-end
- (restore continue)
- (goto (reg continue))
-
- This may seem like a minor change to our previous code for
-evaluation of a sequence: The only difference is that we go through the
-save-restore cycle for the last expression in a sequence as well as for
-the others. The interpreter will still give the same value for any
-expression. But this change is fatal to the tail-recursive
-implementation, because we must now return after evaluating the final
-expression in a sequence in order to undo the (useless) register saves.
-These extra saves will accumulate during a nest of procedure calls.
-Consequently, processes such as `sqrt-iter' will require space
-proportional to the number of iterations rather than requiring constant
-space. This difference can be significant. For example, with tail
-recursion, an infinite loop can be expressed using only the
-procedure-call mechanism:
-
- (define (count n)
- (newline)
- (display n)
- (count (+ n 1)))
-
- Without tail recursion, such a procedure would eventually run out of
-stack space, and expressing a true iteration would require some control
-mechanism other than procedure call.
-
- ---------- Footnotes ----------
-
- (1) We saw in section *Note 5-1:: how to implement such a process
-with a register machine that had no stack; the state of the process was
-stored in a fixed set of registers.
-
- (2) This implementation of tail recursion in `ev-sequence' is one
-variety of a well-known optimization technique used by many compilers.
-In compiling a procedure that ends with a procedure call, one can
-replace the call by a jump to the called procedure's entry point.
-Building this strategy into the interpreter, as we have done in this
-section, provides the optimization uniformly throughout the language.
-
- (3) We can define `no-more-exps?' as follows:
-
- (define (no-more-exps? seq) (null? seq))
-
-\1f
-File: sicp.info, Node: 5-4-3, Next: 5-4-4, Prev: 5-4-2, Up: 5-4
-
-5.4.3 Conditionals, Assignments, and Definitions
-------------------------------------------------
-
-As with the metacircular evaluator, special forms are handled by
-selectively evaluating fragments of the expression. For an `if'
-expression, we must evaluate the predicate and decide, based on the
-value of predicate, whether to evaluate the consequent or the
-alternative.
-
- Before evaluating the predicate, we save the `if' expression itself
-so that we can later extract the consequent or alternative. We also
-save the environment, which we will need later in order to evaluate the
-consequent or the alternative, and we save `continue', which we will
-need later in order to return to the evaluation of the expression that
-is waiting for the value of the `if'.
-
- ev-if
- (save exp) ; save expression for later
- (save env)
- (save continue)
- (assign continue (label ev-if-decide))
- (assign exp (op if-predicate) (reg exp))
- (goto (label eval-dispatch)) ; evaluate the predicate
-
- When we return from evaluating the predicate, we test whether it was
-true or false and, depending on the result, place either the consequent
-or the alternative in `exp' before going to `eval-dispatch'. Notice
-that restoring `env' and `continue' here sets up `eval-dispatch' to
-have the correct environment and to continue at the right place to
-receive the value of the `if' expression.
-
- ev-if-decide
- (restore continue)
- (restore env)
- (restore exp)
- (test (op true?) (reg val))
- (branch (label ev-if-consequent))
-
- ev-if-alternative
- (assign exp (op if-alternative) (reg exp))
- (goto (label eval-dispatch))
- ev-if-consequent
- (assign exp (op if-consequent) (reg exp))
- (goto (label eval-dispatch))
-
-Assignments and definitions
-...........................
-
-Assignments are handled by `ev-assignment', which is reached from
-`eval-dispatch' with the assignment expression in `exp'. The code at
-`ev-assignment' first evaluates the value part of the expression and
-then installs the new value in the environment. `Set-variable-value!'
-is assumed to be available as a machine operation.
-
- ev-assignment
- (assign unev (op assignment-variable) (reg exp))
- (save unev) ; save variable for later
- (assign exp (op assignment-value) (reg exp))
- (save env)
- (save continue)
- (assign continue (label ev-assignment-1))
- (goto (label eval-dispatch)) ; evaluate the assignment value
- ev-assignment-1
- (restore continue)
- (restore env)
- (restore unev)
- (perform
- (op set-variable-value!) (reg unev) (reg val) (reg env))
- (assign val (const ok))
- (goto (reg continue))
-
- Definitions are handled in a similar way:
-
- ev-definition
- (assign unev (op definition-variable) (reg exp))
- (save unev) ; save variable for later
- (assign exp (op definition-value) (reg exp))
- (save env)
- (save continue)
- (assign continue (label ev-definition-1))
- (goto (label eval-dispatch)) ; evaluate the definition value
- ev-definition-1
- (restore continue)
- (restore env)
- (restore unev)
- (perform
- (op define-variable!) (reg unev) (reg val) (reg env))
- (assign val (const ok))
- (goto (reg continue))
-
- *Exercise 5.23:* Extend the evaluator to handle derived
- expressions such as `cond', `let', and so on (section *Note
- 4-1-2::). You may "cheat" and assume that the syntax transformers
- such as `cond->if' are available as machine operations.(1)
-
- *Exercise 5.24:* Implement `cond' as a new basic special form
- without reducing it to `if'. You will have to construct a loop
- that tests the predicates of successive `cond' clauses until you
- find one that is true, and then use `ev-sequence' to evaluate the
- actions of the clause.
-
- *Exercise 5.25:* Modify the evaluator so that it uses normal-order
- evaluation, based on the lazy evaluator of section *Note 4-2::.
-
- ---------- Footnotes ----------
-
- (1) This isn't really cheating. In an actual implementation built
-from scratch, we would use our explicit-control evaluator to interpret
-a Scheme program that performs source-level transformations like
-`cond->if' in a syntax phase that runs before execution.
-
-\1f
-File: sicp.info, Node: 5-4-4, Prev: 5-4-3, Up: 5-4
-
-5.4.4 Running the Evaluator
----------------------------
-
-With the implementation of the explicit-control evaluator we come to
-the end of a development, begun in *Note Chapter 1::, in which we have
-explored successively more precise models of the evaluation process.
-We started with the relatively informal substitution model, then
-extended this in *Note Chapter 3:: to the environment model, which
-enabled us to deal with state and change. In the metacircular
-evaluator of *Note Chapter 4::, we used Scheme itself as a language for
-making more explicit the environment structure constructed during
-evaluation of an expression. Now, with register machines, we have
-taken a close look at the evaluator's mechanisms for storage
-management, argument passing, and control. At each new level of
-description, we have had to raise issues and resolve ambiguities that
-were not apparent at the previous, less precise treatment of
-evaluation. To understand the behavior of the explicit-control
-evaluator, we can simulate it and monitor its performance.
-
- We will install a driver loop in our evaluator machine. This plays
-the role of the `driver-loop' procedure of section *Note 4-1-4::. The
-evaluator will repeatedly print a prompt, read an expression, evaluate
-the expression by going to `eval-dispatch', and print the result. The
-following instructions form the beginning of the explicit-control
-evaluator's controller sequence:(1)
-
- read-eval-print-loop
- (perform (op initialize-stack))
- (perform
- (op prompt-for-input) (const ";;; EC-Eval input:"))
- (assign exp (op read))
- (assign env (op get-global-environment))
- (assign continue (label print-result))
- (goto (label eval-dispatch))
- print-result
- (perform
- (op announce-output) (const ";;; EC-Eval value:"))
- (perform (op user-print) (reg val))
- (goto (label read-eval-print-loop))
-
- When we encounter an error in a procedure (such as the "unknown
-procedure type error" indicated at `apply-dispatch'), we print an error
-message and return to the driver loop.(2)
-
- unknown-expression-type
- (assign val (const unknown-expression-type-error))
- (goto (label signal-error))
-
- unknown-procedure-type
- (restore continue) ; clean up stack (from `apply-dispatch')
- (assign val (const unknown-procedure-type-error))
- (goto (label signal-error))
-
- signal-error
- (perform (op user-print) (reg val))
- (goto (label read-eval-print-loop))
-
- For the purposes of the simulation, we initialize the stack each
-time through the driver loop, since it might not be empty after an
-error (such as an undefined variable) interrupts an evaluation.(3)
-
- If we combine all the code fragments presented in sections *Note
-5-4-1::-*Note 5-4-4::, we can create an evaluator machine model that we
-can run using the register-machine simulator of section *Note 5-2::.
-
- (define eceval
- (make-machine
- '(exp env val proc argl continue unev)
- eceval-operations
- '(
- read-eval-print-loop
- <_entire machine controller as given above_>
- )))
-
- We must define Scheme procedures to simulate the operations used as
-primitives by the evaluator. These are the same procedures we used for
-the metacircular evaluator in section *Note 4-1::, together with the
-few additional ones defined in footnotes throughout section *Note 5-4::.
-
- (define eceval-operations
- (list (list 'self-evaluating? self-evaluating)
- <_complete list of operations for eceval machine_>))
-
- Finally, we can initialize the global environment and run the
-evaluator:
-
- (define the-global-environment (setup-environment))
-
- (start eceval)
-
- ;;; EC-Eval input:
- (define (append x y)
- (if (null? x)
- y
- (cons (car x)
- (append (cdr x) y))))
- ;;; EC-Eval value:
- ok
-
- ;;; EC-Eval input:
- (append '(a b c) '(d e f))
- ;;; EC-Eval value:
- (a b c d e f)
-
- Of course, evaluating expressions in this way will take much longer
-than if we had directly typed them into Scheme, because of the multiple
-levels of simulation involved. Our expressions are evaluated by the
-explicit-control-evaluator machine, which is being simulated by a Scheme
-program, which is itself being evaluated by the Scheme interpreter.
-
-Monitoring the performance of the evaluator
-...........................................
-
-Simulation can be a powerful tool to guide the implementation of
-evaluators. Simulations make it easy not only to explore variations of
-the register-machine design but also to monitor the performance of the
-simulated evaluator. For example, one important factor in performance
-is how efficiently the evaluator uses the stack. We can observe the
-number of stack operations required to evaluate various expressions by
-defining the evaluator register machine with the version of the
-simulator that collects statistics on stack use (section *Note
-5-2-4::), and adding an instruction at the evaluator's `print-result'
-entry point to print the statistics:
-
- print-result
- (perform (op print-stack-statistics)); added instruction
- (perform
- (op announce-output) (const ";;; EC-Eval value:"))
- ... ; same as before
-
- Interactions with the evaluator now look like this:
-
- ;;; EC-Eval input:
- (define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
- (total-pushes = 3 maximum-depth = 3)
- ;;; EC-Eval value:
- ok
-
- ;;; EC-Eval input:
- (factorial 5)
- (total-pushes = 144 maximum-depth = 28)
- ;;; EC-Eval value:
- 120
-
- Note that the driver loop of the evaluator reinitializes the stack
-at the start of each interaction, so that the statistics printed will
-refer only to stack operations used to evaluate the previous expression.
-
- *Exercise 5.26:* Use the monitored stack to explore the
- tail-recursive property of the evaluator (section *Note 5-4-2::).
- Start the evaluator and define the iterative `factorial' procedure
- from section *Note 1-2-1:::
-
- (define (factorial n)
- (define (iter product counter)
- (if (> counter n)
- product
- (iter (* counter product)
- (+ counter 1))))
- (iter 1 1))
-
- Run the procedure with some small values of n. Record the maximum
- stack depth and the number of pushes required to compute n! for
- each of these values.
-
- a. You will find that the maximum depth required to evaluate n!
- is independent of n. What is that depth?
-
- b. Determine from your data a formula in terms of n for the
- total number of push operations used in evaluating n! for any
- n >= 1. Note that the number of operations used is a linear
- function of n and is thus determined by two constants.
-
-
- *Exercise 5.27:* For comparison with *Note Exercise 5-26::,
- explore the behavior of the following procedure for computing
- factorials recursively:
-
- (define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
-
- By running this procedure with the monitored stack, determine, as
- a function of n, the maximum depth of the stack and the total
- number of pushes used in evaluating n! for n >= 1. (Again, these
- functions will be linear.) Summarize your experiments by filling
- in the following table with the appropriate expressions in terms
- of n:
-
- Maximum depth Number of pushes
-
- Recursive
- factorial
-
- Iterative
- factorial
-
- The maximum depth is a measure of the amount of space used by the
- evaluator in carrying out the computation, and the number of
- pushes correlates well with the time required.
-
- *Exercise 5.28:* Modify the definition of the evaluator by
- changing `eval-sequence' as described in section *Note 5-4-2:: so
- that the evaluator is no longer tail-recursive. Rerun your
- experiments from *Note Exercise 5-26:: and *Note Exercise 5-27::
- to demonstrate that both versions of the `factorial' procedure now
- require space that grows linearly with their input.
-
- *Exercise 5.29:* Monitor the stack operations in the
- tree-recursive Fibonacci computation:
-
- (define (fib n)
- (if (< n 2)
- n
- (+ (fib (- n 1)) (fib (- n 2)))))
-
- a. Give a formula in terms of n for the maximum depth of the
- stack required to compute _Fib_(n) for n >= 2. Hint: In
- section *Note 1-2-2:: we argued that the space used by this
- process grows linearly with n.
-
- b. Give a formula for the total number of pushes used to compute
- _Fib_(n) for n >= 2. You should find that the number of
- pushes (which correlates well with the time used) grows
- exponentially with n. Hint: Let S(n) be the number of pushes
- used in computing _Fib_(n). You should be able to argue that
- there is a formula that expresses S(n) in terms of S(n - 1),
- S(n - 2), and some fixed "overhead" constant k that is
- independent of n. Give the formula, and say what k is. Then
- show that S(n) can be expressed as a _Fib_(n + 1) + b and
- give the values of a and b.
-
-
- *Exercise 5.30:* Our evaluator currently catches and signals only
- two kinds of errors--unknown expression types and unknown
- procedure types. Other errors will take us out of the evaluator
- read-eval-print loop. When we run the evaluator using the
- register-machine simulator, these errors are caught by the
- underlying Scheme system. This is analogous to the computer
- crashing when a user program makes an error.(4) It is a large
- project to make a real error system work, but it is well worth the
- effort to understand what is involved here.
-
- a. Errors that occur in the evaluation process, such as an
- attempt to access an unbound variable, could be caught by
- changing the lookup operation to make it return a
- distinguished condition code, which cannot be a possible
- value of any user variable. The evaluator can test for this
- condition code and then do what is necessary to go to
- `signal-error'. Find all of the places in the evaluator
- where such a change is necessary and fix them. This is lots
- of work.
-
- b. Much worse is the problem of handling errors that are
- signaled by applying primitive procedures, such as an attempt
- to divide by zero or an attempt to extract the `car' of a
- symbol. In a professionally written high-quality system,
- each primitive application is checked for safety as part of
- the primitive. For example, every call to `car' could first
- check that the argument is a pair. If the argument is not a
- pair, the application would return a distinguished condition
- code to the evaluator, which would then report the failure.
- We could arrange for this in our register-machine simulator by
- making each primitive procedure check for applicability and
- returning an appropriate distinguished condition code on
- failure. Then the `primitive-apply' code in the evaluator can
- check for the condition code and go to `signal-error' if
- necessary. Build this structure and make it work. This is a
- major project.
-
-
- ---------- Footnotes ----------
-
- (1) We assume here that `read' and the various printing operations
-are available as primitive machine operations, which is useful for our
-simulation, but completely unrealistic in practice. These are actually
-extremely complex operations. In practice, they would be implemented
-using low-level input-output operations such as transferring single
-characters to and from a device.
-
- To support the `get-global-environment' operation we define
-
- (define the-global-environment (setup-environment))
-
- (define (get-global-environment)
- the-global-environment)
-
- (2) There are other errors that we would like the interpreter to
-handle, but these are not so simple. See *Note Exercise 5-30::.
-
- (3) We could perform the stack initialization only after errors, but
-doing it in the driver loop will be convenient for monitoring the
-evaluator's performance, as described below.
-
- (4) Regrettably, this is the normal state of affairs in conventional
-compiler-based language systems such as C. In UNIX(tm) the system
-"dumps core," and in DOS/Windows(tm) it becomes catatonic. The
-Macintosh(tm) displays a picture of an exploding bomb and offers you
-the opportunity to reboot the computer--if you're lucky.
-
-\1f
-File: sicp.info, Node: 5-5, Prev: 5-4, Up: Chapter 5
-
-5.5 Compilation
-===============
-
-The explicit-control evaluator of section *Note 5-4:: is a register
-machine whose controller interprets Scheme programs. In this section
-we will see how to run Scheme programs on a register machine whose
-controller is not a Scheme interpreter.
-
- The explicit-control evaluator machine is universal--it can carry
-out any computational process that can be described in Scheme. The
-evaluator's controller orchestrates the use of its data paths to
-perform the desired computation. Thus, the evaluator's data paths are
-universal: They are sufficient to perform any computation we desire,
-given an appropriate controller.(1)
-
- Commercial general-purpose computers are register machines organized
-around a collection of registers and operations that constitute an
-efficient and convenient universal set of data paths. The controller
-for a general-purpose machine is an interpreter for a register-machine
-language like the one we have been using. This language is called the "native
-language" of the machine, or simply "machine language". Programs
-written in machine language are sequences of instructions that use the
-machine's data paths. For example, the explicit-control evaluator's
-instruction sequence can be thought of as a machine-language program
-for a general-purpose computer rather than as the controller for a
-specialized interpreter machine.
-
- There are two common strategies for bridging the gap between
-higher-level languages and register-machine languages. The
-explicit-control evaluator illustrates the strategy of interpretation.
-An interpreter written in the native language of a machine configures
-the machine to execute programs written in a language (called the "source
-language") that may differ from the native language of the machine
-performing the evaluation. The primitive procedures of the source
-language are implemented as a library of subroutines written in the
-native language of the given machine. A program to be interpreted
-(called the "source program") is represented as a data structure. The
-interpreter traverses this data structure, analyzing the source
-program. As it does so, it simulates the intended behavior of the
-source program by calling appropriate primitive subroutines from the
-library.
-
- In this section, we explore the alternative strategy of "compilation".
-A compiler for a given source language and machine translates a source
-program into an equivalent program (called the "object program")
-written in the machine's native language. The compiler that we
-implement in this section translates programs written in Scheme into
-sequences of instructions to be executed using the explicit-control
-evaluator machine's data paths.(2)
-
- Compared with interpretation, compilation can provide a great
-increase in the efficiency of program execution, as we will explain
-below in the overview of the compiler. On the other hand, an
-interpreter provides a more powerful environment for interactive
-program development and debugging, because the source program being
-executed is available at run time to be examined and modified. In
-addition, because the entire library of primitives is present, new
-programs can be constructed and added to the system during debugging.
-
- In view of the complementary advantages of compilation and
-interpretation, modern program-development environments pursue a mixed
-strategy. Lisp interpreters are generally organized so that
-interpreted procedures and compiled procedures can call each other.
-This enables a programmer to compile those parts of a program that are
-assumed to be debugged, thus gaining the efficiency advantage of
-compilation, while retaining the interpretive mode of execution for
-those parts of the program that are in the flux of interactive
-development and debugging. In section *Note 5-5-7::, after we have
-implemented the compiler, we will show how to interface it with our
-interpreter to produce an integrated interpreter-compiler development
-system.
-
-An overview of the compiler
-...........................
-
-Our compiler is much like our interpreter, both in its structure and in
-the function it performs. Accordingly, the mechanisms used by the
-compiler for analyzing expressions will be similar to those used by the
-interpreter. Moreover, to make it easy to interface compiled and
-interpreted code, we will design the compiler to generate code that
-obeys the same conventions of register usage as the interpreter: The
-environment will be kept in the `env' register, argument lists will be
-accumulated in `argl', a procedure to be applied will be in `proc',
-procedures will return their answers in `val', and the location to
-which a procedure should return will be kept in `continue'. In
-general, the compiler translates a source program into an object
-program that performs essentially the same register operations as would
-the interpreter in evaluating the same source program.
-
- This description suggests a strategy for implementing a rudimentary
-compiler: We traverse the expression in the same way the interpreter
-does. When we encounter a register instruction that the interpreter
-would perform in evaluating the expression, we do not execute the
-instruction but instead accumulate it into a sequence. The resulting
-sequence of instructions will be the object code. Observe the
-efficiency advantage of compilation over interpretation. Each time the
-interpreter evaluates an expression--for example, `(f 84 96)'--it
-performs the work of classifying the expression (discovering that this
-is a procedure application) and testing for the end of the operand list
-(discovering that there are two operands). With a compiler, the
-expression is analyzed only once, when the instruction sequence is
-generated at compile time. The object code produced by the compiler
-contains only the instructions that evaluate the operator and the two
-operands, assemble the argument list, and apply the procedure (in
-`proc') to the arguments (in `argl').
-
- This is the same kind of optimization we implemented in the
-analyzing evaluator of section *Note 4-1-7::. But there are further
-opportunities to gain efficiency in compiled code. As the interpreter
-runs, it follows a process that must be applicable to any expression in
-the language. In contrast, a given segment of compiled code is meant
-to execute some particular expression. This can make a big difference,
-for example in the use of the stack to save registers. When the
-interpreter evaluates an expression, it must be prepared for any
-contingency. Before evaluating a subexpression, the interpreter saves
-all registers that will be needed later, because the subexpression
-might require an arbitrary evaluation. A compiler, on the other hand,
-can exploit the structure of the particular expression it is processing
-to generate code that avoids unnecessary stack operations.
-
- As a case in point, consider the combination `(f 84 96)'. Before the
-interpreter evaluates the operator of the combination, it prepares for
-this evaluation by saving the registers containing the operands and the
-environment, whose values will be needed later. The interpreter then
-evaluates the operator to obtain the result in `val', restores the
-saved registers, and finally moves the result from `val' to `proc'.
-However, in the particular expression we are dealing with, the operator
-is the symbol `f', whose evaluation is accomplished by the machine
-operation `lookup-variable-value', which does not alter any registers.
-The compiler that we implement in this section will take advantage of
-this fact and generate code that evaluates the operator using the
-instruction
-
- (assign proc (op lookup-variable-value) (const f) (reg env))
-
- This code not only avoids the unnecessary saves and restores but
-also assigns the value of the lookup directly to `proc', whereas the
-interpreter would obtain the result in `val' and then move this to
-`proc'.
-
- A compiler can also optimize access to the environment. Having
-analyzed the code, the compiler can in many cases know in which frame a
-particular variable will be located and access that frame directly,
-rather than performing the `lookup-variable-value' search. We will
-discuss how to implement such variable access in section *Note 5-5-6::.
-Until then, however, we will focus on the kind of register and stack
-optimizations described above. There are many other optimizations that
-can be performed by a compiler, such as coding primitive operations "in
-line" instead of using a general `apply' mechanism (see *Note Exercise
-5-38::); but we will not emphasize these here. Our main goal in this
-section is to illustrate the compilation process in a simplified (but
-still interesting) context.
-
-* Menu:
-
-* 5-5-1:: Structure of the Compiler
-* 5-5-2:: Compiling Expressions
-* 5-5-3:: Compiling Combinations
-* 5-5-4:: Combining Instruction Sequences
-* 5-5-5:: An Example of Compiled Code
-* 5-5-6:: Lexical Addressing
-* 5-5-7:: Interfacing Compiled Code to the Evaluator
-
- ---------- Footnotes ----------
-
- (1) This is a theoretical statement. We are not claiming that the
-evaluator's data paths are a particularly convenient or efficient set of
-data paths for a general-purpose computer. For example, they are not
-very good for implementing high-performance floating-point calculations
-or calculations that intensively manipulate bit vectors.
-
- (2) Actually, the machine that runs compiled code can be simpler
-than the interpreter machine, because we won't use the `exp' and `unev'
-registers. The interpreter used these to hold pieces of unevaluated
-expressions. With the compiler, however, these expressions get built
-into the compiled code that the register machine will run. For the same
-reason, we don't need the machine operations that deal with expression
-syntax. But compiled code will use a few additional machine operations
-(to represent compiled procedure objects) that didn't appear in the
-explicit-control evaluator machine.
-
-\1f
-File: sicp.info, Node: 5-5-1, Next: 5-5-2, Prev: 5-5, Up: 5-5
-
-5.5.1 Structure of the Compiler
--------------------------------
-
-In section *Note 4-1-7:: we modified our original metacircular
-interpreter to separate analysis from execution. We analyzed each
-expression to produce an execution procedure that took an environment
-as argument and performed the required operations. In our compiler, we
-will do essentially the same analysis. Instead of producing execution
-procedures, however, we will generate sequences of instructions to be
-run by our register machine.
-
- The procedure `compile' is the top-level dispatch in the compiler.
-It corresponds to the `eval' procedure of section *Note 4-1-1::, the
-`analyze' procedure of section *Note 4-1-7::, and the `eval-dispatch'
-entry point of the explicit-control-evaluator in section *Note 5-4-1::.
-The compiler, like the interpreters, uses the expression-syntax
-procedures defined in section *Note 4-1-2::.(1) `Compile' performs a
-case analysis on the syntactic type of the expression to be compiled.
-For each type of expression, it dispatches to a specialized "code
-generator":
-
- (define (compile exp target linkage)
- (cond ((self-evaluating? exp)
- (compile-self-evaluating exp target linkage))
- ((quoted? exp) (compile-quoted exp target linkage))
- ((variable? exp)
- (compile-variable exp target linkage))
- ((assignment? exp)
- (compile-assignment exp target linkage))
- ((definition? exp)
- (compile-definition exp target linkage))
- ((if? exp) (compile-if exp target linkage))
- ((lambda? exp) (compile-lambda exp target linkage))
- ((begin? exp)
- (compile-sequence (begin-actions exp)
- target
- linkage))
- ((cond? exp) (compile (cond->if exp) target linkage))
- ((application? exp)
- (compile-application exp target linkage))
- (else
- (error "Unknown expression type -- COMPILE" exp))))
-
-Targets and linkages
-....................
-
-`Compile' and the code generators that it calls take two arguments in
-addition to the expression to compile. There is a "target", which
-specifies the register in which the compiled code is to return the
-value of the expression. There is also a "linkage descriptor", which
-describes how the code resulting from the compilation of the expression
-should proceed when it has finished its execution. The linkage
-descriptor can require that the code do one of the following three
-things:
-
- * continue at the next instruction in sequence (this is specified by
- the linkage descriptor `next'),
-
- * return from the procedure being compiled (this is specified by the
- linkage descriptor `return'), or
-
- * jump to a named entry point (this is specified by using the
- designated label as the linkage descriptor).
-
-
- For example, compiling the expression `5' (which is self-evaluating)
-with a target of the `val' register and a linkage of `next' should
-produce the instruction
-
- (assign val (const 5))
-
-Compiling the same expression with a linkage of `return' should produce
-the instructions
-
- (assign val (const 5))
- (goto (reg continue))
-
- In the first case, execution will continue with the next instruction
-in the sequence. In the second case, we will return from a procedure
-call. In both cases, the value of the expression will be placed into
-the target `val' register.
-
-Instruction sequences and stack usage
-.....................................
-
-Each code generator returns an "instruction sequence" containing the
-object code it has generated for the expression. Code generation for a
-compound expression is accomplished by combining the output from
-simpler code generators for component expressions, just as evaluation
-of a compound expression is accomplished by evaluating the component
-expressions.
-
- The simplest method for combining instruction sequences is a
-procedure called `append-instruction-sequences'. It takes as arguments
-any number of instruction sequences that are to be executed
-sequentially; it appends them and returns the combined sequence. That
-is, if <SEQ_1> and <SEQ_2> are sequences of instructions, then
-evaluating
-
- (append-instruction-sequences <SEQ_1> <SEQ_2>)
-
-produces the sequence
-
- <SEQ_1>
- <SEQ_2>
-
- Whenever registers might need to be saved, the compiler's code
-generators use `preserving', which is a more subtle method for
-combining instruction sequences. `Preserving' takes three arguments: a
-set of registers and two instruction sequences that are to be executed
-sequentially. It appends the sequences in such a way that the contents
-of each register in the set is preserved over the execution of the
-first sequence, if this is needed for the execution of the second
-sequence. That is, if the first sequence modifies the register and the
-second sequence actually needs the register's original contents, then
-`preserving' wraps a `save' and a `restore' of the register around the
-first sequence before appending the sequences. Otherwise, `preserving'
-simply returns the appended instruction sequences. Thus, for example,
-
- (preserving (list <REG_1> <REG_2>) <SEQ_1> <SEQ_2>)
-
-produces one of the following four sequences of instructions, depending
-on how <SEQ_1> and <SEQ_2> use <REG_1> and <REG_2>:
-
- <seq_1> | (save <reg_1>) | (save <reg_2>) | (save <reg_2>)
- <seq_2> | <seq_1> | <seq_1> | (save <reg_1>)
- | (restore <reg_1>) | (restore <reg_2>) | <seq_1>
- | <seq_2> | <seq_2> | (restore <reg_1>)
- | | | (restore <reg_2>)
- | | | <seq_2>
-
- By using `preserving' to combine instruction sequences the compiler
-avoids unnecessary stack operations. This also isolates the details of
-whether or not to generate `save' and `restore' instructions within the
-`preserving' procedure, separating them from the concerns that arise in
-writing each of the individual code generators. In fact no `save' or
-`restore' instructions are explicitly produced by the code generators.
-
- In principle, we could represent an instruction sequence simply as a
-list of instructions. `Append-instruction-sequences' could then combine
-instruction sequences by performing an ordinary list `append'. However,
-`preserving' would then be a complex operation, because it would have to
-analyze each instruction sequence to determine how the sequence uses its
-registers. `Preserving' would be inefficient as well as complex,
-because it would have to analyze each of its instruction sequence
-arguments, even though these sequences might themselves have been
-constructed by calls to `preserving', in which case their parts would
-have already been analyzed. To avoid such repetitious analysis we will
-associate with each instruction sequence some information about its
-register use. When we construct a basic instruction sequence we will
-provide this information explicitly, and the procedures that combine
-instruction sequences will derive register-use information for the
-combined sequence from the information associated with the component
-sequences.
-
- An instruction sequence will contain three pieces of information:
-
- * the set of registers that must be initialized before the
- instructions in the sequence are executed (these registers are
- said to be "needed" by the sequence),
-
- * the set of registers whose values are modified by the instructions
- in the sequence, and
-
- * the actual instructions (also called "statements") in the sequence.
-
-
- We will represent an instruction sequence as a list of its three
-parts. The constructor for instruction sequences is thus
-
- (define (make-instruction-sequence needs modifies statements)
- (list needs modifies statements))
-
- For example, the two-instruction sequence that looks up the value of
-the variable `x' in the current environment, assigns the result to
-`val', and then returns, requires registers `env' and `continue' to have
-been initialized, and modifies register `val'. This sequence would
-therefore be constructed as
-
- (make-instruction-sequence '(env continue) '(val)
- '((assign val
- (op lookup-variable-value) (const x) (reg env))
- (goto (reg continue))))
-
- We sometimes need to construct an instruction sequence with no
-statements:
-
- (define (empty-instruction-sequence)
- (make-instruction-sequence '() '() '()))
-
- The procedures for combining instruction sequences are shown in
-section *Note 5-5-4::.
-
- *Exercise 5.31:* In evaluating a procedure application, the
- explicit-control evaluator always saves and restores the `env'
- register around the evaluation of the operator, saves and restores
- `env' around the evaluation of each operand (except the final
- one), saves and restores `argl' around the evaluation of each
- operand, and saves and restores `proc' around the evaluation of
- the operand sequence. For each of the following combinations, say
- which of these `save' and `restore' operations are superfluous and
- thus could be eliminated by the compiler's `preserving' mechanism:
-
- (f 'x 'y)
-
- ((f) 'x 'y)
-
- (f (g 'x) y)
-
- (f (g 'x) 'y)
-
- *Exercise 5.32:* Using the `preserving' mechanism, the compiler
- will avoid saving and restoring `env' around the evaluation of the
- operator of a combination in the case where the operator is a
- symbol. We could also build such optimizations into the
- evaluator. Indeed, the explicit-control evaluator of section
- *Note 5-4:: already performs a similar optimization, by treating
- combinations with no operands as a special case.
-
- a. Extend the explicit-control evaluator to recognize as a
- separate class of expressions combinations whose operator is
- a symbol, and to take advantage of this fact in evaluating
- such expressions.
-
- b. Alyssa P. Hacker suggests that by extending the evaluator to
- recognize more and more special cases we could incorporate
- all the compiler's optimizations, and that this would
- eliminate the advantage of compilation altogether. What do
- you think of this idea?
-
-
- ---------- Footnotes ----------
-
- (1) Notice, however, that our compiler is a Scheme program, and the
-syntax procedures that it uses to manipulate expressions are the actual
-Scheme procedures used with the metacircular evaluator. For the
-explicit-control evaluator, in contrast, we assumed that equivalent
-syntax operations were available as operations for the register
-machine. (Of course, when we simulated the register machine in Scheme,
-we used the actual Scheme procedures in our register machine
-simulation.)
-
-\1f
-File: sicp.info, Node: 5-5-2, Next: 5-5-3, Prev: 5-5-1, Up: 5-5
-
-5.5.2 Compiling Expressions
----------------------------
-
-In this section and the next we implement the code generators to which
-the `compile' procedure dispatches.
-
-Compiling linkage code
-......................
-
-In general, the output of each code generator will end with
-instructions--generated by the procedure `compile-linkage'--that
-implement the required linkage. If the linkage is `return' then we must
-generate the instruction `(goto (reg continue))'. This needs the
-`continue' register and does not modify any registers. If the linkage
-is `next', then we needn't include any additional instructions.
-Otherwise, the linkage is a label, and we generate a `goto' to that
-label, an instruction that does not need or modify any registers.(1)
-
- (define (compile-linkage linkage)
- (cond ((eq? linkage 'return)
- (make-instruction-sequence '(continue) '()
- '((goto (reg continue)))))
- ((eq? linkage 'next)
- (empty-instruction-sequence))
- (else
- (make-instruction-sequence '() '()
- `((goto (label ,linkage)))))))
-
- The linkage code is appended to an instruction sequence by
-`preserving' the `continue' register, since a `return' linkage will
-require the `continue' register: If the given instruction sequence
-modifies `continue' and the linkage code needs it, `continue' will be
-saved and restored.
-
- (define (end-with-linkage linkage instruction-sequence)
- (preserving '(continue)
- instruction-sequence
- (compile-linkage linkage)))
-
-Compiling simple expressions
-............................
-
-The code generators for self-evaluating expressions, quotations, and
-variables construct instruction sequences that assign the required
-value to the target register and then proceed as specified by the
-linkage descriptor.
-
- (define (compile-self-evaluating exp target linkage)
- (end-with-linkage linkage
- (make-instruction-sequence '() (list target)
- `((assign ,target (const ,exp))))))
-
- (define (compile-quoted exp target linkage)
- (end-with-linkage linkage
- (make-instruction-sequence '() (list target)
- `((assign ,target (const ,(text-of-quotation exp)))))))
-
- (define (compile-variable exp target linkage)
- (end-with-linkage linkage
- (make-instruction-sequence '(env) (list target)
- `((assign ,target
- (op lookup-variable-value)
- (const ,exp)
- (reg env))))))
-
- All these assignment instructions modify the target register, and
-the one that looks up a variable needs the `env' register.
-
- Assignments and definitions are handled much as they are in the
-interpreter. We recursively generate code that computes the value to
-be assigned to the variable, and append to it a two-instruction
-sequence that actually sets or defines the variable and assigns the
-value of the whole expression (the symbol `ok') to the target register.
-The recursive compilation has target `val' and linkage `next' so that
-the code will put its result into `val' and continue with the code that
-is appended after it. The appending is done preserving `env', since
-the environment is needed for setting or defining the variable and the
-code for the variable value could be the compilation of a complex
-expression that might modify the registers in arbitrary ways.
-
- (define (compile-assignment exp target linkage)
- (let ((var (assignment-variable exp))
- (get-value-code
- (compile (assignment-value exp) 'val 'next)))
- (end-with-linkage linkage
- (preserving '(env)
- get-value-code
- (make-instruction-sequence '(env val) (list target)
- `((perform (op set-variable-value!)
- (const ,var)
- (reg val)
- (reg env))
- (assign ,target (const ok))))))))
-
- (define (compile-definition exp target linkage)
- (let ((var (definition-variable exp))
- (get-value-code
- (compile (definition-value exp) 'val 'next)))
- (end-with-linkage linkage
- (preserving '(env)
- get-value-code
- (make-instruction-sequence '(env val) (list target)
- `((perform (op define-variable!)
- (const ,var)
- (reg val)
- (reg env))
- (assign ,target (const ok))))))))
-
- The appended two-instruction sequence requires `env' and `val' and
-modifies the target. Note that although we preserve `env' for this
-sequence, we do not preserve `val', because the `get-value-code' is
-designed to explicitly place its result in `val' for use by this
-sequence. (In fact, if we did preserve `val', we would have a bug,
-because this would cause the previous contents of `val' to be restored
-right after the `get-value-code' is run.)
-
-Compiling conditional expressions
-.................................
-
-The code for an `if' expression compiled with a given target and linkage
-has the form
-
- <_compilation of predicate, target `val', linkage `next'_>
- (test (op false?) (reg val))
- (branch (label false-branch))
- true-branch
- <_compilation of consequent with given target and given linkage or `after-if'_>
- false-branch
- <_compilation of alternative with given target and linkage_>
- after-if
-
- To generate this code, we compile the predicate, consequent, and
-alternative, and combine the resulting code with instructions to test
-the predicate result and with newly generated labels to mark the true
-and false branches and the end of the conditional.(2) In this
-arrangement of code, we must branch around the true branch if the test
-is false. The only slight complication is in how the linkage for the
-true branch should be handled. If the linkage for the conditional is
-`return' or a label, then the true and false branches will both use
-this same linkage. If the linkage is `next', the true branch ends with
-a jump around the code for the false branch to the label at the end of
-the conditional.
-
- (define (compile-if exp target linkage)
- (let ((t-branch (make-label 'true-branch))
- (f-branch (make-label 'false-branch))
- (after-if (make-label 'after-if)))
- (let ((consequent-linkage
- (if (eq? linkage 'next) after-if linkage)))
- (let ((p-code (compile (if-predicate exp) 'val 'next))
- (c-code
- (compile
- (if-consequent exp) target consequent-linkage))
- (a-code
- (compile (if-alternative exp) target linkage)))
- (preserving '(env continue)
- p-code
- (append-instruction-sequences
- (make-instruction-sequence '(val) '()
- `((test (op false?) (reg val))
- (branch (label ,f-branch))))
- (parallel-instruction-sequences
- (append-instruction-sequences t-branch c-code)
- (append-instruction-sequences f-branch a-code))
- after-if))))))
-
- `Env' is preserved around the predicate code because it could be
-needed by the true and false branches, and `continue' is preserved
-because it could be needed by the linkage code in those branches. The
-code for the true and false branches (which are not executed
-sequentially) is appended using a special combiner
-`parallel-instruction-sequences' described in section *Note 5-5-4::.
-
- Note that `cond' is a derived expression, so all that the compiler
-needs to do handle it is to apply the `cond->if' transformer (from
-section *Note 4-1-2::) and compile the resulting `if' expression.
-
-Compiling sequences
-...................
-
-The compilation of sequences (from procedure bodies or explicit `begin'
-expressions) parallels their evaluation. Each expression of the
-sequence is compiled--the last expression with the linkage specified
-for the sequence, and the other expressions with linkage `next' (to
-execute the rest of the sequence). The instruction sequences for the
-individual expressions are appended to form a single instruction
-sequence, such that `env' (needed for the rest of the sequence) and
-`continue' (possibly needed for the linkage at the end of the sequence)
-are preserved.
-
- (define (compile-sequence seq target linkage)
- (if (last-exp? seq)
- (compile (first-exp seq) target linkage)
- (preserving '(env continue)
- (compile (first-exp seq) target 'next)
- (compile-sequence (rest-exps seq) target linkage))))
-
-Compiling `lambda' expressions
-..............................
-
-`Lambda' expressions construct procedures. The object code for a
-`lambda' expression must have the form
-
- <_construct procedure object and assign it to target register_>
- <LINKAGE>
-
- When we compile the `lambda' expression, we also generate the code
-for the procedure body. Although the body won't be executed at the
-time of procedure construction, it is convenient to insert it into the
-object code right after the code for the `lambda'. If the linkage for
-the `lambda' expression is a label or `return', this is fine. But if
-the linkage is `next', we will need to skip around the code for the
-procedure body by using a linkage that jumps to a label that is
-inserted after the body. The object code thus has the form
-
- <_construct procedure object and assign it to target register_>
- <_code for given linkage_> _or_ `(goto (label after-lambda))'
- <_compilation of procedure body_>
- after-lambda
-
- `Compile-lambda' generates the code for constructing the procedure
-object followed by the code for the procedure body. The procedure
-object will be constructed at run time by combining the current
-environment (the environment at the point of definition) with the entry
-point to the compiled procedure body (a newly generated label).(3)
-
- (define (compile-lambda exp target linkage)
- (let ((proc-entry (make-label 'entry))
- (after-lambda (make-label 'after-lambda)))
- (let ((lambda-linkage
- (if (eq? linkage 'next) after-lambda linkage)))
- (append-instruction-sequences
- (tack-on-instruction-sequence
- (end-with-linkage lambda-linkage
- (make-instruction-sequence '(env) (list target)
- `((assign ,target
- (op make-compiled-procedure)
- (label ,proc-entry)
- (reg env)))))
- (compile-lambda-body exp proc-entry))
- after-lambda))))
-
- `Compile-lambda' uses the special combiner
-`tack-on-instruction-sequence' (section *Note 5-5-4::) rather than
-`append-instruction-sequences' to append the procedure body to the
-`lambda' expression code, because the body is not part of the sequence
-of instructions that will be executed when the combined sequence is
-entered; rather, it is in the sequence only because that was a
-convenient place to put it.
-
- `Compile-lambda-body' constructs the code for the body of the
-procedure. This code begins with a label for the entry point. Next
-come instructions that will cause the run-time evaluation environment
-to switch to the correct environment for evaluating the procedure
-body--namely, the definition environment of the procedure, extended to
-include the bindings of the formal parameters to the arguments with
-which the procedure is called. After this comes the code for the
-sequence of expressions that makes up the procedure body. The sequence
-is compiled with linkage `return' and target `val' so that it will end
-by returning from the procedure with the procedure result in `val'.
-
- (define (compile-lambda-body exp proc-entry)
- (let ((formals (lambda-parameters exp)))
- (append-instruction-sequences
- (make-instruction-sequence '(env proc argl) '(env)
- `(,proc-entry
- (assign env (op compiled-procedure-env) (reg proc))
- (assign env
- (op extend-environment)
- (const ,formals)
- (reg argl)
- (reg env))))
- (compile-sequence (lambda-body exp) 'val 'return))))
-
- ---------- Footnotes ----------
-
- (1) This procedure uses a feature of Lisp called "backquote" (or "quasiquote")
-that is handy for constructing lists. Preceding a list with a
-backquote symbol is much like quoting it, except that anything in the
-list that is flagged with a comma is evaluated.
-
- For example, if the value of `linkage' is the symbol
-`branch25', then the expression
-``((goto (label ,linkage)))'
-evaluates to the list
-`((goto (label branch25)))'.
-Similarly, if the value of `x' is the list `(a b c)', then
-``(1 2 ,(car x))'
-evaluates to the list
-`(1 2 a)'.
-
- (2) We can't just use the labels `true-branch', `false-branch', and
-`after-if' as shown above, because there might be more than one `if' in
-the program. The compiler uses the procedure `make-label' to generate
-labels. `Make-label' takes a symbol as argument and returns a new
-symbol that begins with the given symbol. For example, successive
-calls to `(make-label 'a)' would return `a1', `a2', and so on.
-`Make-label' can be implemented similarly to the generation of unique
-variable names in the query language, as follows:
-
- (define label-counter 0)
-
- (define (new-label-number)
- (set! label-counter (+ 1 label-counter))
- label-counter)
-
- (define (make-label name)
- (string->symbol
- (string-append (symbol->string name)
- (number->string (new-label-number)))))
-
- (3) [Footnote 38] We need machine operations to implement a data
-structure for representing compiled procedures, analogous to the
-structure for compound procedures described in section *Note 4-1-3:::
-
- (define (make-compiled-procedure entry env)
- (list 'compiled-procedure entry env))
-
- (define (compiled-procedure? proc)
- (tagged-list? proc 'compiled-procedure))
-
- (define (compiled-procedure-entry c-proc) (cadr c-proc))
-
- (define (compiled-procedure-env c-proc) (caddr c-proc))
-
-\1f
-File: sicp.info, Node: 5-5-3, Next: 5-5-4, Prev: 5-5-2, Up: 5-5
-
-5.5.3 Compiling Combinations
-----------------------------
-
-The essence of the compilation process is the compilation of procedure
-applications. The code for a combination compiled with a given target
-and linkage has the form
-
- <_compilation of operator, target `proc', linkage `next'_>
- <_evaluate operands and construct argument list in `argl'_>
- <_compilation of procedure call with given target and linkage_>
-
- The registers `env', `proc', and `argl' may have to be saved and
-restored during evaluation of the operator and operands. Note that
-this is the only place in the compiler where a target other than `val'
-is specified.
-
- The required code is generated by `compile-application'. This
-recursively compiles the operator, to produce code that puts the
-procedure to be applied into `proc', and compiles the operands, to
-produce code that evaluates the individual operands of the application.
-The instruction sequences for the operands are combined (by
-`construct-arglist') with code that constructs the list of arguments in
-`argl', and the resulting argument-list code is combined with the
-procedure code and the code that performs the procedure call (produced
-by `compile-procedure-call'). In appending the code sequences, the
-`env' register must be preserved around the evaluation of the operator
-(since evaluating the operator might modify `env', which will be needed
-to evaluate the operands), and the `proc' register must be preserved
-around the construction of the argument list (since evaluating the
-operands might modify `proc', which will be needed for the actual
-procedure application). `Continue' must also be preserved throughout,
-since it is needed for the linkage in the procedure call.
-
- (define (compile-application exp target linkage)
- (let ((proc-code (compile (operator exp) 'proc 'next))
- (operand-codes
- (map (lambda (operand) (compile operand 'val 'next))
- (operands exp))))
- (preserving '(env continue)
- proc-code
- (preserving '(proc continue)
- (construct-arglist operand-codes)
- (compile-procedure-call target linkage)))))
-
- The code to construct the argument list will evaluate each operand
-into `val' and then `cons' that value onto the argument list being
-accumulated in `argl'. Since we `cons' the arguments onto `argl' in
-sequence, we must start with the last argument and end with the first,
-so that the arguments will appear in order from first to last in the
-resulting list. Rather than waste an instruction by initializing `argl'
-to the empty list to set up for this sequence of evaluations, we make
-the first code sequence construct the initial `argl'. The general form
-of the argument-list construction is thus as follows:
-
- <_compilation of last operand, targeted to `val'_>
- (assign argl (op list) (reg val))
- <_compilation of next operand, targeted to `val'_>
- (assign argl (op cons) (reg val) (reg argl))
- ...
- <_compilation of first operand, targeted to `val'_>
- (assign argl (op cons) (reg val) (reg argl))
-
- `Argl' must be preserved around each operand evaluation except the
-first (so that arguments accumulated so far won't be lost), and `env'
-must be preserved around each operand evaluation except the last (for
-use by subsequent operand evaluations).
-
- Compiling this argument code is a bit tricky, because of the special
-treatment of the first operand to be evaluated and the need to preserve
-`argl' and `env' in different places. The `construct-arglist'
-procedure takes as arguments the code that evaluates the individual
-operands. If there are no operands at all, it simply emits the
-instruction
-
- (assign argl (const ()))
-
- Otherwise, `construct-arglist' creates code that initializes `argl'
-with the last argument, and appends code that evaluates the rest of the
-arguments and adjoins them to `argl' in succession. In order to process
-the arguments from last to first, we must reverse the list of operand
-code sequences from the order supplied by `compile-application'.
-
- (define (construct-arglist operand-codes)
- (let ((operand-codes (reverse operand-codes)))
- (if (null? operand-codes)
- (make-instruction-sequence '() '(argl)
- '((assign argl (const ()))))
- (let ((code-to-get-last-arg
- (append-instruction-sequences
- (car operand-codes)
- (make-instruction-sequence '(val) '(argl)
- '((assign argl (op list) (reg val)))))))
- (if (null? (cdr operand-codes))
- code-to-get-last-arg
- (preserving '(env)
- code-to-get-last-arg
- (code-to-get-rest-args
- (cdr operand-codes))))))))
-
- (define (code-to-get-rest-args operand-codes)
- (let ((code-for-next-arg
- (preserving '(argl)
- (car operand-codes)
- (make-instruction-sequence '(val argl) '(argl)
- '((assign argl
- (op cons) (reg val) (reg argl)))))))
- (if (null? (cdr operand-codes))
- code-for-next-arg
- (preserving '(env)
- code-for-next-arg
- (code-to-get-rest-args (cdr operand-codes))))))
-
-Applying procedures
-...................
-
-After evaluating the elements of a combination, the compiled code must
-apply the procedure in `proc' to the arguments in `argl'. The code
-performs essentially the same dispatch as the `apply' procedure in the
-metacircular evaluator of section *Note 4-1-1:: or the `apply-dispatch'
-entry point in the explicit-control evaluator of section *Note 5-4-1::.
-It checks whether the procedure to be applied is a primitive procedure
-or a compiled procedure. For a primitive procedure, it uses
-`apply-primitive-procedure'; we will see shortly how it handles compiled
-procedures. The procedure-application code has the following form:
-
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch))
- compiled-branch
- <_code to apply compiled procedure with given target and appropriate linkage_>
- primitive-branch
- (assign <TARGET>
- (op apply-primitive-procedure)
- (reg proc)
- (reg argl))
- <LINKAGE>
- after-call
-
- Observe that the compiled branch must skip around the primitive
-branch. Therefore, if the linkage for the original procedure call was
-`next', the compound branch must use a linkage that jumps to a label
-that is inserted after the primitive branch. (This is similar to the
-linkage used for the true branch in `compile-if'.)
-
- (define (compile-procedure-call target linkage)
- (let ((primitive-branch (make-label 'primitive-branch))
- (compiled-branch (make-label 'compiled-branch))
- (after-call (make-label 'after-call)))
- (let ((compiled-linkage
- (if (eq? linkage 'next) after-call linkage)))
- (append-instruction-sequences
- (make-instruction-sequence '(proc) '()
- `((test (op primitive-procedure?) (reg proc))
- (branch (label ,primitive-branch))))
- (parallel-instruction-sequences
- (append-instruction-sequences
- compiled-branch
- (compile-proc-appl target compiled-linkage))
- (append-instruction-sequences
- primitive-branch
- (end-with-linkage linkage
- (make-instruction-sequence '(proc argl)
- (list target)
- `((assign ,target
- (op apply-primitive-procedure)
- (reg proc)
- (reg argl)))))))
- after-call))))
-
- The primitive and compound branches, like the true and false
-branches in `compile-if', are appended using
-`parallel-instruction-sequences' rather than the ordinary
-`append-instruction-sequences', because they will not be executed
-sequentially.
-
-Applying compiled procedures
-............................
-
-The code that handles procedure application is the most subtle part of
-the compiler, even though the instruction sequences it generates are
-very short. A compiled procedure (as constructed by `compile-lambda')
-has an entry point, which is a label that designates where the code for
-the procedure starts. The code at this entry point computes a result
-in `val' and returns by executing the instruction `(goto (reg
-continue))'. Thus, we might expect the code for a compiled-procedure
-application (to be generated by `compile-proc-appl') with a given
-target and linkage to look like this if the linkage is a label
-
- (assign continue (label proc-return))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- proc-return
- (assign <TARGET> (reg val)) ; included if target is not `val'
- (goto (label <LINKAGE>)) ; linkage code
-
-or like this if the linkage is `return'.
-
- (save continue)
- (assign continue (label proc-return))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- proc-return
- (assign <TARGET> (reg val)) ; included if target is not `val'
- (restore continue)
- (goto (reg continue)) ; linkage code
-
- This code sets up `continue' so that the procedure will return to a
-label `proc-return' and jumps to the procedure's entry point. The code
-at `proc-return' transfers the procedure's result from `val' to the
-target register (if necessary) and then jumps to the location specified
-by the linkage. (The linkage is always `return' or a label, because
-`compile-procedure-call' replaces a `next' linkage for the
-compound-procedure branch by an `after-call' label.)
-
- In fact, if the target is not `val', that is exactly the code our
-compiler will generate.(1) Usually, however, the target is `val' (the
-only time the compiler specifies a different register is when targeting
-the evaluation of an operator to `proc'), so the procedure result is
-put directly into the target register and there is no need to return to
-a special location that copies it. Instead, we simplify the code by
-setting up `continue' so that the procedure will "return" directly to
-the place specified by the caller's linkage:
-
- <_set up `continue' for linkage_>
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
-
- If the linkage is a label, we set up `continue' so that the
-procedure will return to that label. (That is, the `(goto (reg
-continue))' the procedure ends with becomes equivalent to the `(goto
-(label <LINKAGE>))' at `proc-return' above.)
-
- (assign continue (label <LINKAGE>))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
-
- If the linkage is `return', we don't need to set up `continue' at
-all: It already holds the desired location. (That is, the `(goto (reg
-continue))' the procedure ends with goes directly to the place where the
-`(goto (reg continue))' at `proc-return' would have gone.)
-
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
-
- With this implementation of the `return' linkage, the compiler
-generates tail-recursive code. Calling a procedure as the final step
-in a procedure body does a direct transfer, without saving any
-information on the stack.
-
- Suppose instead that we had handled the case of a procedure call
-with a linkage of `return' and a target of `val' as shown above for a
-non-`val' target. This would destroy tail recursion. Our system would
-still give the same value for any expression. But each time we called
-a procedure, we would save `continue' and return after the call to undo
-the (useless) save. These extra saves would accumulate during a nest
-of procedure calls.(2)
-
- `Compile-proc-appl' generates the above procedure-application code by
-considering four cases, depending on whether the target for the call is
-`val' and whether the linkage is `return'. Observe that the
-instruction sequences are declared to modify all the registers, since
-executing the procedure body can change the registers in arbitrary
-ways.(3) Also note that the code sequence for the case with target
-`val' and linkage `return' is declared to need `continue': Even though
-`continue' is not explicitly used in the two-instruction sequence, we
-must be sure that `continue' will have the correct value when we enter
-the compiled procedure.
-
- (define (compile-proc-appl target linkage)
- (cond ((and (eq? target 'val) (not (eq? linkage 'return)))
- (make-instruction-sequence '(proc) all-regs
- `((assign continue (label ,linkage))
- (assign val (op compiled-procedure-entry)
- (reg proc))
- (goto (reg val)))))
- ((and (not (eq? target 'val))
- (not (eq? linkage 'return)))
- (let ((proc-return (make-label 'proc-return)))
- (make-instruction-sequence '(proc) all-regs
- `((assign continue (label ,proc-return))
- (assign val (op compiled-procedure-entry)
- (reg proc))
- (goto (reg val))
- ,proc-return
- (assign ,target (reg val))
- (goto (label ,linkage))))))
- ((and (eq? target 'val) (eq? linkage 'return))
- (make-instruction-sequence '(proc continue) all-regs
- '((assign val (op compiled-procedure-entry)
- (reg proc))
- (goto (reg val)))))
- ((and (not (eq? target 'val)) (eq? linkage 'return))
- (error "return linkage, target not val -- COMPILE"
- target))))
-
- ---------- Footnotes ----------
-
- (1) Actually, we signal an error when the target is not `val' and
-the linkage is `return', since the only place we request `return'
-linkages is in compiling procedures, and our convention is that
-procedures return their values in `val'.
-
- (2) Making a compiler generate tail-recursive code might seem like a
-straightforward idea. But most compilers for common languages,
-including C and Pascal, do not do this, and therefore these languages
-cannot represent iterative processes in terms of procedure call alone.
-The difficulty with tail recursion in these languages is that their
-implementations use the stack to store procedure arguments and local
-variables as well as return addresses. The Scheme implementations
-described in this book store arguments and variables in memory to be
-garbage-collected. The reason for using the stack for variables and
-arguments is that it avoids the need for garbage collection in languages
-that would not otherwise require it, and is generally believed to be
-more efficient. Sophisticated Lisp compilers can, in fact, use the
-stack for arguments without destroying tail recursion. (See Hanson
-1990 for a description.) There is also some debate about whether stack
-allocation is actually more efficient than garbage collection in the
-first place, but the details seem to hinge on fine points of computer
-architecture. (See Appel 1987 and Miller and Rozas 1994 for opposing
-views on this issue.)
-
- (3) The variable `all-regs' is bound to the list of names of all the
-registers:
-
- (define all-regs '(env proc val argl continue))
-
-\1f
-File: sicp.info, Node: 5-5-4, Next: 5-5-5, Prev: 5-5-3, Up: 5-5
-
-5.5.4 Combining Instruction Sequences
--------------------------------------
-
-This section describes the details on how instruction sequences are
-represented and combined. Recall from section *Note 5-5-1:: that an
-instruction sequence is represented as a list of the registers needed,
-the registers modified, and the actual instructions. We will also
-consider a label (symbol) to be a degenerate case of an instruction
-sequence, which doesn't need or modify any registers. So to determine
-the registers needed and modified by instruction sequences we use the
-selectors
-
- (define (registers-needed s)
- (if (symbol? s) '() (car s)))
-
- (define (registers-modified s)
- (if (symbol? s) '() (cadr s)))
-
- (define (statements s)
- (if (symbol? s) (list s) (caddr s)))
-
-and to determine whether a given sequence needs or modifies a given
-register we use the predicates
-
- (define (needs-register? seq reg)
- (memq reg (registers-needed seq)))
-
- (define (modifies-register? seq reg)
- (memq reg (registers-modified seq)))
-
- In terms of these predicates and selectors, we can implement the
-various instruction sequence combiners used throughout the compiler.
-
- The basic combiner is `append-instruction-sequences'. This takes as
-arguments an arbitrary number of instruction sequences that are to be
-executed sequentially and returns an instruction sequence whose
-statements are the statements of all the sequences appended together.
-The subtle point is to determine the registers that are needed and
-modified by the resulting sequence. It modifies those registers that
-are modified by any of the sequences; it needs those registers that
-must be initialized before the first sequence can be run (the registers
-needed by the first sequence), together with those registers needed by
-any of the other sequences that are not initialized (modified) by
-sequences preceding it.
-
- The sequences are appended two at a time by `append-2-sequences'.
-This takes two instruction sequences `seq1' and `seq2' and returns the
-instruction sequence whose statements are the statements of `seq1'
-followed by the statements of `seq2', whose modified registers are those
-registers that are modified by either `seq1' or `seq2', and whose
-needed registers are the registers needed by `seq1' together with those
-registers needed by `seq2' that are not modified by `seq1'. (In terms
-of set operations, the new set of needed registers is the union of the
-set of registers needed by `seq1' with the set difference of the
-registers needed by `seq2' and the registers modified by `seq1'.) Thus,
-`append-instruction-sequences' is implemented as follows:
-
- (define (append-instruction-sequences . seqs)
- (define (append-2-sequences seq1 seq2)
- (make-instruction-sequence
- (list-union (registers-needed seq1)
- (list-difference (registers-needed seq2)
- (registers-modified seq1)))
- (list-union (registers-modified seq1)
- (registers-modified seq2))
- (append (statements seq1) (statements seq2))))
- (define (append-seq-list seqs)
- (if (null? seqs)
- (empty-instruction-sequence)
- (append-2-sequences (car seqs)
- (append-seq-list (cdr seqs)))))
- (append-seq-list seqs))
-
- This procedure uses some simple operations for manipulating sets
-represented as lists, similar to the (unordered) set representation
-described in section *Note 2-3-3:::
-
- (define (list-union s1 s2)
- (cond ((null? s1) s2)
- ((memq (car s1) s2) (list-union (cdr s1) s2))
- (else (cons (car s1) (list-union (cdr s1) s2)))))
-
- (define (list-difference s1 s2)
- (cond ((null? s1) '())
- ((memq (car s1) s2) (list-difference (cdr s1) s2))
- (else (cons (car s1)
- (list-difference (cdr s1) s2)))))
-
- `Preserving', the second major instruction sequence combiner, takes
-a list of registers `regs' and two instruction sequences `seq1' and
-`seq2' that are to be executed sequentially. It returns an instruction
-sequence whose statements are the statements of `seq1' followed by the
-statements of `seq2', with appropriate `save' and `restore'
-instructions around `seq1' to protect the registers in `regs' that are
-modified by `seq1' but needed by `seq2'. To accomplish this,
-`preserving' first creates a sequence that has the required `save's
-followed by the statements of `seq1' followed by the required
-`restore's. This sequence needs the registers being saved and restored
-in addition to the registers needed by `seq1', and modifies the
-registers modified by `seq1' except for the ones being saved and
-restored. This augmented sequence and `seq2' are then appended in the
-usual way. The following procedure implements this strategy
-recursively, walking down the list of registers to be preserved:(1)
-
- (define (preserving regs seq1 seq2)
- (if (null? regs)
- (append-instruction-sequences seq1 seq2)
- (let ((first-reg (car regs)))
- (if (and (needs-register? seq2 first-reg)
- (modifies-register? seq1 first-reg))
- (preserving (cdr regs)
- (make-instruction-sequence
- (list-union (list first-reg)
- (registers-needed seq1))
- (list-difference (registers-modified seq1)
- (list first-reg))
- (append `((save ,first-reg))
- (statements seq1)
- `((restore ,first-reg))))
- seq2)
- (preserving (cdr regs) seq1 seq2)))))
-
- Another sequence combiner, `tack-on-instruction-sequence', is used by
-`compile-lambda' to append a procedure body to another sequence.
-Because the procedure body is not "in line" to be executed as part of
-the combined sequence, its register use has no impact on the register
-use of the sequence in which it is embedded. We thus ignore the
-procedure body's sets of needed and modified registers when we tack it
-onto the other sequence.
-
- (define (tack-on-instruction-sequence seq body-seq)
- (make-instruction-sequence
- (registers-needed seq)
- (registers-modified seq)
- (append (statements seq) (statements body-seq))))
-
- `Compile-if' and `compile-procedure-call' use a special combiner
-called `parallel-instruction-sequences' to append the two alternative
-branches that follow a test. The two branches will never be executed
-sequentially; for any particular evaluation of the test, one branch or
-the other will be entered. Because of this, the registers needed by
-the second branch are still needed by the combined sequence, even if
-these are modified by the first branch.
-
- (define (parallel-instruction-sequences seq1 seq2)
- (make-instruction-sequence
- (list-union (registers-needed seq1)
- (registers-needed seq2))
- (list-union (registers-modified seq1)
- (registers-modified seq2))
- (append (statements seq1) (statements seq2))))
-
- ---------- Footnotes ----------
-
- (1) Note that `preserving' calls `append' with three arguments.
-Though the definition of `append' shown in this book accepts only two
-arguments, Scheme standardly provides an `append' procedure that takes
-an arbitrary number of arguments.
-
-\1f
-File: sicp.info, Node: 5-5-5, Next: 5-5-6, Prev: 5-5-4, Up: 5-5
-
-5.5.5 An Example of Compiled Code
----------------------------------
-
-Now that we have seen all the elements of the compiler, let us examine
-an example of compiled code to see how things fit together. We will
-compile the definition of a recursive `factorial' procedure by calling
-`compile':
-
- (compile
- '(define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n)))
- 'val
- 'next)
-
- We have specified that the value of the `define' expression should be
-placed in the `val' register. We don't care what the compiled code does
-after executing the `define', so our choice of `next' as the linkage
-descriptor is arbitrary.
-
- `Compile' determines that the expression is a definition, so it calls
-`compile-definition' to compile code to compute the value to be assigned
-(targeted to `val'), followed by code to install the definition,
-followed by code to put the value of the `define' (which is the symbol
-`ok') into the target register, followed finally by the linkage code.
-`Env' is preserved around the computation of the value, because it is
-needed in order to install the definition. Because the linkage is
-`next', there is no linkage code in this case. The skeleton of the
-compiled code is thus
-
- <_save `env' if modified by code to compute value_>
- <_compilation of definition value, target `val', linkage `next'_>
- <_restore `env' if saved above_>
- (perform (op define-variable!)
- (const factorial)
- (reg val)
- (reg env))
- (assign val (const ok))
-
- The expression that is to be compiled to produce the value for the
-variable `factorial' is a `lambda' expression whose value is the
-procedure that computes factorials. `Compile' handles this by calling
-`compile-lambda', which compiles the procedure body, labels it as a new
-entry point, and generates the instruction that will combine the
-procedure body at the new entry point with the run-time environment and
-assign the result to `val'. The sequence then skips around the
-compiled procedure code, which is inserted at this point. The
-procedure code itself begins by extending the procedure's definition
-environment by a frame that binds the formal parameter `n' to the
-procedure argument. Then comes the actual procedure body. Since this
-code for the value of the variable doesn't modify the `env' register,
-the optional `save' and `restore' shown above aren't generated. (The
-procedure code at `entry2' isn't executed at this point, so its use of
-`env' is irrelevant.) Therefore, the skeleton for the compiled code
-becomes
-
- (assign val (op make-compiled-procedure)
- (label entry2)
- (reg env))
- (goto (label after-lambda1))
- entry2
- (assign env (op compiled-procedure-env) (reg proc))
- (assign env (op extend-environment)
- (const (n))
- (reg argl)
- (reg env))
- <_compilation of procedure body_>
- after-lambda1
- (perform (op define-variable!)
- (const factorial)
- (reg val)
- (reg env))
- (assign val (const ok))
-
- A procedure body is always compiled (by `compile-lambda-body') as a
-sequence with target `val' and linkage `return'. The sequence in this
-case consists of a single `if' expression:
-
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n))
-
- `Compile-if' generates code that first computes the predicate
-(targeted to `val'), then checks the result and branches around the
-true branch if the predicate is false. `Env' and `continue' are
-preserved around the predicate code, since they may be needed for the
-rest of the `if' expression. Since the `if' expression is the final
-expression (and only expression) in the sequence making up the
-procedure body, its target is `val' and its linkage is `return', so the
-true and false branches are both compiled with target `val' and linkage
-`return'. (That is, the value of the conditional, which is the value
-computed by either of its branches, is the value of the procedure.)
-
- <_save `continue', `env' if modified by predicate and needed by branches_>
- <_compilation of predicate, target `val', linkage `next'_>
- <_restore `continue', `env' if saved above_>
- (test (op false?) (reg val))
- (branch (label false-branch4))
- true-branch5
- <_compilation of true branch, target `val', linkage `return'_>
- false-branch4
- <_compilation of false branch, target `val', linkage `return'_>
- after-if3
-
- The predicate `(= n 1)' is a procedure call. This looks up the
-operator (the symbol `=') and places this value in `proc'. It then
-assembles the arguments `1' and the value of `n' into `argl'. Then it
-tests whether `proc' contains a primitive or a compound procedure, and
-dispatches to a primitive branch or a compound branch accordingly. Both
-branches resume at the `after-call' label. The requirements to preserve
-registers around the evaluation of the operator and operands don't
-result in any saving of registers, because in this case those
-evaluations don't modify the registers in question.
-
- (assign proc
- (op lookup-variable-value) (const =) (reg env))
- (assign val (const 1))
- (assign argl (op list) (reg val))
- (assign val (op lookup-variable-value) (const n) (reg env))
- (assign argl (op cons) (reg val) (reg argl))
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch17))
- compiled-branch16
- (assign continue (label after-call15))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch17
- (assign val (op apply-primitive-procedure)
- (reg proc)
- (reg argl))
- after-call15
-
- The true branch, which is the constant 1, compiles (with target
-`val' and linkage `return') to
-
- (assign val (const 1))
- (goto (reg continue))
-
- The code for the false branch is another a procedure call, where the
-procedure is the value of the symbol `*', and the arguments are `n' and
-the result of another procedure call (a call to `factorial'). Each of
-these calls sets up `proc' and `argl' and its own primitive and compound
-branches. *Note Figure 5-17:: shows the complete compilation of the
-definition of the `factorial' procedure. Notice that the possible
-`save' and `restore' of `continue' and `env' around the predicate, shown
-above, are in fact generated, because these registers are modified by
-the procedure call in the predicate and needed for the procedure call
-and the `return' linkage in the branches.
-
- *Exercise 5.33:* Consider the following definition of a factorial
- procedure, which is slightly different from the one given above:
-
- (define (factorial-alt n)
- (if (= n 1)
- 1
- (* n (factorial-alt (- n 1)))))
-
- Compile this procedure and compare the resulting code with that
- produced for `factorial'. Explain any differences you find. Does
- either program execute more efficiently than the other?
-
- *Exercise 5.34:* Compile the iterative factorial procedure
-
- (define (factorial n)
- (define (iter product counter)
- (if (> counter n)
- product
- (iter (* counter product)
- (+ counter 1))))
- (iter 1 1))
-
- Annotate the resulting code, showing the essential difference
- between the code for iterative and recursive versions of
- `factorial' that makes one process build up stack space and the
- other run in constant stack space.
-
- *Figure 5.17:* Compilation of the definition of the `factorial'
- procedure
-
- ;; construct the procedure and skip over code for the procedure body
- (assign val
- (op make-compiled-procedure) (label entry2) (reg env))
- (goto (label after-lambda1))
-
- entry2 ; calls to `factorial' will enter here
- (assign env (op compiled-procedure-env) (reg proc))
- (assign env
- (op extend-environment) (const (n)) (reg argl) (reg env))
- ;; begin actual procedure body
- (save continue)
- (save env)
-
- ;; compute `(= n 1)'
- (assign proc (op lookup-variable-value) (const =) (reg env))
- (assign val (const 1))
- (assign argl (op list) (reg val))
- (assign val (op lookup-variable-value) (const n) (reg env))
- (assign argl (op cons) (reg val) (reg argl))
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch17))
- compiled-branch16
- (assign continue (label after-call15))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch17
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
-
- after-call15 ; `val' now contains result of `(= n 1)'
- (restore env)
- (restore continue)
- (test (op false?) (reg val))
- (branch (label false-branch4))
- true-branch5 ; return 1
- (assign val (const 1))
- (goto (reg continue))
-
- false-branch4
- ;; compute and return `(* (factorial (- n 1)) n)'
- (assign proc (op lookup-variable-value) (const *) (reg env))
- (save continue)
- (save proc) ; save `*' procedure
- (assign val (op lookup-variable-value) (const n) (reg env))
- (assign argl (op list) (reg val))
- (save argl) ; save partial argument list for `*'
-
- ;; compute `(factorial (- n 1))', which is the other argument for `*'
- (assign proc
- (op lookup-variable-value) (const factorial) (reg env))
- (save proc) ; save `factorial' procedure
-
- ;; compute `(- n 1)', which is the argument for `factorial'
- (assign proc (op lookup-variable-value) (const -) (reg env))
- (assign val (const 1))
- (assign argl (op list) (reg val))
- (assign val (op lookup-variable-value) (const n) (reg env))
- (assign argl (op cons) (reg val) (reg argl))
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch8))
- compiled-branch7
- (assign continue (label after-call6))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch8
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
-
- after-call6 ; `val' now contains result of `(- n 1)'
- (assign argl (op list) (reg val))
- (restore proc) ; restore `factorial'
- ;; apply `factorial'
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch11))
- compiled-branch10
- (assign continue (label after-call9))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch11
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
-
- after-call9 ; `val' now contains result of `(factorial (- n 1))'
- (restore argl) ; restore partial argument list for `*'
- (assign argl (op cons) (reg val) (reg argl))
- (restore proc) ; restore `*'
- (restore continue)
- ;; apply `*' and return its value
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch14))
- compiled-branch13
- ;; note that a compound procedure here is called tail-recursively
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch14
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
- (goto (reg continue))
- after-call12
- after-if3
- after-lambda1
- ;; assign the procedure to the variable `factorial'
- (perform
- (op define-variable!) (const factorial) (reg val) (reg env))
- (assign val (const ok))
-
-
- *Exercise 5.35:* What expression was compiled to produce the code
- shown in *Note Figure 5-18::?
-
- *Figure 5.18:* An example of compiler output. See *Note Exercise
- 5-35::.
-
- (assign val (op make-compiled-procedure) (label entry16)
- (reg env))
- (goto (label after-lambda15))
- entry16
- (assign env (op compiled-procedure-env) (reg proc))
- (assign env
- (op extend-environment) (const (x)) (reg argl) (reg env))
- (assign proc (op lookup-variable-value) (const +) (reg env))
- (save continue)
- (save proc)
- (save env)
- (assign proc (op lookup-variable-value) (const g) (reg env))
- (save proc)
- (assign proc (op lookup-variable-value) (const +) (reg env))
- (assign val (const 2))
- (assign argl (op list) (reg val))
- (assign val (op lookup-variable-value) (const x) (reg env))
- (assign argl (op cons) (reg val) (reg argl))
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch19))
- compiled-branch18
- (assign continue (label after-call17))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch19
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
- after-call17
- (assign argl (op list) (reg val))
- (restore proc)
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch22))
- compiled-branch21
- (assign continue (label after-call20))
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch22
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
-
- after-call20
- (assign argl (op list) (reg val))
- (restore env)
- (assign val (op lookup-variable-value) (const x) (reg env))
- (assign argl (op cons) (reg val) (reg argl))
- (restore proc)
- (restore continue)
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-branch25))
- compiled-branch24
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
- primitive-branch25
- (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
- (goto (reg continue))
- after-call23
- after-lambda15
- (perform (op define-variable!) (const f) (reg val) (reg env))
- (assign val (const ok))
-
-
- *Exercise 5.36:* What order of evaluation does our compiler
- produce for operands of a combination? Is it left-to-right,
- right-to-left, or some other order? Where in the compiler is this
- order determined? Modify the compiler so that it produces some
- other order of evaluation. (See the discussion of order of
- evaluation for the explicit-control evaluator in section *Note
- 5-4-1::.) How does changing the order of operand evaluation
- affect the efficiency of the code that constructs the argument
- list?
-
- *Exercise 5.37:* One way to understand the compiler's `preserving'
- mechanism for optimizing stack usage is to see what extra
- operations would be generated if we did not use this idea. Modify
- `preserving' so that it always generates the `save' and `restore'
- operations. Compile some simple expressions and identify the
- unnecessary stack operations that are generated. Compare the code
- to that generated with the `preserving' mechanism intact.
-
- *Exercise 5.38:* Our compiler is clever about avoiding unnecessary
- stack operations, but it is not clever at all when it comes to
- compiling calls to the primitive procedures of the language in
- terms of the primitive operations supplied by the machine. For
- example, consider how much code is compiled to compute `(+ a 1)':
- The code sets up an argument list in `argl', puts the primitive
- addition procedure (which it finds by looking up the symbol `+' in
- the environment) into `proc', and tests whether the procedure is
- primitive or compound. The compiler always generates code to
- perform the test, as well as code for primitive and compound
- branches (only one of which will be executed). We have not shown
- the part of the controller that implements primitives, but we
- presume that these instructions make use of primitive arithmetic
- operations in the machine's data paths. Consider how much less
- code would be generated if the compiler could "open-code"
- primitives--that is, if it could generate code to directly use
- these primitive machine operations. The expression `(+ a 1)'
- might be compiled into something as simple as (1)
-
- (assign val (op lookup-variable-value) (const a) (reg env))
- (assign val (op +) (reg val) (const 1))
-
- In this exercise we will extend our compiler to support open
- coding of selected primitives. Special-purpose code will be
- generated for calls to these primitive procedures instead of the
- general procedure-application code. In order to support this, we
- will augment our machine with special argument registers `arg1'
- and `arg2'. The primitive arithmetic operations of the machine
- will take their inputs from `arg1' and `arg2'. The results may be
- put into `val', `arg1', or `arg2'.
-
- The compiler must be able to recognize the application of an
- open-coded primitive in the source program. We will augment the
- dispatch in the `compile' procedure to recognize the names of
- these primitives in addition to the reserved words (the special
- forms) it currently recognizes.(2) For each special form our
- compiler has a code generator. In this exercise we will construct
- a family of code generators for the open-coded primitives.
-
- a. The open-coded primitives, unlike the special forms, all need
- their operands evaluated. Write a code generator
- `spread-arguments' for use by all the open-coding code
- generators. `Spread-arguments' should take an operand list
- and compile the given operands targeted to successive
- argument registers. Note that an operand may contain a call
- to an open-coded primitive, so argument registers will have
- to be preserved during operand evaluation.
-
- b. For each of the primitive procedures `=', `*', `-', and `+',
- write a code generator that takes a combination with that
- operator, together with a target and a linkage descriptor,
- and produces code to spread the arguments into the registers
- and then perform the operation targeted to the given target
- with the given linkage. You need only handle expressions
- with two operands. Make `compile' dispatch to these code
- generators.
-
- c. Try your new compiler on the `factorial' example. Compare
- the resulting code with the result produced without open
- coding.
-
- d. Extend your code generators for `+' and `*' so that they can
- handle expressions with arbitrary numbers of operands. An
- expression with more than two operands will have to be
- compiled into a sequence of operations, each with only two
- inputs.
-
-
- ---------- Footnotes ----------
-
- (1) We have used the same symbol `+' here to denote both the
-source-language procedure and the machine operation. In general there
-will not be a one-to-one correspondence between primitives of the
-source language and primitives of the machine.
-
- (2) Making the primitives into reserved words is in general a bad
-idea, since a user cannot then rebind these names to different
-procedures. Moreover, if we add reserved words to a compiler that is
-in use, existing programs that define procedures with these names will
-stop working. See *Note Exercise 5-44:: for ideas on how to avoid this
-problem.
-
-\1f
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-
-5.5.6 Lexical Addressing
-------------------------
-
-One of the most common optimizations performed by compilers is the
-optimization of variable lookup. Our compiler, as we have implemented
-it so far, generates code that uses the `lookup-variable-value'
-operation of the evaluator machine. This searches for a variable by
-comparing it with each variable that is currently bound, working frame
-by frame outward through the run-time environment. This search can be
-expensive if the frames are deeply nested or if there are many
-variables. For example, consider the problem of looking up the value
-of `x' while evaluating the expression `(* x y z)' in an application of
-the procedure that is returned by
-
- (let ((x 3) (y 4))
- (lambda (a b c d e)
- (let ((y (* a b x))
- (z (+ c d x)))
- (* x y z))))
-
- Since a `let' expression is just syntactic sugar for a `lambda'
-combination, this expression is equivalent to
-
- ((lambda (x y)
- (lambda (a b c d e)
- ((lambda (y z) (* x y z))
- (* a b x)
- (+ c d x))))
- 3
- 4)
-
- Each time `lookup-variable-value' searches for `x', it must determine
-that the symbol `x' is not `eq?' to `y' or `z' (in the first frame),
-nor to `a', `b', `c', `d', or `e' (in the second frame). We will
-assume, for the moment, that our programs do not use `define'--that
-variables are bound only with `lambda'. Because our language is
-lexically scoped, the run-time environment for any expression will have
-a structure that parallels the lexical structure of the program in
-which the expression appears.(1) Thus, the compiler can know, when it
-analyzes the above expression, that each time the procedure is applied
-the variable `x' in `(* x y z)' will be found two frames out from the
-current frame and will be the first variable in that frame.
-
- We can exploit this fact by inventing a new kind of variable-lookup
-operation, `lexical-address-lookup', that takes as arguments an
-environment and a "lexical address" that consists of two numbers: a number
-"frame number", which specifies how many frames to pass over, and a "displacement
-number", which specifies how many variables to pass over in that frame.
-`Lexical-address-lookup' will produce the value of the variable stored
-at that lexical address relative to the current environment. If we add
-the `lexical-address-lookup' operation to our machine, we can make the
-compiler generate code that references variables using this operation,
-rather than `lookup-variable-value'. Similarly, our compiled code can
-use a new `lexical-address-set!' operation instead of
-`set-variable-value!'.
-
- In order to generate such code, the compiler must be able to
-determine the lexical address of a variable it is about to compile a
-reference to. The lexical address of a variable in a program depends
-on where one is in the code. For example, in the following program,
-the address of `x' in expression <E1> is (2,0)--two frames back and the
-first variable in the frame. At that point `y' is at address (0,0) and
-`c' is at address (1,2). In expression <E2>, `x' is at (1,0), `y' is
-at (1,1), and `c' is at (0,2).
-
- ((lambda (x y)
- (lambda (a b c d e)
- ((lambda (y z) <E1>)
- <E2>
- (+ c d x))))
- 3
- 4)
-
- One way for the compiler to produce code that uses lexical
-addressing is to maintain a data structure called a "compile-time
-environment". This keeps track of which variables will be at which
-positions in which frames in the run-time environment when a particular
-variable-access operation is executed. The compile-time environment is
-a list of frames, each containing a list of variables. (There will of
-course be no values bound to the variables, since values are not
-computed at compile time.) The compile-time environment becomes an
-additional argument to `compile' and is passed along to each code
-generator. The top-level call to `compile' uses an empty compile-time
-environment. When a `lambda' body is compiled, `compile-lambda-body'
-extends the compile-time environment by a frame containing the
-procedure's parameters, so that the sequence making up the body is
-compiled with that extended environment. At each point in the
-compilation, `compile-variable' and `compile-assignment' use the
-compile-time environment in order to generate the appropriate lexical
-addresses.
-
- *Note Exercise 5-39:: through *Note Exercise 5-43:: describe how to
-complete this sketch of the lexical-addressing strategy in order to
-incorporate lexical lookup into the compiler. *Note Exercise 5-44::
-describes another use for the compile-time environment.
-
- *Exercise 5.39:* Write a procedure `lexical-address-lookup' that
- implements the new lookup operation. It should take two
- arguments--a lexical address and a run-time environment--and
- return the value of the variable stored at the specified lexical
- address. `Lexical-address-lookup' should signal an error if the
- value of the variable is the symbol `*unassigned*'.(2) Also write
- a procedure `lexical-address-set!' that implements the operation
- that changes the value of the variable at a specified lexical
- address.
-
- *Exercise 5.40:* Modify the compiler to maintain the compile-time
- environment as described above. That is, add a
- compile-time-environment argument to `compile' and the various code
- generators, and extend it in `compile-lambda-body'.
-
- *Exercise 5.41:* Write a procedure `find-variable' that takes as
- arguments a variable and a compile-time environment and returns
- the lexical address of the variable with respect to that
- environment. For example, in the program fragment that is shown
- above, the compile-time environment during the compilation of
- expression <E1> is `((y z) (a b c d e) (x y))'. `Find-variable'
- should produce
-
- (find-variable 'c '((y z) (a b c d e) (x y)))
- (1 2)
-
- (find-variable 'x '((y z) (a b c d e) (x y)))
- (2 0)
-
- (find-variable 'w '((y z) (a b c d e) (x y)))
- not-found
-
- *Exercise 5.42:* Using `find-variable' from *Note Exercise 5-41::,
- rewrite `compile-variable' and `compile-assignment' to output
- lexical-address instructions. In cases where `find-variable'
- returns `not-found' (that is, where the variable is not in the
- compile-time environment), you should have the code generators use
- the evaluator operations, as before, to search for the binding.
- (The only place a variable that is not found at compile time can
- be is in the global environment, which is part of the run-time
- environment but is not part of the compile-time environment.(3)
- Thus, if you wish, you may have the evaluator operations look
- directly in the global environment, which can be obtained with the
- operation `(op get-global-environment)', instead of having them
- search the whole run-time environment found in `env'.) Test the
- modified compiler on a few simple cases, such as the nested
- `lambda' combination at the beginning of this section.
-
- *Exercise 5.43:* We argued in section *Note 4-1-6:: that internal
- definitions for block structure should not be considered "real"
- `define's. Rather, a procedure body should be interpreted as if
- the internal variables being defined were installed as ordinary
- `lambda' variables initialized to their correct values using
- `set!'. Section *Note 4-1-6:: and *Note Exercise 4-16:: showed
- how to modify the metacircular interpreter to accomplish this by
- scanning out internal definitions. Modify the compiler to perform
- the same transformation before it compiles a procedure body.
-
- *Exercise 5.44:* In this section we have focused on the use of the
- compile-time environment to produce lexical addresses. But there
- are other uses for compile-time environments. For instance, in
- *Note Exercise 5-38:: we increased the efficiency of compiled code
- by open-coding primitive procedures. Our implementation treated
- the names of open-coded procedures as reserved words. If a
- program were to rebind such a name, the mechanism described in
- *Note Exercise 5-38:: would still open-code it as a primitive,
- ignoring the new binding. For example, consider the procedure
-
- (lambda (+ * a b x y)
- (+ (* a x) (* b y)))
-
- which computes a linear combination of `x' and `y'. We might call
- it with arguments `+matrix', `*matrix', and four matrices, but the
- open-coding compiler would still open-code the `+' and the `*' in
- `(+ (* a x) (* b y))' as primitive `+' and `*'. Modify the
- open-coding compiler to consult the compile-time environment in
- order to compile the correct code for expressions involving the
- names of primitive procedures. (The code will work correctly as
- long as the program does not `define' or `set!' these names.)
-
- ---------- Footnotes ----------
-
- (1) This is not true if we allow internal definitions, unless we
-scan them out. See *Note Exercise 5-43::.
-
- (2) This is the modification to variable lookup required if we
-implement the scanning method to eliminate internal definitions (*Note
-Exercise 5-43::). We will need to eliminate these definitions in order
-for lexical addressing to work.
-
- (3) Lexical addresses cannot be used to access variables in the
-global environment, because these names can be defined and redefined
-interactively at any time. With internal definitions scanned out, as
-in *Note Exercise 5-43::, the only definitions the compiler sees are
-those at top level, which act on the global environment. Compilation
-of a definition does not cause the defined name to be entered in the
-compile-time environment.
-
-\1f
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-
-5.5.7 Interfacing Compiled Code to the Evaluator
-------------------------------------------------
-
-We have not yet explained how to load compiled code into the evaluator
-machine or how to run it. We will assume that the
-explicit-control-evaluator machine has been defined as in section *Note
-5-4-4::, with the additional operations specified in footnote *Note
-Footnote 38::. We will implement a procedure `compile-and-go' that
-compiles a Scheme expression, loads the resulting object code into the
-evaluator machine, and causes the machine to run the code in the
-evaluator global environment, print the result, and enter the
-evaluator's driver loop. We will also modify the evaluator so that
-interpreted expressions can call compiled procedures as well as
-interpreted ones. We can then put a compiled procedure into the
-machine and use the evaluator to call it:
-
- (compile-and-go
- '(define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n))))
- ;;; EC-Eval value:
- ok
-
- ;;; EC-Eval input:
- (factorial 5)
- ;;; EC-Eval value:
- 120
-
- To allow the evaluator to handle compiled procedures (for example,
-to evaluate the call to `factorial' above), we need to change the code
-at `apply-dispatch' (section *Note 5-4-1::) so that it recognizes
-compiled procedures (as distinct from compound or primitive procedures)
-and transfers control directly to the entry point of the compiled
-code:(1)
-
- apply-dispatch
- (test (op primitive-procedure?) (reg proc))
- (branch (label primitive-apply))
- (test (op compound-procedure?) (reg proc))
- (branch (label compound-apply))
- (test (op compiled-procedure?) (reg proc))
- (branch (label compiled-apply))
- (goto (label unknown-procedure-type))
-
- compiled-apply
- (restore continue)
- (assign val (op compiled-procedure-entry) (reg proc))
- (goto (reg val))
-
- Note the restore of `continue' at `compiled-apply'. Recall that the
-evaluator was arranged so that at `apply-dispatch', the continuation
-would be at the top of the stack. The compiled code entry point, on
-the other hand, expects the continuation to be in `continue', so
-`continue' must be restored before the compiled code is executed.
-
- To enable us to run some compiled code when we start the evaluator
-machine, we add a `branch' instruction at the beginning of the
-evaluator machine, which causes the machine to go to a new entry point
-if the `flag' register is set.(2)
-
- (branch (label external-entry)) ; branches if `flag' is set
- read-eval-print-loop
- (perform (op initialize-stack))
- ...
-
- `External-entry' assumes that the machine is started with `val'
-containing the location of an instruction sequence that puts a result
-into `val' and ends with `(goto (reg continue))'. Starting at this
-entry point jumps to the location designated by `val', but first assigns
-`continue' so that execution will return to `print-result', which
-prints the value in `val' and then goes to the beginning of the
-evaluator's read-eval-print loop.(3)
-
- external-entry
- (perform (op initialize-stack))
- (assign env (op get-global-environment))
- (assign continue (label print-result))
- (goto (reg val))
-
- Now we can use the following procedure to compile a procedure
-definition, execute the compiled code, and run the read-eval-print loop
-so we can try the procedure. Because we want the compiled code to
-return to the location in `continue' with its result in `val', we
-compile the expression with a target of `val' and a linkage of
-`return'. In order to transform the object code produced by the
-compiler into executable instructions for the evaluator register
-machine, we use the procedure `assemble' from the register-machine
-simulator (section *Note 5-2-2::). We then initialize the `val'
-register to point to the list of instructions, set the `flag' so that
-the evaluator will go to `external-entry', and start the evaluator.
-
- (define (compile-and-go expression)
- (let ((instructions
- (assemble (statements
- (compile expression 'val 'return))
- eceval)))
- (set! the-global-environment (setup-environment))
- (set-register-contents! eceval 'val instructions)
- (set-register-contents! eceval 'flag true)
- (start eceval)))
-
- If we have set up stack monitoring, as at the end of section *Note
-5-4-4::, we can examine the stack usage of compiled code:
-
- (compile-and-go
- '(define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n))))
-
- (total-pushes = 0 maximum-depth = 0)
- ;;; EC-Eval value:
- ok
-
- ;;; EC-Eval input:
- (factorial 5)
- (total-pushes = 31 maximum-depth = 14)
- ;;; EC-Eval value:
- 120
-
- Compare this example with the evaluation of `(factorial 5)' using the
-interpreted version of the same procedure, shown at the end of section
-*Note 5-4-4::. The interpreted version required 144 pushes and a
-maximum stack depth of 28. This illustrates the optimization that
-results from our compilation strategy.
-
-Interpretation and compilation
-..............................
-
-With the programs in this section, we can now experiment with the
-alternative execution strategies of interpretation and compilation.(4)
-An interpreter raises the machine to the level of the user program; a
-compiler lowers the user program to the level of the machine language.
-We can regard the Scheme language (or any programming language) as a
-coherent family of abstractions erected on the machine language.
-Interpreters are good for interactive program development and debugging
-because the steps of program execution are organized in terms of these
-abstractions, and are therefore more intelligible to the programmer.
-Compiled code can execute faster, because the steps of program
-execution are organized in terms of the machine language, and the
-compiler is free to make optimizations that cut across the higher-level
-abstractions.(5)
-
- The alternatives of interpretation and compilation also lead to
-different strategies for porting languages to new computers. Suppose
-that we wish to implement Lisp for a new machine. One strategy is to
-begin with the explicit-control evaluator of section *Note 5-4:: and
-translate its instructions to instructions for the new machine. A
-different strategy is to begin with the compiler and change the code
-generators so that they generate code for the new machine. The second
-strategy allows us to run any Lisp program on the new machine by first
-compiling it with the compiler running on our original Lisp system, and
-linking it with a compiled version of the run-time library.(6) Better
-yet, we can compile the compiler itself, and run this on the new
-machine to compile other Lisp programs.(7) Or we can compile one of
-the interpreters of section *Note 4-1:: to produce an interpreter that
-runs on the new machine.
-
- *Exercise 5.45:* By comparing the stack operations used by
- compiled code to the stack operations used by the evaluator for the
- same computation, we can determine the extent to which the
- compiler optimizes use of the stack, both in speed (reducing the
- total number of stack operations) and in space (reducing the
- maximum stack depth). Comparing this optimized stack use to the
- performance of a special-purpose machine for the same computation
- gives some indication of the quality of the compiler.
-
- a. *Note Exercise 5-27:: asked you to determine, as a function
- of n, the number of pushes and the maximum stack depth needed
- by the evaluator to compute n! using the recursive factorial
- procedure given above. *Note Exercise 5-14:: asked you to do
- the same measurements for the special-purpose factorial
- machine shown in *Note Figure 5-11::. Now perform the same
- analysis using the compiled `factorial' procedure.
-
- Take the ratio of the number of pushes in the compiled
- version to the number of pushes in the interpreted version,
- and do the same for the maximum stack depth. Since the
- number of operations and the stack depth used to compute n!
- are linear in n, these ratios should approach constants as n
- becomes large. What are these constants? Similarly, find
- the ratios of the stack usage in the special-purpose machine
- to the usage in the interpreted version.
-
- Compare the ratios for special-purpose versus interpreted
- code to the ratios for compiled versus interpreted code. You
- should find that the special-purpose machine does much better
- than the compiled code, since the hand-tailored controller
- code should be much better than what is produced by our
- rudimentary general-purpose compiler.
-
- b. Can you suggest improvements to the compiler that would help
- it generate code that would come closer in performance to the
- hand-tailored version?
-
-
- *Exercise 5.46:* Carry out an analysis like the one in *Note
- Exercise 5-45:: to determine the effectiveness of compiling the
- tree-recursive Fibonacci procedure
-
- (define (fib n)
- (if (< n 2)
- n
- (+ (fib (- n 1)) (fib (- n 2)))))
-
- compared to the effectiveness of using the special-purpose
- Fibonacci machine of *Note Figure 5-12::. (For measurement of the
- interpreted performance, see *Note Exercise 5-29::.) For
- Fibonacci, the time resource used is not linear in n; hence the
- ratios of stack operations will not approach a limiting value that
- is independent of n.
-
- *Exercise 5.47:* This section described how to modify the
- explicit-control evaluator so that interpreted code can call
- compiled procedures. Show how to modify the compiler so that
- compiled procedures can call not only primitive procedures and
- compiled procedures, but interpreted procedures as well. This
- requires modifying `compile-procedure-call' to handle the case of
- compound (interpreted) procedures. Be sure to handle all the same
- `target' and `linkage' combinations as in `compile-proc-appl'. To
- do the actual procedure application, the code needs to jump to the
- evaluator's `compound-apply' entry point. This label cannot be
- directly referenced in object code (since the assembler requires
- that all labels referenced by the code it is assembling be defined
- there), so we will add a register called `compapp' to the
- evaluator machine to hold this entry point, and add an instruction
- to initialize it:
-
- (assign compapp (label compound-apply))
- (branch (label external-entry)) ; branches if `flag' is set
- read-eval-print-loop
- ...
-
- To test your code, start by defining a procedure `f' that calls a
- procedure `g'. Use `compile-and-go' to compile the definition of
- `f' and start the evaluator. Now, typing at the evaluator, define
- `g' and try to call `f'.
-
- *Exercise 5.48:* The `compile-and-go' interface implemented in
- this section is awkward, since the compiler can be called only
- once (when the evaluator machine is started). Augment the
- compiler-interpreter interface by providing a `compile-and-run'
- primitive that can be called from within the explicit-control
- evaluator as follows:
-
- ;;; EC-Eval input:
- (compile-and-run
- '(define (factorial n)
- (if (= n 1)
- 1
- (* (factorial (- n 1)) n))))
- ;;; EC-Eval value:
- ok
-
- ;;; EC-Eval input:
- (factorial 5)
- ;;; EC-Eval value:
- 120
-
- *Exercise 5.49:* As an alternative to using the explicit-control
- evaluator's read-eval-print loop, design a register machine that
- performs a read-compile-execute-print loop. That is, the machine
- should run a loop that reads an expression, compiles it, assembles
- and executes the resulting code, and prints the result. This is
- easy to run in our simulated setup, since we can arrange to call
- the procedures `compile' and `assemble' as "register-machine
- operations."
-
- *Exercise 5.50:* Use the compiler to compile the metacircular
- evaluator of section *Note 4-1:: and run this program using the
- register-machine simulator. (To compile more than one definition
- at a time, you can package the definitions in a `begin'.) The
- resulting interpreter will run very slowly because of the multiple
- levels of interpretation, but getting all the details to work is
- an instructive exercise.
-
- *Exercise 5.51:* Develop a rudimentary implementation of Scheme in
- C (or some other low-level language of your choice) by translating
- the explicit-control evaluator of section *Note 5-4:: into C. In
- order to run this code you will need to also provide appropriate
- storage-allocation routines and other run-time support.
-
- *Exercise 5.52:* As a counterpoint to exercise *Note Exercise
- 5-51::, modify the compiler so that it compiles Scheme procedures
- into sequences of C instructions. Compile the metacircular
- evaluator of section *Note 4-1:: to produce a Scheme interpreter
- written in C.
-
- ---------- Footnotes ----------
-
- (1) Of course, compiled procedures as well as interpreted procedures
-are compound (nonprimitive). For compatibility with the terminology
-used in the explicit-control evaluator, in this section we will use
-"compound" to mean interpreted (as opposed to compiled).
-
- (2) Now that the evaluator machine starts with a `branch', we must
-always initialize the `flag' register before starting the evaluator
-machine. To start the machine at its ordinary read-eval-print loop, we
-could use
-
- (define (start-eceval)
- (set! the-global-environment (setup-environment))
- (set-register-contents! eceval 'flag false)
- (start eceval))
-
- (3) Since a compiled procedure is an object that the system may try
-to print, we also modify the system print operation `user-print' (from
-section *Note 4-1-4::) so that it will not attempt to print the
-components of a compiled procedure:
-
- (define (user-print object)
- (cond ((compound-procedure? object)
- (display (list 'compound-procedure
- (procedure-parameters object)
- (procedure-body object)
- '<procedure-env>)))
- ((compiled-procedure? object)
- (display '<compiled-procedure>))
- (else (display object))))
-
- (4) We can do even better by extending the compiler to allow
-compiled code to call interpreted procedures. See *Note Exercise
-5-47::.
-
- (5) Independent of the strategy of execution, we incur significant
-overhead if we insist that errors encountered in execution of a user
-program be detected and signaled, rather than being allowed to kill the
-system or produce wrong answers. For example, an out-of-bounds array
-reference can be detected by checking the validity of the reference
-before performing it. The overhead of checking, however, can be many
-times the cost of the array reference itself, and a programmer should
-weigh speed against safety in determining whether such a check is
-desirable. A good compiler should be able to produce code with such
-checks, should avoid redundant checks, and should allow programmers to
-control the extent and type of error checking in the compiled code.
-
- Compilers for popular languages, such as C and C++, put hardly any
-error-checking operations into running code, so as to make things run
-as fast as possible. As a result, it falls to programmers to
-explicitly provide error checking. Unfortunately, people often neglect
-to do this, even in critical applications where speed is not a
-constraint. Their programs lead fast and dangerous lives. For
-example, the notorious "Worm" that paralyzed the Internet in 1988
-exploited the UNIX(tm) operating system's failure to check whether the
-input buffer has overflowed in the finger daemon. (See Spafford 1989.)
-
- (6) Of course, with either the interpretation or the compilation
-strategy we must also implement for the new machine storage allocation,
-input and output, and all the various operations that we took as
-"primitive" in our discussion of the evaluator and compiler. One
-strategy for minimizing work here is to write as many of these
-operations as possible in Lisp and then compile them for the new
-machine. Ultimately, everything reduces to a small kernel (such as
-garbage collection and the mechanism for applying actual machine
-primitives) that is hand-coded for the new machine.
-
- (7) This strategy leads to amusing tests of correctness of the
-compiler, such as checking whether the compilation of a program on the
-new machine, using the compiled compiler, is identical with the
-compilation of the program on the original Lisp system. Tracking down
-the source of differences is fun but often frustrating, because the
-results are extremely sensitive to minuscule details.
-
-\1f
-File: sicp.info, Node: References, Next: Index, Prev: Chapter 5, Up: Top
-
-References
-**********
-
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-
-\1f
-File: sicp.info, Node: Index, Prev: References, Up: Top
-
-Index
-*****
-
-\0\b[index\0\b]
-* Menu:
-
-* abstract: 2-1-3. (line 139)
-* abstract syntax: 4-1-1. (line 44)
-* abstraction: 2-1-2. (line 16)
-* abstraction barriers: Chapter 2. (line 118)
-* accumulator <1>: 3-1-1. (line 216)
-* accumulator: 2-2-3. (line 76)
-* acquired: 3-4-2. (line 388)
-* action: 5-1-1. (line 164)
-* additive: 2-4-3. (line 26)
-* additively <1>: 2-4. (line 51)
-* additively: Chapter 2. (line 158)
-* address: 5-3-1. (line 8)
-* address arithmetic: 5-3-1. (line 16)
-* agenda: 3-3-4. (line 345)
-* algebraic specification: 2-1-3. (line 149)
-* aliasing: 3-1-3. (line 313)
-* and-gate: 3-3-4. (line 30)
-* applicative-order: 4-2-1. (line 7)
-* applicative-order evaluation: 1-1-5. (line 127)
-* arbiter: 3-4-2. (line 633)
-* arguments: 1-1-1. (line 47)
-* assembler: 5-2-1. (line 20)
-* assertions: 4-4-1. (line 20)
-* assignment: 3-1. (line 37)
-* atomically: 3-4-2. (line 447)
-* automatic storage allocation: 5-3. (line 30)
-* average damping: 1-3-3. (line 180)
-* B-trees: 2-3-3. (line 459)
-* backbone: 3-3-3. (line 19)
-* backquote: 5-5-2. (line 279)
-* backtracks: 4-3-1. (line 82)
-* balanced: 2-2-2. (line 195)
-* barrier synchronization: 3-4-2. (line 652)
-* base: 5-3-1. (line 35)
-* Bertrand's: 3-5-2. (line 411)
-* bignum: 5-3-1. (line 97)
-* bindings: 3-2. (line 24)
-* binds: 1-1-8. (line 109)
-* binomial coefficients: 1-2-2. (line 251)
-* block structure: 1-1-8. (line 182)
-* bound variable: 1-1-8. (line 108)
-* box-and-pointer notation: 2-2. (line 9)
-* breakpoint: 5-2-4. (line 109)
-* broken heart: 5-3-2. (line 125)
-* bugs: Chapter 1. (line 38)
-* cache-coherence: 3-4-1. (line 222)
-* call-by-name <1>: 3-5-1. (line 405)
-* call-by-name: 4-2-2. (line 383)
-* call-by-name thunks: 3-5-1. (line 412)
-* call-by-need <1>: 3-5-1. (line 411)
-* call-by-need: 4-2-2. (line 383)
-* call-by-need thunks: 3-5-1. (line 413)
-* capturing: 1-1-8. (line 123)
-* Carmichael numbers: 1-2-6. (line 305)
-* case analysis: 1-1-6. (line 20)
-* cell: 3-4-2. (line 409)
-* chronological: 4-3-1. (line 86)
-* Church numerals: 2-1-3. (line 126)
-* Church-Turing thesis: 4-1-5. (line 147)
-* clauses: 1-1-6. (line 41)
-* closed world assumption: 4-4-3. (line 153)
-* closure: Chapter 2. (line 130)
-* closure property: 2-2. (line 56)
-* code generator: 5-5-1. (line 21)
-* coerce: 2-5-2. (line 270)
-* coercion: 2-5-2. (line 63)
-* combinations: 1-1-1. (line 44)
-* comments: 2-2-3. (line 671)
-* compacting: 5-3-2. (line 309)
-* compilation: 5-5. (line 45)
-* compile-time environment: 5-5-6. (line 77)
-* composition: 1-3-4. (line 233)
-* compound data: Chapter 2. (line 37)
-* compound data object: Chapter 2. (line 64)
-* compound procedure: 1-1-4. (line 34)
-* computability: 4-1-5. (line 142)
-* computational process: Chapter 1. (line 18)
-* concurrently: 3-4. (line 38)
-* congruent modulo: 1-2-6. (line 56)
-* connectors: 3-3-5. (line 35)
-* consequent: 1-1-6. (line 48)
-* constraint networks: 3-3-5. (line 35)
-* constructors: 2-1. (line 27)
-* continuation procedures: 4-3-3. (line 27)
-* continued fraction: 1-3-3. (line 199)
-* control: 4-4-3. (line 11)
-* controller: 5-1. (line 7)
-* conventional interfaces <1>: Chapter 2. (line 133)
-* conventional interfaces: 2-2-3. (line 11)
-* current time: 3-3-4. (line 491)
-* data <1>: 2-1-3. (line 13)
-* data: Chapter 1. (line 20)
-* data abstraction <1>: Chapter 2. (line 75)
-* data abstraction: 2-1. (line 15)
-* data paths: 5-1. (line 6)
-* data-directed <1>: Chapter 2. (line 156)
-* data-directed <2>: 2-4-3. (line 42)
-* data-directed: 2-4. (line 61)
-* deadlock: 3-4-2. (line 497)
-* deadlock-recovery: 3-4-2. (line 646)
-* debug: Chapter 1. (line 53)
-* deep binding: 4-1-3. (line 227)
-* deferred: 1-2-1. (line 96)
-* delayed: 3-5-1. (line 137)
-* delayed argument: 3-5-4. (line 73)
-* delayed evaluation <1>: Chapter 3. (line 64)
-* delayed evaluation: 3-5. (line 40)
-* dense: 2-5-3. (line 252)
-* dependency-directed backtracking: 4-3-1. (line 215)
-* depth-first search: 4-3-1. (line 86)
-* deque: 3-3-2. (line 253)
-* derived expressions: 4-1-2. (line 230)
-* digital signals: 3-3-4. (line 19)
-* dispatching on type: 2-4-3. (line 7)
-* displacement number: 5-5-6. (line 49)
-* dotted-tail: 2-2-1. (line 239)
-* driver loop: 4-1-4. (line 77)
-* empty list: 2-2-1. (line 90)
-* encapsulated: 3-1-1. (line 300)
-* enclosing: 3-2. (line 26)
-* entry points: 5-1-1. (line 23)
-* enumerator: 2-2-3. (line 71)
-* environment: 1-1-2. (line 55)
-* environment model: Chapter 3. (line 56)
-* environments: 3-2. (line 21)
-* Euclid's: 1-2-5. (line 38)
-* Euclidean ring: 2-5-3. (line 684)
-* evaluating: 1-1-1. (line 9)
-* evaluator: Chapter 4. (line 56)
-* event-driven simulation: 3-3-4. (line 13)
-* evlis tail recursion: 5-4-1. (line 288)
-* execution procedure: 4-1-7. (line 30)
-* explicit-control evaluator: 5-4. (line 11)
-* expression: 1-1-1. (line 8)
-* failure continuation: 4-3-3. (line 31)
-* FIFO: 3-3-2. (line 17)
-* filter <1>: 2-2-3. (line 73)
-* filter: 1-3-1. (line 218)
-* first-class: 1-3-4. (line 200)
-* fixed point: 1-3-3. (line 102)
-* fixed-length: 2-3-4. (line 28)
-* forcing: 4-2-2. (line 22)
-* forwarding address: 5-3-2. (line 126)
-* frame <1>: 4-4-2. (line 47)
-* frame: 5-5-6. (line 48)
-* frame coordinate map: 2-2-4. (line 265)
-* framed-stack: 5-4-1. (line 266)
-* frames: 3-2. (line 23)
-* free: 1-1-8. (line 112)
-* free list: 5-3-1. (line 290)
-* front: 3-3-2. (line 13)
-* full-adder: 3-3-4. (line 127)
-* function boxes: 3-3-4. (line 21)
-* functional programming: 3-1-3. (line 18)
-* functional programming languages: 3-5-5. (line 172)
-* garbage: 5-3-2. (line 12)
-* garbage collection <1>: 5-3. (line 35)
-* garbage collection: 5-3-2. (line 27)
-* garbage collector: 3-3-1. (line 454)
-* garbage-collected: 4-2-2. (line 398)
-* generic operations: Chapter 2. (line 152)
-* generic procedures <1>: 2-3-4. (line 220)
-* generic procedures: 2-4. (line 56)
-* glitches: Chapter 1. (line 38)
-* global <1>: 3-2. (line 28)
-* global: 1-2. (line 35)
-* global environment: 1-1-2. (line 55)
-* golden ratio: 1-2-2. (line 70)
-* grammar: 4-3-2. (line 148)
-* half-adder: 3-3-4. (line 52)
-* half-interval method: 1-3-3. (line 20)
-* Halting Theorem: 4-1-5. (line 185)
-* headed list: 3-3-3. (line 21)
-* hiding principle: 3-1-1. (line 301)
-* hierarchical: 2-2. (line 60)
-* hierarchy of types: 2-5-2. (line 159)
-* higher-order procedures: 1-3. (line 39)
-* Horner's rule: 2-2-3. (line 285)
-* imperative programming: 3-1-3. (line 223)
-* indeterminates: 2-5-3. (line 39)
-* index: 5-3-1. (line 36)
-* indexing: 4-4-2. (line 421)
-* instantiated with: 4-4-1. (line 202)
-* instruction: 5-2-4. (line 80)
-* instruction execution: 5-2-1. (line 119)
-* instruction sequence: 5-5-1. (line 88)
-* instruction tracing: 5-2-4. (line 86)
-* instructions <1>: Chapter 5. (line 37)
-* instructions: 5-1-1. (line 22)
-* integerizing factor: 2-5-3. (line 534)
-* integers: 1-1. (line 54)
-* integrator: 3-5-3. (line 421)
-* interning: 5-3-1. (line 117)
-* interpreter <1>: Chapter 1. (line 74)
-* interpreter: Chapter 4. (line 56)
-* invariant quantity: 1-2-4. (line 95)
-* inverter: 3-3-4. (line 25)
-* iterative improvement: 1-3-4. (line 290)
-* iterative process: 1-2-1. (line 110)
-* k-term: 1-3-3. (line 215)
-* key: 2-3-3. (line 389)
-* labels: 5-1-1. (line 23)
-* lazy: 4-2-1. (line 13)
-* lexical address: 5-5-6. (line 48)
-* lexical addressing: 4-1-3. (line 228)
-* lexical scoping: 1-1-8. (line 192)
-* linear iterative process: 1-2-1. (line 117)
-* linear recursive process: 1-2-1. (line 105)
-* linkage descriptor: 5-5-1. (line 52)
-* list: 2-2-1. (line 404)
-* list structure: 2-2-1. (line 405)
-* list-structured: 2-1-1. (line 121)
-* list-structured memory: 5-3. (line 8)
-* local evolution: 1-2. (line 32)
-* local state variables: 3-1. (line 29)
-* location: 5-3-1. (line 8)
-* logic-programming: Chapter 4. (line 129)
-* logical and: 3-3-4. (line 33)
-* logical deductions: 4-4-1. (line 474)
-* logical or: 3-3-4. (line 37)
-* machine language: 5-5. (line 23)
-* macro: 4-1-2. (line 401)
-* map: 2-2-3. (line 74)
-* mark-sweep: 5-3-2. (line 298)
-* means of abstraction: 1-1. (line 21)
-* means of combination: 1-1. (line 18)
-* Memoization: 3-3-3. (line 258)
-* memoization: 1-2-2. (line 245)
-* memoize: 4-2-2. (line 27)
-* merge: 3-5-5. (line 206)
-* message passing <1>: 2-4-3. (line 359)
-* message passing: 2-1-3. (line 91)
-* message-passing: 3-1-1. (line 187)
-* metacircular: 4-1. (line 12)
-* Metalinguistic abstraction: Chapter 4. (line 52)
-* Miller-Rabin test: 1-2-6. (line 253)
-* modular: Chapter 3. (line 24)
-* modulo: 1-2-6. (line 59)
-* modus ponens: 4-4-3. (line 257)
-* moments in time: 3-4. (line 30)
-* Monte Carlo: 3-1-2. (line 43)
-* Monte Carlo integration: 3-1-2. (line 140)
-* mutable data: 3-3. (line 26)
-* mutators: 3-3. (line 18)
-* mutex: 3-4-2. (line 387)
-* mutual exclusion: 3-4-2. (line 593)
-* n-fold smoothed function: 1-3-4. (line 267)
-* native language: 5-5. (line 22)
-* needed: 5-5-1. (line 162)
-* networks: Chapter 4. (line 147)
-* Newton's method: 1-3-4. (line 67)
-* nil: 2-2-1. (line 90)
-* non-computable: 4-1-5. (line 186)
-* non-strict: 4-2-1. (line 44)
-* nondeterministic: 3-4-1. (line 237)
-* nondeterministic choice point: 4-3-1. (line 59)
-* nondeterministic computing <1>: Chapter 4. (line 120)
-* nondeterministic computing: 4-3. (line 7)
-* normal-order: 4-2-1. (line 9)
-* normal-order evaluation <1>: 1-1-5. (line 125)
-* normal-order evaluation: Chapter 4. (line 116)
-* obarray: 5-3-1. (line 109)
-* object program: 5-5. (line 47)
-* objects: Chapter 3. (line 45)
-* open-code: 5-5-5. (line 383)
-* operands: 1-1-1. (line 45)
-* operator <1>: 4-1-6. (line 302)
-* operator: 1-1-1. (line 45)
-* or-gate: 3-3-4. (line 35)
-* order of growth: 1-2-3. (line 8)
-* ordinary: 2-5-1. (line 25)
-* output prompt: 4-1-4. (line 81)
-* package: 2-4-3. (line 97)
-* painter: 2-2-4. (line 31)
-* pair: 2-1-1. (line 85)
-* parse: 4-3-2. (line 135)
-* Pascal's triangle: 1-2-2. (line 211)
-* pattern: 4-4-1. (line 124)
-* pattern matcher: 4-4-2. (line 37)
-* pattern matching: 4-4-2. (line 23)
-* pattern variable: 4-4-1. (line 128)
-* pipelining: 3-4. (line 73)
-* pointer: 2-2. (line 10)
-* poly: 2-5-3. (line 77)
-* power series: 3-5-2. (line 293)
-* predicate: 1-1-6. (line 41)
-* prefix: 2-3-4. (line 56)
-* prefix code: 2-3-4. (line 57)
-* prefix notation: 1-1-1. (line 51)
-* pretty-printing: 1-1-1. (line 89)
-* primitive constraints: 3-3-5. (line 27)
-* primitive expressions: 1-1. (line 14)
-* probabilistic algorithms: 1-2-6. (line 144)
-* procedural abstraction: 1-1-8. (line 45)
-* procedural epistemology: Preface 1e. (line 66)
-* procedure: 1-2-1. (line 132)
-* procedure definitions: 1-1-4. (line 18)
-* procedures: Chapter 1. (line 119)
-* process: 1-2-1. (line 131)
-* program: Chapter 1. (line 21)
-* programming languages: Chapter 1. (line 31)
-* prompt: 4-1-4. (line 79)
-* pseudo-random: 3-1-2. (line 200)
-* pseudodivision: 2-5-3. (line 538)
-* pseudoremainder: 2-5-3. (line 539)
-* quasiquote: 5-5-2. (line 279)
-* queries: 4-4. (line 99)
-* query language: 4-4. (line 98)
-* queue: 3-3-2. (line 12)
-* quote: 2-3-1. (line 20)
-* Ramanujan numbers: 3-5-3. (line 398)
-* rational: 2-5-3. (line 426)
-* RC circuit: 3-5-3. (line 465)
-* read-eval-print loop: 1-1-1. (line 97)
-* reader macro characters: 4-4-4-7. (line 128)
-* real numbers: 1-1. (line 55)
-* rear: 3-3-2. (line 13)
-* Recursion: 4-1-5. (line 162)
-* recursion equations: Chapter 1. (line 68)
-* recursive <1>: 1-1-8. (line 7)
-* recursive: 1-1-3. (line 23)
-* recursive process: 1-2-1. (line 100)
-* red-black trees: 2-3-3. (line 459)
-* referentially transparent: 3-1-3. (line 156)
-* register: Chapter 5. (line 36)
-* register table: 5-2-1. (line 105)
-* registers: Chapter 5. (line 38)
-* released: 3-4-2. (line 388)
-* remainder of: 1-2-6. (line 58)
-* resolution: 4-4. (line 134)
-* ripple-carry adder: 3-3-4. (line 238)
-* robust: 2-2-4. (line 530)
-* RSA algorithm: 1-2-6. (line 322)
-* rules <1>: 4-4-1. (line 334)
-* rules: 4-4. (line 113)
-* satisfy: 4-4-1. (line 201)
-* scope: 1-1-8. (line 113)
-* selectors: 2-1. (line 27)
-* semaphore: 3-4-2. (line 596)
-* separator code: 2-3-4. (line 53)
-* sequence: 2-2-1. (line 18)
-* sequence accelerator: 3-5-3. (line 97)
-* sequences: 1-3-1. (line 254)
-* serializer: 3-4-2. (line 30)
-* serializers: 3-4-2. (line 91)
-* series RLC circuit: 3-5-4. (line 161)
-* shadow: 3-2. (line 62)
-* shared: 3-3-1. (line 248)
-* side-effect bugs: 3-1-3. (line 320)
-* sieve of: 3-5-2. (line 55)
-* smoothing: 1-3-4. (line 259)
-* source language: 5-5. (line 34)
-* source program: 5-5. (line 39)
-* sparse: 2-5-3. (line 254)
-* special: 1-1-3. (line 96)
-* stack <1>: 5-1-4. (line 82)
-* stack: 1-2-1. (line 233)
-* state variables <1>: 1-2-1. (line 112)
-* state variables: 3-1. (line 11)
-* statements: 5-5-1. (line 167)
-* stop-and-copy: 5-3-2. (line 36)
-* stratified design: 2-2-4. (line 499)
-* stream processing: 1-1-5. (line 157)
-* streams <1>: 3-5. (line 11)
-* streams: Chapter 3. (line 48)
-* strict: 4-2-1. (line 46)
-* subroutine: 5-1-3. (line 75)
-* substitution: 1-1-5. (line 153)
-* substitution model: 1-1-5. (line 58)
-* subtype: 2-5-2. (line 159)
-* success continuation: 4-3-3. (line 29)
-* summation of a series: 1-3-1. (line 50)
-* summer: 3-5-3. (line 422)
-* supertype: 2-5-2. (line 162)
-* symbolic expressions: Chapter 2. (line 139)
-* syntactic sugar: 1-1-3. (line 118)
-* syntax: 4-1. (line 42)
-* systematically search: 4-3-1. (line 75)
-* systems: Chapter 4. (line 148)
-* tableau: 3-5-3. (line 138)
-* tabulation <1>: 1-2-2. (line 245)
-* tabulation: 3-3-3. (line 258)
-* tagged: 5-3-1. (line 274)
-* tail-recursive <1>: 1-2-1. (line 155)
-* tail-recursive: 5-4-2. (line 82)
-* target: 5-5-1. (line 50)
-* thrashing: UTF. (line 22)
-* thunk: 4-2-2. (line 369)
-* thunks: 4-2-2. (line 16)
-* time: 3-4. (line 14)
-* time segments: 3-3-4. (line 473)
-* tower: 2-5-2. (line 165)
-* tree accumulation: 1-1-3. (line 46)
-* tree recursion: 1-2-2. (line 6)
-* trees: 2-2-2. (line 36)
-* truth maintenance: 4-3-1. (line 220)
-* Turing machine: 4-1-5. (line 145)
-* type: 2-4. (line 59)
-* type field: 5-3-1. (line 268)
-* type tag: 2-4-2. (line 25)
-* type-inferencing: 3-5-4. (line 310)
-* typed pointers: 5-3-1. (line 51)
-* unbound: 3-2. (line 32)
-* unification <1>: 4-4-2. (line 24)
-* unification: 4-4. (line 43)
-* unification algorithm: 4-4. (line 133)
-* univariate polynomials: 2-5-3. (line 41)
-* universal: 4-1-5. (line 73)
-* upward-compatible extension: 4-2-2. (line 341)
-* value: 1-1-2. (line 8)
-* value of a variable: 3-2. (line 28)
-* values: 2-3-1. (line 22)
-* variable: 1-1-2. (line 8)
-* variable-length: 2-3-4. (line 30)
-* vector: 5-3-1. (line 19)
-* width: 2-1-4. (line 87)
-* wires: 3-3-4. (line 19)
-* wishful thinking: 2-1-1. (line 24)
-* zero crossings: 3-5-3. (line 506)
-
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-Node: 4-4-3\7f1076881
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-Node: 5-1\7f1144169
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-Node: References\7f1416871
-Node: Index\7f1432991
-\1f
-End Tag Table
--- /dev/null
+This is the file .../info/dir, which contains the
+topmost node of the Info hierarchy, called (dir)Top.
+The first time you invoke Info you start off looking at this node.
+\1f
+File: dir, Node: Top This is the top of the INFO tree
+
+ This (the Directory node) gives a menu of major topics.
+ Typing "q" exits, "?" lists all Info commands, "d" returns here,
+ "h" gives a primer for first-timers,
+ "mEmacs<Return>" visits the Emacs manual, etc.
+
+ In Emacs, you can click mouse button 2 on a menu item or cross reference
+ to select it.
+
+* Menu:
+
+The Algorithmic Language Scheme
+* SICP: (sicp). Structure and Interpretation of Computer
+ Programs
--- /dev/null
+;;; sicp-autoloads.el --- automatically extracted autoloads (do not edit) -*- lexical-binding: t -*-
+;; Generated by the `loaddefs-generate' function.
+
+;; This file is part of GNU Emacs.
+
+;;; Code:
+
+(add-to-list 'load-path (or (and load-file-name (directory-file-name (file-name-directory load-file-name))) (car load-path)))
+
+\f
+;;; End of scraped data
+
+(provide 'sicp-autoloads)
+
+;; Local Variables:
+;; version-control: never
+;; no-byte-compile: t
+;; no-update-autoloads: t
+;; no-native-compile: t
+;; coding: utf-8-emacs-unix
+;; End:
+
+;;; sicp-autoloads.el ends here
--- /dev/null
+;; -*- no-byte-compile: t; lexical-binding: nil -*-
+(define-package "sicp" "20240826.1844"
+ "Structure and Interpretation of Computer Programs in info format."
+ ()
+ :url "https://github.com/webframp/sicp-info"
+ :commit "552b44fd873b5cadde4e76d10cef4d6e21a4287a"
+ :revdesc "552b44fd873b")
--- /dev/null
+;;; sicp.el --- Structure and Interpretation of Computer Programs in info format
+
+;;; Copyright (C) 1993 Hal Abelson, Jerry Sussman, and Julie Sussman
+
+;; Author: Hal Abelson
+;; Jerry Sussman
+;; Julie Sussman
+;; Package-Version: 20240826.1844
+;; Package-Revision: 552b44fd873b
+;; Homepage: https://mitpress.mit.edu/sicp
+
+;; This file is not part of GNU Emacs.
+
+;; This program is free software; you can redistribute it and/or modify
+;; it under the terms of the GNU General Public License as published by
+;; the Free Software Foundation, either version 3 of the License, or
+;; (at your option) any later version.
+
+;; This program is distributed in the hope that it will be useful,
+;; but WITHOUT ANY WARRANTY; without even the implied warranty of
+;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+;; GNU General Public License for more details.
+
+;; You should have received a copy of the GNU General Public License
+;; along with this program. If not, see <http://www.gnu.org/licenses/>.
+
+;; The included sicp.info and sicp.texi files reatain their
+;; origin Creative Commons Attribution-ShareAlike 4.0
+;; International Public License
+;; See http://creativecommons.org/licenses/by-sa/4.0/
+;; and http://creativecommons.org/licenses/by-sa/4.0/legalcode.
+
+
+;;; Commentary:
+
+;; The second edition of the book Structure and Interpretation of
+;; Computer Programs (SICP) in info format.
+
+;; This library provides the feature `sicp' and does nothing else.
+;; This allows making the info file available on Melpa. The texi
+;; file was taken from http://www.neilvandyke.org/sicp-texi. The
+;; html version can be found at https://mitpress.mit.edu/sicp.
+
+;; If you want to recreate the info file, you can do so using
+;;
+;; makeinfo -–no-split sicp.texi -o sicp.info
+
+;;; Code:
+
+(provide 'sicp)
+
+;;; sicp.el ends here
--- /dev/null
+This is sicp.info, produced by makeinfo version 4.8 from sicp.texi.
+
+INFO-DIR-SECTION The Algorithmic Language Scheme
+START-INFO-DIR-ENTRY
+* SICP: (sicp). Structure and Interpretation of Computer Programs
+END-INFO-DIR-ENTRY
+
+\1f
+File: sicp.info, Node: Top, Next: UTF, Prev: (dir), Up: (dir)
+
+Structure and Interpretation of Computer Programs
+=================================================
+
+Second Edition
+by Harold Abelson and Gerald Jay Sussman, with Julie Sussman
+foreword by Alan J. Perlis
+(C) 1996 Massachusetts Institute of Technology
+
+Unofficial Texinfo Format version 2.neilvandyke4 (January 10, 2007)
+
+* Menu:
+
+* UTF:: Unofficial Texinfo Format
+* Dedication:: Dedication
+* Foreword:: Foreword
+* Preface:: Preface to the Second Edition
+* Preface 1e:: Preface to the First Edition
+* Acknowledgements:: Acknowledgements
+* Chapter 1:: Building Abstractions with Procedures
+* Chapter 2:: Building Abstractions with Data
+* Chapter 3:: Modularity, Objects, and State
+* Chapter 4:: Metalinguistic Abstraction
+* Chapter 5:: Computing with Register Machines
+* References:: References
+* Index:: Index
+
+ --- The Detailed Node Listing ---
+
+Programming in Lisp
+
+* 1-1:: The Elements of Programming
+* 1-2:: Procedures and the Processes They Generate
+* 1-3:: Formulating Abstractions with Higher-Order Procedures
+
+The Elements of Programming
+
+* 1-1-1:: Expressions
+* 1-1-2:: Naming and the Environment
+* 1-1-3:: Evaluating Combinations
+* 1-1-4:: Compound Procedures
+* 1-1-5:: The Substitution Model for Procedure Application
+* 1-1-6:: Conditional Expressions and Predicates
+* 1-1-7:: Example: Square Roots by Newton's Method
+* 1-1-8:: Procedures as Black-Box Abstractions
+
+Procedures and the Processes They Generate
+
+* 1-2-1:: Linear Recursion and Iteration
+* 1-2-2:: Tree Recursion
+* 1-2-3:: Orders of Growth
+* 1-2-4:: Exponentiation
+* 1-2-5:: Greatest Common Divisors
+* 1-2-6:: Example: Testing for Primality
+
+Formulating Abstractions with Higher-Order Procedures
+
+* 1-3-1:: Procedures as Arguments
+* 1-3-2:: Constructing Procedures Using `Lambda'
+* 1-3-3:: Procedures as General Methods
+* 1-3-4:: Procedures as Returned Values
+
+Building Abstractions with Data
+
+* 2-1:: Introduction to Data Abstraction
+* 2-2:: Hierarchical Data and the Closure Property
+* 2-3:: Symbolic Data
+* 2-4:: Multiple Representations for Abstract Data
+* 2-5:: Systems with Generic Operations
+
+Introduction to Data Abstraction
+
+* 2-1-1:: Example: Arithmetic Operations for Rational Numbers
+* 2-1-2:: Abstraction Barriers
+* 2-1-3:: What Is Meant by Data?
+* 2-1-4:: Extended Exercise: Interval Arithmetic
+
+Hierarchical Data and the Closure Property
+
+* 2-2-1:: Representing Sequences
+* 2-2-2:: Hierarchical Structures
+* 2-2-3:: Sequences as Conventional Interfaces
+* 2-2-4:: Example: A Picture Language
+
+Symbolic Data
+
+* 2-3-1:: Quotation
+* 2-3-2:: Example: Symbolic Differentiation
+* 2-3-3:: Example: Representing Sets
+* 2-3-4:: Example: Huffman Encoding Trees
+
+Multiple Representations for Abstract Data
+
+* 2-4-1:: Representations for Complex Numbers
+* 2-4-2:: Tagged data
+* 2-4-3:: Data-Directed Programming and Additivity
+
+Systems with Generic Operations
+
+* 2-5-1:: Generic Arithmetic Operations
+* 2-5-2:: Combining Data of Different Types
+* 2-5-3:: Example: Symbolic Algebra
+
+Modularity, Objects, and State
+
+* 3-1:: Assignment and Local State
+* 3-2:: The Environment Model of Evaluation
+* 3-3:: Modeling with Mutable Data
+* 3-4:: Concurrency: Time Is of the Essence
+* 3-5:: Streams
+
+Assignment and Local State
+
+* 3-1-1:: Local State Variables
+* 3-1-2:: The Benefits of Introducing Assignment
+* 3-1-3:: The Costs of Introducing Assignment
+
+The Environment Model of Evaluation
+
+* 3-2-1:: The Rules for Evaluation
+* 3-2-2:: Applying Simple Procedures
+* 3-2-3:: Frames as the Repository of Local State
+* 3-2-4:: Internal Definitions
+
+Modeling with Mutable Data
+
+* 3-3-1:: Mutable List Structure
+* 3-3-2:: Representing Queues
+* 3-3-3:: Representing Tables
+* 3-3-4:: A Simulator for Digital Circuits
+* 3-3-5:: Propagation of Constraints
+
+Concurrency: Time Is of the Essence
+
+* 3-4-1:: The Nature of Time in Concurrent Systems
+* 3-4-2:: Mechanisms for Controlling Concurrency
+
+Streams
+
+* 3-5-1:: Streams Are Delayed Lists
+* 3-5-2:: Infinite Streams
+* 3-5-3:: Exploiting the Stream Paradigm
+* 3-5-4:: Streams and Delayed Evaluation
+* 3-5-5:: Modularity of Functional Programs and Modularity of
+ Objects
+
+Metalinguistic Abstraction
+
+* 4-1:: The Metacircular Evaluator
+* 4-2:: Variations on a Scheme -- Lazy Evaluation
+* 4-3:: Variations on a Scheme -- Nondeterministic Computing
+* 4-4:: Logic Programming
+
+The Metacircular Evaluator
+
+* 4-1-1:: The Core of the Evaluator
+* 4-1-2:: Representing Expressions
+* 4-1-3:: Evaluator Data Structures
+* 4-1-4:: Running the Evaluator as a Program
+* 4-1-5:: Data as Programs
+* 4-1-6:: Internal Definitions
+* 4-1-7:: Separating Syntactic Analysis from Execution
+
+Variations on a Scheme -- Lazy Evaluation
+
+* 4-2-1:: Normal Order and Applicative Order
+* 4-2-2:: An Interpreter with Lazy Evaluation
+* 4-2-3:: Streams as Lazy Lists
+
+Variations on a Scheme -- Nondeterministic Computing
+
+* 4-3-1:: Amb and Search
+* 4-3-2:: Examples of Nondeterministic Programs
+* 4-3-3:: Implementing the `Amb' Evaluator
+
+Logic Programming
+
+* 4-4-1:: Deductive Information Retrieval
+* 4-4-2:: How the Query System Works
+* 4-4-3:: Is Logic Programming Mathematical Logic?
+* 4-4-4:: Implementing the Query System
+
+Implementing the Query System
+
+* 4-4-4-1:: The Driver Loop and Instantiation
+* 4-4-4-2:: The Evaluator
+* 4-4-4-3:: Finding Assertions by Pattern Matching
+* 4-4-4-4:: Rules and Unification
+* 4-4-4-5:: Maintaining the Data Base
+* 4-4-4-6:: Stream Operations
+* 4-4-4-7:: Query Syntax Procedures
+* 4-4-4-8:: Frames and Bindings
+
+Computing with Register Machines
+
+* 5-1:: Designing Register Machines
+* 5-2:: A Register-Machine Simulator
+* 5-3:: Storage Allocation and Garbage Collection
+* 5-4:: The Explicit-Control Evaluator
+* 5-5:: Compilation
+
+Designing Register Machines
+
+* 5-1-1:: A Language for Describing Register Machines
+* 5-1-2:: Abstraction in Machine Design
+* 5-1-3:: Subroutines
+* 5-1-4:: Using a Stack to Implement Recursion
+* 5-1-5:: Instruction Summary
+
+A Register-Machine Simulator
+
+* 5-2-1:: The Machine Model
+* 5-2-2:: The Assembler
+* 5-2-3:: Generating Execution Procedures for Instructions
+* 5-2-4:: Monitoring Machine Performance
+
+Storage Allocation and Garbage Collection
+
+* 5-3-1:: Memory as Vectors
+* 5-3-2:: Maintaining the Illusion of Infinite Memory
+
+Registers and operations
+
+* 5-4-1:: The Core of the Explicit-Control Evaluator
+* 5-4-2:: Sequence Evaluation and Tail Recursion
+* 5-4-3:: Conditionals, Assignments, and Definitions
+* 5-4-4:: Running the Evaluator
+
+An overview of the compiler
+
+* 5-5-1:: Structure of the Compiler
+* 5-5-2:: Compiling Expressions
+* 5-5-3:: Compiling Combinations
+* 5-5-4:: Combining Instruction Sequences
+* 5-5-5:: An Example of Compiled Code
+* 5-5-6:: Lexical Addressing
+* 5-5-7:: Interfacing Compiled Code to the Evaluator
+
+\1f
+File: sicp.info, Node: UTF, Next: Dedication, Prev: Top, Up: Top
+
+Unofficial Texinfo Format
+*************************
+
+This is the second edition SICP book, from Unofficial Texinfo Format.
+
+ You are probably reading it in an Info hypertext browser, such as
+the Info mode of Emacs. You might alternatively be reading it
+TeX-formatted on your screen or printer, though that would be silly.
+And, if printed, expensive.
+
+ The freely-distributed official HTML-and-GIF format was first
+converted personually to Unofficial Texinfo Format (UTF) version 1 by
+Lyssa Ayth during a long Emacs lovefest weekend in April, 2001.
+
+ The UTF is easier to search than the HTML format. It is also much
+more accessible to people running on modest computers, such as donated
+'386-based PCs. A 386 can, in theory, run Linux, Emacs, and a Scheme
+interpreter simultaneously, but most 386s probably can't also run both
+Netscape and the necessary X Window System without prematurely
+introducing budding young underfunded hackers to the concept of "thrashing".
+UTF can also fit uncompressed on a 1.44MB floppy diskette, which may
+come in handy for installing UTF on PCs that do not have Internet or
+LAN access.
+
+ The Texinfo conversion has been a straight transliteration, to the
+extent possible. Like the TeX-to-HTML conversion, this was not without
+some introduction of breakage. In the case of Unofficial Texinfo
+Format, figures have suffered an amateurish resurrection of the lost
+art of ASCII art. Also, it's quite possible that some errors of
+ambiguity were introduced during the conversion of some of the copious
+superscripts (`^') and subscripts (`_'). Divining _which_ has been
+left as an exercise to the reader. But at least we don't put our brave
+astronauts at risk by encoding the _greater-than-or-equal_ symbol as
+`<u>></u>'.
+
+ If you modify `sicp.texi' to correct errors or improve the ASCII
+art, then update the `@set utfversion 2.neilvandyke4' line to reflect
+your delta. For example, if you started with Lytha's version `1', and
+your name is Bob, then you could name your successive versions
+`1.bob1', `1.bob2', ... `1.bobn'. Also update `utfversiondate'. If
+you want to distribute your version on the Web, then embedding the
+string "sicp.texi" somewhere in the file or Web page will make it
+easier for people to find with Web search engines.
+
+ It is believed that the Unofficial Texinfo Format is in keeping with
+the spirit of the graciously freely-distributed HTML version. But you
+never know when someone's armada of lawyers might need something to do,
+and get their shorts all in a knot over some benign little thing, so
+think twice before you use your full name or distribute Info, DVI,
+PostScript, or PDF formats that might embed your account or machine
+name.
+
+Peath,
+
+Lytha Ayth
+
+ Addendum: See also the SICP video lectures by Abelson and Sussman:
+`http://www.swiss.ai.mit.edu/classes/6.001/abelson-sussman-lectures/'
+
+\1f
+File: sicp.info, Node: Dedication, Next: Foreword, Prev: UTF, Up: Top
+
+Dedication
+**********
+
+This book is dedicated, in respect and admiration, to the spirit that
+lives in the computer.
+
+ "I think that it's extraordinarily important that we in computer
+ science keep fun in computing. When it started out, it was an
+ awful lot of fun. Of course, the paying customers got shafted
+ every now and then, and after a while we began to take their
+ complaints seriously. We began to feel as if we really were
+ responsible for the successful, error-free perfect use of these
+ machines. I don't think we are. I think we're responsible for
+ stretching them, setting them off in new directions, and keeping
+ fun in the house. I hope the field of computer science never
+ loses its sense of fun. Above all, I hope we don't become
+ missionaries. Don't feel as if you're Bible salesmen. The world
+ has too many of those already. What you know about computing
+ other people will learn. Don't feel as if the key to successful
+ computing is only in your hands. What's in your hands, I think
+ and hope, is intelligence: the ability to see the machine as more
+ than when you were first led up to it, that you can make it more."
+
+ --Alan J. Perlis (April 1, 1922 February 7, 1990)
+
+\1f
+File: sicp.info, Node: Foreword, Next: Preface, Prev: Dedication, Up: Top
+
+Foreword
+********
+
+Educators, generals, dieticians, psychologists, and parents program.
+Armies, students, and some societies are programmed. An assault on
+large problems employs a succession of programs, most of which spring
+into existence en route. These programs are rife with issues that
+appear to be particular to the problem at hand. To appreciate
+programming as an intellectual activity in its own right you must turn
+to computer programming; you must read and write computer
+programs--many of them. It doesn't matter much what the programs are
+about or what applications they serve. What does matter is how well
+they perform and how smoothly they fit with other programs in the
+creation of still greater programs. The programmer must seek both
+perfection of part and adequacy of collection. In this book the use of
+"program" is focused on the creation, execution, and study of programs
+written in a dialect of Lisp for execution on a digital computer.
+Using Lisp we restrict or limit not what we may program, but only the
+notation for our program descriptions.
+
+ Our traffic with the subject matter of this book involves us with
+three foci of phenomena: the human mind, collections of computer
+programs, and the computer. Every computer program is a model, hatched
+in the mind, of a real or mental process. These processes, arising
+from human experience and thought, are huge in number, intricate in
+detail, and at any time only partially understood. They are modeled to
+our permanent satisfaction rarely by our computer programs. Thus even
+though our programs are carefully handcrafted discrete collections of
+symbols, mosaics of interlocking functions, they continually evolve: we
+change them as our perception of the model deepens, enlarges,
+generalizes until the model ultimately attains a metastable place
+within still another model with which we struggle. The source of the
+exhilaration associated with computer programming is the continual
+unfolding within the mind and on the computer of mechanisms expressed
+as programs and the explosion of perception they generate. If art
+interprets our dreams, the computer executes them in the guise of
+programs!
+
+ For all its power, the computer is a harsh taskmaster. Its programs
+must be correct, and what we wish to say must be said accurately in
+every detail. As in every other symbolic activity, we become convinced
+of program truth through argument. Lisp itself can be assigned a
+semantics (another model, by the way), and if a program's function can
+be specified, say, in the predicate calculus, the proof methods of
+logic can be used to make an acceptable correctness argument.
+Unfortunately, as programs get large and complicated, as they almost
+always do, the adequacy, consistency, and correctness of the
+specifications themselves become open to doubt, so that complete formal
+arguments of correctness seldom accompany large programs. Since large
+programs grow from small ones, it is crucial that we develop an arsenal
+of standard program structures of whose correctness we have become
+sure--we call them idioms--and learn to combine them into larger
+structures using organizational techniques of proven value. These
+techniques are treated at length in this book, and understanding them
+is essential to participation in the Promethean enterprise called
+programming. More than anything else, the uncovering and mastery of
+powerful organizational techniques accelerates our ability to create
+large, significant programs. Conversely, since writing large programs
+is very taxing, we are stimulated to invent new methods of reducing the
+mass of function and detail to be fitted into large programs.
+
+ Unlike programs, computers must obey the laws of physics. If they
+wish to perform rapidly--a few nanoseconds per state change--they must
+transmit electrons only small distances (at most 11 over 2 feet). The
+heat generated by the huge number of devices so concentrated in space
+has to be removed. An exquisite engineering art has been developed
+balancing between multiplicity of function and density of devices. In
+any event, hardware always operates at a level more primitive than that
+at which we care to program. The processes that transform our Lisp
+programs to "machine" programs are themselves abstract models which we
+program. Their study and creation give a great deal of insight into
+the organizational programs associated with programming arbitrary
+models. Of course the computer itself can be so modeled. Think of it:
+the behavior of the smallest physical switching element is modeled by
+quantum mechanics described by differential equations whose detailed
+behavior is captured by numerical approximations represented in
+computer programs executing on computers composed of ...!
+
+ It is not merely a matter of tactical convenience to separately
+identify the three foci. Even though, as they say, it's all in the
+head, this logical separation induces an acceleration of symbolic
+traffic between these foci whose richness, vitality, and potential is
+exceeded in human experience only by the evolution of life itself. At
+best, relationships between the foci are metastable. The computers are
+never large enough or fast enough. Each breakthrough in hardware
+technology leads to more massive programming enterprises, new
+organizational principles, and an enrichment of abstract models. Every
+reader should ask himself periodically "Toward what end, toward what
+end?"--but do not ask it too often lest you pass up the fun of
+programming for the constipation of bittersweet philosophy.
+
+ Among the programs we write, some (but never enough) perform a
+precise mathematical function such as sorting or finding the maximum of
+a sequence of numbers, determining primality, or finding the square
+root. We call such programs algorithms, and a great deal is known of
+their optimal behavior, particularly with respect to the two important
+parameters of execution time and data storage requirements. A
+programmer should acquire good algorithms and idioms. Even though some
+programs resist precise specifications, it is the responsibility of the
+programmer to estimate, and always to attempt to improve, their
+performance.
+
+ Lisp is a survivor, having been in use for about a quarter of a
+century. Among the active programming languages only Fortran has had a
+longer life. Both languages have supported the programming needs of
+important areas of application, Fortran for scientific and engineering
+computation and Lisp for artificial intelligence. These two areas
+continue to be important, and their programmers are so devoted to these
+two languages that Lisp and Fortran may well continue in active use for
+at least another quarter-century.
+
+ Lisp changes. The Scheme dialect used in this text has evolved from
+the original Lisp and differs from the latter in several important
+ways, including static scoping for variable binding and permitting
+functions to yield functions as values. In its semantic structure
+Scheme is as closely akin to Algol 60 as to early Lisps. Algol 60,
+never to be an active language again, lives on in the genes of Scheme
+and Pascal. It would be difficult to find two languages that are the
+communicating coin of two more different cultures than those gathered
+around these two languages. Pascal is for building pyramids--imposing,
+breathtaking, static structures built by armies pushing heavy blocks
+into place. Lisp is for building organisms--imposing, breathtaking,
+dynamic structures built by squads fitting fluctuating myriads of
+simpler organisms into place. The organizing principles used are the
+same in both cases, except for one extraordinarily important
+difference: The discretionary exportable functionality entrusted to the
+individual Lisp programmer is more than an order of magnitude greater
+than that to be found within Pascal enterprises. Lisp programs inflate
+libraries with functions whose utility transcends the application that
+produced them. The list, Lisp's native data structure, is largely
+responsible for such growth of utility. The simple structure and
+natural applicability of lists are reflected in functions that are
+amazingly nonidiosyncratic. In Pascal the plethora of declarable data
+structures induces a specialization within functions that inhibits and
+penalizes casual cooperation. It is better to have 100 functions
+operate on one data structure than to have 10 functions operate on 10
+data structures. As a result the pyramid must stand unchanged for a
+millennium; the organism must evolve or perish.
+
+ To illustrate this difference, compare the treatment of material and
+exercises within this book with that in any first-course text using
+Pascal. Do not labor under the illusion that this is a text digestible
+at MIT only, peculiar to the breed found there. It is precisely what a
+serious book on programming Lisp must be, no matter who the student is
+or where it is used.
+
+ Note that this is a text about programming, unlike most Lisp books,
+which are used as a preparation for work in artificial intelligence.
+After all, the critical programming concerns of software engineering
+and artificial intelligence tend to coalesce as the systems under
+investigation become larger. This explains why there is such growing
+interest in Lisp outside of artificial intelligence.
+
+ As one would expect from its goals, artificial intelligence research
+generates many significant programming problems. In other programming
+cultures this spate of problems spawns new languages. Indeed, in any
+very large programming task a useful organizing principle is to control
+and isolate traffic within the task modules via the invention of
+language. These languages tend to become less primitive as one
+approaches the boundaries of the system where we humans interact most
+often. As a result, such systems contain complex language-processing
+functions replicated many times. Lisp has such a simple syntax and
+semantics that parsing can be treated as an elementary task. Thus
+parsing technology plays almost no role in Lisp programs, and the
+construction of language processors is rarely an impediment to the rate
+of growth and change of large Lisp systems. Finally, it is this very
+simplicity of syntax and semantics that is responsible for the burden
+and freedom borne by all Lisp programmers. No Lisp program of any size
+beyond a few lines can be written without being saturated with
+discretionary functions. Invent and fit; have fits and reinvent! We
+toast the Lisp programmer who pens his thoughts within nests of
+parentheses.
+
+Alan J. Perlis
+New Haven, Connecticut
+
+\1f
+File: sicp.info, Node: Preface, Next: Preface 1e, Prev: Foreword, Up: Top
+
+Preface to the Second Edition
+*****************************
+
+ Is it possible that software is not like anything else, that it is
+ meant to be discarded: that the whole point is to always see it as
+ a soap bubble?
+
+ --Alan J. Perlis
+
+ The material in this book has been the basis of MIT's entry-level
+computer science subject since 1980. We had been teaching this
+material for four years when the first edition was published, and
+twelve more years have elapsed until the appearance of this second
+edition. We are pleased that our work has been widely adopted and
+incorporated into other texts. We have seen our students take the
+ideas and programs in this book and build them in as the core of new
+computer systems and languages. In literal realization of an ancient
+Talmudic pun, our students have become our builders. We are lucky to
+have such capable students and such accomplished builders.
+
+ In preparing this edition, we have incorporated hundreds of
+clarifications suggested by our own teaching experience and the
+comments of colleagues at MIT and elsewhere. We have redesigned most
+of the major programming systems in the book, including the
+generic-arithmetic system, the interpreters, the register-machine
+simulator, and the compiler; and we have rewritten all the program
+examples to ensure that any Scheme implementation conforming to the
+IEEE Scheme standard (IEEE 1990) will be able to run the code.
+
+ This edition emphasizes several new themes. The most important of
+these is the central role played by different approaches to dealing
+with time in computational models: objects with state, concurrent
+programming, functional programming, lazy evaluation, and
+nondeterministic programming. We have included new sections on
+concurrency and nondeterminism, and we have tried to integrate this
+theme throughout the book.
+
+ The first edition of the book closely followed the syllabus of our
+MIT one-semester subject. With all the new material in the second
+edition, it will not be possible to cover everything in a single
+semester, so the instructor will have to pick and choose. In our own
+teaching, we sometimes skip the section on logic programming (section
+*Note 4-4::), we have students use the register-machine simulator but
+we do not cover its implementation (section *Note 5-2::), and we give
+only a cursory overview of the compiler (section *Note 5-5::). Even
+so, this is still an intense course. Some instructors may wish to
+cover only the first three or four chapters, leaving the other material
+for subsequent courses.
+
+ The World-Wide-Web site `http://mitpress.mit.edu/sicp/' provides
+support for users of this book. This includes programs from the book,
+sample programming assignments, supplementary materials, and
+downloadable implementations of the Scheme dialect of Lisp.
+
+\1f
+File: sicp.info, Node: Preface 1e, Next: Acknowledgements, Prev: Preface, Up: Top
+
+Preface to the First Edition
+****************************
+
+ A computer is like a violin. You can imagine a novice trying
+ first a phonograph and then a violin. The latter, he says, sounds
+ terrible. That is the argument we have heard from our humanists
+ and most of our computer scientists. Computer programs are good,
+ they say, for particular purposes, but they aren't flexible.
+ Neither is a violin, or a typewriter, until you learn how to use
+ it.
+
+ --Marvin Minsky, "Why Programming Is a Good Medium for Expressing
+ Poorly-Understood and Sloppily-Formulated Ideas"
+
+ "The Structure and Interpretation of Computer Programs" is the
+entry-level subject in computer science at the Massachusetts Institute
+of Technology. It is required of all students at MIT who major in
+electrical engineering or in computer science, as one-fourth of the
+"common core curriculum," which also includes two subjects on circuits
+and linear systems and a subject on the design of digital systems. We
+have been involved in the development of this subject since 1978, and
+we have taught this material in its present form since the fall of 1980
+to between 600 and 700 students each year. Most of these students have
+had little or no prior formal training in computation, although many
+have played with computers a bit and a few have had extensive
+programming or hardware-design experience.
+
+ Our design of this introductory computer-science subject reflects
+two major concerns. First, we want to establish the idea that a
+computer language is not just a way of getting a computer to perform
+operations but rather that it is a novel formal medium for expressing
+ideas about methodology. Thus, programs must be written for people to
+read, and only incidentally for machines to execute. Second, we
+believe that the essential material to be addressed by a subject at
+this level is not the syntax of particular programming-language
+constructs, nor clever algorithms for computing particular functions
+efficiently, nor even the mathematical analysis of algorithms and the
+foundations of computing, but rather the techniques used to control the
+intellectual complexity of large software systems.
+
+ Our goal is that students who complete this subject should have a
+good feel for the elements of style and the aesthetics of programming.
+They should have command of the major techniques for controlling
+complexity in a large system. They should be capable of reading a
+50-page-long program, if it is written in an exemplary style. They
+should know what not to read, and what they need not understand at any
+moment. They should feel secure about modifying a program, retaining
+the spirit and style of the original author.
+
+ These skills are by no means unique to computer programming. The
+techniques we teach and draw upon are common to all of engineering
+design. We control complexity by building abstractions that hide
+details when appropriate. We control complexity by establishing
+conventional interfaces that enable us to construct systems by
+combining standard, well-understood pieces in a "mix and match" way.
+We control complexity by establishing new languages for describing a
+design, each of which emphasizes particular aspects of the design and
+deemphasizes others.
+
+ Underlying our approach to this subject is our conviction that
+"computer science" is not a science and that its significance has
+little to do with computers. The computer revolution is a revolution
+in the way we think and in the way we express what we think. The
+essence of this change is the emergence of what might best be called "procedural
+epistemology"--the study of the structure of knowledge from an
+imperative point of view, as opposed to the more declarative point of
+view taken by classical mathematical subjects. Mathematics provides a
+framework for dealing precisely with notions of "what is." Computation
+provides a framework for dealing precisely with notions of "how to."
+
+ In teaching our material we use a dialect of the programming
+language Lisp. We never formally teach the language, because we don't
+have to. We just use it, and students pick it up in a few days. This
+is one great advantage of Lisp-like languages: They have very few ways
+of forming compound expressions, and almost no syntactic structure.
+All of the formal properties can be covered in an hour, like the rules
+of chess. After a short time we forget about syntactic details of the
+language (because there are none) and get on with the real
+issues--figuring out what we want to compute, how we will decompose
+problems into manageable parts, and how we will work on the parts.
+Another advantage of Lisp is that it supports (but does not enforce)
+more of the large-scale strategies for modular decomposition of
+programs than any other language we know. We can make procedural and
+data abstractions, we can use higher-order functions to capture common
+patterns of usage, we can model local state using assignment and data
+mutation, we can link parts of a program with streams and delayed
+evaluation, and we can easily implement embedded languages. All of
+this is embedded in an interactive environment with excellent support
+for incremental program design, construction, testing, and debugging.
+We thank all the generations of Lisp wizards, starting with John
+McCarthy, who have fashioned a fine tool of unprecedented power and
+elegance.
+
+ Scheme, the dialect of Lisp that we use, is an attempt to bring
+together the power and elegance of Lisp and Algol. From Lisp we take
+the metalinguistic power that derives from the simple syntax, the
+uniform representation of programs as data objects, and the
+garbage-collected heap-allocated data. From Algol we take lexical
+scoping and block structure, which are gifts from the pioneers of
+programming-language design who were on the Algol committee. We wish
+to cite John Reynolds and Peter Landin for their insights into the
+relationship of Church's [lambda] calculus to the structure of
+programming languages. We also recognize our debt to the
+mathematicians who scouted out this territory decades before computers
+appeared on the scene. These pioneers include Alonzo Church, Barkley
+Rosser, Stephen Kleene, and Haskell Curry.
+
+\1f
+File: sicp.info, Node: Acknowledgements, Next: Chapter 1, Prev: Preface 1e, Up: Top
+
+Acknowledgements
+****************
+
+We would like to thank the many people who have helped us develop this
+book and this curriculum.
+
+ Our subject is a clear intellectual descendant of "6.231," a
+wonderful subject on programming linguistics and the [lambda] calculus
+taught at MIT in the late 1960s by Jack Wozencraft and Arthur Evans, Jr.
+
+ We owe a great debt to Robert Fano, who reorganized MIT's
+introductory curriculum in electrical engineering and computer science
+to emphasize the principles of engineering design. He led us in
+starting out on this enterprise and wrote the first set of subject
+notes from which this book evolved.
+
+ Much of the style and aesthetics of programming that we try to teach
+were developed in conjunction with Guy Lewis Steele Jr., who
+collaborated with Gerald Jay Sussman in the initial development of the
+Scheme language. In addition, David Turner, Peter Henderson, Dan
+Friedman, David Wise, and Will Clinger have taught us many of the
+techniques of the functional programming community that appear in this
+book.
+
+ Joel Moses taught us about structuring large systems. His
+experience with the Macsyma system for symbolic computation provided
+the insight that one should avoid complexities of control and
+concentrate on organizing the data to reflect the real structure of the
+world being modeled.
+
+ Marvin Minsky and Seymour Papert formed many of our attitudes about
+programming and its place in our intellectual lives. To them we owe
+the understanding that computation provides a means of expression for
+exploring ideas that would otherwise be too complex to deal with
+precisely. They emphasize that a student's ability to write and modify
+programs provides a powerful medium in which exploring becomes a
+natural activity.
+
+ We also strongly agree with Alan Perlis that programming is lots of
+fun and we had better be careful to support the joy of programming.
+Part of this joy derives from observing great masters at work. We are
+fortunate to have been apprentice programmers at the feet of Bill
+Gosper and Richard Greenblatt.
+
+ It is difficult to identify all the people who have contributed to
+the development of our curriculum. We thank all the lecturers,
+recitation instructors, and tutors who have worked with us over the
+past fifteen years and put in many extra hours on our subject,
+especially Bill Siebert, Albert Meyer, Joe Stoy, Randy Davis, Louis
+Braida, Eric Grimson, Rod Brooks, Lynn Stein, and Peter Szolovits. We
+would like to specially acknowledge the outstanding teaching
+contributions of Franklyn Turbak, now at Wellesley; his work in
+undergraduate instruction set a standard that we can all aspire to. We
+are grateful to Jerry Saltzer and Jim Miller for helping us grapple
+with the mysteries of concurrency, and to Peter Szolovits and David
+McAllester for their contributions to the exposition of
+nondeterministic evaluation in *Note Chapter 4::.
+
+ Many people have put in significant effort presenting this material
+at other universities. Some of the people we have worked closely with
+are Jacob Katzenelson at the Technion, Hardy Mayer at the University of
+California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue, and
+Jan Komorowski at the Norwegian University of Science and Technology.
+We are exceptionally proud of our colleagues who have received major
+teaching awards for their adaptations of this subject at other
+universities, including Kenneth Yip at Yale, Brian Harvey at the
+University of California at Berkeley, and Dan Huttenlocher at Cornell.
+
+ Al Moye' arranged for us to teach this material to engineers at
+Hewlett-Packard, and for the production of videotapes of these
+lectures. We would like to thank the talented instructors--in
+particular Jim Miller, Bill Siebert, and Mike Eisenberg--who have
+designed continuing education courses incorporating these tapes and
+taught them at universities and industry all over the world.
+
+ Many educators in other countries have put in significant work
+translating the first edition. Michel Briand, Pierre Chamard, and
+Andre' Pic produced a French edition; Susanne Daniels-Herold produced a
+German edition; and Fumio Motoyoshi produced a Japanese edition. We do
+not know who produced the Chinese edition, but we consider it an honor
+to have been selected as the subject of an "unauthorized" translation.
+
+ It is hard to enumerate all the people who have made technical
+contributions to the development of the Scheme systems we use for
+instructional purposes. In addition to Guy Steele, principal wizards
+have included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas,
+and Stephen Adams. Others who have put in significant time are Richard
+Stallman, Alan Bawden, Kent Pitman, Jon Taft, Neil Mayle, John Lamping,
+Gwyn Osnos, Tracy Larrabee, George Carrette, Soma Chaudhuri, Bill
+Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss, Todd Cass, Patrick
+O'Donnell, Kevin Theobald, Daniel Weise, Kenneth Sinclair, Anthony
+Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth Shyu.
+
+ Beyond the MIT implementation, we would like to thank the many people
+who worked on the IEEE Scheme standard, including William Clinger and
+Jonathan Rees, who edited the R^4RS, and Chris Haynes, David Bartley,
+Chris Hanson, and Jim Miller, who prepared the IEEE standard.
+
+ Dan Friedman has been a long-time leader of the Scheme community.
+The community's broader work goes beyond issues of language design to
+encompass significant educational innovations, such as the high-school
+curriculum based on EdScheme by Schemer's Inc., and the wonderful books
+by Mike Eisenberg and by Brian Harvey and Matthew Wright.
+
+ We appreciate the work of those who contributed to making this a
+real book, especially Terry Ehling, Larry Cohen, and Paul Bethge at the
+MIT Press. Ella Mazel found the wonderful cover image. For the second
+edition we are particularly grateful to Bernard and Ella Mazel for help
+with the book design, and to David Jones, TeX wizard extraordinaire.
+We also are indebted to those readers who made penetrating comments on
+the new draft: Jacob Katzenelson, Hardy Mayer, Jim Miller, and
+especially Brian Harvey, who did unto this book as Julie did unto his
+book `Simply Scheme'.
+
+ Finally, we would like to acknowledge the support of the
+organizations that have encouraged this work over the years, including
+suppport from Hewlett-Packard, made possible by Ira Goldstein and Joel
+Birnbaum, and support from DARPA, made possible by Bob Kahn.
+
+\1f
+File: sicp.info, Node: Chapter 1, Next: Chapter 2, Prev: Acknowledgements, Up: Top
+
+1 Building Abstractions with Procedures
+***************************************
+
+ The acts of the mind, wherein it exerts its power over simple
+ ideas, are chiefly these three: 1. Combining several simple ideas
+ into one compound one, and thus all complex ideas are made. 2.
+ The second is bringing two ideas, whether simple or complex,
+ together, and setting them by one another so as to take a view of
+ them at once, without uniting them into one, by which it gets all
+ its ideas of relations. 3. The third is separating them from all
+ other ideas that accompany them in their real existence: this is
+ called abstraction, and thus all its general ideas are made.
+
+ --John Locke, _An Essay Concerning Human Understanding_ (1690)
+
+ We are about to study the idea of a "computational process".
+Computational processes are abstract beings that inhabit computers. As
+they evolve, processes manipulate other abstract things called "data".
+The evolution of a process is directed by a pattern of rules called a "program".
+People create programs to direct processes. In effect, we conjure the
+spirits of the computer with our spells.
+
+ A computational process is indeed much like a sorcerer's idea of a
+spirit. It cannot be seen or touched. It is not composed of matter at
+all. However, it is very real. It can perform intellectual work. It
+can answer questions. It can affect the world by disbursing money at a
+bank or by controlling a robot arm in a factory. The programs we use
+to conjure processes are like a sorcerer's spells. They are carefully
+composed from symbolic expressions in arcane and esoteric "programming
+languages" that prescribe the tasks we want our processes to perform.
+
+ A computational process, in a correctly working computer, executes
+programs precisely and accurately. Thus, like the sorcerer's
+apprentice, novice programmers must learn to understand and to
+anticipate the consequences of their conjuring. Even small errors
+(usually called "bugs" or "glitches") in programs can have complex and
+unanticipated consequences.
+
+ Fortunately, learning to program is considerably less dangerous than
+learning sorcery, because the spirits we deal with are conveniently
+contained in a secure way. Real-world programming, however, requires
+care, expertise, and wisdom. A small bug in a computer-aided design
+program, for example, can lead to the catastrophic collapse of an
+airplane or a dam or the self-destruction of an industrial robot.
+
+ Master software engineers have the ability to organize programs so
+that they can be reasonably sure that the resulting processes will
+perform the tasks intended. They can visualize the behavior of their
+systems in advance. They know how to structure programs so that
+unanticipated problems do not lead to catastrophic consequences, and
+when problems do arise, they can "debug" their programs. Well-designed
+computational systems, like well-designed automobiles or nuclear
+reactors, are designed in a modular manner, so that the parts can be
+constructed, replaced, and debugged separately.
+
+Programming in Lisp
+...................
+
+We need an appropriate language for describing processes, and we will
+use for this purpose the programming language Lisp. Just as our
+everyday thoughts are usually expressed in our natural language (such
+as English, French, or Japanese), and descriptions of quantitative
+phenomena are expressed with mathematical notations, our procedural
+thoughts will be expressed in Lisp. Lisp was invented in the late
+1950s as a formalism for reasoning about the use of certain kinds of
+logical expressions, called "recursion equations", as a model for
+computation. The language was conceived by John McCarthy and is based
+on his paper "Recursive Functions of Symbolic Expressions and Their
+Computation by Machine" (McCarthy 1960).
+
+ Despite its inception as a mathematical formalism, Lisp is a
+practical programming language. A Lisp "interpreter" is a machine that
+carries out processes described in the Lisp language. The first Lisp
+interpreter was implemented by McCarthy with the help of colleagues and
+students in the Artificial Intelligence Group of the MIT Research
+Laboratory of Electronics and in the MIT Computation Center.(1) Lisp,
+whose name is an acronym for LISt Processing, was designed to provide
+symbol-manipulating capabilities for attacking programming problems
+such as the symbolic differentiation and integration of algebraic
+expressions. It included for this purpose new data objects known as
+atoms and lists, which most strikingly set it apart from all other
+languages of the period.
+
+ Lisp was not the product of a concerted design effort. Instead, it
+evolved informally in an experimental manner in response to users'
+needs and to pragmatic implementation considerations. Lisp's informal
+evolution has continued through the years, and the community of Lisp
+users has traditionally resisted attempts to promulgate any "official"
+definition of the language. This evolution, together with the
+flexibility and elegance of the initial conception, has enabled Lisp,
+which is the second oldest language in widespread use today (only
+Fortran is older), to continually adapt to encompass the most modern
+ideas about program design. Thus, Lisp is by now a family of dialects,
+which, while sharing most of the original features, may differ from one
+another in significant ways. The dialect of Lisp used in this book is
+called Scheme.(2)
+
+ Because of its experimental character and its emphasis on symbol
+manipulation, Lisp was at first very inefficient for numerical
+computations, at least in comparison with Fortran. Over the years,
+however, Lisp compilers have been developed that translate programs
+into machine code that can perform numerical computations reasonably
+efficiently. And for special applications, Lisp has been used with
+great effectiveness.(3) Although Lisp has not yet overcome its old
+reputation as hopelessly inefficient, Lisp is now used in many
+applications where efficiency is not the central concern. For example,
+Lisp has become a language of choice for operating-system shell
+languages and for extension languages for editors and computer-aided
+design systems.
+
+ If Lisp is not a mainstream language, why are we using it as the
+framework for our discussion of programming? Because the language
+possesses unique features that make it an excellent medium for studying
+important programming constructs and data structures and for relating
+them to the linguistic features that support them. The most
+significant of these features is the fact that Lisp descriptions of
+processes, called "procedures", can themselves be represented and
+manipulated as Lisp data. The importance of this is that there are
+powerful program-design techniques that rely on the ability to blur the
+traditional distinction between "passive" data and "active" processes.
+As we shall discover, Lisp's flexibility in handling procedures as data
+makes it one of the most convenient languages in existence for
+exploring these techniques. The ability to represent procedures as
+data also makes Lisp an excellent language for writing programs that
+must manipulate other programs as data, such as the interpreters and
+compilers that support computer languages. Above and beyond these
+considerations, programming in Lisp is great fun.
+
+* Menu:
+
+* 1-1:: The Elements of Programming
+* 1-2:: Procedures and the Processes They Generate
+* 1-3:: Formulating Abstractions with Higher-Order Procedures
+
+ ---------- Footnotes ----------
+
+ (1) The `Lisp 1 Programmer's Manual' appeared in 1960, and the `Lisp
+1.5 Programmer's Manual' (McCarthy 1965) was published in 1962. The
+early history of Lisp is described in McCarthy 1978.
+
+ (2) The two dialects in which most major Lisp programs of the 1970s
+were written are MacLisp (Moon 1978; Pitman 1983), developed at the MIT
+Project MAC, and Interlisp (Teitelman 1974), developed at Bolt Beranek
+and Newman Inc. and the Xerox Palo Alto Research Center. Portable
+Standard Lisp (Hearn 1969; Griss 1981) was a Lisp dialect designed to
+be easily portable between different machines. MacLisp spawned a
+number of subdialects, such as Franz Lisp, which was developed at the
+University of California at Berkeley, and Zetalisp (Moon 1981), which
+was based on a special-purpose processor designed at the MIT Artificial
+Intelligence Laboratory to run Lisp very efficiently. The Lisp dialect
+used in this book, called Scheme (Steele 1975), was invented in 1975 by
+Guy Lewis Steele Jr. and Gerald Jay Sussman of the MIT Artificial
+Intelligence Laboratory and later reimplemented for instructional use
+at MIT. Scheme became an IEEE standard in 1990 (IEEE 1990). The
+Common Lisp dialect (Steele 1982, Steele 1990) was developed by the
+Lisp community to combine features from the earlier Lisp dialects to
+make an industrial standard for Lisp. Common Lisp became an ANSI
+standard in 1994 (ANSI 1994).
+
+ (3) One such special application was a breakthrough computation of
+scientific importance--an integration of the motion of the Solar System
+that extended previous results by nearly two orders of magnitude, and
+demonstrated that the dynamics of the Solar System is chaotic. This
+computation was made possible by new integration algorithms, a
+special-purpose compiler, and a special-purpose computer all
+implemented with the aid of software tools written in Lisp (Abelson et
+al. 1992; Sussman and Wisdom 1992).
+
+\1f
+File: sicp.info, Node: 1-1, Next: 1-2, Prev: Chapter 1, Up: Chapter 1
+
+1.1 The Elements of Programming
+===============================
+
+A powerful programming language is more than just a means for
+instructing a computer to perform tasks. The language also serves as a
+framework within which we organize our ideas about processes. Thus,
+when we describe a language, we should pay particular attention to the
+means that the language provides for combining simple ideas to form
+more complex ideas. Every powerful language has three mechanisms for
+accomplishing this:
+
+"primitive expressions"
+ which represent the simplest entities the language is concerned
+ with,
+
+"means of combination"
+ by which compound elements are built from simpler ones, and
+
+"means of abstraction"
+ by which compound elements can be named and manipulated as units.
+
+
+ In programming, we deal with two kinds of elements: procedures and
+data. (Later we will discover that they are really not so distinct.)
+Informally, data is "stuff" that we want to manipulate, and procedures
+are descriptions of the rules for manipulating the data. Thus, any
+powerful programming language should be able to describe primitive data
+and primitive procedures and should have methods for combining and
+abstracting procedures and data.
+
+ In this chapter we will deal only with simple numerical data so that
+we can focus on the rules for building procedures.(1) In later chapters
+we will see that these same rules allow us to build procedures to
+manipulate compound data as well.
+
+* Menu:
+
+* 1-1-1:: Expressions
+* 1-1-2:: Naming and the Environment
+* 1-1-3:: Evaluating Combinations
+* 1-1-4:: Compound Procedures
+* 1-1-5:: The Substitution Model for Procedure Application
+* 1-1-6:: Conditional Expressions and Predicates
+* 1-1-7:: Example: Square Roots by Newton's Method
+* 1-1-8:: Procedures as Black-Box Abstractions
+
+ ---------- Footnotes ----------
+
+ (1) The characterization of numbers as "simple data" is a barefaced
+bluff. In fact, the treatment of numbers is one of the trickiest and
+most confusing aspects of any programming language. Some typical
+issues involved are these: Some computer systems distinguish "integers",
+such as 2, from "real numbers", such as 2.71. Is the real number 2.00
+different from the integer 2? Are the arithmetic operations used for
+integers the same as the operations used for real numbers? Does 6
+divided by 2 produce 3, or 3.0? How large a number can we represent?
+How many decimal places of accuracy can we represent? Is the range of
+integers the same as the range of real numbers? Above and beyond these
+questions, of course, lies a collection of issues concerning roundoff
+and truncation errors - the entire science of numerical analysis.
+Since our focus in this book is on large-scale program design rather
+than on numerical techniques, we are going to ignore these problems.
+The numerical examples in this chapter will exhibit the usual roundoff
+behavior that one observes when using arithmetic operations that
+preserve a limited number of decimal places of accuracy in noninteger
+operations.
+
+\1f
+File: sicp.info, Node: 1-1-1, Next: 1-1-2, Prev: 1-1, Up: 1-1
+
+1.1.1 Expressions
+-----------------
+
+One easy way to get started at programming is to examine some typical
+interactions with an interpreter for the Scheme dialect of Lisp.
+Imagine that you are sitting at a computer terminal. You type an "expression",
+and the interpreter responds by displaying the result of its "evaluating"
+that expression.
+
+ One kind of primitive expression you might type is a number. (More
+precisely, the expression that you type consists of the numerals that
+represent the number in base 10.) If you present Lisp with a number
+
+ 486
+
+the interpreter will respond by printing (1)
+
+ 486
+
+ Expressions representing numbers may be combined with an expression
+representing a primitive procedure (such as `+' or `*') to form a
+compound expression that represents the application of the procedure to
+those numbers. For example:
+
+ (+ 137 349)
+ 486
+
+ (- 1000 334)
+ 666
+
+ (* 5 99)
+ 495
+
+ (/ 10 5)
+ 2
+
+ (+ 2.7 10)
+ 12.7
+
+ Expressions such as these, formed by delimiting a list of
+expressions within parentheses in order to denote procedure
+application, are called "combinations". The leftmost element in the
+list is called the "operator", and the other elements are called "operands".
+The value of a combination is obtained by applying the procedure
+specified by the operator to the "arguments" that are the values of the
+operands.
+
+ The convention of placing the operator to the left of the operands
+is known as "prefix notation", and it may be somewhat confusing at
+first because it departs significantly from the customary mathematical
+convention. Prefix notation has several advantages, however. One of
+them is that it can accommodate procedures that may take an arbitrary
+number of arguments, as in the following examples:
+
+ (+ 21 35 12 7)
+ 75
+
+ (* 25 4 12)
+ 1200
+
+ No ambiguity can arise, because the operator is always the leftmost
+element and the entire combination is delimited by the parentheses.
+
+ A second advantage of prefix notation is that it extends in a
+straightforward way to allow combinations to be nested, that is, to
+have combinations whose elements are themselves combinations:
+
+ (+ (* 3 5) (- 10 6))
+ 19
+
+ There is no limit (in principle) to the depth of such nesting and to
+the overall complexity of the expressions that the Lisp interpreter can
+evaluate. It is we humans who get confused by still relatively simple
+expressions such as
+
+ (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))
+
+which the interpreter would readily evaluate to be 57. We can help
+ourselves by writing such an expression in the form
+
+ (+ (* 3
+ (+ (* 2 4)
+ (+ 3 5)))
+ (+ (- 10 7)
+ 6))
+
+following a formatting convention known as "pretty-printing", in which
+each long combination is written so that the operands are aligned
+vertically. The resulting indentations display clearly the structure
+of the expression.(2)
+
+ Even with complex expressions, the interpreter always operates in
+the same basic cycle: It reads an expression from the terminal,
+evaluates the expression, and prints the result. This mode of
+operation is often expressed by saying that the interpreter runs in a "read-eval-print
+loop". Observe in particular that it is not necessary to explicitly
+instruct the interpreter to print the value of the expression.(3)
+
+ ---------- Footnotes ----------
+
+ (1) Throughout this book, when we wish to emphasize the distinction
+between the input typed by the user and the response printed by the
+interpreter, we will show the latter in slanted characters.
+
+ (2) Lisp systems typically provide features to aid the user in
+formatting expressions. Two especially useful features are one that
+automatically indents to the proper pretty-print position whenever a
+new line is started and one that highlights the matching left
+parenthesis whenever a right parenthesis is typed.
+
+ (3) Lisp obeys the convention that every expression has a value.
+This convention, together with the old reputation of Lisp as an
+inefficient language, is the source of the quip by Alan Perlis
+(paraphrasing Oscar Wilde) that "Lisp programmers know the value of
+everything but the cost of nothing."
+
+\1f
+File: sicp.info, Node: 1-1-2, Next: 1-1-3, Prev: 1-1-1, Up: 1-1
+
+1.1.2 Naming and the Environment
+--------------------------------
+
+A critical aspect of a programming language is the means it provides
+for using names to refer to computational objects. We say that the
+name identifies a "variable" whose "value" is the object.
+
+ In the Scheme dialect of Lisp, we name things with `define'. Typing
+
+ (define size 2)
+
+causes the interpreter to associate the value 2 with the name
+`size'.(1) Once the name `size' has been associated with the number 2,
+we can refer to the value 2 by name:
+
+ size
+ 2
+
+ (* 5 size)
+ 10
+
+ Here are further examples of the use of `define':
+
+ (define pi 3.14159)
+
+ (define radius 10)
+
+ (* pi (* radius radius))
+ 314.159
+
+ (define circumference (* 2 pi radius))
+
+ circumference
+ 62.8318
+
+ `Define' is our language's simplest means of abstraction, for it
+allows us to use simple names to refer to the results of compound
+operations, such as the `circumference' computed above. In general,
+computational objects may have very complex structures, and it would be
+extremely inconvenient to have to remember and repeat their details
+each time we want to use them. Indeed, complex programs are
+constructed by building, step by step, computational objects of
+increasing complexity. The interpreter makes this step-by-step program
+construction particularly convenient because name-object associations
+can be created incrementally in successive interactions. This feature
+encourages the incremental development and testing of programs and is
+largely responsible for the fact that a Lisp program usually consists
+of a large number of relatively simple procedures.
+
+ It should be clear that the possibility of associating values with
+symbols and later retrieving them means that the interpreter must
+maintain some sort of memory that keeps track of the name-object pairs.
+This memory is called the "environment" (more precisely the "global
+environment", since we will see later that a computation may involve a
+number of different environments).(2)
+
+ ---------- Footnotes ----------
+
+ (1) In this book, we do not show the interpreter's response to
+evaluating definitions, since this is highly implementation-dependent.
+
+ (2) *Note Chapter 3:: will show that this notion of environment is
+crucial, both for understanding how the interpreter works and for
+implementing interpreters.
+
+\1f
+File: sicp.info, Node: 1-1-3, Next: 1-1-4, Prev: 1-1-2, Up: 1-1
+
+1.1.3 Evaluating Combinations
+-----------------------------
+
+One of our goals in this chapter is to isolate issues about thinking
+procedurally. As a case in point, let us consider that, in evaluating
+combinations, the interpreter is itself following a procedure.
+
+ To evaluate a combination, do the following:
+
+ 1. Evaluate the subexpressions of the combination.
+
+ 2. Apply the procedure that is the value of the leftmost
+ subexpression (the operator) to the arguments that are the
+ values of the other subexpressions (the operands).
+
+
+ Even this simple rule illustrates some important points about
+processes in general. First, observe that the first step dictates that
+in order to accomplish the evaluation process for a combination we must
+first perform the evaluation process on each element of the
+combination. Thus, the evaluation rule is "recursive" in nature; that
+is, it includes, as one of its steps, the need to invoke the rule
+itself.(1)
+
+ Notice how succinctly the idea of recursion can be used to express
+what, in the case of a deeply nested combination, would otherwise be
+viewed as a rather complicated process. For example, evaluating
+
+ (* (+ 2 (* 4 6))
+ (+ 3 5 7))
+
+requires that the evaluation rule be applied to four different
+combinations. We can obtain a picture of this process by representing
+the combination in the form of a tree, as shown in *Note Figure 1-1::.
+Each combination is represented by a node with branches corresponding
+to the operator and the operands of the combination stemming from it.
+The terminal nodes (that is, nodes with no branches stemming from them)
+represent either operators or numbers. Viewing evaluation in terms of
+the tree, we can imagine that the values of the operands percolate
+upward, starting from the terminal nodes and then combining at higher
+and higher levels. In general, we shall see that recursion is a very
+powerful technique for dealing with hierarchical, treelike objects. In
+fact, the "percolate values upward" form of the evaluation rule is an
+example of a general kind of process known as "tree accumulation".
+
+ *Figure 1.1:* Tree representation, showing the value of each
+ subcombination.
+
+ 390
+ /|\____________
+ / | \
+ * 26 15
+ /|\ |
+ / | \ // \\
+ + 2 24 / | | \
+ /|\ + 3 5 7
+ / | \
+ * 4 6
+
+ Next, observe that the repeated application of the first step brings
+us to the point where we need to evaluate, not combinations, but
+primitive expressions such as numerals, built-in operators, or other
+names. We take care of the primitive cases by stipulating that
+
+ * the values of numerals are the numbers that they name,
+
+ * the values of built-in operators are the machine instruction
+ sequences that carry out the corresponding operations, and
+
+ * the values of other names are the objects associated with those
+ names in the environment.
+
+
+ We may regard the second rule as a special case of the third one by
+stipulating that symbols such as `+' and `*' are also included in the
+global environment, and are associated with the sequences of machine
+instructions that are their "values." The key point to notice is the
+role of the environment in determining the meaning of the symbols in
+expressions. In an interactive language such as Lisp, it is
+meaningless to speak of the value of an expression such as `(+ x 1)'
+without specifying any information about the environment that would
+provide a meaning for the symbol `x' (or even for the symbol `+'). As
+we shall see in *Note Chapter 3::, the general notion of the
+environment as providing a context in which evaluation takes place will
+play an important role in our understanding of program execution.
+
+ Notice that the evaluation rule given above does not handle
+definitions. For instance, evaluating `(define x 3)' does not apply
+`define' to two arguments, one of which is the value of the symbol `x'
+and the other of which is 3, since the purpose of the `define' is
+precisely to associate `x' with a value. (That is, `(define x 3)' is
+not a combination.)
+
+ Such exceptions to the general evaluation rule are called forms
+"special forms". `Define' is the only example of a special form that
+we have seen so far, but we will meet others shortly. Each special
+form has its own evaluation rule. The various kinds of expressions
+(each with its associated evaluation rule) constitute the syntax of the
+programming language. In comparison with most other programming
+languages, Lisp has a very simple syntax; that is, the evaluation rule
+for expressions can be described by a simple general rule together with
+specialized rules for a small number of special forms.(2)
+
+ ---------- Footnotes ----------
+
+ (1) It may seem strange that the evaluation rule says, as part of
+the first step, that we should evaluate the leftmost element of a
+combination, since at this point that can only be an operator such as
+`+' or `*' representing a built-in primitive procedure such as addition
+or multiplication. We will see later that it is useful to be able to
+work with combinations whose operators are themselves compound
+expressions.
+
+ (2) Special syntactic forms that are simply convenient alternative
+surface structures for things that can be written in more uniform ways
+are sometimes called "syntactic sugar", to use a phrase coined by Peter
+Landin. In comparison with users of other languages, Lisp programmers,
+as a rule, are less concerned with matters of syntax. (By contrast,
+examine any Pascal manual and notice how much of it is devoted to
+descriptions of syntax.) This disdain for syntax is due partly to the
+flexibility of Lisp, which makes it easy to change surface syntax, and
+partly to the observation that many "convenient" syntactic constructs,
+which make the language less uniform, end up causing more trouble than
+they are worth when programs become large and complex. In the words of
+Alan Perlis, "Syntactic sugar causes cancer of the semicolon."
+
+\1f
+File: sicp.info, Node: 1-1-4, Next: 1-1-5, Prev: 1-1-3, Up: 1-1
+
+1.1.4 Compound Procedures
+-------------------------
+
+We have identified in Lisp some of the elements that must appear in any
+powerful programming language:
+
+ * Numbers and arithmetic operations are primitive data and
+ procedures.
+
+ * Nesting of combinations provides a means of combining operations.
+
+ * Definitions that associate names with values provide a limited
+ means of abstraction.
+
+
+ Now we will learn about "procedure definitions", a much more powerful
+abstraction technique by which a compound operation can be given a name
+and then referred to as a unit.
+
+ We begin by examining how to express the idea of "squaring." We
+might say, "To square something, multiply it by itself." This is
+expressed in our language as
+
+ (define (square x) (* x x))
+
+ We can understand this in the following way:
+
+ (define (square x) (* x x))
+ | | | | | |
+ To square something, multiply it by itself.
+
+ We have here a "compound procedure", which has been given the name
+`square'. The procedure represents the operation of multiplying
+something by itself. The thing to be multiplied is given a local name,
+`x', which plays the same role that a pronoun plays in natural
+language. Evaluating the definition creates this compound procedure
+and associates it with the name `square'.(1)
+
+ The general form of a procedure definition is
+
+ (define (<NAME> <FORMAL PARAMETERS>) <BODY>)
+
+ The <NAME> is a symbol to be associated with the procedure
+definition in the environment.(2) The <FORMAL PARAMETERS> are the names
+used within the body of the procedure to refer to the corresponding
+arguments of the procedure. The <BODY> is an expression that will
+yield the value of the procedure application when the formal parameters
+are replaced by the actual arguments to which the procedure is
+applied.(3) The <NAME> and the <FORMAL PARAMETERS> are grouped within
+parentheses, just as they would be in an actual call to the procedure
+being defined.
+
+ Having defined `square', we can now use it:
+
+ (square 21)
+ 441
+
+ (square (+ 2 5))
+ 49
+
+ (square (square 3))
+ 81
+
+ We can also use `square' as a building block in defining other
+procedures. For example, x^2 + y^2 can be expressed as
+
+ (+ (square x) (square y))
+
+ We can easily define a procedure `sum-of-squares' that, given any two
+numbers as arguments, produces the sum of their squares:
+
+ (define (sum-of-squares x y)
+ (+ (square x) (square y)))
+ (sum-of-squares 3 4)
+ 25
+
+ Now we can use `sum-of-squares' as a building block in constructing
+further procedures:
+
+ (define (f a)
+ (sum-of-squares (+ a 1) (* a 2)))
+
+ (f 5)
+ 136
+
+ Compound procedures are used in exactly the same way as primitive
+procedures. Indeed, one could not tell by looking at the definition of
+`sum-of-squares' given above whether `square' was built into the
+interpreter, like `+' and `*', or defined as a compound procedure.
+
+ ---------- Footnotes ----------
+
+ (1) Observe that there are two different operations being combined
+here: we are creating the procedure, and we are giving it the name
+`square'. It is possible, indeed important, to be able to separate
+these two notions--to create procedures without naming them, and to
+give names to procedures that have already been created. We will see
+how to do this in section *Note 1-3-2::.
+
+ (2) Throughout this book, we will describe the general syntax of
+expressions by using italic symbols delimited by angle brackets--e.g.,
+<NAME>--to denote the "slots" in the expression to be filled in when
+such an expression is actually used.
+
+ (3) More generally, the body of the procedure can be a sequence of
+expressions. In this case, the interpreter evaluates each expression
+in the sequence in turn and returns the value of the final expression
+as the value of the procedure application.
+
+\1f
+File: sicp.info, Node: 1-1-5, Next: 1-1-6, Prev: 1-1-4, Up: 1-1
+
+1.1.5 The Substitution Model for Procedure Application
+------------------------------------------------------
+
+To evaluate a combination whose operator names a compound procedure, the
+interpreter follows much the same process as for combinations whose
+operators name primitive procedures, which we described in section
+*Note 1-1-3::. That is, the interpreter evaluates the elements of the
+combination and applies the procedure (which is the value of the
+operator of the combination) to the arguments (which are the values of
+the operands of the combination).
+
+ We can assume that the mechanism for applying primitive procedures
+to arguments is built into the interpreter. For compound procedures,
+the application process is as follows:
+
+ To apply a compound procedure to arguments, evaluate the body of
+ the procedure with each formal parameter replaced by the
+ corresponding argument.
+
+ To illustrate this process, let's evaluate the combination
+
+ (f 5)
+
+where `f' is the procedure defined in section *Note 1-1-4::. We begin
+by retrieving the body of `f':
+
+ (sum-of-squares (+ a 1) (* a 2))
+
+ Then we replace the formal parameter `a' by the argument 5:
+
+ (sum-of-squares (+ 5 1) (* 5 2))
+
+ Thus the problem reduces to the evaluation of a combination with two
+operands and an operator `sum-of-squares'. Evaluating this combination
+involves three subproblems. We must evaluate the operator to get the
+procedure to be applied, and we must evaluate the operands to get the
+arguments. Now `(+ 5 1)' produces 6 and `(* 5 2)' produces 10, so we
+must apply the `sum-of-squares' procedure to 6 and 10. These values
+are substituted for the formal parameters `x' and `y' in the body of
+`sum-of-squares', reducing the expression to
+
+ (+ (square 6) (square 10))
+
+ If we use the definition of `square', this reduces to
+
+ (+ (* 6 6) (* 10 10))
+
+which reduces by multiplication to
+
+ (+ 36 100)
+
+and finally to
+
+ 136
+
+ The process we have just described is called the "substitution model"
+for procedure application. It can be taken as a model that determines
+the "meaning" of procedure application, insofar as the procedures in
+this chapter are concerned. However, there are two points that should
+be stressed:
+
+ * The purpose of the substitution is to help us think about procedure
+ application, not to provide a description of how the interpreter
+ really works. Typical interpreters do not evaluate procedure
+ applications by manipulating the text of a procedure to substitute
+ values for the formal parameters. In practice, the "substitution"
+ is accomplished by using a local environment for the formal
+ parameters. We will discuss this more fully in *Note Chapter 3::
+ and *Note Chapter 4:: when we examine the implementation of an
+ interpreter in detail.
+
+ * Over the course of this book, we will present a sequence of
+ increasingly elaborate models of how interpreters work,
+ culminating with a complete implementation of an interpreter and
+ compiler in *Note Chapter 5::. The substitution model is only the
+ first of these models--a way to get started thinking formally
+ about the evaluation process. In general, when modeling phenomena
+ in science and engineering, we begin with simplified, incomplete
+ models. As we examine things in greater detail, these simple
+ models become inadequate and must be replaced by more refined
+ models. The substitution model is no exception. In particular,
+ when we address in *Note Chapter 3:: the use of procedures with
+ "mutable data," we will see that the substitution model breaks
+ down and must be replaced by a more complicated model of procedure
+ application.(1)
+
+
+Applicative order versus normal order
+.....................................
+
+According to the description of evaluation given in section *Note
+1-1-3::, the interpreter first evaluates the operator and operands and
+then applies the resulting procedure to the resulting arguments. This
+is not the only way to perform evaluation. An alternative evaluation
+model would not evaluate the operands until their values were needed.
+Instead it would first substitute operand expressions for parameters
+until it obtained an expression involving only primitive operators, and
+would then perform the evaluation. If we used this method, the
+evaluation of `(f 5)' would proceed according to the sequence of
+expansions
+
+ (sum-of-squares (+ 5 1) (* 5 2))
+
+ (+ (square (+ 5 1)) (square (* 5 2)) )
+
+ (+ (* (+ 5 1) (+ 5 1)) (* (* 5 2) (* 5 2)))
+
+followed by the reductions
+
+ (+ (* 6 6) (* 10 10))
+
+ (+ 36 100)
+
+ 136
+
+ This gives the same answer as our previous evaluation model, but the
+process is different. In particular, the evaluations of `(+ 5 1)' and
+`(* 5 2)' are each performed twice here, corresponding to the reduction
+of the expression `(* x x)' with `x' replaced respectively by `(+ 5 1)'
+and `(* 5 2)'.
+
+ This alternative "fully expand and then reduce" evaluation method is
+known as "normal-order evaluation", in contrast to the "evaluate the
+arguments and then apply" method that the interpreter actually uses,
+which is called "applicative-order evaluation". It can be shown that,
+for procedure applications that can be modeled using substitution
+(including all the procedures in the first two chapters of this book)
+and that yield legitimate values, normal-order and applicative-order
+evaluation produce the same value. (See *Note Exercise 1-5:: for an
+instance of an "illegitimate" value where normal-order and
+applicative-order evaluation do not give the same result.)
+
+ Lisp uses applicative-order evaluation, partly because of the
+additional efficiency obtained from avoiding multiple evaluations of
+expressions such as those illustrated with `(+ 5 1)' and `(* 5 2)'
+above and, more significantly, because normal-order evaluation becomes
+much more complicated to deal with when we leave the realm of
+procedures that can be modeled by substitution. On the other hand,
+normal-order evaluation can be an extremely valuable tool, and we will
+investigate some of its implications in *Note Chapter 3:: and *Note
+Chapter 4::.(2)
+
+ ---------- Footnotes ----------
+
+ (1) Despite the simplicity of the substitution idea, it turns out to
+be surprisingly complicated to give a rigorous mathematical definition
+of the substitution process. The problem arises from the possibility of
+confusion between the names used for the formal parameters of a
+procedure and the (possibly identical) names used in the expressions to
+which the procedure may be applied. Indeed, there is a long history of
+erroneous definitions of "substitution" in the literature of logic and
+programming semantics. See Stoy 1977 for a careful discussion of
+substitution.
+
+ (2) In *Note Chapter 3:: we will introduce "stream processing",
+which is a way of handling apparently "infinite" data structures by
+incorporating a limited form of normal-order evaluation. In section
+*Note 4-2:: we will modify the Scheme interpreter to produce a
+normal-order variant of Scheme.
+
+\1f
+File: sicp.info, Node: 1-1-6, Next: 1-1-7, Prev: 1-1-5, Up: 1-1
+
+1.1.6 Conditional Expressions and Predicates
+--------------------------------------------
+
+The expressive power of the class of procedures that we can define at
+this point is very limited, because we have no way to make tests and to
+perform different operations depending on the result of a test. For
+instance, we cannot define a procedure that computes the absolute value
+of a number by testing whether the number is positive, negative, or
+zero and taking different actions in the different cases according to
+the rule
+
+ /
+ | x if x > 0
+ |x| = < 0 if x = 0
+ | -x if x < 0
+ \
+
+ This construct is called a "case analysis", and there is a special
+form in Lisp for notating such a case analysis. It is called `cond'
+(which stands for "conditional"), and it is used as follows:
+
+ (define (abs x)
+ (cond ((> x 0) x)
+ ((= x 0) 0)
+ ((< x 0) (- x))))
+
+ The general form of a conditional expression is
+
+ (cond (<P1> <E1>)
+ (<P2> <E2>)
+ ...
+ (<PN> <EN>))
+
+consisting of the symbol `cond' followed by parenthesized pairs of
+expressions
+
+ (<P> <E>)
+
+called "clauses". The first expression in each pair is a "predicate"--that
+is, an expression whose value is interpreted as either true or false.(1)
+
+ Conditional expressions are evaluated as follows. The predicate
+<P1> is evaluated first. If its value is false, then <P2> is
+evaluated. If <P2>'s value is also false, then <P3> is evaluated.
+This process continues until a predicate is found whose value is true,
+in which case the interpreter returns the value of the corresponding expression
+"consequent expression" <E> of the clause as the value of the
+conditional expression. If none of the <P>'s is found to be true, the
+value of the `cond' is undefined.
+
+ The word "predicate" is used for procedures that return true or
+false, as well as for expressions that evaluate to true or false. The
+absolute-value procedure `abs' makes use of the primitive predicates
+`>', `<', and `='.(2) These take two numbers as arguments and test
+whether the first number is, respectively, greater than, less than, or
+equal to the second number, returning true or false accordingly.
+
+ Another way to write the absolute-value procedure is
+
+ (define (abs x)
+ (cond ((< x 0) (- x))
+ (else x)))
+
+which could be expressed in English as "If x is less than zero return -
+x; otherwise return x." `Else' is a special symbol that can be used in
+place of the <P> in the final clause of a `cond'. This causes the
+`cond' to return as its value the value of the corresponding <E>
+whenever all previous clauses have been bypassed. In fact, any
+expression that always evaluates to a true value could be used as the
+<P> here.
+
+ Here is yet another way to write the absolute-value procedure:
+
+ (define (abs x)
+ (if (< x 0)
+ (- x)
+ x))
+
+ This uses the special form `if', a restricted type of conditional
+that can be used when there are precisely two cases in the case
+analysis. The general form of an `if' expression is
+
+ (if <PREDICATE> <CONSEQUENT> <ALTERNATIVE>)
+
+ To evaluate an `if' expression, the interpreter starts by evaluating
+the <PREDICATE> part of the expression. If the <PREDICATE> evaluates
+to a true value, the interpreter then evaluates the <CONSEQUENT> and
+returns its value. Otherwise it evaluates the <ALTERNATIVE> and returns
+its value.(3)
+
+ In addition to primitive predicates such as `<', `=', and `>', there
+are logical composition operations, which enable us to construct
+compound predicates. The three most frequently used are these:
+
+ * `(and <E1> ... <EN>)'
+
+ The interpreter evaluates the expressions <E> one at a time, in
+ left-to-right order. If any <E> evaluates to false, the value of
+ the `and' expression is false, and the rest of the <E>'s are not
+ evaluated. If all <E>'s evaluate to true values, the value of the
+ `and' expression is the value of the last one.
+
+ * `(or <E1> ... <EN>)'
+
+ The interpreter evaluates the expressions <E> one at a time, in
+ left-to-right order. If any <E> evaluates to a true value, that
+ value is returned as the value of the `or' expression, and the
+ rest of the <E>'s are not evaluated. If all <E>'s evaluate to
+ false, the value of the `or' expression is false.
+
+ * `(not <E>)'
+
+ The value of a `not' expression is true when the expression <E>
+ evaluates to false, and false otherwise.
+
+
+ Notice that `and' and `or' are special forms, not procedures, because
+the subexpressions are not necessarily all evaluated. `Not' is an
+ordinary procedure.
+
+ As an example of how these are used, the condition that a number x
+be in the range 5 < x < 10 may be expressed as
+
+ (and (> x 5) (< x 10))
+
+ As another example, we can define a predicate to test whether one
+number is greater than or equal to another as
+
+ (define (>= x y)
+ (or (> x y) (= x y)))
+
+or alternatively as
+
+ (define (>= x y)
+ (not (< x y)))
+
+ *Exercise 1.1:* Below is a sequence of expressions. What is the
+ result printed by the interpreter in response to each expression?
+ Assume that the sequence is to be evaluated in the order in which
+ it is presented.
+
+ 10
+
+ (+ 5 3 4)
+
+ (- 9 1)
+
+ (/ 6 2)
+
+ (+ (* 2 4) (- 4 6))
+
+ (define a 3)
+
+ (define b (+ a 1))
+
+ (+ a b (* a b))
+
+ (= a b)
+
+ (if (and (> b a) (< b (* a b)))
+ b
+ a)
+
+ (cond ((= a 4) 6)
+ ((= b 4) (+ 6 7 a))
+ (else 25))
+
+ (+ 2 (if (> b a) b a))
+
+ (* (cond ((> a b) a)
+ ((< a b) b)
+ (else -1))
+ (+ a 1))
+
+ *Exercise 1.2:* Translate the following expression into prefix
+ form.
+
+ 5 + 4 + (2 - (3 - (6 + 4/5)))
+ -----------------------------
+ 3(6 - 2)(2 - 7)
+
+ *Exercise 1.3:* Define a procedure that takes three numbers as
+ arguments and returns the sum of the squares of the two larger
+ numbers.
+
+ *Exercise 1.4:* Observe that our model of evaluation allows for
+ combinations whose operators are compound expressions. Use this
+ observation to describe the behavior of the following procedure:
+
+ (define (a-plus-abs-b a b)
+ ((if (> b 0) + -) a b))
+
+ *Exercise 1.5:* Ben Bitdiddle has invented a test to determine
+ whether the interpreter he is faced with is using
+ applicative-order evaluation or normal-order evaluation. He
+ defines the following two procedures:
+
+ (define (p) (p))
+
+ (define (test x y)
+ (if (= x 0)
+ 0
+ y))
+
+ Then he evaluates the expression
+
+ (test 0 (p))
+
+ What behavior will Ben observe with an interpreter that uses
+ applicative-order evaluation? What behavior will he observe with
+ an interpreter that uses normal-order evaluation? Explain your
+ answer. (Assume that the evaluation rule for the special form
+ `if' is the same whether the interpreter is using normal or
+ applicative order: The predicate expression is evaluated first,
+ and the result determines whether to evaluate the consequent or
+ the alternative expression.)
+
+ ---------- Footnotes ----------
+
+ (1) "Interpreted as either true or false" means this: In Scheme,
+there are two distinguished values that are denoted by the constants
+`#t' and `#f'. When the interpreter checks a predicate's value, it
+interprets `#f' as false. Any other value is treated as true. (Thus,
+providing `#t' is logically unnecessary, but it is convenient.) In
+this book we will use names `true' and `false', which are associated
+with the values `#t' and `#f' respectively.
+
+ (2) `Abs' also uses the "minus" operator `-', which, when used with
+a single operand, as in `(- x)', indicates negation.
+
+ (3) A minor difference between `if' and `cond' is that the <E> part
+of each `cond' clause may be a sequence of expressions. If the
+corresponding <P> is found to be true, the expressions <E> are
+evaluated in sequence and the value of the final expression in the
+sequence is returned as the value of the `cond'. In an `if'
+expression, however, the <CONSEQUENT> and <ALTERNATIVE> must be single
+expressions.
+
+\1f
+File: sicp.info, Node: 1-1-7, Next: 1-1-8, Prev: 1-1-6, Up: 1-1
+
+1.1.7 Example: Square Roots by Newton's Method
+----------------------------------------------
+
+Procedures, as introduced above, are much like ordinary mathematical
+functions. They specify a value that is determined by one or more
+parameters. But there is an important difference between mathematical
+functions and computer procedures. Procedures must be effective.
+
+ As a case in point, consider the problem of computing square roots.
+We can define the square-root function as
+
+ sqrt(x) = the y such that y >= 0 and y^2 = x
+
+ This describes a perfectly legitimate mathematical function. We
+could use it to recognize whether one number is the square root of
+another, or to derive facts about square roots in general. On the
+other hand, the definition does not describe a procedure. Indeed, it
+tells us almost nothing about how to actually find the square root of a
+given number. It will not help matters to rephrase this definition in
+pseudo-Lisp:
+
+ (define (sqrt x)
+ (the y (and (>= y 0)
+ (= (square y) x))))
+
+ This only begs the question.
+
+ The contrast between function and procedure is a reflection of the
+general distinction between describing properties of things and
+describing how to do things, or, as it is sometimes referred to, the
+distinction between declarative knowledge and imperative knowledge. In
+mathematics we are usually concerned with declarative (what is)
+descriptions, whereas in computer science we are usually concerned with
+imperative (how to) descriptions.(1)
+
+ How does one compute square roots? The most common way is to use
+Newton's method of successive approximations, which says that whenever
+we have a guess y for the value of the square root of a number x, we
+can perform a simple manipulation to get a better guess (one closer to
+the actual square root) by averaging y with x/y.(2) For example, we can
+compute the square root of 2 as follows. Suppose our initial guess is
+1:
+
+ Guess Quotient Average
+ 1 (2/1) = 2 ((2 + 1)/2) = 1.5
+ 1.5 (2/1.5) = 1.3333 ((1.3333 + 1.5)/2) = 1.4167
+ 1.4167 (2/1.4167) = 1.4118 ((1.4167 + 1.4118)/2) = 1.4142
+ 1.4142 ... ...
+
+Continuing this process, we obtain better and better approximations to
+the square root.
+
+ Now let's formalize the process in terms of procedures. We start
+with a value for the radicand (the number whose square root we are
+trying to compute) and a value for the guess. If the guess is good
+enough for our purposes, we are done; if not, we must repeat the
+process with an improved guess. We write this basic strategy as a
+procedure:
+
+ (define (sqrt-iter guess x)
+ (if (good-enough? guess x)
+ guess
+ (sqrt-iter (improve guess x)
+ x)))
+
+ A guess is improved by averaging it with the quotient of the
+radicand and the old guess:
+
+ (define (improve guess x)
+ (average guess (/ x guess)))
+
+where
+
+ (define (average x y)
+ (/ (+ x y) 2))
+
+ We also have to say what we mean by "good enough." The following
+will do for illustration, but it is not really a very good test. (See
+exercise *Note Exercise 1-7::.) The idea is to improve the answer
+until it is close enough so that its square differs from the radicand
+by less than a predetermined tolerance (here 0.001):(3)
+
+ (define (good-enough? guess x)
+ (< (abs (- (square guess) x)) 0.001))
+
+ Finally, we need a way to get started. For instance, we can always
+guess that the square root of any number is 1:(4)
+
+ (define (sqrt x)
+ (sqrt-iter 1.0 x))
+
+ If we type these definitions to the interpreter, we can use `sqrt'
+just as we can use any procedure:
+
+ (sqrt 9)
+ 3.00009155413138
+
+ (sqrt (+ 100 37))
+ 11.704699917758145
+
+ (sqrt (+ (sqrt 2) (sqrt 3)))
+ 1.7739279023207892
+
+ (square (sqrt 1000))
+ 1000.000369924366
+
+ The `sqrt' program also illustrates that the simple procedural
+language we have introduced so far is sufficient for writing any purely
+numerical program that one could write in, say, C or Pascal. This
+might seem surprising, since we have not included in our language any
+iterative (looping) constructs that direct the computer to do something
+over and over again. `Sqrt-iter', on the other hand, demonstrates how
+iteration can be accomplished using no special construct other than the
+ordinary ability to call a procedure.(5)
+
+ *Exercise 1.6:* Alyssa P. Hacker doesn't see why `if' needs to be
+ provided as a special form. "Why can't I just define it as an
+ ordinary procedure in terms of `cond'?" she asks. Alyssa's friend
+ Eva Lu Ator claims this can indeed be done, and she defines a new
+ version of `if':
+
+ (define (new-if predicate then-clause else-clause)
+ (cond (predicate then-clause)
+ (else else-clause)))
+
+ Eva demonstrates the program for Alyssa:
+
+ (new-if (= 2 3) 0 5)
+ 5
+
+ (new-if (= 1 1) 0 5)
+ 0
+
+ Delighted, Alyssa uses `new-if' to rewrite the square-root program:
+
+ (define (sqrt-iter guess x)
+ (new-if (good-enough? guess x)
+ guess
+ (sqrt-iter (improve guess x)
+ x)))
+
+ What happens when Alyssa attempts to use this to compute square
+ roots? Explain.
+
+ *Exercise 1.7:* The `good-enough?' test used in computing square
+ roots will not be very effective for finding the square roots of
+ very small numbers. Also, in real computers, arithmetic operations
+ are almost always performed with limited precision. This makes
+ our test inadequate for very large numbers. Explain these
+ statements, with examples showing how the test fails for small and
+ large numbers. An alternative strategy for implementing
+ `good-enough?' is to watch how `guess' changes from one iteration
+ to the next and to stop when the change is a very small fraction
+ of the guess. Design a square-root procedure that uses this kind
+ of end test. Does this work better for small and large numbers?
+
+ *Exercise 1.8:* Newton's method for cube roots is based on the
+ fact that if y is an approximation to the cube root of x, then a
+ better approximation is given by the value
+
+ x/y^2 + 2y
+ ----------
+ 3
+
+ Use this formula to implement a cube-root procedure analogous to
+ the square-root procedure. (In section *Note 1-3-4:: we will see
+ how to implement Newton's method in general as an abstraction of
+ these square-root and cube-root procedures.)
+
+ ---------- Footnotes ----------
+
+ (1) Declarative and imperative descriptions are intimately related,
+as indeed are mathematics and computer science. For instance, to say
+that the answer produced by a program is "correct" is to make a
+declarative statement about the program. There is a large amount of
+research aimed at establishing techniques for proving that programs are
+correct, and much of the technical difficulty of this subject has to do
+with negotiating the transition between imperative statements (from
+which programs are constructed) and declarative statements (which can be
+used to deduce things). In a related vein, an important current area in
+programming-language design is the exploration of so-called very
+high-level languages, in which one actually programs in terms of
+declarative statements. The idea is to make interpreters sophisticated
+enough so that, given "what is" knowledge specified by the programmer,
+they can generate "how to" knowledge automatically. This cannot be
+done in general, but there are important areas where progress has been
+made. We shall revisit this idea in *Note Chapter 4::.
+
+ (2) This square-root algorithm is actually a special case of
+Newton's method, which is a general technique for finding roots of
+equations. The square-root algorithm itself was developed by Heron of
+Alexandria in the first century A.D. We will see how to express the
+general Newton's method as a Lisp procedure in section *Note 1-3-4::.
+
+ (3) We will usually give predicates names ending with question
+marks, to help us remember that they are predicates. This is just a
+stylistic convention. As far as the interpreter is concerned, the
+question mark is just an ordinary character.
+
+ (4) Observe that we express our initial guess as 1.0 rather than 1.
+This would not make any difference in many Lisp implementations. MIT
+Scheme, however, distinguishes between exact integers and decimal
+values, and dividing two integers produces a rational number rather
+than a decimal. For example, dividing 10 by 6 yields 5/3, while
+dividing 10.0 by 6.0 yields 1.6666666666666667. (We will learn how to
+implement arithmetic on rational numbers in section *Note 2-1-1::.) If
+we start with an initial guess of 1 in our square-root program, and x
+is an exact integer, all subsequent values produced in the square-root
+computation will be rational numbers rather than decimals. Mixed
+operations on rational numbers and decimals always yield decimals, so
+starting with an initial guess of 1.0 forces all subsequent values to
+be decimals.
+
+ (5) Readers who are worried about the efficiency issues involved in
+using procedure calls to implement iteration should note the remarks on
+"tail recursion" in section *Note 1-2-1::.
+
+\1f
+File: sicp.info, Node: 1-1-8, Prev: 1-1-7, Up: 1-1
+
+1.1.8 Procedures as Black-Box Abstractions
+------------------------------------------
+
+`Sqrt' is our first example of a process defined by a set of mutually
+defined procedures. Notice that the definition of `sqrt-iter' is "recursive";
+that is, the procedure is defined in terms of itself. The idea of
+being able to define a procedure in terms of itself may be disturbing;
+it may seem unclear how such a "circular" definition could make sense
+at all, much less specify a well-defined process to be carried out by a
+computer. This will be addressed more carefully in section *Note
+1-2::. But first let's consider some other important points
+illustrated by the `sqrt' example.
+
+ Observe that the problem of computing square roots breaks up
+naturally into a number of subproblems: how to tell whether a guess is
+good enough, how to improve a guess, and so on. Each of these tasks is
+accomplished by a separate procedure. The entire `sqrt' program can be
+viewed as a cluster of procedures (shown in *Note Figure 1-2::) that
+mirrors the decomposition of the problem into subproblems.
+
+ *Figure 1.2:* Procedural decomposition of the `sqrt' program.
+
+ sqrt
+ |
+ sqrt-iter
+ / \
+ good-enough improve
+ / \ |
+ square abs average
+
+ The importance of this decomposition strategy is not simply that one
+is dividing the program into parts. After all, we could take any large
+program and divide it into parts--the first ten lines, the next ten
+lines, the next ten lines, and so on. Rather, it is crucial that each
+procedure accomplishes an identifiable task that can be used as a
+module in defining other procedures. For example, when we define the
+`good-enough?' procedure in terms of `square', we are able to regard
+the `square' procedure as a "black box." We are not at that moment
+concerned with _how_ the procedure computes its result, only with the
+fact that it computes the square. The details of how the square is
+computed can be suppressed, to be considered at a later time. Indeed,
+as far as the `good-enough?' procedure is concerned, `square' is not
+quite a procedure but rather an abstraction of a procedure, a so-called "procedural
+abstraction". At this level of abstraction, any procedure that
+computes the square is equally good.
+
+ Thus, considering only the values they return, the following two
+procedures for squaring a number should be indistinguishable. Each
+takes a numerical argument and produces the square of that number as
+the value.(1)
+
+ (define (square x) (* x x))
+
+ (define (square x)
+ (exp (double (log x))))
+
+ (define (double x) (+ x x))
+
+ So a procedure definition should be able to suppress detail. The
+users of the procedure may not have written the procedure themselves,
+but may have obtained it from another programmer as a black box. A
+user should not need to know how the procedure is implemented in order
+to use it.
+
+Local names
+...........
+
+One detail of a procedure's implementation that should not matter to
+the user of the procedure is the implementer's choice of names for the
+procedure's formal parameters. Thus, the following procedures should
+not be distinguishable:
+
+ (define (square x) (* x x))
+
+ (define (square y) (* y y))
+
+ This principle--that the meaning of a procedure should be
+independent of the parameter names used by its author--seems on the
+surface to be self-evident, but its consequences are profound. The
+simplest consequence is that the parameter names of a procedure must be
+local to the body of the procedure. For example, we used `square' in
+the definition of `good-enough?' in our square-root procedure:
+
+ (define (good-enough? guess x)
+ (< (abs (- (square guess) x)) 0.001))
+
+ The intention of the author of `good-enough?' is to determine if the
+square of the first argument is within a given tolerance of the second
+argument. We see that the author of `good-enough?' used the name
+`guess' to refer to the first argument and `x' to refer to the second
+argument. The argument of `square' is `guess'. If the author of
+`square' used `x' (as above) to refer to that argument, we see that the
+`x' in `good-enough?' must be a different `x' than the one in `square'.
+Running the procedure `square' must not affect the value of `x' that
+is used by `good-enough?', because that value of `x' may be needed by
+`good-enough?' after `square' is done computing.
+
+ If the parameters were not local to the bodies of their respective
+procedures, then the parameter `x' in `square' could be confused with
+the parameter `x' in `good-enough?', and the behavior of `good-enough?'
+would depend upon which version of `square' we used. Thus, `square'
+would not be the black box we desired.
+
+ A formal parameter of a procedure has a very special role in the
+procedure definition, in that it doesn't matter what name the formal
+parameter has. Such a name is called a "bound variable", and we say
+that the procedure definition "binds" its formal parameters. The
+meaning of a procedure definition is unchanged if a bound variable is
+consistently renamed throughout the definition.(2) If a variable is
+not bound, we say that it is "free". The set of expressions for which
+a binding defines a name is called the "scope" of that name. In a
+procedure definition, the bound variables declared as the formal
+parameters of the procedure have the body of the procedure as their
+scope.
+
+ In the definition of `good-enough?' above, `guess' and `x' are bound
+variables but `<', `-', `abs', and `square' are free. The meaning of
+`good-enough?' should be independent of the names we choose for `guess'
+and `x' so long as they are distinct and different from `<', `-',
+`abs', and `square'. (If we renamed `guess' to `abs' we would have
+introduced a bug by "capturing" the variable `abs'. It would have
+changed from free to bound.) The meaning of `good-enough?' is not
+independent of the names of its free variables, however. It surely
+depends upon the fact (external to this definition) that the symbol
+`abs' names a procedure for computing the absolute value of a number.
+`Good-enough?' will compute a different function if we substitute `cos'
+for `abs' in its definition.
+
+Internal definitions and block structure
+........................................
+
+We have one kind of name isolation available to us so far: The formal
+parameters of a procedure are local to the body of the procedure. The
+square-root program illustrates another way in which we would like to
+control the use of names. The existing program consists of separate
+procedures:
+
+ (define (sqrt x)
+ (sqrt-iter 1.0 x))
+
+ (define (sqrt-iter guess x)
+ (if (good-enough? guess x)
+ guess
+ (sqrt-iter (improve guess x) x)))
+
+ (define (good-enough? guess x)
+ (< (abs (- (square guess) x)) 0.001))
+
+ (define (improve guess x)
+ (average guess (/ x guess)))
+
+ The problem with this program is that the only procedure that is
+important to users of `sqrt' is `sqrt'. The other procedures
+(`sqrt-iter', `good-enough?', and `improve') only clutter up their
+minds. They may not define any other procedure called `good-enough?'
+as part of another program to work together with the square-root
+program, because `sqrt' needs it. The problem is especially severe in
+the construction of large systems by many separate programmers. For
+example, in the construction of a large library of numerical
+procedures, many numerical functions are computed as successive
+approximations and thus might have procedures named `good-enough?' and
+`improve' as auxiliary procedures. We would like to localize the
+subprocedures, hiding them inside `sqrt' so that `sqrt' could coexist
+with other successive approximations, each having its own private
+`good-enough?' procedure. To make this possible, we allow a procedure
+to have internal definitions that are local to that procedure. For
+example, in the square-root problem we can write
+
+ (define (sqrt x)
+ (define (good-enough? guess x)
+ (< (abs (- (square guess) x)) 0.001))
+ (define (improve guess x)
+ (average guess (/ x guess)))
+ (define (sqrt-iter guess x)
+ (if (good-enough? guess x)
+ guess
+ (sqrt-iter (improve guess x) x)))
+ (sqrt-iter 1.0 x))
+
+ Such nesting of definitions, called "block structure", is basically
+the right solution to the simplest name-packaging problem. But there
+is a better idea lurking here. In addition to internalizing the
+definitions of the auxiliary procedures, we can simplify them. Since
+`x' is bound in the definition of `sqrt', the procedures
+`good-enough?', `improve', and `sqrt-iter', which are defined
+internally to `sqrt', are in the scope of `x'. Thus, it is not
+necessary to pass `x' explicitly to each of these procedures. Instead,
+we allow `x' to be a free variable in the internal definitions, as
+shown below. Then `x' gets its value from the argument with which the
+enclosing procedure `sqrt' is called. This discipline is called "lexical
+scoping".(3)
+
+ (define (sqrt x)
+ (define (good-enough? guess)
+ (< (abs (- (square guess) x)) 0.001))
+ (define (improve guess)
+ (average guess (/ x guess)))
+ (define (sqrt-iter guess)
+ (if (good-enough? guess)
+ guess
+ (sqrt-iter (improve guess))))
+ (sqrt-iter 1.0))
+
+ We will use block structure extensively to help us break up large
+programs into tractable pieces.(4) The idea of block structure
+originated with the programming language Algol 60. It appears in most
+advanced programming languages and is an important tool for helping to
+organize the construction of large programs.
+
+ ---------- Footnotes ----------
+
+ (1) It is not even clear which of these procedures is a more
+efficient implementation. This depends upon the hardware available.
+There are machines for which the "obvious" implementation is the less
+efficient one. Consider a machine that has extensive tables of
+logarithms and antilogarithms stored in a very efficient manner.
+
+ (2) The concept of consistent renaming is actually subtle and
+difficult to define formally. Famous logicians have made embarrassing
+errors here.
+
+ (3) [Footnote 28] Lexical scoping dictates that free variables in a
+procedure are taken to refer to bindings made by enclosing procedure
+definitions; that is, they are looked up in the environment in which the
+procedure was defined. We will see how this works in detail in chapter
+3 when we study environments and the detailed behavior of the
+interpreter.
+
+ (4) Embedded definitions must come first in a procedure body. The
+management is not responsible for the consequences of running programs
+that intertwine definition and use.
+
+\1f
+File: sicp.info, Node: 1-2, Next: 1-3, Prev: 1-1, Up: Chapter 1
+
+1.2 Procedures and the Processes They Generate
+==============================================
+
+We have now considered the elements of programming: We have used
+primitive arithmetic operations, we have combined these operations, and
+we have abstracted these composite operations by defining them as
+compound procedures. But that is not enough to enable us to say that
+we know how to program. Our situation is analogous to that of someone
+who has learned the rules for how the pieces move in chess but knows
+nothing of typical openings, tactics, or strategy. Like the novice
+chess player, we don't yet know the common patterns of usage in the
+domain. We lack the knowledge of which moves are worth making (which
+procedures are worth defining). We lack the experience to predict the
+consequences of making a move (executing a procedure).
+
+ The ability to visualize the consequences of the actions under
+consideration is crucial to becoming an expert programmer, just as it
+is in any synthetic, creative activity. In becoming an expert
+photographer, for example, one must learn how to look at a scene and
+know how dark each region will appear on a print for each possible
+choice of exposure and development conditions. Only then can one
+reason backward, planning framing, lighting, exposure, and development
+to obtain the desired effects. So it is with programming, where we are
+planning the course of action to be taken by a process and where we
+control the process by means of a program. To become experts, we must
+learn to visualize the processes generated by various types of
+procedures. Only after we have developed such a skill can we learn to
+reliably construct programs that exhibit the desired behavior.
+
+ A procedure is a pattern for the "local evolution" of a computational
+process. It specifies how each stage of the process is built upon the
+previous stage. We would like to be able to make statements about the
+overall, or "global", behavior of a process whose local evolution has
+been specified by a procedure. This is very difficult to do in
+general, but we can at least try to describe some typical patterns of
+process evolution.
+
+ In this section we will examine some common "shapes" for processes
+generated by simple procedures. We will also investigate the rates at
+which these processes consume the important computational resources of
+time and space. The procedures we will consider are very simple.
+Their role is like that played by test patterns in photography: as
+oversimplified prototypical patterns, rather than practical examples in
+their own right.
+
+* Menu:
+
+* 1-2-1:: Linear Recursion and Iteration
+* 1-2-2:: Tree Recursion
+* 1-2-3:: Orders of Growth
+* 1-2-4:: Exponentiation
+* 1-2-5:: Greatest Common Divisors
+* 1-2-6:: Example: Testing for Primality
+
+\1f
+File: sicp.info, Node: 1-2-1, Next: 1-2-2, Prev: 1-2, Up: 1-2
+
+1.2.1 Linear Recursion and Iteration
+------------------------------------
+
+ *Figure 1.3:* A linear recursive process for computing 6!.
+
+ (factorial 6) ------------------------.
+ (* 6 (factorial 5)) |
+ (* 6 (* 5 (factorial 4))) |
+ (* 6 (* 5 (* 4 (factorial 3)))) |
+ (* 6 (* 5 (* 4 (* 3 (factorial 2))))) |
+ (* 6 (* 5 (* 4 (* 3 (* 2 (factorial 1)))))) |
+ (* 6 (* 5 (* 4 (* 3 (* 2 1))))) |
+ (* 6 (* 5 (* 4 (* 3 2)))) |
+ (* 6 (* 5 (* 4 6))) |
+ (* 6 (* 5 24)) |
+ (* 6 120) |
+ 720 <-------------------------------'
+
+We begin by considering the factorial function, defined by
+
+ n! = n * (n - 1) * (n - 2) ... 3 * 2 * 1
+
+ There are many ways to compute factorials. One way is to make use
+of the observation that n! is equal to n times (n - 1)! for any positive
+integer n:
+
+ n! = n * [(n - 1) * (n - 2) ... 3 * 2 * 1] = n * (n - 1)!
+
+ Thus, we can compute n! by computing (n - 1)! and multiplying the
+result by n. If we add the stipulation that 1! is equal to 1, this
+observation translates directly into a procedure:
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* n (factorial (- n 1)))))
+
+ We can use the substitution model of section *Note 1-1-5:: to watch
+this procedure in action computing 6!, as shown in *Note Figure 1-3::.
+
+ Now let's take a different perspective on computing factorials. We
+could describe a rule for computing n! by specifying that we first
+multiply 1 by 2, then multiply the result by 3, then by 4, and so on
+until we reach n. More formally, we maintain a running product,
+together with a counter that counts from 1 up to n. We can describe
+the computation by saying that the counter and the product
+simultaneously change from one step to the next according to the rule
+
+ product <- counter ... product
+
+ counter <- counter + 1
+
+and stipulating that n! is the value of the product when the counter
+exceeds n.
+
+ *Figure 1.4:* A linear iterative process for computing 6!.
+
+ (factorial 6) -----.
+ (fact-iter 1 1 6) |
+ (fact-iter 1 2 6) |
+ (fact-iter 2 3 6) |
+ (fact-iter 6 4 6) |
+ (fact-iter 24 5 6) |
+ (fact-iter 120 6 6) |
+ (fact-iter 720 7 6) V
+ 720
+
+ Once again, we can recast our description as a procedure for
+computing factorials:(1)
+
+ (define (factorial n)
+ (fact-iter 1 1 n))
+
+ (define (fact-iter product counter max-count)
+ (if (> counter max-count)
+ product
+ (fact-iter (* counter product)
+ (+ counter 1)
+ max-count)))
+
+ As before, we can use the substitution model to visualize the
+process of computing 6!, as shown in *Note Figure 1-4::.
+
+ Compare the two processes. From one point of view, they seem hardly
+different at all. Both compute the same mathematical function on the
+same domain, and each requires a number of steps proportional to n to
+compute n!. Indeed, both processes even carry out the same sequence of
+multiplications, obtaining the same sequence of partial products. On
+the other hand, when we consider the "shapes" of the two processes, we
+find that they evolve quite differently.
+
+ Consider the first process. The substitution model reveals a shape
+of expansion followed by contraction, indicated by the arrow in *Note
+Figure 1-3::. The expansion occurs as the process builds up a chain of operations
+"deferred operations" (in this case, a chain of multiplications). The
+contraction occurs as the operations are actually performed. This type
+of process, characterized by a chain of deferred operations, is called
+a "recursive process". Carrying out this process requires that the
+interpreter keep track of the operations to be performed later on. In
+the computation of n!, the length of the chain of deferred
+multiplications, and hence the amount of information needed to keep
+track of it, grows linearly with n (is proportional to n), just like
+the number of steps. Such a process is called a "linear recursive
+process".
+
+ By contrast, the second process does not grow and shrink. At each
+step, all we need to keep track of, for any n, are the current values
+of the variables `product', `counter', and `max-count'. We call this an "iterative
+process". In general, an iterative process is one whose state can be
+summarized by a fixed number of "state variables", together with a
+fixed rule that describes how the state variables should be updated as
+the process moves from state to state and an (optional) end test that
+specifies conditions under which the process should terminate. In
+computing n!, the number of steps required grows linearly with n. Such
+a process is called a "linear iterative process".
+
+ The contrast between the two processes can be seen in another way.
+In the iterative case, the program variables provide a complete
+description of the state of the process at any point. If we stopped
+the computation between steps, all we would need to do to resume the
+computation is to supply the interpreter with the values of the three
+program variables. Not so with the recursive process. In this case
+there is some additional "hidden" information, maintained by the
+interpreter and not contained in the program variables, which indicates
+"where the process is" in negotiating the chain of deferred operations.
+The longer the chain, the more information must be maintained.(2)
+
+ In contrasting iteration and recursion, we must be careful not to
+confuse the notion of a recursive "process" with the notion of a
+recursive "procedure". When we describe a procedure as recursive, we
+are referring to the syntactic fact that the procedure definition
+refers (either directly or indirectly) to the procedure itself. But
+when we describe a process as following a pattern that is, say,
+linearly recursive, we are speaking about how the process evolves, not
+about the syntax of how a procedure is written. It may seem disturbing
+that we refer to a recursive procedure such as `fact-iter' as
+generating an iterative process. However, the process really is
+iterative: Its state is captured completely by its three state
+variables, and an interpreter need keep track of only three variables
+in order to execute the process.
+
+ One reason that the distinction between process and procedure may be
+confusing is that most implementations of common languages (including
+Ada, Pascal, and C) are designed in such a way that the interpretation
+of any recursive procedure consumes an amount of memory that grows with
+the number of procedure calls, even when the process described is, in
+principle, iterative. As a consequence, these languages can describe
+iterative processes only by resorting to special-purpose "looping
+constructs" such as `do', `repeat', `until', `for', and `while'. The
+implementation of Scheme we shall consider in *Note Chapter 5:: does
+not share this defect. It will execute an iterative process in
+constant space, even if the iterative process is described by a
+recursive procedure. An implementation with this property is called "tail-recursive".
+With a tail-recursive implementation, iteration can be expressed using
+the ordinary procedure call mechanism, so that special iteration
+constructs are useful only as syntactic sugar.(3)
+
+ *Exercise 1.9:* Each of the following two procedures defines a
+ method for adding two positive integers in terms of the procedures
+ `inc', which increments its argument by 1, and `dec', which
+ decrements its argument by 1.
+
+ (define (+ a b)
+ (if (= a 0)
+ b
+ (inc (+ (dec a) b))))
+
+ (define (+ a b)
+ (if (= a 0)
+ b
+ (+ (dec a) (inc b))))
+
+ Using the substitution model, illustrate the process generated by
+ each procedure in evaluating `(+ 4 5)'. Are these processes
+ iterative or recursive?
+
+ *Exercise 1.10:* The following procedure computes a mathematical
+ function called Ackermann's function.
+
+ (define (A x y)
+ (cond ((= y 0) 0)
+ ((= x 0) (* 2 y))
+ ((= y 1) 2)
+ (else (A (- x 1)
+ (A x (- y 1))))))
+
+ What are the values of the following expressions?
+
+ (A 1 10)
+
+ (A 2 4)
+
+ (A 3 3)
+
+ Consider the following procedures, where `A' is the procedure
+ defined above:
+
+ (define (f n) (A 0 n))
+
+ (define (g n) (A 1 n))
+
+ (define (h n) (A 2 n))
+
+ (define (k n) (* 5 n n))
+
+ Give concise mathematical definitions for the functions computed
+ by the procedures `f', `g', and `h' for positive integer values of
+ n. For example, `(k n)' computes 5n^2.
+
+ ---------- Footnotes ----------
+
+ (1) In a real program we would probably use the block structure
+introduced in the last section to hide the definition of `fact-iter':
+
+ (define (factorial n)
+ (define (iter product counter)
+ (if (> counter n)
+ product
+ (iter (* counter product)
+ (+ counter 1))))
+ (iter 1 1))
+
+ We avoided doing this here so as to minimize the number of things to
+think about at once.
+
+ (2) When we discuss the implementation of procedures on register
+machines in *Note Chapter 5::, we will see that any iterative process
+can be realized "in hardware" as a machine that has a fixed set of
+registers and no auxiliary memory. In contrast, realizing a recursive
+process requires a machine that uses an auxiliary data structure known
+as a "stack".
+
+ (3) Tail recursion has long been known as a compiler optimization
+trick. A coherent semantic basis for tail recursion was provided by
+Carl Hewitt (1977), who explained it in terms of the "message-passing"
+model of computation that we shall discuss in *Note Chapter 3::.
+Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see
+Steele 1975) constructed a tail-recursive interpreter for Scheme.
+Steele later showed how tail recursion is a consequence of the natural
+way to compile procedure calls (Steele 1977). The IEEE standard for
+Scheme requires that Scheme implementations be tail-recursive.
+
+\1f
+File: sicp.info, Node: 1-2-2, Next: 1-2-3, Prev: 1-2-1, Up: 1-2
+
+1.2.2 Tree Recursion
+--------------------
+
+Another common pattern of computation is called "tree recursion". As
+an example, consider computing the sequence of Fibonacci numbers, in
+which each number is the sum of the preceding two:
+
+ 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
+
+ In general, the Fibonacci numbers can be defined by the rule
+
+ /
+ | 0 if n = 0
+ Fib(n) = < 1 if n = 1
+ | Fib(n - 1) + Fib(n - 2) otherwise
+ \
+
+ We can immediately translate this definition into a recursive
+procedure for computing Fibonacci numbers:
+
+ (define (fib n)
+ (cond ((= n 0) 0)
+ ((= n 1) 1)
+ (else (+ (fib (- n 1))
+ (fib (- n 2))))))
+
+ *Figure 1.5:* The tree-recursive process generated in computing
+ `(fib 5)'.
+
+ ..<............ fib5 <..........
+ ... ___________/ \___________ .
+ ... / . ..... \ .
+ .. fib4 . . . . . fib3 .
+ .. ____/. \____ .. . __/ \__ .
+ .. / . . .. \ . .. / . . \ .
+ .. fib3 . . fib2 . . fib2 . . fib1 .
+ .. / . \ . . / \ . . / \ ... . | .
+ .. / . . \ . . / . \ . . / . \ . . 1 .
+ . fib2 . . fib1. .fib1 . fib0 . .fib1. . fib0 . . .
+ . / \ . . | . . | . . | . . | . . | . .>
+ V / . \ . 1 . . 1 . . 0 . . 1 . . 0 ..
+ . fib1 .. fib0.. . . . . . V . .. .
+ . | . . | . .> .>. . . ..>. .>
+ . 1 . . 0 .
+ . . . .
+ .>. ..
+
+ Consider the pattern of this computation. To compute `(fib 5)', we
+compute `(fib 4)' and `(fib 3)'. To compute `(fib 4)', we compute
+`(fib 3)' and `(fib 2)'. In general, the evolved process looks like a
+tree, as shown in *Note Figure 1-5::. Notice that the branches split
+into two at each level (except at the bottom); this reflects the fact
+that the `fib' procedure calls itself twice each time it is invoked.
+
+ This procedure is instructive as a prototypical tree recursion, but
+it is a terrible way to compute Fibonacci numbers because it does so
+much redundant computation. Notice in *Note Figure 1-5:: that the
+entire computation of `(fib 3)'--almost half the work--is duplicated.
+In fact, it is not hard to show that the number of times the procedure
+will compute `(fib 1)' or `(fib 0)' (the number of leaves in the above
+tree, in general) is precisely _Fib_(n + 1). To get an idea of how bad
+this is, one can show that the value of _Fib_(n) grows exponentially
+with n. More precisely (see *Note Exercise 1-13::), _Fib_(n) is the
+closest integer to [phi]^n /[sqrt](5), where
+
+ [phi] = (1 + [sqrt]5)/2 ~= 1.6180
+
+is the "golden ratio", which satisfies the equation
+
+ [phi]^2 = [phi] + 1
+
+ Thus, the process uses a number of steps that grows exponentially
+with the input. On the other hand, the space required grows only
+linearly with the input, because we need keep track only of which nodes
+are above us in the tree at any point in the computation. In general,
+the number of steps required by a tree-recursive process will be
+proportional to the number of nodes in the tree, while the space
+required will be proportional to the maximum depth of the tree.
+
+ We can also formulate an iterative process for computing the
+Fibonacci numbers. The idea is to use a pair of integers a and b,
+initialized to _Fib_(1) = 1 and _Fib_(0) = 0, and to repeatedly apply
+the simultaneous transformations
+
+ a <- a + b
+ b <- a
+
+It is not hard to show that, after applying this transformation n times,
+a and b will be equal, respectively, to _Fib_(n + 1) and _Fib_(n).
+Thus, we can compute Fibonacci numbers iteratively using the procedure
+
+ (define (fib n)
+ (fib-iter 1 0 n))
+
+ (define (fib-iter a b count)
+ (if (= count 0)
+ b
+ (fib-iter (+ a b) a (- count 1))))
+
+ This second method for computing _Fib_(n) is a linear iteration. The
+difference in number of steps required by the two methods--one linear in
+n, one growing as fast as _Fib_(n) itself--is enormous, even for small
+inputs.
+
+ One should not conclude from this that tree-recursive processes are
+useless. When we consider processes that operate on hierarchically
+structured data rather than numbers, we will find that tree recursion
+is a natural and powerful tool.(1) But even in numerical operations,
+tree-recursive processes can be useful in helping us to understand and
+design programs. For instance, although the first `fib' procedure is
+much less efficient than the second one, it is more straightforward,
+being little more than a translation into Lisp of the definition of the
+Fibonacci sequence. To formulate the iterative algorithm required
+noticing that the computation could be recast as an iteration with
+three state variables.
+
+Example: Counting change
+........................
+
+It takes only a bit of cleverness to come up with the iterative
+Fibonacci algorithm. In contrast, consider the following problem: How
+many different ways can we make change of $ 1.00, given half-dollars,
+quarters, dimes, nickels, and pennies? More generally, can we write a
+procedure to compute the number of ways to change any given amount of
+money?
+
+ This problem has a simple solution as a recursive procedure.
+Suppose we think of the types of coins available as arranged in some
+order. Then the following relation holds:
+
+ The number of ways to change amount a using n kinds of coins equals
+
+ * the number of ways to change amount a using all but the first kind
+ of coin, plus
+
+ * the number of ways to change amount a - d using all n kinds of
+ coins, where d is the denomination of the first kind of coin.
+
+
+ To see why this is true, observe that the ways to make change can be
+divided into two groups: those that do not use any of the first kind of
+coin, and those that do. Therefore, the total number of ways to make
+change for some amount is equal to the number of ways to make change
+for the amount without using any of the first kind of coin, plus the
+number of ways to make change assuming that we do use the first kind of
+coin. But the latter number is equal to the number of ways to make
+change for the amount that remains after using a coin of the first kind.
+
+ Thus, we can recursively reduce the problem of changing a given
+amount to the problem of changing smaller amounts using fewer kinds of
+coins. Consider this reduction rule carefully, and convince yourself
+that we can use it to describe an algorithm if we specify the following
+degenerate cases:(2)
+
+ * If a is exactly 0, we should count that as 1 way to make change.
+
+ * If a is less than 0, we should count that as 0 ways to make change.
+
+ * If n is 0, we should count that as 0 ways to make change.
+
+
+ We can easily translate this description into a recursive procedure:
+
+ (define (count-change amount)
+ (cc amount 5))
+
+ (define (cc amount kinds-of-coins)
+ (cond ((= amount 0) 1)
+ ((or (< amount 0) (= kinds-of-coins 0)) 0)
+ (else (+ (cc amount
+ (- kinds-of-coins 1))
+ (cc (- amount
+ (first-denomination kinds-of-coins))
+ kinds-of-coins)))))
+
+ (define (first-denomination kinds-of-coins)
+ (cond ((= kinds-of-coins 1) 1)
+ ((= kinds-of-coins 2) 5)
+ ((= kinds-of-coins 3) 10)
+ ((= kinds-of-coins 4) 25)
+ ((= kinds-of-coins 5) 50)))
+
+ (The `first-denomination' procedure takes as input the number of
+kinds of coins available and returns the denomination of the first
+kind. Here we are thinking of the coins as arranged in order from
+largest to smallest, but any order would do as well.) We can now
+answer our original question about changing a dollar:
+
+ (count-change 100)
+ 292
+
+ `Count-change' generates a tree-recursive process with redundancies
+similar to those in our first implementation of `fib'. (It will take
+quite a while for that 292 to be computed.) On the other hand, it is
+not obvious how to design a better algorithm for computing the result,
+and we leave this problem as a challenge. The observation that a
+tree-recursive process may be highly inefficient but often easy to
+specify and understand has led people to propose that one could get the
+best of both worlds by designing a "smart compiler" that could
+transform tree-recursive procedures into more efficient procedures that
+compute the same result.(3)
+
+ *Exercise 1.11:* A function f is defined by the rule that f(n) = n
+ if n<3 and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n>= 3.
+ Write a procedure that computes f by means of a recursive process.
+ Write a procedure that computes f by means of an iterative
+ process.
+
+ *Exercise 1.12:* The following pattern of numbers is called "Pascal's
+ triangle".
+
+ 1
+ 1 1
+ 1 2 1
+ 1 3 3 1
+ 1 4 6 4 1
+
+ The numbers at the edge of the triangle are all 1, and each number
+ inside the triangle is the sum of the two numbers above it.(4)
+ Write a procedure that computes elements of Pascal's triangle by
+ means of a recursive process.
+
+ *Exercise 1.13:* Prove that _Fib_(n) is the closest integer to
+ [phi]^n/[sqrt](5), where [phi] = (1 + [sqrt](5))/2. Hint: Let
+ [illegiblesymbol] = (1 - [sqrt](5))/2. Use induction and the
+ definition of the Fibonacci numbers (see section *Note 1-2-2::) to
+ prove that _Fib_(n) = ([phi]^n - [illegiblesymbol]^n)/[sqrt](5).
+
+ ---------- Footnotes ----------
+
+ (1) An example of this was hinted at in section *Note 1-1-3::: The
+interpreter itself evaluates expressions using a tree-recursive process.
+
+ (2) For example, work through in detail how the reduction rule
+applies to the problem of making change for 10 cents using pennies and
+nickels.
+
+ (3) One approach to coping with redundant computations is to arrange
+matters so that we automatically construct a table of values as they
+are computed. Each time we are asked to apply the procedure to some
+argument, we first look to see if the value is already stored in the
+table, in which case we avoid performing the redundant computation.
+This strategy, known as "tabulation" or "memoization", can be
+implemented in a straightforward way. Tabulation can sometimes be used
+to transform processes that require an exponential number of steps
+(such as `count-change') into processes whose space and time
+requirements grow linearly with the input. See *Note Exercise 3-27::.
+
+ (4) The elements of Pascal's triangle are called the "binomial
+coefficients", because the nth row consists of the coefficients of the
+terms in the expansion of (x + y)^n. This pattern for computing the
+coefficients appeared in Blaise Pascal's 1653 seminal work on
+probability theory, `Traite' du triangle arithme'tique'. According to
+Knuth (1973), the same pattern appears in the `Szu-yuen Yu"-chien'
+("The Precious Mirror of the Four Elements"), published by the Chinese
+mathematician Chu Shih-chieh in 1303, in the works of the
+twelfth-century Persian poet and mathematician Omar Khayyam, and in the
+works of the twelfth-century Hindu mathematician Bha'scara A'cha'rya.
+
+\1f
+File: sicp.info, Node: 1-2-3, Next: 1-2-4, Prev: 1-2-2, Up: 1-2
+
+1.2.3 Orders of Growth
+----------------------
+
+The previous examples illustrate that processes can differ considerably
+in the rates at which they consume computational resources. One
+convenient way to describe this difference is to use the notion of "order
+of growth" to obtain a gross measure of the resources required by a
+process as the inputs become larger.
+
+ Let n be a parameter that measures the size of the problem, and let
+R(n) be the amount of resources the process requires for a problem of
+size n. In our previous examples we took n to be the number for which
+a given function is to be computed, but there are other possibilities.
+For instance, if our goal is to compute an approximation to the square
+root of a number, we might take n to be the number of digits accuracy
+required. For matrix multiplication we might take n to be the number
+of rows in the matrices. In general there are a number of properties
+of the problem with respect to which it will be desirable to analyze a
+given process. Similarly, R(n) might measure the number of internal
+storage registers used, the number of elementary machine operations
+performed, and so on. In computers that do only a fixed number of
+operations at a time, the time required will be proportional to the
+number of elementary machine operations performed.
+
+ We say that R(n) has order of growth [theta](f(n)), written R(n) =
+[theta](f(n)) (pronounced "theta of f(n)"), if there are positive
+constants k_1 and k_2 independent of n such that
+
+ k_1 f(n) <= R(n) <= k_2 f(n)
+
+for any sufficiently large value of n. (In other words, for large n,
+the value R(n) is sandwiched between k_1f(n) and k_2f(n).)
+
+ For instance, with the linear recursive process for computing
+factorial described in section *Note 1-2-1:: the number of steps grows
+proportionally to the input n. Thus, the steps required for this
+process grows as [theta](n). We also saw that the space required grows
+as [theta](n). For the iterative factorial, the number of steps is
+still [theta](n) but the space is [theta](1)--that is, constant.(1) The
+tree-recursive Fibonacci computation requires [theta]([phi]^n) steps
+and space [theta](n), where [phi] is the golden ratio described in
+section *Note 1-2-2::.
+
+ Orders of growth provide only a crude description of the behavior of
+a process. For example, a process requiring n^2 steps and a process
+requiring 1000n^2 steps and a process requiring 3n^2 + 10n + 17 steps
+all have [theta](n^2) order of growth. On the other hand, order of
+growth provides a useful indication of how we may expect the behavior
+of the process to change as we change the size of the problem. For a
+[theta](n) (linear) process, doubling the size will roughly double the
+amount of resources used. For an exponential process, each increment
+in problem size will multiply the resource utilization by a constant
+factor. In the remainder of section *Note 1-2:: we will examine two
+algorithms whose order of growth is logarithmic, so that doubling the
+problem size increases the resource requirement by a constant amount.
+
+ *Exercise 1.14:* Draw the tree illustrating the process generated
+ by the `count-change' procedure of section *Note 1-2-2:: in making
+ change for 11 cents. What are the orders of growth of the space
+ and number of steps used by this process as the amount to be
+ changed increases?
+
+ *Exercise 1.15:* The sine of an angle (specified in radians) can
+ be computed by making use of the approximation `sin' xapprox x if
+ x is sufficiently small, and the trigonometric identity
+
+ x x
+ sin x = 3 sin --- - 4 sin^3 ---
+ 3 3
+
+ to reduce the size of the argument of `sin'. (For purposes of this
+ exercise an angle is considered "sufficiently small" if its
+ magnitude is not greater than 0.1 radians.) These ideas are
+ incorporated in the following procedures:
+
+ (define (cube x) (* x x x))
+
+ (define (p x) (- (* 3 x) (* 4 (cube x))))
+
+ (define (sine angle)
+ (if (not (> (abs angle) 0.1))
+ angle
+ (p (sine (/ angle 3.0)))))
+
+ a. How many times is the procedure `p' applied when `(sine
+ 12.15)' is evaluated?
+
+ b. What is the order of growth in space and number of steps (as
+ a function of a) used by the process generated by the `sine'
+ procedure when `(sine a)' is evaluated?
+
+
+ ---------- Footnotes ----------
+
+ (1) These statements mask a great deal of oversimplification. For
+instance, if we count process steps as "machine operations" we are
+making the assumption that the number of machine operations needed to
+perform, say, a multiplication is independent of the size of the
+numbers to be multiplied, which is false if the numbers are
+sufficiently large. Similar remarks hold for the estimates of space.
+Like the design and description of a process, the analysis of a process
+can be carried out at various levels of abstraction.
+
+\1f
+File: sicp.info, Node: 1-2-4, Next: 1-2-5, Prev: 1-2-3, Up: 1-2
+
+1.2.4 Exponentiation
+--------------------
+
+Consider the problem of computing the exponential of a given number.
+We would like a procedure that takes as arguments a base b and a
+positive integer exponent n and computes b^n. One way to do this is
+via the recursive definition
+
+ b^n = b * b^(n - 1)
+ b^0 = 1
+
+which translates readily into the procedure
+
+ (define (expt b n)
+ (if (= n 0)
+ 1
+ (* b (expt b (- n 1)))))
+
+ This is a linear recursive process, which requires [theta](n) steps
+and [theta](n) space. Just as with factorial, we can readily formulate
+an equivalent linear iteration:
+
+ (define (expt b n)
+ (expt-iter b n 1))
+
+ (define (expt-iter b counter product)
+ (if (= counter 0)
+ product
+ (expt-iter b
+ (- counter 1)
+ (* b product))))
+
+ This version requires [theta](n) steps and [theta](1) space.
+
+ We can compute exponentials in fewer steps by using successive
+squaring. For instance, rather than computing b^8 as
+
+ b * (b * (b * (b * (b * (b * (b * b))))))
+
+we can compute it using three multiplications:
+
+ b^2 = b * b
+ b^4 = b^2 * b^2
+ b^8 = b^4 * b^4
+
+ This method works fine for exponents that are powers of 2. We can
+also take advantage of successive squaring in computing exponentials in
+general if we use the rule
+
+ b^n = (b^(b/2))^2 if n is even
+ b^n = b * b^(n - 1) if n is odd
+
+ We can express this method as a procedure:
+
+ (define (fast-expt b n)
+ (cond ((= n 0) 1)
+ ((even? n) (square (fast-expt b (/ n 2))))
+ (else (* b (fast-expt b (- n 1))))))
+
+where the predicate to test whether an integer is even is defined in
+terms of the primitive procedure `remainder' by
+
+ (define (even? n)
+ (= (remainder n 2) 0))
+
+ The process evolved by `fast-expt' grows logarithmically with n in
+both space and number of steps. To see this, observe that computing
+b^(2n) using `fast-expt' requires only one more multiplication than
+computing b^n. The size of the exponent we can compute therefore
+doubles (approximately) with every new multiplication we are allowed.
+Thus, the number of multiplications required for an exponent of n grows
+about as fast as the logarithm of n to the base 2. The process has
+[theta](`log' n) growth.(1)
+
+ The difference between [theta](`log' n) growth and [theta](n) growth
+becomes striking as n becomes large. For example, `fast-expt' for n =
+1000 requires only 14 multiplications.(2) It is also possible to use
+the idea of successive squaring to devise an iterative algorithm that
+computes exponentials with a logarithmic number of steps (see *Note
+Exercise 1-16::), although, as is often the case with iterative
+algorithms, this is not written down so straightforwardly as the
+recursive algorithm.(3)
+
+ *Exercise 1.16:* Design a procedure that evolves an iterative
+ exponentiation process that uses successive squaring and uses a
+ logarithmic number of steps, as does `fast-expt'. (Hint: Using the
+ observation that (b^(n/2))^2 = (b^2)^(n/2), keep, along with the
+ exponent n and the base b, an additional state variable a, and
+ define the state transformation in such a way that the product a
+ b^n is unchanged from state to state. At the beginning of the
+ process a is taken to be 1, and the answer is given by the value
+ of a at the end of the process. In general, the technique of
+ defining an "invariant quantity" that remains unchanged from state
+ to state is a powerful way to think about the design of iterative
+ algorithms.)
+
+ *Exercise 1.17:* The exponentiation algorithms in this section are
+ based on performing exponentiation by means of repeated
+ multiplication. In a similar way, one can perform integer
+ multiplication by means of repeated addition. The following
+ multiplication procedure (in which it is assumed that our language
+ can only add, not multiply) is analogous to the `expt' procedure:
+
+ (define (* a b)
+ (if (= b 0)
+ 0
+ (+ a (* a (- b 1)))))
+
+ This algorithm takes a number of steps that is linear in `b'. Now
+ suppose we include, together with addition, operations `double',
+ which doubles an integer, and `halve', which divides an (even)
+ integer by 2. Using these, design a multiplication procedure
+ analogous to `fast-expt' that uses a logarithmic number of steps.
+
+ *Exercise 1.18:* Using the results of *Note Exercise 1-16:: and
+ *Note Exercise 1-17::, devise a procedure that generates an
+ iterative process for multiplying two integers in terms of adding,
+ doubling, and halving and uses a logarithmic number of steps.(4)
+
+ *Exercise 1.19:* There is a clever algorithm for computing the
+ Fibonacci numbers in a logarithmic number of steps. Recall the
+ transformation of the state variables a and b in the `fib-iter'
+ process of section *Note 1-2-2::: a <- a + b and b <- a. Call
+ this transformation T, and observe that applying T over and over
+ again n times, starting with 1 and 0, produces the pair _Fib_(n +
+ 1) and _Fib_(n). In other words, the Fibonacci numbers are
+ produced by applying T^n, the nth power of the transformation T,
+ starting with the pair (1,0). Now consider T to be the special
+ case of p = 0 and q = 1 in a family of transformations T_(pq),
+ where T_(pq) transforms the pair (a,b) according to a <- bq + aq +
+ ap and b <- bp + aq. Show that if we apply such a transformation
+ T_(pq) twice, the effect is the same as using a single
+ transformation T_(p'q') of the same form, and compute p' and q' in
+ terms of p and q. This gives us an explicit way to square these
+ transformations, and thus we can compute T^n using successive
+ squaring, as in the `fast-expt' procedure. Put this all together
+ to complete the following procedure, which runs in a logarithmic
+ number of steps:(5)
+
+ (define (fib n)
+ (fib-iter 1 0 0 1 n))
+
+ (define (fib-iter a b p q count)
+ (cond ((= count 0) b)
+ ((even? count)
+ (fib-iter a
+ b
+ <??> ; compute p'
+ <??> ; compute q'
+ (/ count 2)))
+ (else (fib-iter (+ (* b q) (* a q) (* a p))
+ (+ (* b p) (* a q))
+ p
+ q
+ (- count 1)))))
+
+ ---------- Footnotes ----------
+
+ (1) More precisely, the number of multiplications required is equal
+to 1 less than the log base 2 of n plus the number of ones in the
+binary representation of n. This total is always less than twice the
+log base 2 of n. The arbitrary constants k_1 and k_2 in the definition
+of order notation imply that, for a logarithmic process, the base to
+which logarithms are taken does not matter, so all such processes are
+described as [theta](`log' n).
+
+ (2) You may wonder why anyone would care about raising numbers to
+the 1000th power. See section *Note 1-2-6::.
+
+ (3) This iterative algorithm is ancient. It appears in the
+`Chandah-sutra' by A'cha'rya Pingala, written before 200 B.C. See Knuth
+1981, section 4.6.3, for a full discussion and analysis of this and
+other methods of exponentiation.
+
+ (4) This algorithm, which is sometimes known as the "Russian peasant
+method" of multiplication, is ancient. Examples of its use are found
+in the Rhind Papyrus, one of the two oldest mathematical documents in
+existence, written about 1700 B.C. (and copied from an even older
+document) by an Egyptian scribe named A'h-mose.
+
+ (5) This exercise was suggested to us by Joe Stoy, based on an
+example in Kaldewaij 1990.
+
+\1f
+File: sicp.info, Node: 1-2-5, Next: 1-2-6, Prev: 1-2-4, Up: 1-2
+
+1.2.5 Greatest Common Divisors
+------------------------------
+
+The greatest common divisor (GCD) of two integers a and b is defined to
+be the largest integer that divides both a and b with no remainder.
+For example, the GCD of 16 and 28 is 4. In *Note Chapter 2::, when we
+investigate how to implement rational-number arithmetic, we will need
+to be able to compute GCDs in order to reduce rational numbers to
+lowest terms. (To reduce a rational number to lowest terms, we must
+divide both the numerator and the denominator by their GCD. For
+example, 16/28 reduces to 4/7.) One way to find the GCD of two
+integers is to factor them and search for common factors, but there is
+a famous algorithm that is much more efficient.
+
+ The idea of the algorithm is based on the observation that, if r is
+the remainder when a is divided by b, then the common divisors of a and
+b are precisely the same as the common divisors of b and r. Thus, we
+can use the equation
+
+ GCD(a,b) = GCD(b,r)
+
+to successively reduce the problem of computing a GCD to the problem of
+computing the GCD of smaller and smaller pairs of integers. For
+example,
+
+ GCD(206,40) = GCD(40,6)
+ = GCD(6,4)
+ = GCD(4,2)
+ = GCD(2,0)
+ = 2
+
+reduces GCD(206,40) to GCD(2,0), which is 2. It is possible to show
+that starting with any two positive integers and performing repeated
+reductions will always eventually produce a pair where the second
+number is 0. Then the GCD is the other number in the pair. This
+method for computing the GCD is known as Algorithm "Euclid's
+Algorithm".(1)
+
+ It is easy to express Euclid's Algorithm as a procedure:
+
+ (define (gcd a b)
+ (if (= b 0)
+ a
+ (gcd b (remainder a b))))
+
+ This generates an iterative process, whose number of steps grows as
+the logarithm of the numbers involved.
+
+ The fact that the number of steps required by Euclid's Algorithm has
+logarithmic growth bears an interesting relation to the Fibonacci
+numbers:
+
+ *Lame''s Theorem:* If Euclid's Algorithm requires k steps to
+ compute the GCD of some pair, then the smaller number in the pair
+ must be greater than or equal to the kth Fibonacci number.(2)
+
+ We can use this theorem to get an order-of-growth estimate for
+Euclid's Algorithm. Let n be the smaller of the two inputs to the
+procedure. If the process takes k steps, then we must have n>= _Fib_(k)
+approx [phi]^k/[sqrt](5). Therefore the number of steps k grows as the
+logarithm (to the base [phi]) of n. Hence, the order of growth is
+[theta](`log' n).
+
+ *Exercise 1.20:* The process that a procedure generates is of
+ course dependent on the rules used by the interpreter. As an
+ example, consider the iterative `gcd' procedure given above.
+ Suppose we were to interpret this procedure using normal-order
+ evaluation, as discussed in section *Note 1-1-5::. (The
+ normal-order-evaluation rule for `if' is described in *Note
+ Exercise 1-5::.) Using the substitution method (for normal
+ order), illustrate the process generated in evaluating `(gcd 206
+ 40)' and indicate the `remainder' operations that are actually
+ performed. How many `remainder' operations are actually performed
+ in the normal-order evaluation of `(gcd 206 40)'? In the
+ applicative-order evaluation?
+
+ ---------- Footnotes ----------
+
+ (1) Euclid's Algorithm is so called because it appears in Euclid's
+`Elements' (Book 7, ca. 300 B.C.). According to Knuth (1973), it can
+be considered the oldest known nontrivial algorithm. The ancient
+Egyptian method of multiplication (*Note Exercise 1-18::) is surely
+older, but, as Knuth explains, Euclid's algorithm is the oldest known
+to have been presented as a general algorithm, rather than as a set of
+illustrative examples.
+
+ (2) This theorem was proved in 1845 by Gabriel Lame', a French
+mathematician and engineer known chiefly for his contributions to
+mathematical physics. To prove the theorem, we consider pairs (a_k
+,b_k), where a_k>= b_k, for which Euclid's Algorithm terminates in k
+steps. The proof is based on the claim that, if (a_(k+1), b_(k+1)) ->
+(a_k, b_k) -> (a_(k-1), b_(k-1)) are three successive pairs in the
+reduction process, then we must have b_(k+1)>= b_k + b_(k-1). To
+verify the claim, consider that a reduction step is defined by applying
+the transformation a_(k-1) = b_k, b_(k-1) = remainder of a_k divided by
+b_k. The second equation means that a_k = qb_k + b_(k-1) for some
+positive integer q. And since q must be at least 1 we have a_k = qb_k
++ b_(k-1) >= b_k + b_(k-1). But in the previous reduction step we have
+b_(k+1) = a_k. Therefore, b_(k+1) = a_k>= b_k + b_(k-1). This verifies
+the claim. Now we can prove the theorem by induction on k, the number
+of steps that the algorithm requires to terminate. The result is true
+for k = 1, since this merely requires that b be at least as large as
+_Fib_(1) = 1. Now, assume that the result is true for all integers
+less than or equal to k and establish the result for k + 1. Let
+(a_(k+1), b_(k+1)) -> (a_k, b_k) -> (a_k-1), b_(k-1)) be successive
+pairs in the reduction process. By our induction hypotheses, we have
+b_(k-1) >= _Fib_(k - 1) and b_k >= _Fib_(k). Thus, applying the claim
+we just proved together with the definition of the Fibonacci numbers
+gives b_(k+1) >= b_k + b_(k-1) >= _Fib_(k) + _Fib_(k - 1) = _Fib_(k +
+1), which completes the proof of Lame''s Theorem.
+
+\1f
+File: sicp.info, Node: 1-2-6, Prev: 1-2-5, Up: 1-2
+
+1.2.6 Example: Testing for Primality
+------------------------------------
+
+This section describes two methods for checking the primality of an
+integer n, one with order of growth [theta](_[sqrt]_(n)), and a
+"probabilistic" algorithm with order of growth [theta](`log' n). The
+exercises at the end of this section suggest programming projects based
+on these algorithms.
+
+Searching for divisors
+......................
+
+Since ancient times, mathematicians have been fascinated by problems
+concerning prime numbers, and many people have worked on the problem of
+determining ways to test if numbers are prime. One way to test if a
+number is prime is to find the number's divisors. The following
+program finds the smallest integral divisor (greater than 1) of a given
+number n. It does this in a straightforward way, by testing n for
+divisibility by successive integers starting with 2.
+
+ (define (smallest-divisor n)
+ (find-divisor n 2))
+
+ (define (find-divisor n test-divisor)
+ (cond ((> (square test-divisor) n) n)
+ ((divides? test-divisor n) test-divisor)
+ (else (find-divisor n (+ test-divisor 1)))))
+
+ (define (divides? a b)
+ (= (remainder b a) 0))
+
+ We can test whether a number is prime as follows: n is prime if and
+only if n is its own smallest divisor.
+
+ (define (prime? n)
+ (= n (smallest-divisor n)))
+
+ The end test for `find-divisor' is based on the fact that if n is not
+prime it must have a divisor less than or equal to _[sqrt]_(n).(1)
+This means that the algorithm need only test divisors between 1 and
+_[sqrt]_(n). Consequently, the number of steps required to identify n
+as prime will have order of growth [theta](_[sqrt]_(n)).
+
+The Fermat test
+...............
+
+The [theta](`log' n) primality test is based on a result from number
+theory known as Fermat's Little Theorem.(2)
+
+ *Fermat's Little Theorem:* If n is a prime number and a is any
+ positive integer less than n, then a raised to the nth power is
+ congruent to a modulo n.
+
+ (Two numbers are said to be "congruent modulo" n if they both have
+the same remainder when divided by n. The remainder of a number a when
+divided by n is also referred to as the "remainder of" a "modulo" n, or
+simply as a "modulo" n.)
+
+ If n is not prime, then, in general, most of the numbers a< n will
+not satisfy the above relation. This leads to the following algorithm
+for testing primality: Given a number n, pick a random number a < n and
+compute the remainder of a^n modulo n. If the result is not equal to
+a, then n is certainly not prime. If it is a, then chances are good
+that n is prime. Now pick another random number a and test it with the
+same method. If it also satisfies the equation, then we can be even
+more confident that n is prime. By trying more and more values of a,
+we can increase our confidence in the result. This algorithm is known
+as the Fermat test.
+
+ To implement the Fermat test, we need a procedure that computes the
+exponential of a number modulo another number:
+
+ (define (expmod base exp m)
+ (cond ((= exp 0) 1)
+ ((even? exp)
+ (remainder (square (expmod base (/ exp 2) m))
+ m))
+ (else
+ (remainder (* base (expmod base (- exp 1) m))
+ m))))
+
+ This is very similar to the `fast-expt' procedure of section *Note
+1-2-4::. It uses successive squaring, so that the number of steps
+grows logarithmically with the exponent.(3)
+
+ The Fermat test is performed by choosing at random a number a
+between 1 and n - 1 inclusive and checking whether the remainder modulo
+n of the nth power of a is equal to a. The random number a is chosen
+using the procedure `random', which we assume is included as a primitive
+in Scheme. `Random' returns a nonnegative integer less than its integer
+input. Hence, to obtain a random number between 1 and n - 1, we call
+`random' with an input of n - 1 and add 1 to the result:
+
+ (define (fermat-test n)
+ (define (try-it a)
+ (= (expmod a n n) a))
+ (try-it (+ 1 (random (- n 1)))))
+
+ The following procedure runs the test a given number of times, as
+specified by a parameter. Its value is true if the test succeeds every
+time, and false otherwise.
+
+ (define (fast-prime? n times)
+ (cond ((= times 0) true)
+ ((fermat-test n) (fast-prime? n (- times 1)))
+ (else false)))
+
+Probabilistic methods
+.....................
+
+The Fermat test differs in character from most familiar algorithms, in
+which one computes an answer that is guaranteed to be correct. Here,
+the answer obtained is only probably correct. More precisely, if n
+ever fails the Fermat test, we can be certain that n is not prime. But
+the fact that n passes the test, while an extremely strong indication,
+is still not a guarantee that n is prime. What we would like to say is
+that for any number n, if we perform the test enough times and find
+that n always passes the test, then the probability of error in our
+primality test can be made as small as we like.
+
+ Unfortunately, this assertion is not quite correct. There do exist
+numbers that fool the Fermat test: numbers n that are not prime and yet
+have the property that a^n is congruent to a modulo n for all integers
+a < n. Such numbers are extremely rare, so the Fermat test is quite
+reliable in practice.(4)
+
+ There are variations of the Fermat test that cannot be fooled. In
+these tests, as with the Fermat method, one tests the primality of an
+integer n by choosing a random integer a<n and checking some condition
+that depends upon n and a. (See *Note Exercise 1-28:: for an example
+of such a test.) On the other hand, in contrast to the Fermat test,
+one can prove that, for any n, the condition does not hold for most of
+the integers a<n unless n is prime. Thus, if n passes the test for
+some random choice of a, the chances are better than even that n is
+prime. If n passes the test for two random choices of a, the chances
+are better than 3 out of 4 that n is prime. By running the test with
+more and more randomly chosen values of a we can make the probability
+of error as small as we like.
+
+ The existence of tests for which one can prove that the chance of
+error becomes arbitrarily small has sparked interest in algorithms of
+this type, which have come to be known as "probabilistic algorithms".
+There is a great deal of research activity in this area, and
+probabilistic algorithms have been fruitfully applied to many fields.(5)
+
+ *Exercise 1.21:* Use the `smallest-divisor' procedure to find the
+ smallest divisor of each of the following numbers: 199, 1999,
+ 19999.
+
+ *Exercise 1.22:* Most Lisp implementations include a primitive
+ called `runtime' that returns an integer that specifies the amount
+ of time the system has been running (measured, for example, in
+ microseconds). The following `timed-prime-test' procedure, when
+ called with an integer n, prints n and checks to see if n is
+ prime. If n is prime, the procedure prints three asterisks
+ followed by the amount of time used in performing the test.
+
+ (define (timed-prime-test n)
+ (newline)
+ (display n)
+ (start-prime-test n (runtime)))
+
+ (define (start-prime-test n start-time)
+ (if (prime? n)
+ (report-prime (- (runtime) start-time))))
+
+ (define (report-prime elapsed-time)
+ (display " *** ")
+ (display elapsed-time))
+
+ Using this procedure, write a procedure `search-for-primes' that
+ checks the primality of consecutive odd integers in a specified
+ range. Use your procedure to find the three smallest primes
+ larger than 1000; larger than 10,000; larger than 100,000; larger
+ than 1,000,000. Note the time needed to test each prime. Since
+ the testing algorithm has order of growth of [theta](_[sqrt]_(n)),
+ you should expect that testing for primes around 10,000 should
+ take about _[sqrt]_(10) times as long as testing for primes around
+ 1000. Do your timing data bear this out? How well do the data
+ for 100,000 and 1,000,000 support the _[sqrt]_(n) prediction? Is
+ your result compatible with the notion that programs on your
+ machine run in time proportional to the number of steps required
+ for the computation?
+
+ *Exercise 1.23:* The `smallest-divisor' procedure shown at the
+ start of this section does lots of needless testing: After it
+ checks to see if the number is divisible by 2 there is no point in
+ checking to see if it is divisible by any larger even numbers.
+ This suggests that the values used for `test-divisor' should not
+ be 2, 3, 4, 5, 6, ..., but rather 2, 3, 5, 7, 9, .... To
+ implement this change, define a procedure `next' that returns 3 if
+ its input is equal to 2 and otherwise returns its input plus 2.
+ Modify the `smallest-divisor' procedure to use `(next
+ test-divisor)' instead of `(+ test-divisor 1)'. With
+ `timed-prime-test' incorporating this modified version of
+ `smallest-divisor', run the test for each of the 12 primes found in
+ *Note Exercise 1-22::. Since this modification halves the number
+ of test steps, you should expect it to run about twice as fast.
+ Is this expectation confirmed? If not, what is the observed ratio
+ of the speeds of the two algorithms, and how do you explain the
+ fact that it is different from 2?
+
+ *Exercise 1.24:* Modify the `timed-prime-test' procedure of *Note
+ Exercise 1-22:: to use `fast-prime?' (the Fermat method), and test
+ each of the 12 primes you found in that exercise. Since the
+ Fermat test has [theta](`log' n) growth, how would you expect the
+ time to test primes near 1,000,000 to compare with the time needed
+ to test primes near 1000? Do your data bear this out? Can you
+ explain any discrepancy you find?
+
+ *Exercise 1.25:* Alyssa P. Hacker complains that we went to a lot
+ of extra work in writing `expmod'. After all, she says, since we
+ already know how to compute exponentials, we could have simply
+ written
+
+ (define (expmod base exp m)
+ (remainder (fast-expt base exp) m))
+
+ Is she correct? Would this procedure serve as well for our fast
+ prime tester? Explain.
+
+ *Exercise 1.26:* Louis Reasoner is having great difficulty doing
+ *Note Exercise 1-24::. His `fast-prime?' test seems to run more
+ slowly than his `prime?' test. Louis calls his friend Eva Lu Ator
+ over to help. When they examine Louis's code, they find that he
+ has rewritten the `expmod' procedure to use an explicit
+ multiplication, rather than calling `square':
+
+ (define (expmod base exp m)
+ (cond ((= exp 0) 1)
+ ((even? exp)
+ (remainder (* (expmod base (/ exp 2) m)
+ (expmod base (/ exp 2) m))
+ m))
+ (else
+ (remainder (* base (expmod base (- exp 1) m))
+ m))))
+
+ "I don't see what difference that could make," says Louis. "I
+ do." says Eva. "By writing the procedure like that, you have
+ transformed the [theta](`log' n) process into a [theta](n)
+ process." Explain.
+
+ *Exercise 1.27:* Demonstrate that the Carmichael numbers listed in
+ *Note Footnote 1-47:: really do fool the Fermat test. That is,
+ write a procedure that takes an integer n and tests whether a^n is
+ congruent to a modulo n for every a<n, and try your procedure on
+ the given Carmichael numbers.
+
+ *Exercise 1.28:* One variant of the Fermat test that cannot be
+ fooled is called the "Miller-Rabin test" (Miller 1976; Rabin
+ 1980). This starts from an alternate form of Fermat's Little
+ Theorem, which states that if n is a prime number and a is any
+ positive integer less than n, then a raised to the (n - 1)st power
+ is congruent to 1 modulo n. To test the primality of a number n
+ by the Miller-Rabin test, we pick a random number a<n and raise a
+ to the (n - 1)st power modulo n using the `expmod' procedure.
+ However, whenever we perform the squaring step in `expmod', we
+ check to see if we have discovered a "nontrivial square root of 1
+ modulo n," that is, a number not equal to 1 or n - 1 whose square
+ is equal to 1 modulo n. It is possible to prove that if such a
+ nontrivial square root of 1 exists, then n is not prime. It is
+ also possible to prove that if n is an odd number that is not
+ prime, then, for at least half the numbers a<n, computing a^(n-1)
+ in this way will reveal a nontrivial square root of 1 modulo n.
+ (This is why the Miller-Rabin test cannot be fooled.) Modify the
+ `expmod' procedure to signal if it discovers a nontrivial square
+ root of 1, and use this to implement the Miller-Rabin test with a
+ procedure analogous to `fermat-test'. Check your procedure by
+ testing various known primes and non-primes. Hint: One convenient
+ way to make `expmod' signal is to have it return 0.
+
+ ---------- Footnotes ----------
+
+ (1) If d is a divisor of n, then so is n/d. But d and n/d cannot
+both be greater than _[sqrt]_(n).
+
+ (2) Pierre de Fermat (1601-1665) is considered to be the founder of
+modern number theory. He obtained many important number-theoretic
+results, but he usually announced just the results, without providing
+his proofs. Fermat's Little Theorem was stated in a letter he wrote in
+1640. The first published proof was given by Euler in 1736 (and an
+earlier, identical proof was discovered in the unpublished manuscripts
+of Leibniz). The most famous of Fermat's results--known as Fermat's
+Last Theorem--was jotted down in 1637 in his copy of the book
+`Arithmetic' (by the third-century Greek mathematician Diophantus) with
+the remark "I have discovered a truly remarkable proof, but this margin
+is too small to contain it." Finding a proof of Fermat's Last Theorem
+became one of the most famous challenges in number theory. A complete
+solution was finally given in 1995 by Andrew Wiles of Princeton
+University.
+
+ (3) The reduction steps in the cases where the exponent e is greater
+than 1 are based on the fact that, for any integers x, y, and m, we can
+find the remainder of x times y modulo m by computing separately the
+remainders of x modulo m and y modulo m, multiplying these, and then
+taking the remainder of the result modulo m. For instance, in the case
+where e is even, we compute the remainder of b^(e/2) modulo m, square
+this, and take the remainder modulo m. This technique is useful
+because it means we can perform our computation without ever having to
+deal with numbers much larger than m. (Compare *Note Exercise 1-25::.)
+
+ (4) [Footnote 1.47] Numbers that fool the Fermat test are called "Carmichael
+numbers", and little is known about them other than that they are
+extremely rare. There are 255 Carmichael numbers below 100,000,000.
+The smallest few are 561, 1105, 1729, 2465, 2821, and 6601. In testing
+primality of very large numbers chosen at random, the chance of
+stumbling upon a value that fools the Fermat test is less than the
+chance that cosmic radiation will cause the computer to make an error
+in carrying out a "correct" algorithm. Considering an algorithm to be
+inadequate for the first reason but not for the second illustrates the
+difference between mathematics and engineering.
+
+ (5) One of the most striking applications of probabilistic prime
+testing has been to the field of cryptography. Although it is now
+computationally infeasible to factor an arbitrary 200-digit number, the
+primality of such a number can be checked in a few seconds with the
+Fermat test. This fact forms the basis of a technique for constructing
+"unbreakable codes" suggested by Rivest, Shamir, and Adleman (1977).
+The resulting "RSA algorithm" has become a widely used technique for
+enhancing the security of electronic communications. Because of this
+and related developments, the study of prime numbers, once considered
+the epitome of a topic in "pure" mathematics to be studied only for its
+own sake, now turns out to have important practical applications to
+cryptography, electronic funds transfer, and information retrieval.
+
+\1f
+File: sicp.info, Node: 1-3, Prev: 1-2, Up: Chapter 1
+
+1.3 Formulating Abstractions with Higher-Order Procedures
+=========================================================
+
+We have seen that procedures are, in effect, abstractions that describe
+compound operations on numbers independent of the particular numbers.
+For example, when we
+
+ (define (cube x) (* x x x))
+
+we are not talking about the cube of a particular number, but rather
+about a method for obtaining the cube of any number. Of course we
+could get along without ever defining this procedure, by always writing
+expressions such as
+
+ (* 3 3 3)
+ (* x x x)
+ (* y y y)
+
+and never mentioning `cube' explicitly. This would place us at a
+serious disadvantage, forcing us to work always at the level of the
+particular operations that happen to be primitives in the language
+(multiplication, in this case) rather than in terms of higher-level
+operations. Our programs would be able to compute cubes, but our
+language would lack the ability to express the concept of cubing. One
+of the things we should demand from a powerful programming language is
+the ability to build abstractions by assigning names to common patterns
+and then to work in terms of the abstractions directly. Procedures
+provide this ability. This is why all but the most primitive
+programming languages include mechanisms for defining procedures.
+
+ Yet even in numerical processing we will be severely limited in our
+ability to create abstractions if we are restricted to procedures whose
+parameters must be numbers. Often the same programming pattern will be
+used with a number of different procedures. To express such patterns
+as concepts, we will need to construct procedures that can accept
+procedures as arguments or return procedures as values. Procedures
+that manipulate procedures are called "higher-order procedures". This
+section shows how higher-order procedures can serve as powerful
+abstraction mechanisms, vastly increasing the expressive power of our
+language.
+
+* Menu:
+
+* 1-3-1:: Procedures as Arguments
+* 1-3-2:: Constructing Procedures Using `Lambda'
+* 1-3-3:: Procedures as General Methods
+* 1-3-4:: Procedures as Returned Values
+
+\1f
+File: sicp.info, Node: 1-3-1, Next: 1-3-2, Prev: 1-3, Up: 1-3
+
+1.3.1 Procedures as Arguments
+-----------------------------
+
+Consider the following three procedures. The first computes the sum of
+the integers from `a' through `b':
+
+ (define (sum-integers a b)
+ (if (> a b)
+ 0
+ (+ a (sum-integers (+ a 1) b))))
+
+ The second computes the sum of the cubes of the integers in the
+given range:
+
+ (define (sum-cubes a b)
+ (if (> a b)
+ 0
+ (+ (cube a) (sum-cubes (+ a 1) b))))
+
+ The third computes the sum of a sequence of terms in the series
+
+ 1 1 1
+ ----- + ----- + ------ + ...
+ 1 * 3 5 * 7 9 * 11
+
+which converges to [pi]/8 (very slowly):(1)
+
+ (define (pi-sum a b)
+ (if (> a b)
+ 0
+ (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b))))
+
+ These three procedures clearly share a common underlying pattern.
+They are for the most part identical, differing only in the name of the
+procedure, the function of `a' used to compute the term to be added,
+and the function that provides the next value of `a'. We could
+generate each of the procedures by filling in slots in the same
+template:
+
+ (define (<NAME> a b)
+ (if (> a b)
+ 0
+ (+ (<TERM> a)
+ (<NAME> (<NEXT> a) b))))
+
+ The presence of such a common pattern is strong evidence that there
+is a useful abstraction waiting to be brought to the surface. Indeed,
+mathematicians long ago identified the abstraction of "summation of a
+series" and invented "sigma notation," for example
+
+ b
+ ---
+ > f(n) = f(a) + ... + f(b)
+ ---
+ n=a
+
+to express this concept. The power of sigma notation is that it allows
+mathematicians to deal with the concept of summation itself rather than
+only with particular sums--for example, to formulate general results
+about sums that are independent of the particular series being summed.
+
+ Similarly, as program designers, we would like our language to be
+powerful enough so that we can write a procedure that expresses the
+concept of summation itself rather than only procedures that compute
+particular sums. We can do so readily in our procedural language by
+taking the common template shown above and transforming the "slots"
+into formal parameters:
+
+ (define (sum term a next b)
+ (if (> a b)
+ 0
+ (+ (term a)
+ (sum term (next a) next b))))
+
+ Notice that `sum' takes as its arguments the lower and upper bounds
+`a' and `b' together with the procedures `term' and `next'. We can use
+`sum' just as we would any procedure. For example, we can use it
+(along with a procedure `inc' that increments its argument by 1) to
+define `sum-cubes':
+
+ (define (inc n) (+ n 1))
+
+ (define (sum-cubes a b)
+ (sum cube a inc b))
+
+ Using this, we can compute the sum of the cubes of the integers from
+1 to 10:
+
+ (sum-cubes 1 10)
+ 3025
+
+ With the aid of an identity procedure to compute the term, we can
+define `sum-integers' in terms of `sum':
+
+ (define (identity x) x)
+
+ (define (sum-integers a b)
+ (sum identity a inc b))
+
+ Then we can add up the integers from 1 to 10:
+
+ (sum-integers 1 10)
+ 55
+
+ We can also define `pi-sum' in the same way:(2)
+
+ (define (pi-sum a b)
+ (define (pi-term x)
+ (/ 1.0 (* x (+ x 2))))
+ (define (pi-next x)
+ (+ x 4))
+ (sum pi-term a pi-next b))
+
+ Using these procedures, we can compute an approximation to [pi]:
+
+ (* 8 (pi-sum 1 1000))
+ 3.139592655589783
+
+ Once we have `sum', we can use it as a building block in formulating
+further concepts. For instance, the definite integral of a function f
+between the limits a and b can be approximated numerically using the
+formula
+
+ /b / / dx \ / dx \ / dx \ \
+ | f = | f| a + -- | + f| a + dx + -- | + f| a + 2dx + -- | + ...| dx
+ /a \ \ 2 / \ 2 / \ 2 / /
+
+for small values of dx. We can express this directly as a procedure:
+
+ (define (integral f a b dx)
+ (define (add-dx x) (+ x dx))
+ (* (sum f (+ a (/ dx 2.0)) add-dx b)
+ dx))
+
+ (integral cube 0 1 0.01)
+ .24998750000000042
+
+ (integral cube 0 1 0.001)
+ .249999875000001
+
+(The exact value of the integral of `cube' between 0 and 1 is 1/4.)
+
+ *Exercise 1.29:* Simpson's Rule is a more accurate method of
+ numerical integration than the method illustrated above. Using
+ Simpson's Rule, the integral of a function f between a and b is
+ approximated as
+
+ h
+ - (y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + ... + 2y_(n-2) + 4y_(n-1) + y_n)
+ 3
+
+ where h = (b - a)/n, for some even integer n, and y_k = f(a + kh).
+ (Increasing n increases the accuracy of the approximation.)
+ Define a procedure that takes as arguments f, a, b, and n and
+ returns the value of the integral, computed using Simpson's Rule.
+ Use your procedure to integrate `cube' between 0 and 1 (with n =
+ 100 and n = 1000), and compare the results to those of the
+ `integral' procedure shown above.
+
+ *Exercise 1.30:* The `sum' procedure above generates a linear
+ recursion. The procedure can be rewritten so that the sum is
+ performed iteratively. Show how to do this by filling in the
+ missing expressions in the following definition:
+
+ (define (sum term a next b)
+ (define (iter a result)
+ (if <??>
+ <??>
+ (iter <??> <??>)))
+ (iter <??> <??>))
+
+ *Exercise 1.31:*
+ a. The `sum' procedure is only the simplest of a vast number of
+ similar abstractions that can be captured as higher-order
+ procedures.(3) Write an analogous procedure called `product'
+ that returns the product of the values of a function at
+ points over a given range. Show how to define `factorial' in
+ terms of `product'. Also use `product' to compute
+ approximations to [pi] using the formula(4)
+
+ pi 2 * 4 * 4 * 6 * 6 * 8 ...
+ -- = -------------------------
+ 4 3 * 3 * 5 * 5 * 7 * 7 ...
+
+ b. If your `product' procedure generates a recursive process,
+ write one that generates an iterative process. If it
+ generates an iterative process, write one that generates a
+ recursive process.
+
+
+ *Exercise 1.32:*
+ a. Show that `sum' and `product' (*Note Exercise 1-31::) are
+ both special cases of a still more general notion called
+ `accumulate' that combines a collection of terms, using some
+ general accumulation function:
+
+ (accumulate combiner null-value term a next b)
+
+ `Accumulate' takes as arguments the same term and range
+ specifications as `sum' and `product', together with a
+ `combiner' procedure (of two arguments) that specifies how
+ the current term is to be combined with the accumulation of
+ the preceding terms and a `null-value' that specifies what
+ base value to use when the terms run out. Write `accumulate'
+ and show how `sum' and `product' can both be defined as
+ simple calls to `accumulate'.
+
+ b. If your `accumulate' procedure generates a recursive process,
+ write one that generates an iterative process. If it
+ generates an iterative process, write one that generates a
+ recursive process.
+
+
+ *Exercise 1.33:* You can obtain an even more general version of
+ `accumulate' (*Note Exercise 1-32::) by introducing the notion of
+ a "filter" on the terms to be combined. That is, combine only
+ those terms derived from values in the range that satisfy a
+ specified condition. The resulting `filtered-accumulate'
+ abstraction takes the same arguments as accumulate, together with
+ an additional predicate of one argument that specifies the filter.
+ Write `filtered-accumulate' as a procedure. Show how to express
+ the following using `filtered-accumulate':
+
+ a. the sum of the squares of the prime numbers in the interval a
+ to b (assuming that you have a `prime?' predicate already
+ written)
+
+ b. the product of all the positive integers less than n that are
+ relatively prime to n (i.e., all positive integers i < n such
+ that GCD(i,n) = 1).
+
+
+ ---------- Footnotes ----------
+
+ (1) This series, usually written in the equivalent form ([pi]/4) = 1
+- (1/3) + (1/5) - (1/7) + ..., is due to Leibniz. We'll see how to use
+this as the basis for some fancy numerical tricks in section *Note
+3-5-3::.
+
+ (2) Notice that we have used block structure (section *Note 1-1-8::)
+to embed the definitions of `pi-next' and `pi-term' within `pi-sum',
+since these procedures are unlikely to be useful for any other purpose.
+We will see how to get rid of them altogether in section *Note 1-3-2::.
+
+ (3) The intent of *Note Exercise 1-31:: through *Note Exercise
+1-33:: is to demonstrate the expressive power that is attained by using
+an appropriate abstraction to consolidate many seemingly disparate
+operations. However, though accumulation and filtering are elegant
+ideas, our hands are somewhat tied in using them at this point since we
+do not yet have data structures to provide suitable means of
+combination for these abstractions. We will return to these ideas in
+section *Note 2-2-3:: when we show how to use "sequences" as interfaces
+for combining filters and accumulators to build even more powerful
+abstractions. We will see there how these methods really come into
+their own as a powerful and elegant approach to designing programs.
+
+ (4) This formula was discovered by the seventeenth-century English
+mathematician John Wallis.
+
+\1f
+File: sicp.info, Node: 1-3-2, Next: 1-3-3, Prev: 1-3-1, Up: 1-3
+
+1.3.2 Constructing Procedures Using `Lambda'
+--------------------------------------------
+
+In using `sum' as in section *Note 1-3-1::, it seems terribly awkward to
+have to define trivial procedures such as `pi-term' and `pi-next' just
+so we can use them as arguments to our higher-order procedure. Rather
+than define `pi-next' and `pi-term', it would be more convenient to
+have a way to directly specify "the procedure that returns its input
+incremented by 4" and "the procedure that returns the reciprocal of its
+input times its input plus 2." We can do this by introducing the
+special form `lambda', which creates procedures. Using `lambda' we can
+describe what we want as
+
+ (lambda (x) (+ x 4))
+
+and
+
+ (lambda (x) (/ 1.0 (* x (+ x 2))))
+
+ Then our `pi-sum' procedure can be expressed without defining any
+auxiliary procedures as
+
+ (define (pi-sum a b)
+ (sum (lambda (x) (/ 1.0 (* x (+ x 2))))
+ a
+ (lambda (x) (+ x 4))
+ b))
+
+ Again using `lambda', we can write the `integral' procedure without
+having to define the auxiliary procedure `add-dx':
+
+ (define (integral f a b dx)
+ (* (sum f
+ (+ a (/ dx 2.0))
+ (lambda (x) (+ x dx))
+ b)
+ dx))
+
+ In general, `lambda' is used to create procedures in the same way as
+`define', except that no name is specified for the procedure:
+
+ (lambda (<FORMAL-PARAMETERS>) <BODY>)
+
+ The resulting procedure is just as much a procedure as one that is
+created using `define'. The only difference is that it has not been
+associated with any name in the environment. In fact,
+
+ (define (plus4 x) (+ x 4))
+
+is equivalent to
+
+ (define plus4 (lambda (x) (+ x 4)))
+
+ We can read a `lambda' expression as follows:
+
+ (lambda (x) (+ x 4))
+ | | | | |
+ the procedure of an argument x that adds x and 4
+
+ Like any expression that has a procedure as its value, a `lambda'
+expression can be used as the operator in a combination such as
+
+ ((lambda (x y z) (+ x y (square z))) 1 2 3)
+ 12
+
+or, more generally, in any context where we would normally use a
+procedure name.(1)
+
+Using `let' to create local variables
+.....................................
+
+Another use of `lambda' is in creating local variables. We often need
+local variables in our procedures other than those that have been bound
+as formal parameters. For example, suppose we wish to compute the
+function
+
+ f(x,y) = x(1 + xy)^2 + y(1 - y) + (1 + xy)(1 - y)
+
+which we could also express as
+
+ a = 1 + xy
+ b = 1 - y
+ f(x,y) = xa^2 + yb + ab
+
+ In writing a procedure to compute f, we would like to include as
+local variables not only x and y but also the names of intermediate
+quantities like a and b. One way to accomplish this is to use an
+auxiliary procedure to bind the local variables:
+
+ (define (f x y)
+ (define (f-helper a b)
+ (+ (* x (square a))
+ (* y b)
+ (* a b)))
+ (f-helper (+ 1 (* x y))
+ (- 1 y)))
+
+ Of course, we could use a `lambda' expression to specify an anonymous
+procedure for binding our local variables. The body of `f' then
+becomes a single call to that procedure:
+
+ (define (f x y)
+ ((lambda (a b)
+ (+ (* x (square a))
+ (* y b)
+ (* a b)))
+ (+ 1 (* x y))
+ (- 1 y)))
+
+ This construct is so useful that there is a special form called
+`let' to make its use more convenient. Using `let', the `f' procedure
+could be written as
+
+ (define (f x y)
+ (let ((a (+ 1 (* x y)))
+ (b (- 1 y)))
+ (+ (* x (square a))
+ (* y b)
+ (* a b))))
+
+ The general form of a `let' expression is
+
+ (let ((<VAR1> <EXP1>)
+ (<VAR2> <EXP2>)
+ ...
+ (<VARN> <VARN>))
+ <BODY>)
+
+which can be thought of as saying
+
+ let <VAR_1> have the value <EXP_1> and
+ <VAR_2> have the value <EXP_2> and
+ ...
+ <VAR_N> have the value <EXP_N>
+ in <BODY>
+
+ The first part of the `let' expression is a list of name-expression
+pairs. When the `let' is evaluated, each name is associated with the
+value of the corresponding expression. The body of the `let' is
+evaluated with these names bound as local variables. The way this
+happens is that the `let' expression is interpreted as an alternate
+syntax for
+
+ ((lambda (<VAR_1> ... <VAR_N>)
+ <BODY>)
+ <EXP_1>
+ ...
+ <EXP_N>)
+
+ No new mechanism is required in the interpreter in order to provide
+local variables. A `let' expression is simply syntactic sugar for the
+underlying `lambda' application.
+
+ We can see from this equivalence that the scope of a variable
+specified by a `let' expression is the body of the `let'. This implies
+that:
+
+ * `Let' allows one to bind variables as locally as possible to where
+ they are to be used. For example, if the value of `x' is 5, the
+ value of the expression
+
+ (+ (let ((x 3))
+ (+ x (* x 10)))
+ x)
+
+ is 38. Here, the `x' in the body of the `let' is 3, so the value
+ of the `let' expression is 33. On the other hand, the `x' that is
+ the second argument to the outermost `+' is still 5.
+
+ * The variables' values are computed outside the `let'. This
+ matters when the expressions that provide the values for the local
+ variables depend upon variables having the same names as the local
+ variables themselves. For example, if the value of `x' is 2, the
+ expression
+
+ (let ((x 3)
+ (y (+ x 2)))
+ (* x y))
+
+ will have the value 12 because, inside the body of the `let', `x'
+ will be 3 and `y' will be 4 (which is the outer `x' plus 2).
+
+
+ Sometimes we can use internal definitions to get the same effect as
+with `let'. For example, we could have defined the procedure `f' above
+as
+
+ (define (f x y)
+ (define a (+ 1 (* x y)))
+ (define b (- 1 y))
+ (+ (* x (square a))
+ (* y b)
+ (* a b)))
+
+ We prefer, however, to use `let' in situations like this and to use
+internal `define' only for internal procedures.(2)
+
+ *Exercise 1.34:* Suppose we define the procedure
+
+ (define (f g)
+ (g 2))
+
+ Then we have
+
+ (f square)
+ 4
+
+ (f (lambda (z) (* z (+ z 1))))
+ 6
+
+ What happens if we (perversely) ask the interpreter to evaluate
+ the combination `(f f)'? Explain.
+
+ ---------- Footnotes ----------
+
+ (1) It would be clearer and less intimidating to people learning
+Lisp if a name more obvious than `lambda', such as `make-procedure',
+were used. But the convention is firmly entrenched. The notation is
+adopted from the [lambda] calculus, a mathematical formalism introduced
+by the mathematical logician Alonzo Church (1941). Church developed
+the [lambda] calculus to provide a rigorous foundation for studying the
+notions of function and function application. The [lambda] calculus
+has become a basic tool for mathematical investigations of the
+semantics of programming languages.
+
+ (2) Understanding internal definitions well enough to be sure a
+program means what we intend it to mean requires a more elaborate model
+of the evaluation process than we have presented in this chapter. The
+subtleties do not arise with internal definitions of procedures,
+however. We will return to this issue in section *Note 4-1-6::, after
+we learn more about evaluation.
+
+\1f
+File: sicp.info, Node: 1-3-3, Next: 1-3-4, Prev: 1-3-2, Up: 1-3
+
+1.3.3 Procedures as General Methods
+-----------------------------------
+
+We introduced compound procedures in section *Note 1-1-4:: as a
+mechanism for abstracting patterns of numerical operations so as to
+make them independent of the particular numbers involved. With
+higher-order procedures, such as the `integral' procedure of section
+*Note 1-3-1::, we began to see a more powerful kind of abstraction:
+procedures used to express general methods of computation, independent
+of the particular functions involved. In this section we discuss two
+more elaborate examples--general methods for finding zeros and fixed
+points of functions--and show how these methods can be expressed
+directly as procedures.
+
+Finding roots of equations by the half-interval method
+......................................................
+
+The "half-interval method" is a simple but powerful technique for
+finding roots of an equation f(x) = 0, where f is a continuous
+function. The idea is that, if we are given points a and b such that
+f(a) < 0 < f(b), then f must have at least one zero between a and b.
+To locate a zero, let x be the average of a and b and compute f(x). If
+f(x) > 0, then f must have a zero between a and x. If f(x) < 0, then f
+must have a zero between x and b. Continuing in this way, we can
+identify smaller and smaller intervals on which f must have a zero.
+When we reach a point where the interval is small enough, the process
+stops. Since the interval of uncertainty is reduced by half at each
+step of the process, the number of steps required grows as
+[theta](`log'( L/T)), where L is the length of the original interval
+and T is the error tolerance (that is, the size of the interval we will
+consider "small enough"). Here is a procedure that implements this
+strategy:
+
+ (define (search f neg-point pos-point)
+ (let ((midpoint (average neg-point pos-point)))
+ (if (close-enough? neg-point pos-point)
+ midpoint
+ (let ((test-value (f midpoint)))
+ (cond ((positive? test-value)
+ (search f neg-point midpoint))
+ ((negative? test-value)
+ (search f midpoint pos-point))
+ (else midpoint))))))
+
+ We assume that we are initially given the function f together with
+points at which its values are negative and positive. We first compute
+the midpoint of the two given points. Next we check to see if the
+given interval is small enough, and if so we simply return the midpoint
+as our answer. Otherwise, we compute as a test value the value of f at
+the midpoint. If the test value is positive, then we continue the
+process with a new interval running from the original negative point to
+the midpoint. If the test value is negative, we continue with the
+interval from the midpoint to the positive point. Finally, there is
+the possibility that the test value is 0, in which case the midpoint is
+itself the root we are searching for.
+
+ To test whether the endpoints are "close enough" we can use a
+procedure similar to the one used in section *Note 1-1-7:: for
+computing square roots:(1)
+
+ (define (close-enough? x y)
+ (< (abs (- x y)) 0.001))
+
+ `Search' is awkward to use directly, because we can accidentally
+give it points at which f's values do not have the required sign, in
+which case we get a wrong answer. Instead we will use `search' via the
+following procedure, which checks to see which of the endpoints has a
+negative function value and which has a positive value, and calls the
+`search' procedure accordingly. If the function has the same sign on
+the two given points, the half-interval method cannot be used, in which
+case the procedure signals an error.(2)
+
+ (define (half-interval-method f a b)
+ (let ((a-value (f a))
+ (b-value (f b)))
+ (cond ((and (negative? a-value) (positive? b-value))
+ (search f a b))
+ ((and (negative? b-value) (positive? a-value))
+ (search f b a))
+ (else
+ (error "Values are not of opposite sign" a b)))))
+
+ The following example uses the half-interval method to approximate
+[pi] as the root between 2 and 4 of `sin' x = 0:
+
+ (half-interval-method sin 2.0 4.0)
+ 3.14111328125
+
+ Here is another example, using the half-interval method to search
+for a root of the equation x^3 - 2x - 3 = 0 between 1 and 2:
+
+ (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))
+ 1.0
+ 2.0)
+ 1.89306640625
+
+Finding fixed points of functions
+.................................
+
+A number x is called a "fixed point" of a function f if x satisfies the
+equation f(x) = x. For some functions f we can locate a fixed point by
+beginning with an initial guess and applying f repeatedly,
+
+ f(x), f(f(x), (f(f(f(x))))
+
+until the value does not change very much. Using this idea, we can
+devise a procedure `fixed-point' that takes as inputs a function and an
+initial guess and produces an approximation to a fixed point of the
+function. We apply the function repeatedly until we find two
+successive values whose difference is less than some prescribed
+tolerance:
+
+ (define tolerance 0.00001)
+
+ (define (fixed-point f first-guess)
+ (define (close-enough? v1 v2)
+ (< (abs (- v1 v2)) tolerance))
+ (define (try guess)
+ (let ((next (f guess)))
+ (if (close-enough? guess next)
+ next
+ (try next))))
+ (try first-guess))
+
+ For example, we can use this method to approximate the fixed point
+of the cosine function, starting with 1 as an initial approximation:(3)
+
+ (fixed-point cos 1.0)
+ .7390822985224023
+
+ Similarly, we can find a solution to the equation y = `sin' y +
+`cos' y:
+
+ (fixed-point (lambda (y) (+ (sin y) (cos y)))
+ 1.0)
+ 1.2587315962971173
+
+ The fixed-point process is reminiscent of the process we used for
+finding square roots in section *Note 1-1-7::. Both are based on the
+idea of repeatedly improving a guess until the result satisfies some
+criterion. In fact, we can readily formulate the square-root
+computation as a fixed-point search. Computing the square root of some
+number x requires finding a y such that y^2 = x. Putting this equation
+into the equivalent form y = x/y, we recognize that we are looking for
+a fixed point of the function(4) y |-> x/y, and we can therefore try to
+compute square roots as
+
+ (define (sqrt x)
+ (fixed-point (lambda (y) (/ x y))
+ 1.0))
+
+ Unfortunately, this fixed-point search does not converge. Consider
+an initial guess y_1. The next guess is y_2 = x/y_1 and the next guess
+is y_3 = x/y_2 = x/(x/y_1) = y_1. This results in an infinite loop in
+which the two guesses y_1 and y_2 repeat over and over, oscillating
+about the answer.
+
+ One way to control such oscillations is to prevent the guesses from
+changing so much. Since the answer is always between our guess y and
+x/y, we can make a new guess that is not as far from y as x/y by
+averaging y with x/y, so that the next guess after y is (1/2)(y + x/y)
+instead of x/y. The process of making such a sequence of guesses is
+simply the process of looking for a fixed point of y |-> (1/2)(y + x/y):
+
+ (define (sqrt x)
+ (fixed-point (lambda (y) (average y (/ x y)))
+ 1.0))
+
+ (Note that y = (1/2)(y + x/y) is a simple transformation of the
+equation y = x/y; to derive it, add y to both sides of the equation and
+divide by 2.)
+
+ With this modification, the square-root procedure works. In fact,
+if we unravel the definitions, we can see that the sequence of
+approximations to the square root generated here is precisely the same
+as the one generated by our original square-root procedure of section
+*Note 1-1-7::. This approach of averaging successive approximations to
+a solution, a technique we that we call "average damping", often aids
+the convergence of fixed-point searches.
+
+ *Exercise 1.35:* Show that the golden ratio [phi] (section *Note
+ 1-2-2::) is a fixed point of the transformation x |-> 1 + 1/x, and
+ use this fact to compute [phi] by means of the `fixed-point'
+ procedure.
+
+ *Exercise 1.36:* Modify `fixed-point' so that it prints the
+ sequence of approximations it generates, using the `newline' and
+ `display' primitives shown in *Note Exercise 1-22::. Then find a
+ solution to x^x = 1000 by finding a fixed point of x |->
+ `log'(1000)/`log'(x). (Use Scheme's primitive `log' procedure,
+ which computes natural logarithms.) Compare the number of steps
+ this takes with and without average damping. (Note that you
+ cannot start `fixed-point' with a guess of 1, as this would cause
+ division by `log'(1) = 0.)
+
+ *Exercise 1.37:*
+ a. An infinite "continued fraction" is an expression of the form
+
+ N_1
+ f = ---------------------
+ N_2
+ D_1 + ---------------
+ N_3
+ D_2 + ---------
+ D_3 + ...
+
+ As an example, one can show that the infinite continued
+ fraction expansion with the n_i and the D_i all equal to 1
+ produces 1/[phi], where [phi] is the golden ratio (described
+ in section *Note 1-2-2::). One way to approximate an
+ infinite continued fraction is to truncate the expansion
+ after a given number of terms. Such a truncation--a
+ so-called finite continued fraction "k-term finite continued
+ fraction"--has the form
+
+ N_1
+ -----------------
+ N_2
+ D_1 + -----------
+ ... N_K
+ + -----
+ D_K
+
+ Suppose that `n' and `d' are procedures of one argument (the
+ term index i) that return the n_i and D_i of the terms of the
+ continued fraction. Define a procedure `cont-frac' such that
+ evaluating `(cont-frac n d k)' computes the value of the
+ k-term finite continued fraction. Check your procedure by
+ approximating 1/[phi] using
+
+ (cont-frac (lambda (i) 1.0)
+ (lambda (i) 1.0)
+ k)
+
+ for successive values of `k'. How large must you make `k' in
+ order to get an approximation that is accurate to 4 decimal
+ places?
+
+ b. If your `cont-frac' procedure generates a recursive process,
+ write one that generates an iterative process. If it
+ generates an iterative process, write one that generates a
+ recursive process.
+
+
+ *Exercise 1.38:* In 1737, the Swiss mathematician Leonhard Euler
+ published a memoir `De Fractionibus Continuis', which included a
+ continued fraction expansion for e - 2, where e is the base of the
+ natural logarithms. In this fraction, the n_i are all 1, and the
+ D_i are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, .... Write
+ a program that uses your `cont-frac' procedure from *Note Exercise
+ 1-37:: to approximate e, based on Euler's expansion.
+
+ *Exercise 1.39:* A continued fraction representation of the
+ tangent function was published in 1770 by the German mathematician
+ J.H. Lambert:
+
+ x
+ tan x = ---------------
+ x^2
+ 1 - -----------
+ x^2
+ 3 - -------
+ 5 - ...
+
+ where x is in radians. Define a procedure `(tan-cf x k)' that
+ computes an approximation to the tangent function based on
+ Lambert's formula. `K' specifies the number of terms to compute,
+ as in *Note Exercise 1-37::.
+
+ ---------- Footnotes ----------
+
+ (1) We have used 0.001 as a representative "small" number to
+indicate a tolerance for the acceptable error in a calculation. The
+appropriate tolerance for a real calculation depends upon the problem
+to be solved and the limitations of the computer and the algorithm.
+This is often a very subtle consideration, requiring help from a
+numerical analyst or some other kind of magician.
+
+ (2) This can be accomplished using `error', which takes as arguments
+a number of items that are printed as error messages.
+
+ (3) Try this during a boring lecture: Set your calculator to radians
+mode and then repeatedly press the `cos' button until you obtain the
+fixed point.
+
+ (4) |-> (pronounced "maps to") is the mathematician's way of writing
+`lambda'. y |-> x/y means `(lambda(y) (/ x y))', that is, the function
+whose value at y is x/y.
+
+\1f
+File: sicp.info, Node: 1-3-4, Prev: 1-3-3, Up: 1-3
+
+1.3.4 Procedures as Returned Values
+-----------------------------------
+
+The above examples demonstrate how the ability to pass procedures as
+arguments significantly enhances the expressive power of our
+programming language. We can achieve even more expressive power by
+creating procedures whose returned values are themselves procedures.
+
+ We can illustrate this idea by looking again at the fixed-point
+example described at the end of section *Note 1-3-3::. We formulated a
+new version of the square-root procedure as a fixed-point search,
+starting with the observation that [sqrt]x is a fixed-point of the
+function y |-> x/y. Then we used average damping to make the
+approximations converge. Average damping is a useful general technique
+in itself. Namely, given a function f, we consider the function whose
+value at x is equal to the average of x and f(x).
+
+ We can express the idea of average damping by means of the following
+procedure:
+
+ (define (average-damp f)
+ (lambda (x) (average x (f x))))
+
+ `Average-damp' is a procedure that takes as its argument a procedure
+`f' and returns as its value a procedure (produced by the `lambda')
+that, when applied to a number `x', produces the average of `x' and `(f
+x)'. For example, applying `average-damp' to the `square' procedure
+produces a procedure whose value at some number x is the average of x
+and x^2. Applying this resulting procedure to 10 returns the average
+of 10 and 100, or 55:(1)
+
+ ((average-damp square) 10)
+ 55
+
+ Using `average-damp', we can reformulate the square-root procedure as
+follows:
+
+ (define (sqrt x)
+ (fixed-point (average-damp (lambda (y) (/ x y)))
+ 1.0))
+
+ Notice how this formulation makes explicit the three ideas in the
+method: fixed-point search, average damping, and the function y |-> x/y.
+It is instructive to compare this formulation of the square-root method
+with the original version given in section *Note 1-1-7::. Bear in mind
+that these procedures express the same process, and notice how much
+clearer the idea becomes when we express the process in terms of these
+abstractions. In general, there are many ways to formulate a process
+as a procedure. Experienced programmers know how to choose procedural
+formulations that are particularly perspicuous, and where useful
+elements of the process are exposed as separate entities that can be
+reused in other applications. As a simple example of reuse, notice
+that the cube root of x is a fixed point of the function y |-> x/y^2,
+so we can immediately generalize our square-root procedure to one that
+extracts cube roots:(2)
+
+ (define (cube-root x)
+ (fixed-point (average-damp (lambda (y) (/ x (square y))))
+ 1.0))
+
+Newton's method
+...............
+
+When we first introduced the square-root procedure, in section *Note
+1-1-7::, we mentioned that this was a special case of "Newton's
+method". If x |-> g(x) is a differentiable function, then a solution
+of the equation g(x) = 0 is a fixed point of the function x |-> f(x)
+where
+
+ g(x)
+ f(x) = x - -----
+ Dg(x)
+
+and Dg(x) is the derivative of g evaluated at x. Newton's method is
+the use of the fixed-point method we saw above to approximate a
+solution of the equation by finding a fixed point of the function f.(3)
+
+ For many functions g and for sufficiently good initial guesses for x,
+Newton's method converges very rapidly to a solution of g(x) = 0.(4)
+
+ In order to implement Newton's method as a procedure, we must first
+express the idea of derivative. Note that "derivative," like average
+damping, is something that transforms a function into another function.
+For instance, the derivative of the function x |-> x^3 is the function
+x |-> 3x^2. In general, if g is a function and dx is a small number,
+then the derivative Dg of g is the function whose value at any number x
+is given (in the limit of small dx) by
+
+ g(x + dx) - g(x)
+ Dg(c) = ----------------
+ dx
+
+Thus, we can express the idea of derivative (taking dx to be, say,
+0.00001) as the procedure
+
+ (define (deriv g)
+ (lambda (x)
+ (/ (- (g (+ x dx)) (g x))
+ dx)))
+
+along with the definition
+
+ (define dx 0.00001)
+
+ Like `average-damp', `deriv' is a procedure that takes a procedure as
+argument and returns a procedure as value. For example, to approximate
+the derivative of x |-> x^3 at 5 (whose exact value is 75) we can
+evaluate
+
+ (define (cube x) (* x x x))
+
+ ((deriv cube) 5)
+ 75.00014999664018
+
+ With the aid of `deriv', we can express Newton's method as a
+fixed-point process:
+
+ (define (newton-transform g)
+ (lambda (x)
+ (- x (/ (g x) ((deriv g) x)))))
+
+ (define (newtons-method g guess)
+ (fixed-point (newton-transform g) guess))
+
+ The `newton-transform' procedure expresses the formula at the
+beginning of this section, and `newtons-method' is readily defined in
+terms of this. It takes as arguments a procedure that computes the
+function for which we want to find a zero, together with an initial
+guess. For instance, to find the square root of x, we can use Newton's
+method to find a zero of the function y |-> y^2 - x starting with an
+initial guess of 1.(5)
+
+ This provides yet another form of the square-root procedure:
+
+ (define (sqrt x)
+ (newtons-method (lambda (y) (- (square y) x))
+ 1.0))
+
+Abstractions and first-class procedures
+.......................................
+
+We've seen two ways to express the square-root computation as an
+instance of a more general method, once as a fixed-point search and
+once using Newton's method. Since Newton's method was itself expressed
+as a fixed-point process, we actually saw two ways to compute square
+roots as fixed points. Each method begins with a function and finds a
+fixed point of some transformation of the function. We can express
+this general idea itself as a procedure:
+
+ (define (fixed-point-of-transform g transform guess)
+ (fixed-point (transform g) guess))
+
+ This very general procedure takes as its arguments a procedure `g'
+that computes some function, a procedure that transforms `g', and an
+initial guess. The returned result is a fixed point of the transformed
+function.
+
+ Using this abstraction, we can recast the first square-root
+computation from this section (where we look for a fixed point of the
+average-damped version of y |-> x/y) as an instance of this general
+method:
+
+ (define (sqrt x)
+ (fixed-point-of-transform (lambda (y) (/ x y))
+ average-damp
+ 1.0))
+
+ Similarly, we can express the second square-root computation from
+this section (an instance of Newton's method that finds a fixed point
+of the Newton transform of y |-> y^2 - x) as
+
+ (define (sqrt x)
+ (fixed-point-of-transform (lambda (y) (- (square y) x))
+ newton-transform
+ 1.0))
+
+ We began section *Note 1-3:: with the observation that compound
+procedures are a crucial abstraction mechanism, because they permit us
+to express general methods of computing as explicit elements in our
+programming language. Now we've seen how higher-order procedures
+permit us to manipulate these general methods to create further
+abstractions.
+
+ As programmers, we should be alert to opportunities to identify the
+underlying abstractions in our programs and to build upon them and
+generalize them to create more powerful abstractions. This is not to
+say that one should always write programs in the most abstract way
+possible; expert programmers know how to choose the level of
+abstraction appropriate to their task. But it is important to be able
+to think in terms of these abstractions, so that we can be ready to
+apply them in new contexts. The significance of higher-order
+procedures is that they enable us to represent these abstractions
+explicitly as elements in our programming language, so that they can be
+handled just like other computational elements.
+
+ In general, programming languages impose restrictions on the ways in
+which computational elements can be manipulated. Elements with the
+fewest restrictions are said to have "first-class" status. Some of the
+"rights and privileges" of first-class elements are:(6)
+
+ * They may be named by variables.
+
+ * They may be passed as arguments to procedures.
+
+ * They may be returned as the results of procedures.
+
+ * They may be included in data structures.(7)
+
+
+ Lisp, unlike other common programming languages, awards procedures
+full first-class status. This poses challenges for efficient
+implementation, but the resulting gain in expressive power is
+enormous.(8)
+
+ *Exercise 1.40:* Define a procedure `cubic' that can be used
+ together with the `newtons-method' procedure in expressions of the
+ form
+
+ (newtons-method (cubic a b c) 1)
+
+ to approximate zeros of the cubic x^3 + ax^2 + bx + c.
+
+ *Exercise 1.41:* Define a procedure `double' that takes a
+ procedure of one argument as argument and returns a procedure that
+ applies the original procedure twice. For example, if `inc' is a
+ procedure that adds 1 to its argument, then `(double inc)' should
+ be a procedure that adds 2. What value is returned by
+
+ (((double (double double)) inc) 5)
+
+ *Exercise 1.42:* Let f and g be two one-argument functions. The "composition"
+ f after g is defined to be the function x |-> f(g(x)). Define a
+ procedure `compose' that implements composition. For example, if
+ `inc' is a procedure that adds 1 to its argument,
+
+ ((compose square inc) 6)
+ 49
+
+ *Exercise 1.43:* If f is a numerical function and n is a positive
+ integer, then we can form the nth repeated application of f, which
+ is defined to be the function whose value at x is
+ f(f(...(f(x))...)). For example, if f is the function x |-> x +
+ 1, then the nth repeated application of f is the function x |-> x
+ + n. If f is the operation of squaring a number, then the nth
+ repeated application of f is the function that raises its argument
+ to the 2^nth power. Write a procedure that takes as inputs a
+ procedure that computes f and a positive integer n and returns the
+ procedure that computes the nth repeated application of f. Your
+ procedure should be able to be used as follows:
+
+ ((repeated square 2) 5)
+ 625
+
+ Hint: You may find it convenient to use `compose' from *Note
+ Exercise 1-42::.
+
+ *Exercise 1.44:* The idea of "smoothing" a function is an
+ important concept in signal processing. If f is a function and dx
+ is some small number, then the smoothed version of f is the
+ function whose value at a point x is the average of f(x - dx),
+ f(x), and f(x + dx). Write a procedure `smooth' that takes as
+ input a procedure that computes f and returns a procedure that
+ computes the smoothed f. It is sometimes valuable to repeatedly
+ smooth a function (that is, smooth the smoothed function, and so
+ on) to obtained the "n-fold smoothed function". Show how to
+ generate the n-fold smoothed function of any given function using
+ `smooth' and `repeated' from *Note Exercise 1-43::.
+
+ *Exercise 1.45:* We saw in section *Note 1-3-3:: that attempting
+ to compute square roots by naively finding a fixed point of y |->
+ x/y does not converge, and that this can be fixed by average
+ damping. The same method works for finding cube roots as fixed
+ points of the average-damped y |-> x/y^2. Unfortunately, the
+ process does not work for fourth roots--a single average damp is
+ not enough to make a fixed-point search for y |-> x/y^3 converge.
+ On the other hand, if we average damp twice (i.e., use the average
+ damp of the average damp of y |-> x/y^3) the fixed-point search
+ does converge. Do some experiments to determine how many average
+ damps are required to compute nth roots as a fixed-point search
+ based upon repeated average damping of y |-> x/y^(n-1). Use this
+ to implement a simple procedure for computing nth roots using
+ `fixed-point', `average-damp', and the `repeated' procedure of
+ *Note Exercise 1-43::. Assume that any arithmetic operations you
+ need are available as primitives.
+
+ *Exercise 1.46:* Several of the numerical methods described in
+ this chapter are instances of an extremely general computational
+ strategy known as "iterative improvement". Iterative improvement
+ says that, to compute something, we start with an initial guess
+ for the answer, test if the guess is good enough, and otherwise
+ improve the guess and continue the process using the improved
+ guess as the new guess. Write a procedure `iterative-improve'
+ that takes two procedures as arguments: a method for telling
+ whether a guess is good enough and a method for improving a guess.
+ `Iterative-improve' should return as its value a procedure that
+ takes a guess as argument and keeps improving the guess until it
+ is good enough. Rewrite the `sqrt' procedure of section *Note
+ 1-1-7:: and the `fixed-point' procedure of section *Note 1-3-3::
+ in terms of `iterative-improve'.
+
+ ---------- Footnotes ----------
+
+ (1) Observe that this is a combination whose operator is itself a
+combination. *Note Exercise 1-4:: already demonstrated the ability to
+form such combinations, but that was only a toy example. Here we begin
+to see the real need for such combinations--when applying a procedure
+that is obtained as the value returned by a higher-order procedure.
+
+ (2) See *Note Exercise 1-45:: for a further generalization.
+
+ (3) Elementary calculus books usually describe Newton's method in
+terms of the sequence of approximations x_(n+1) = x_n - g(x_n)/Dg(x_n).
+Having language for talking about processes and using the idea of
+fixed points simplifies the description of the method.
+
+ (4) Newton's method does not always converge to an answer, but it can
+be shown that in favorable cases each iteration doubles the
+number-of-digits accuracy of the approximation to the solution. In
+such cases, Newton's method will converge much more rapidly than the
+half-interval method.
+
+ (5) For finding square roots, Newton's method converges rapidly to
+the correct solution from any starting point.
+
+ (6) The notion of first-class status of programming-language
+elements is due to the British computer scientist Christopher Strachey
+(1916-1975).
+
+ (7) We'll see examples of this after we introduce data structures in
+*Note Chapter 2::.
+
+ (8) The major implementation cost of first-class procedures is that
+allowing procedures to be returned as values requires reserving storage
+for a procedure's free variables even while the procedure is not
+executing. In the Scheme implementation we will study in section *Note
+4-1::, these variables are stored in the procedure's environment.
+
+\1f
+File: sicp.info, Node: Chapter 2, Next: Chapter 3, Prev: Chapter 1, Up: Top
+
+2 Building Abstractions with Data
+*********************************
+
+ We now come to the decisive step of mathematical abstraction: we
+ forget about what the symbols stand for. ...[The mathematician]
+ need not be idle; there are many operations which he may carry out
+ with these symbols, without ever having to look at the things they
+ stand for.
+
+ --Hermann Weyl, `The Mathematical Way of Thinking'
+
+ We concentrated in *Note Chapter 1:: on computational processes and
+on the role of procedures in program design. We saw how to use
+primitive data (numbers) and primitive operations (arithmetic
+operations), how to combine procedures to form compound procedures
+through composition, conditionals, and the use of parameters, and how
+to abstract procedures by using `define'. We saw that a procedure can
+be regarded as a pattern for the local evolution of a process, and we
+classified, reasoned about, and performed simple algorithmic analyses of
+some common patterns for processes as embodied in procedures. We also
+saw that higher-order procedures enhance the power of our language by
+enabling us to manipulate, and thereby to reason in terms of, general
+methods of computation. This is much of the essence of programming.
+
+ In this chapter we are going to look at more complex data. All the
+procedures in *Note Chapter 1:: operate on simple numerical data, and
+simple data are not sufficient for many of the problems we wish to
+address using computation. Programs are typically designed to model
+complex phenomena, and more often than not one must construct
+computational objects that have several parts in order to model
+real-world phenomena that have several aspects. Thus, whereas our
+focus in *Note Chapter 1:: was on building abstractions by combining
+procedures to form compound procedures, we turn in this chapter to
+another key aspect of any programming language: the means it provides
+for building abstractions by combining data objects to form "compound
+data".
+
+ Why do we want compound data in a programming language? For the
+same reasons that we want compound procedures: to elevate the
+conceptual level at which we can design our programs, to increase the
+modularity of our designs, and to enhance the expressive power of our
+language. Just as the ability to define procedures enables us to deal
+with processes at a higher conceptual level than that of the primitive
+operations of the language, the ability to construct compound data
+objects enables us to deal with data at a higher conceptual level than
+that of the primitive data objects of the language.
+
+ Consider the task of designing a system to perform arithmetic with
+rational numbers. We could imagine an operation `add-rat' that takes
+two rational numbers and produces their sum. In terms of simple data,
+a rational number can be thought of as two integers: a numerator and a
+denominator. Thus, we could design a program in which each rational
+number would be represented by two integers (a numerator and a
+denominator) and where `add-rat' would be implemented by two procedures
+(one producing the numerator of the sum and one producing the
+denominator). But this would be awkward, because we would then need to
+explicitly keep track of which numerators corresponded to which
+denominators. In a system intended to perform many operations on many
+rational numbers, such bookkeeping details would clutter the programs
+substantially, to say nothing of what they would do to our minds. It
+would be much better if we could "glue together" a numerator and
+denominator to form a pair--a "compound data object"--that our programs
+could manipulate in a way that would be consistent with regarding a
+rational number as a single conceptual unit.
+
+ The use of compound data also enables us to increase the modularity
+of our programs. If we can manipulate rational numbers directly as
+objects in their own right, then we can separate the part of our
+program that deals with rational numbers per se from the details of how
+rational numbers may be represented as pairs of integers. The general
+technique of isolating the parts of a program that deal with how data
+objects are represented from the parts of a program that deal with how
+data objects are used is a powerful design methodology called "data
+abstraction". We will see how data abstraction makes programs much
+easier to design, maintain, and modify.
+
+ The use of compound data leads to a real increase in the expressive
+power of our programming language. Consider the idea of forming a
+"linear combination" ax + by. We might like to write a procedure that
+would accept a, b, x, and y as arguments and return the value of ax +
+by. This presents no difficulty if the arguments are to be numbers,
+because we can readily define the procedure
+
+ (define (linear-combination a b x y)
+ (+ (* a x) (* b y)))
+
+ But suppose we are not concerned only with numbers. Suppose we
+would like to express, in procedural terms, the idea that one can form
+linear combinations whenever addition and multiplication are
+defined--for rational numbers, complex numbers, polynomials, or
+whatever. We could express this as a procedure of the form
+
+ (define (linear-combination a b x y)
+ (add (mul a x) (mul b y)))
+
+where `add' and `mul' are not the primitive procedures `+' and `*' but
+rather more complex things that will perform the appropriate operations
+for whatever kinds of data we pass in as the arguments `a', `b', `x',
+and `y'. The key point is that the only thing `linear-combination'
+should need to know about `a', `b', `x', and `y' is that the procedures
+`add' and `mul' will perform the appropriate manipulations. From the
+perspective of the procedure `linear-combination', it is irrelevant
+what `a', `b', `x', and `y' are and even more irrelevant how they might
+happen to be represented in terms of more primitive data. This same
+example shows why it is important that our programming language provide
+the ability to manipulate compound objects directly: Without this,
+there is no way for a procedure such as `linear-combination' to pass
+its arguments along to `add' and `mul' without having to know their
+detailed structure.(1)
+
+ We begin this chapter by implementing the rational-number arithmetic
+system mentioned above. This will form the background for our
+discussion of compound data and data abstraction. As with compound
+procedures, the main issue to be addressed is that of abstraction as a
+technique for coping with complexity, and we will see how data
+abstraction enables us to erect suitable "abstraction barriers" between
+different parts of a program.
+
+ We will see that the key to forming compound data is that a
+programming language should provide some kind of "glue" so that data
+objects can be combined to form more complex data objects. There are
+many possible kinds of glue. Indeed, we will discover how to form
+compound data using no special "data" operations at all, only
+procedures. This will further blur the distinction between "procedure"
+and "data," which was already becoming tenuous toward the end of *Note
+Chapter 1::. We will also explore some conventional techniques for
+representing sequences and trees. One key idea in dealing with
+compound data is the notion of "closure"--that the glue we use for
+combining data objects should allow us to combine not only primitive
+data objects, but compound data objects as well. Another key idea is
+that compound data objects can serve as "conventional interfaces" for
+combining program modules in mix-and-match ways. We illustrate some of
+these ideas by presenting a simple graphics language that exploits
+closure.
+
+ We will then augment the representational power of our language by
+introducing "symbolic expressions"--data whose elementary parts can be
+arbitrary symbols rather than only numbers. We explore various
+alternatives for representing sets of objects. We will find that, just
+as a given numerical function can be computed by many different
+computational processes, there are many ways in which a given data
+structure can be represented in terms of simpler objects, and the
+choice of representation can have significant impact on the time and
+space requirements of processes that manipulate the data. We will
+investigate these ideas in the context of symbolic differentiation, the
+representation of sets, and the encoding of information.
+
+ Next we will take up the problem of working with data that may be
+represented differently by different parts of a program. This leads to
+the need to implement "generic operations", which must handle many
+different types of data. Maintaining modularity in the presence of
+generic operations requires more powerful abstraction barriers than can
+be erected with simple data abstraction alone. In particular, we
+introduce programming "data-directed programming" as a technique that
+allows individual data representations to be designed in isolation and
+then combined "additively" (i.e., without modification). To illustrate
+the power of this approach to system design, we close the chapter by
+applying what we have learned to the implementation of a package for
+performing symbolic arithmetic on polynomials, in which the
+coefficients of the polynomials can be integers, rational numbers,
+complex numbers, and even other polynomials.
+
+* Menu:
+
+* 2-1:: Introduction to Data Abstraction
+* 2-2:: Hierarchical Data and the Closure Property
+* 2-3:: Symbolic Data
+* 2-4:: Multiple Representations for Abstract Data
+* 2-5:: Systems with Generic Operations
+
+ ---------- Footnotes ----------
+
+ (1) The ability to directly manipulate procedures provides an
+analogous increase in the expressive power of a programming language.
+For example, in section *Note 1-3-1:: we introduced the `sum'
+procedure, which takes a procedure `term' as an argument and computes
+the sum of the values of `term' over some specified interval. In order
+to define `sum', it is crucial that we be able to speak of a procedure
+such as `term' as an entity in its own right, without regard for how
+`term' might be expressed with more primitive operations. Indeed, if
+we did not have the notion of "a procedure," it is doubtful that we
+would ever even think of the possibility of defining an operation such
+as `sum'. Moreover, insofar as performing the summation is concerned,
+the details of how `term' may be constructed from more primitive
+operations are irrelevant.
+
+\1f
+File: sicp.info, Node: 2-1, Next: 2-2, Prev: Chapter 2, Up: Chapter 2
+
+2.1 Introduction to Data Abstraction
+====================================
+
+In section *Note 1-1-8::, we noted that a procedure used as an element
+in creating a more complex procedure could be regarded not only as a
+collection of particular operations but also as a procedural
+abstraction. That is, the details of how the procedure was implemented
+could be suppressed, and the particular procedure itself could be
+replaced by any other procedure with the same overall behavior. In
+other words, we could make an abstraction that would separate the way
+the procedure would be used from the details of how the procedure would
+be implemented in terms of more primitive procedures. The analogous
+notion for compound data is called "data abstraction". Data
+abstraction is a methodology that enables us to isolate how a compound
+data object is used from the details of how it is constructed from more
+primitive data objects.
+
+ The basic idea of data abstraction is to structure the programs that
+are to use compound data objects so that they operate on "abstract
+data." That is, our programs should use data in such a way as to make
+no assumptions about the data that are not strictly necessary for
+performing the task at hand. At the same time, a "concrete" data
+representation is defined independent of the programs that use the
+data. The interface between these two parts of our system will be a
+set of procedures, called "selectors" and "constructors", that
+implement the abstract data in terms of the concrete representation. To
+illustrate this technique, we will consider how to design a set of
+procedures for manipulating rational numbers.
+
+* Menu:
+
+* 2-1-1:: Example: Arithmetic Operations for Rational Numbers
+* 2-1-2:: Abstraction Barriers
+* 2-1-3:: What Is Meant by Data?
+* 2-1-4:: Extended Exercise: Interval Arithmetic
+
+\1f
+File: sicp.info, Node: 2-1-1, Next: 2-1-2, Prev: 2-1, Up: 2-1
+
+2.1.1 Example: Arithmetic Operations for Rational Numbers
+---------------------------------------------------------
+
+Suppose we want to do arithmetic with rational numbers. We want to be
+able to add, subtract, multiply, and divide them and to test whether
+two rational numbers are equal.
+
+ Let us begin by assuming that we already have a way of constructing
+a rational number from a numerator and a denominator. We also assume
+that, given a rational number, we have a way of extracting (or
+selecting) its numerator and its denominator. Let us further assume
+that the constructor and selectors are available as procedures:
+
+ * `(make-rat <N> <D>)' returns therational number whose numerator is
+ the integer `<N>' and whose denominator is the integer `<D>'.
+
+ * `(numer <X>)' returns the numerator of the rational number `<X>'.
+
+ * `(denom <X>)' returns the denominator of the rational number `<X>'.
+
+
+ We are using here a powerful strategy of synthesis: "wishful
+thinking". We haven't yet said how a rational number is represented,
+or how the procedures `numer', `denom', and `make-rat' should be
+implemented. Even so, if we did have these three procedures, we could
+then add, subtract, multiply, divide, and test equality by using the
+following relations:
+
+ n_1 n_2 n_1 d_2 + n_2 d_1
+ --- + --- = -----------------
+ d_1 d_2 d_1 d_2
+
+ n_1 n_2 n_1 d_2 - n_2 d_1
+ --- - --- = -----------------
+ d_1 d_2 d_1 d_2
+
+ n_1 n_2 n_1 n_2
+ --- * --- = -------
+ d_1 d_2 d_1 d_2
+
+ n_1 / d_1 n_1 d_2
+ --------- = -------
+ n_2 / d_2 d_1 n_2
+
+ n_1 n_2
+ --- = --- if and only if n_1 d_2 = n_2 d_1
+ d_1 d_2
+
+We can express these rules as procedures:
+
+ (define (add-rat x y)
+ (make-rat (+ (* (numer x) (denom y))
+ (* (numer y) (denom x)))
+ (* (denom x) (denom y))))
+
+ (define (sub-rat x y)
+ (make-rat (- (* (numer x) (denom y))
+ (* (numer y) (denom x)))
+ (* (denom x) (denom y))))
+
+ (define (mul-rat x y)
+ (make-rat (* (numer x) (numer y))
+ (* (denom x) (denom y))))
+
+ (define (div-rat x y)
+ (make-rat (* (numer x) (denom y))
+ (* (denom x) (numer y))))
+
+ (define (equal-rat? x y)
+ (= (* (numer x) (denom y))
+ (* (numer y) (denom x))))
+
+ Now we have the operations on rational numbers defined in terms of
+the selector and constructor procedures `numer', `denom', and
+`make-rat'. But we haven't yet defined these. What we need is some
+way to glue together a numerator and a denominator to form a rational
+number.
+
+Pairs
+.....
+
+To enable us to implement the concrete level of our data abstraction,
+our language provides a compound structure called a "pair", which can be
+constructed with the primitive procedure `cons'. This procedure takes
+two arguments and returns a compound data object that contains the two
+arguments as parts. Given a pair, we can extract the parts using the
+primitive procedures `car' and `cdr'.(1) Thus, we can use `cons',
+`car', and `cdr' as follows:
+
+ (define x (cons 1 2))
+
+ (car x)
+ 1
+
+ (cdr x)
+ 2
+
+ Notice that a pair is a data object that can be given a name and
+manipulated, just like a primitive data object. Moreover, `cons' can
+be used to form pairs whose elements are pairs, and so on:
+
+ (define x (cons 1 2))
+
+ (define y (cons 3 4))
+
+ (define z (cons x y))
+
+ (car (car z))
+ 1
+
+ (car (cdr z))
+ 3
+
+ In section *Note 2-2:: we will see how this ability to combine pairs
+means that pairs can be used as general-purpose building blocks to
+create all sorts of complex data structures. The single compound-data
+primitive "pair", implemented by the procedures `cons', `car', and
+`cdr', is the only glue we need. Data objects constructed from pairs
+are called "list-structured" data.
+
+Representing rational numbers
+.............................
+
+Pairs offer a natural way to complete the rational-number system.
+Simply represent a rational number as a pair of two integers: a
+numerator and a denominator. Then `make-rat', `numer', and `denom' are
+readily implemented as follows:(2)
+
+ (define (make-rat n d) (cons n d))
+
+ (define (numer x) (car x))
+
+ (define (denom x) (cdr x))
+
+ Also, in order to display the results of our computations, we can
+print rational numbers by printing the numerator, a slash, and the
+denominator:(3)
+
+ (define (print-rat x)
+ (newline)
+ (display (numer x))
+ (display "/")
+ (display (denom x)))
+
+ Now we can try our rational-number procedures:
+
+ (define one-half (make-rat 1 2))
+
+ (print-rat one-half)
+ 1/2
+
+ (define one-third (make-rat 1 3))
+
+ (print-rat (add-rat one-half one-third))
+ 5/6
+
+ (print-rat (mul-rat one-half one-third))
+ 1/6
+
+ (print-rat (add-rat one-third one-third))
+ 6/9
+
+ As the final example shows, our rational-number implementation does
+not reduce rational numbers to lowest terms. We can remedy this by
+changing `make-rat'. If we have a `gcd' procedure like the one in
+section *Note 1-2-5:: that produces the greatest common divisor of two
+integers, we can use `gcd' to reduce the numerator and the denominator
+to lowest terms before constructing the pair:
+
+ (define (make-rat n d)
+ (let ((g (gcd n d)))
+ (cons (/ n g) (/ d g))))
+
+ Now we have
+
+ (print-rat (add-rat one-third one-third))
+ 2/3
+
+as desired. This modification was accomplished by changing the
+constructor `make-rat' without changing any of the procedures (such as
+`add-rat' and `mul-rat') that implement the actual operations.
+
+ *Exercise 2.1:* Define a better version of `make-rat' that handles
+ both positive and negative arguments. `Make-rat' should normalize
+ the sign so that if the rational number is positive, both the
+ numerator and denominator are positive, and if the rational number
+ is negative, only the numerator is negative.
+
+ ---------- Footnotes ----------
+
+ (1) The name `cons' stands for "construct." The names `car' and
+`cdr' derive from the original implementation of Lisp on the IBM 704.
+That machine had an addressing scheme that allowed one to reference the
+"address" and "decrement" parts of a memory location. `Car' stands for
+"Contents of Address part of Register" and `cdr' (pronounced
+"could-er") stands for "Contents of Decrement part of Register."
+
+ (2) Another way to define the selectors and constructor is
+
+ (define make-rat cons)
+ (define numer car)
+ (define denom cdr)
+
+ The first definition associates the name `make-rat' with the value
+of the expression `cons', which is the primitive procedure that
+constructs pairs. Thus `make-rat' and `cons' are names for the same
+primitive constructor.
+
+ Defining selectors and constructors in this way is efficient:
+Instead of `make-rat' _calling_ `cons', `make-rat' _is_ `cons', so
+there is only one procedure called, not two, when `make-rat' is called.
+On the other hand, doing this defeats debugging aids that trace
+procedure calls or put breakpoints on procedure calls: You may want to
+watch `make-rat' being called, but you certainly don't want to watch
+every call to `cons'.
+
+ We have chosen not to use this style of definition in this book.
+
+ (3) `Display' is the Scheme primitive for printing data. The Scheme
+primitive `newline' starts a new line for printing. Neither of these
+procedures returns a useful value, so in the uses of `print-rat' below,
+we show only what `print-rat' prints, not what the interpreter prints
+as the value returned by `print-rat'.
+
+\1f
+File: sicp.info, Node: 2-1-2, Next: 2-1-3, Prev: 2-1-1, Up: 2-1
+
+2.1.2 Abstraction Barriers
+--------------------------
+
+Before continuing with more examples of compound data and data
+abstraction, let us consider some of the issues raised by the
+rational-number example. We defined the rational-number operations in
+terms of a constructor `make-rat' and selectors `numer' and `denom'.
+In general, the underlying idea of data abstraction is to identify for
+each type of data object a basic set of operations in terms of which
+all manipulations of data objects of that type will be expressed, and
+then to use only those operations in manipulating the data.
+
+ We can envision the structure of the rational-number system as shown
+in figure *Note Figure 2-1::. The horizontal lines represent barriers
+"abstraction barriers" that isolate different "levels" of the system.
+At each level, the barrier separates the programs (above) that use the
+data abstraction from the programs (below) that implement the data
+abstraction. Programs that use rational numbers manipulate them solely
+in terms of the procedures supplied "for public use" by the
+rational-number package: `add-rat', `sub-rat', `mul-rat', `div-rat',
+and `equal-rat?'. These, in turn, are implemented solely in terms of
+the constructor and selectors `make-rat', `numer', and `denom', which
+themselves are implemented in terms of pairs. The details of how pairs
+are implemented are irrelevant to the rest of the rational-number
+package so long as pairs can be manipulated by the use of `cons',
+`car', and `cdr'. In effect, procedures at each level are the
+interfaces that define the abstraction barriers and connect the
+different levels.
+
+ *Figure 2.1:* Data-abstraction barriers in the rational-number
+ package.
+
+ +------------------------------------+
+ --------| Programs that use rational numbers |--------
+ +------------------------------------+
+ Rational numbers in promblem domain
+ +---------------------------+
+ ------------| add-rat sub-rat ... |-------------
+ +---------------------------+
+ Rational numbers as numerators and denominators
+ +------------------------+
+ --------------| make-rat numer denom |--------------
+ +------------------------+
+ Rational numbers as pairs
+ +----------------+
+ ------------------| cons car cdr |------------------
+ +----------------+
+ However pairs are implemented
+
+ This simple idea has many advantages. One advantage is that it
+makes programs much easier to maintain and to modify. Any complex data
+structure can be represented in a variety of ways with the primitive
+data structures provided by a programming language. Of course, the
+choice of representation influences the programs that operate on it;
+thus, if the representation were to be changed at some later time, all
+such programs might have to be modified accordingly. This task could
+be time-consuming and expensive in the case of large programs unless
+the dependence on the representation were to be confined by design to a
+very few program modules.
+
+ For example, an alternate way to address the problem of reducing
+rational numbers to lowest terms is to perform the reduction whenever
+we access the parts of a rational number, rather than when we construct
+it. This leads to different constructor and selector procedures:
+
+ (define (make-rat n d)
+ (cons n d))
+
+ (define (numer x)
+ (let ((g (gcd (car x) (cdr x))))
+ (/ (car x) g)))
+
+ (define (denom x)
+ (let ((g (gcd (car x) (cdr x))))
+ (/ (cdr x) g)))
+
+ The difference between this implementation and the previous one lies
+in when we compute the `gcd'. If in our typical use of rational
+numbers we access the numerators and denominators of the same rational
+numbers many times, it would be preferable to compute the `gcd' when
+the rational numbers are constructed. If not, we may be better off
+waiting until access time to compute the `gcd'. In any case, when we
+change from one representation to the other, the procedures `add-rat',
+`sub-rat', and so on do not have to be modified at all.
+
+ Constraining the dependence on the representation to a few interface
+procedures helps us design programs as well as modify them, because it
+allows us to maintain the flexibility to consider alternate
+implementations. To continue with our simple example, suppose we are
+designing a rational-number package and we can't decide initially
+whether to perform the `gcd' at construction time or at selection time.
+The data-abstraction methodology gives us a way to defer that decision
+without losing the ability to make progress on the rest of the system.
+
+ *Exercise 2.2:* Consider the problem of representing line segments
+ in a plane. Each segment is represented as a pair of points: a
+ starting point and an ending point. Define a constructor
+ `make-segment' and selectors `start-segment' and `end-segment'
+ that define the representation of segments in terms of points.
+ Furthermore, a point can be represented as a pair of numbers: the
+ x coordinate and the y coordinate. Accordingly, specify a
+ constructor `make-point' and selectors `x-point' and `y-point'
+ that define this representation. Finally, using your selectors
+ and constructors, define a procedure `midpoint-segment' that takes
+ a line segment as argument and returns its midpoint (the point
+ whose coordinates are the average of the coordinates of the
+ endpoints). To try your procedures, you'll need a way to print
+ points:
+
+ (define (print-point p)
+ (newline)
+ (display "(")
+ (display (x-point p))
+ (display ",")
+ (display (y-point p))
+ (display ")"))
+
+ *Exercise 2.3:* Implement a representation for rectangles in a
+ plane. (Hint: You may want to make use of *Note Exercise 2-2::.)
+ In terms of your constructors and selectors, create procedures
+ that compute the perimeter and the area of a given rectangle. Now
+ implement a different representation for rectangles. Can you
+ design your system with suitable abstraction barriers, so that the
+ same perimeter and area procedures will work using either
+ representation?
+
+\1f
+File: sicp.info, Node: 2-1-3, Next: 2-1-4, Prev: 2-1-2, Up: 2-1
+
+2.1.3 What Is Meant by Data?
+----------------------------
+
+We began the rational-number implementation in section *Note 2-1-1:: by
+implementing the rational-number operations `add-rat', `sub-rat', and
+so on in terms of three unspecified procedures: `make-rat', `numer',
+and `denom'. At that point, we could think of the operations as being
+defined in terms of data objects--numerators, denominators, and rational
+numbers--whose behavior was specified by the latter three procedures.
+
+ But exactly what is meant by "data"? It is not enough to say
+"whatever is implemented by the given selectors and constructors."
+Clearly, not every arbitrary set of three procedures can serve as an
+appropriate basis for the rational-number implementation. We need to
+guarantee that, if we construct a rational number `x' from a pair of
+integers `n' and `d', then extracting the `numer' and the `denom' of
+`x' and dividing them should yield the same result as dividing `n' by
+`d'. In other words, `make-rat', `numer', and `denom' must satisfy the
+condition that, for any integer `n' and any non-zero integer `d', if
+`x' is (`make-rat n d'), then
+
+ (numer x) n
+ --------- = ---
+ (denom x) d
+
+ In fact, this is the only condition `make-rat', `numer', and `denom'
+must fulfill in order to form a suitable basis for a rational-number
+representation. In general, we can think of data as defined by some
+collection of selectors and constructors, together with specified
+conditions that these procedures must fulfill in order to be a valid
+representation.(1)
+
+ This point of view can serve to define not only "high-level" data
+objects, such as rational numbers, but lower-level objects as well.
+Consider the notion of a pair, which we used in order to define our
+rational numbers. We never actually said what a pair was, only that
+the language supplied procedures `cons', `car', and `cdr' for operating
+on pairs. But the only thing we need to know about these three
+operations is that if we glue two objects together using `cons' we can
+retrieve the objects using `car' and `cdr'. That is, the operations
+satisfy the condition that, for any objects `x' and `y', if `z' is
+`(cons x y)' then `(car z)' is `x' and `(cdr z)' is `y'. Indeed, we
+mentioned that these three procedures are included as primitives in our
+language. However, any triple of procedures that satisfies the above
+condition can be used as the basis for implementing pairs. This point
+is illustrated strikingly by the fact that we could implement `cons',
+`car', and `cdr' without using any data structures at all but only
+using procedures. Here are the definitions:
+
+ (define (cons x y)
+ (define (dispatch m)
+ (cond ((= m 0) x)
+ ((= m 1) y)
+ (else (error "Argument not 0 or 1 -- CONS" m))))
+ dispatch)
+
+ (define (car z) (z 0))
+
+ (define (cdr z) (z 1))
+
+ This use of procedures corresponds to nothing like our intuitive
+notion of what data should be. Nevertheless, all we need to do to show
+that this is a valid way to represent pairs is to verify that these
+procedures satisfy the condition given above.
+
+ The subtle point to notice is that the value returned by `(cons x
+y)' is a procedure--namely the internally defined procedure `dispatch',
+which takes one argument and returns either `x' or `y' depending on
+whether the argument is 0 or 1. Correspondingly, `(car z)' is defined
+to apply `z' to 0. Hence, if `z' is the procedure formed by `(cons x
+y)', then `z' applied to 0 will yield `x'. Thus, we have shown that
+`(car (cons x y))' yields `x', as desired. Similarly, `(cdr (cons x
+y))' applies the procedure returned by `(cons x y)' to 1, which returns
+`y'. Therefore, this procedural implementation of pairs is a valid
+implementation, and if we access pairs using only `cons', `car', and
+`cdr' we cannot distinguish this implementation from one that uses
+"real" data structures.
+
+ The point of exhibiting the procedural representation of pairs is
+not that our language works this way (Scheme, and Lisp systems in
+general, implement pairs directly, for efficiency reasons) but that it
+could work this way. The procedural representation, although obscure,
+is a perfectly adequate way to represent pairs, since it fulfills the
+only conditions that pairs need to fulfill. This example also
+demonstrates that the ability to manipulate procedures as objects
+automatically provides the ability to represent compound data. This
+may seem a curiosity now, but procedural representations of data will
+play a central role in our programming repertoire. This style of
+programming is often called "message passing", and we will be using it
+as a basic tool in *Note Chapter 3:: when we address the issues of
+modeling and simulation.
+
+ *Exercise 2.4:* Here is an alternative procedural representation
+ of pairs. For this representation, verify that `(car (cons x y))'
+ yields `x' for any objects `x' and `y'.
+
+ (define (cons x y)
+ (lambda (m) (m x y)))
+
+ (define (car z)
+ (z (lambda (p q) p)))
+
+ What is the corresponding definition of `cdr'? (Hint: To verify
+ that this works, make use of the substitution model of section
+ *Note 1-1-5::.)
+
+ *Exercise 2.5:* Show that we can represent pairs of nonnegative
+ integers using only numbers and arithmetic operations if we
+ represent the pair a and b as the integer that is the product 2^a
+ 3^b. Give the corresponding definitions of the procedures `cons',
+ `car', and `cdr'.
+
+ *Exercise 2.6:* In case representing pairs as procedures wasn't
+ mind-boggling enough, consider that, in a language that can
+ manipulate procedures, we can get by without numbers (at least
+ insofar as nonnegative integers are concerned) by implementing 0
+ and the operation of adding 1 as
+
+ (define zero (lambda (f) (lambda (x) x)))
+
+ (define (add-1 n)
+ (lambda (f) (lambda (x) (f ((n f) x)))))
+
+ This representation is known as "Church numerals", after its
+ inventor, Alonzo Church, the logician who invented the [lambda]
+ calculus.
+
+ Define `one' and `two' directly (not in terms of `zero' and
+ `add-1'). (Hint: Use substitution to evaluate `(add-1 zero)').
+ Give a direct definition of the addition procedure `+' (not in
+ terms of repeated application of `add-1').
+
+ ---------- Footnotes ----------
+
+ (1) Surprisingly, this idea is very difficult to formulate
+rigorously. There are two approaches to giving such a formulation. One,
+pioneered by C. A. R. Hoare (1972), is known as the method of models
+"abstract models". It formalizes the "procedures plus conditions"
+specification as outlined in the rational-number example above. Note
+that the condition on the rational-number representation was stated in
+terms of facts about integers (equality and division). In general,
+abstract models define new kinds of data objects in terms of previously
+defined types of data objects. Assertions about data objects can
+therefore be checked by reducing them to assertions about previously
+defined data objects. Another approach, introduced by Zilles at MIT,
+by Goguen, Thatcher, Wagner, and Wright at IBM (see Thatcher, Wagner,
+and Wright 1978), and by Guttag at Toronto (see Guttag 1977), is called "algebraic
+specification". It regards the "procedures" as elements of an abstract
+algebraic system whose behavior is specified by axioms that correspond
+to our "conditions," and uses the techniques of abstract algebra to
+check assertions about data objects. Both methods are surveyed in the
+paper by Liskov and Zilles (1975).
+
+\1f
+File: sicp.info, Node: 2-1-4, Prev: 2-1-3, Up: 2-1
+
+2.1.4 Extended Exercise: Interval Arithmetic
+--------------------------------------------
+
+Alyssa P. Hacker is designing a system to help people solve engineering
+problems. One feature she wants to provide in her system is the
+ability to manipulate inexact quantities (such as measured parameters
+of physical devices) with known precision, so that when computations
+are done with such approximate quantities the results will be numbers
+of known precision.
+
+ Electrical engineers will be using Alyssa's system to compute
+electrical quantities. It is sometimes necessary for them to compute
+the value of a parallel equivalent resistance R_p of two resistors R_1
+and R_2 using the formula
+
+ 1
+ R_p = -------------
+ 1/R_1 + 1/R_2
+
+ Resistance values are usually known only up to some tolerance
+guaranteed by the manufacturer of the resistor. For example, if you
+buy a resistor labeled "6.8 ohms with 10% tolerance" you can only be
+sure that the resistor has a resistance between 6.8 - 0.68 = 6.12 and
+6.8 + 0.68 = 7.48 ohms. Thus, if you have a 6.8-ohm 10% resistor in
+parallel with a 4.7-ohm 5% resistor, the resistance of the combination
+can range from about 2.58 ohms (if the two resistors are at the lower
+bounds) to about 2.97 ohms (if the two resistors are at the upper
+bounds).
+
+ Alyssa's idea is to implement "interval arithmetic" as a set of
+arithmetic operations for combining "intervals" (objects that represent
+the range of possible values of an inexact quantity). The result of
+adding, subtracting, multiplying, or dividing two intervals is itself
+an interval, representing the range of the result.
+
+ Alyssa postulates the existence of an abstract object called an
+"interval" that has two endpoints: a lower bound and an upper bound.
+She also presumes that, given the endpoints of an interval, she can
+construct the interval using the data constructor `make-interval'.
+Alyssa first writes a procedure for adding two intervals. She reasons
+that the minimum value the sum could be is the sum of the two lower
+bounds and the maximum value it could be is the sum of the two upper
+bounds:
+
+ (define (add-interval x y)
+ (make-interval (+ (lower-bound x) (lower-bound y))
+ (+ (upper-bound x) (upper-bound y))))
+
+ Alyssa also works out the product of two intervals by finding the
+minimum and the maximum of the products of the bounds and using them as
+the bounds of the resulting interval. (`Min' and `max' are primitives
+that find the minimum or maximum of any number of arguments.)
+
+ (define (mul-interval x y)
+ (let ((p1 (* (lower-bound x) (lower-bound y)))
+ (p2 (* (lower-bound x) (upper-bound y)))
+ (p3 (* (upper-bound x) (lower-bound y)))
+ (p4 (* (upper-bound x) (upper-bound y))))
+ (make-interval (min p1 p2 p3 p4)
+ (max p1 p2 p3 p4))))
+
+ To divide two intervals, Alyssa multiplies the first by the
+reciprocal of the second. Note that the bounds of the reciprocal
+interval are the reciprocal of the upper bound and the reciprocal of
+the lower bound, in that order.
+
+ (define (div-interval x y)
+ (mul-interval x
+ (make-interval (/ 1.0 (upper-bound y))
+ (/ 1.0 (lower-bound y)))))
+
+ *Exercise 2.7:* Alyssa's program is incomplete because she has not
+ specified the implementation of the interval abstraction. Here is
+ a definition of the interval constructor:
+
+ (define (make-interval a b) (cons a b))
+
+ Define selectors `upper-bound' and `lower-bound' to complete the
+ implementation.
+
+ *Exercise 2.8:* Using reasoning analogous to Alyssa's, describe
+ how the difference of two intervals may be computed. Define a
+ corresponding subtraction procedure, called `sub-interval'.
+
+ *Exercise 2.9:* The "width" of an interval is half of the
+ difference between its upper and lower bounds. The width is a
+ measure of the uncertainty of the number specified by the
+ interval. For some arithmetic operations the width of the result
+ of combining two intervals is a function only of the widths of the
+ argument intervals, whereas for others the width of the
+ combination is not a function of the widths of the argument
+ intervals. Show that the width of the sum (or difference) of two
+ intervals is a function only of the widths of the intervals being
+ added (or subtracted). Give examples to show that this is not
+ true for multiplication or division.
+
+ *Exercise 2.10:* Ben Bitdiddle, an expert systems programmer,
+ looks over Alyssa's shoulder and comments that it is not clear what
+ it means to divide by an interval that spans zero. Modify
+ Alyssa's code to check for this condition and to signal an error
+ if it occurs.
+
+ *Exercise 2.11:* In passing, Ben also cryptically comments: "By
+ testing the signs of the endpoints of the intervals, it is
+ possible to break `mul-interval' into nine cases, only one of which
+ requires more than two multiplications." Rewrite this procedure
+ using Ben's suggestion.
+
+ After debugging her program, Alyssa shows it to a potential user,
+ who complains that her program solves the wrong problem. He wants
+ a program that can deal with numbers represented as a center value
+ and an additive tolerance; for example, he wants to work with
+ intervals such as 3.5 +/- 0.15 rather than [3.35, 3.65]. Alyssa
+ returns to her desk and fixes this problem by supplying an
+ alternate constructor and alternate selectors:
+
+ (define (make-center-width c w)
+ (make-interval (- c w) (+ c w)))
+
+ (define (center i)
+ (/ (+ (lower-bound i) (upper-bound i)) 2))
+
+ (define (width i)
+ (/ (- (upper-bound i) (lower-bound i)) 2))
+
+ Unfortunately, most of Alyssa's users are engineers. Real
+ engineering situations usually involve measurements with only a
+ small uncertainty, measured as the ratio of the width of the
+ interval to the midpoint of the interval. Engineers usually
+ specify percentage tolerances on the parameters of devices, as in
+ the resistor specifications given earlier.
+
+ *Exercise 2.12:* Define a constructor `make-center-percent' that
+ takes a center and a percentage tolerance and produces the desired
+ interval. You must also define a selector `percent' that produces
+ the percentage tolerance for a given interval. The `center'
+ selector is the same as the one shown above.
+
+ *Exercise 2.13:* Show that under the assumption of small
+ percentage tolerances there is a simple formula for the approximate
+ percentage tolerance of the product of two intervals in terms of
+ the tolerances of the factors. You may simplify the problem by
+ assuming that all numbers are positive.
+
+ After considerable work, Alyssa P. Hacker delivers her finished
+ system. Several years later, after she has forgotten all about
+ it, she gets a frenzied call from an irate user, Lem E. Tweakit.
+ It seems that Lem has noticed that the formula for parallel
+ resistors can be written in two algebraically equivalent ways:
+
+ R_1 R_2
+ ---------
+ R_1 + R_2
+
+ and
+
+ 1
+ -------------
+ 1/R_1 + 1/R_2
+
+ He has written the following two programs, each of which computes
+ the parallel-resistors formula differently:
+
+ (define (par1 r1 r2)
+ (div-interval (mul-interval r1 r2)
+ (add-interval r1 r2)))
+
+ (define (par2 r1 r2)
+ (let ((one (make-interval 1 1)))
+ (div-interval one
+ (add-interval (div-interval one r1)
+ (div-interval one r2)))))
+
+ Lem complains that Alyssa's program gives different answers for
+ the two ways of computing. This is a serious complaint.
+
+ *Exercise 2.14:* Demonstrate that Lem is right. Investigate the
+ behavior of the system on a variety of arithmetic expressions.
+ Make some intervals A and B, and use them in computing the
+ expressions A/A and A/B. You will get the most insight by using
+ intervals whose width is a small percentage of the center value.
+ Examine the results of the computation in center-percent form (see
+ *Note Exercise 2-12::).
+
+ *Exercise 2.15:* Eva Lu Ator, another user, has also noticed the
+ different intervals computed by different but algebraically
+ equivalent expressions. She says that a formula to compute with
+ intervals using Alyssa's system will produce tighter error bounds
+ if it can be written in such a form that no variable that
+ represents an uncertain number is repeated. Thus, she says,
+ `par2' is a "better" program for parallel resistances than `par1'.
+ Is she right? Why?
+
+ *Exercise 2.16:* Explain, in general, why equivalent algebraic
+ expressions may lead to different answers. Can you devise an
+ interval-arithmetic package that does not have this shortcoming,
+ or is this task impossible? (Warning: This problem is very
+ difficult.)
+
+\1f
+File: sicp.info, Node: 2-2, Next: 2-3, Prev: 2-1, Up: Chapter 2
+
+2.2 Hierarchical Data and the Closure Property
+==============================================
+
+As we have seen, pairs provide a primitive "glue" that we can use to
+construct compound data objects. *Note Figure 2-2:: shows a standard
+way to visualize a pair--in this case, the pair formed by `(cons 1 2)'.
+In this representation, which is called "box-and-pointer notation",
+each object is shown as a "pointer" to a box. The box for a primitive
+object contains a representation of the object. For example, the box
+for a number contains a numeral. The box for a pair is actually a
+double box, the left part containing (a pointer to) the `car' of the
+pair and the right part containing the `cdr'.
+
+ *Figure 2.2:* Box-and-pointer representation of `(cons 1 2)'.
+
+ +---+---+ +---+
+ ---->| * | *-+---->| 2 |
+ +-|-+---+ +---+
+ |
+ V
+ +---+
+ | 1 |
+ +---+
+
+ We have already seen that `cons' can be used to combine not only
+numbers but pairs as well. (You made use of this fact, or should have,
+in doing *Note Exercise 2-2:: and *Note Exercise 2-3::.) As a
+consequence, pairs provide a universal building block from which we can
+construct all sorts of data structures. *Note Figure 2-3:: shows two
+ways to use pairs to combine the numbers 1, 2, 3, and 4.
+
+ *Figure 2.3:* Two ways to combine 1, 2, 3, and 4 using pairs.
+
+ +---+---+ +---+---+ +---+---+ +---+
+ ---->| * | *-+---->| * | * | ---->| * | *-+---->| 4 |
+ +-|-+---+ +-|-+-|-+ +-|-+---+ +---+
+ | | | |
+ V V V V
+ +---+---+ +---+ +---+ +---+---+ +---+---+
+ | * | * | | 3 | | 4 | | * | *-+---->| * | * |
+ +-|-+-|-+ +---+ +---+ +-|-+---+ +-|-+-|-+
+ | | | | |
+ V V V V V
+ +---+ +---+ +---+ +---+ +---+
+ | 1 | | 2 | | 1 | | 2 | | 3 |
+ +---+ +---+ +---+ +---+ +---+
+
+ (cons (cons 1 2) (cons (cons 1
+ (cons 3 4)) (cons 2 3))
+ 4)
+
+ The ability to create pairs whose elements are pairs is the essence
+of list structure's importance as a representational tool. We refer to
+this ability as the "closure property" of `cons'. In general, an
+operation for combining data objects satisfies the closure property if
+the results of combining things with that operation can themselves be
+combined using the same operation.(1) Closure is the key to power in
+any means of combination because it permits us to create "hierarchical"
+structures--structures made up of parts, which themselves are made up
+of parts, and so on.
+
+ From the outset of *Note Chapter 1::, we've made essential use of
+closure in dealing with procedures, because all but the very simplest
+programs rely on the fact that the elements of a combination can
+themselves be combinations. In this section, we take up the
+consequences of closure for compound data. We describe some
+conventional techniques for using pairs to represent sequences and
+trees, and we exhibit a graphics language that illustrates closure in a
+vivid way.(2)
+
+* Menu:
+
+* 2-2-1:: Representing Sequences
+* 2-2-2:: Hierarchical Structures
+* 2-2-3:: Sequences as Conventional Interfaces
+* 2-2-4:: Example: A Picture Language
+
+ ---------- Footnotes ----------
+
+ (1) The use of the word "closure" here comes from abstract algebra,
+where a set of elements is said to be closed under an operation if
+applying the operation to elements in the set produces an element that
+is again an element of the set. The Lisp community also
+(unfortunately) uses the word "closure" to describe a totally unrelated
+concept: A closure is an implementation technique for representing
+procedures with free variables. We do not use the word "closure" in
+this second sense in this book.
+
+ (2) The notion that a means of combination should satisfy closure is
+a straightforward idea. Unfortunately, the data combiners provided in
+many popular programming languages do not satisfy closure, or make
+closure cumbersome to exploit. In Fortran or Basic, one typically
+combines data elements by assembling them into arrays--but one cannot
+form arrays whose elements are themselves arrays. Pascal and C admit
+structures whose elements are structures. However, this requires that
+the programmer manipulate pointers explicitly, and adhere to the
+restriction that each field of a structure can contain only elements of
+a prespecified form. Unlike Lisp with its pairs, these languages have
+no built-in general-purpose glue that makes it easy to manipulate
+compound data in a uniform way. This limitation lies behind Alan
+Perlis's comment in his foreword to this book: "In Pascal the plethora
+of declarable data structures induces a specialization within functions
+that inhibits and penalizes casual cooperation. It is better to have
+100 functions operate on one data structure than to have 10 functions
+operate on 10 data structures."
+
+\1f
+File: sicp.info, Node: 2-2-1, Next: 2-2-2, Prev: 2-2, Up: 2-2
+
+2.2.1 Representing Sequences
+----------------------------
+
+ *Figure 2.4:* The sequence 1, 2, 3, 4 represented as a chain of
+ pairs.
+
+ +---+---+ +---+---+ +---+---+ +---+---+
+ ---->| * | *-+---->| * | *-+---->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
+ | | | |
+ V V V V
+ +---+ +---+ +---+ +---+
+ | 1 | | 2 | | 3 | | 4 |
+ +---+ +---+ +---+ +---+
+
+One of the useful structures we can build with pairs is a "sequence"--an
+ordered collection of data objects. There are, of course, many ways to
+represent sequences in terms of pairs. One particularly
+straightforward representation is illustrated in *Note Figure 2-4::,
+where the sequence 1, 2, 3, 4 is represented as a chain of pairs. The
+`car' of each pair is the corresponding item in the chain, and the
+`cdr' of the pair is the next pair in the chain. The `cdr' of the
+final pair signals the end of the sequence by pointing to a
+distinguished value that is not a pair, represented in box-and-pointer
+diagrams as a diagonal line and in programs as the value of the
+variable `nil'. The entire sequence is constructed by nested `cons'
+operations:
+
+ (cons 1
+ (cons 2
+ (cons 3
+ (cons 4 nil))))
+
+ Such a sequence of pairs, formed by nested `cons'es, is called a "list",
+and Scheme provides a primitive called `list' to help in constructing
+lists.(1) The above sequence could be produced by `(list 1 2 3 4)'.
+In general,
+
+ (list <A_1> <A_2> ... <A_N>)
+
+is equivalent to
+
+ (cons <A_1>
+ (cons <A_2>
+ (cons ...
+ (cons <A_N>
+ nil)
+ ...)))
+
+ Lisp systems conventionally print lists by printing the sequence of
+elements, enclosed in parentheses. Thus, the data object in *Note
+Figure 2-4:: is printed as `(1 2 3 4)':
+
+ (define one-through-four (list 1 2 3 4))
+
+ one-through-four
+ (1 2 3 4)
+
+ Be careful not to confuse the expression `(list 1 2 3 4)' with the
+list `(1 2 3 4)', which is the result obtained when the expression is
+evaluated. Attempting to evaluate the expression `(1 2 3 4)' will
+signal an error when the interpreter tries to apply the procedure `1' to
+arguments `2', `3', and `4'.
+
+ We can think of `car' as selecting the first item in the list, and of
+`cdr' as selecting the sublist consisting of all but the first item.
+Nested applications of `car' and `cdr' can be used to extract the
+second, third, and subsequent items in the list.(2) The constructor
+`cons' makes a list like the original one, but with an additional item
+at the beginning.
+
+ (car one-through-four)
+ 1
+
+ (cdr one-through-four)
+ (2 3 4)
+
+ (car (cdr one-through-four))
+ 2
+
+ (cons 10 one-through-four)
+ (10 1 2 3 4)
+
+ (cons 5 one-through-four)
+ (5 1 2 3 4)
+
+ The value of `nil', used to terminate the chain of pairs, can be
+thought of as a sequence of no elements, the "empty list". The word "nil"
+is a contraction of the Latin word _nihil_, which means "nothing."(3)
+
+List operations
+...............
+
+The use of pairs to represent sequences of elements as lists is
+accompanied by conventional programming techniques for manipulating
+lists by successively "`cdr'ing down" the lists. For example, the
+procedure `list-ref' takes as arguments a list and a number n and
+returns the nth item of the list. It is customary to number the
+elements of the list beginning with 0. The method for computing
+`list-ref' is the following:
+
+ * For n = 0, `list-ref' should return the `car' of the list.
+
+ * Otherwise, `list-ref' should return the (n - 1)st item of the
+ `cdr' of the list.
+
+
+ (define (list-ref items n)
+ (if (= n 0)
+ (car items)
+ (list-ref (cdr items) (- n 1))))
+
+ (define squares (list 1 4 9 16 25))
+
+ (list-ref squares 3)
+ 16
+
+ Often we `cdr' down the whole list. To aid in this, Scheme includes
+a primitive predicate `null?', which tests whether its argument is the
+empty list. The procedure `length', which returns the number of items
+in a list, illustrates this typical pattern of use:
+
+ (define (length items)
+ (if (null? items)
+ 0
+ (+ 1 (length (cdr items)))))
+
+ (define odds (list 1 3 5 7))
+
+ (length odds)
+ 4
+
+ The `length' procedure implements a simple recursive plan. The
+reduction step is:
+
+ * The `length' of any list is 1 plus the `length' of the `cdr' of
+ the list.
+
+
+ This is applied successively until we reach the base case:
+
+ * The `length' of the empty list is 0.
+
+
+ We could also compute `length' in an iterative style:
+
+ (define (length items)
+ (define (length-iter a count)
+ (if (null? a)
+ count
+ (length-iter (cdr a) (+ 1 count))))
+ (length-iter items 0))
+
+ Another conventional programming technique is to "`cons' up" an
+answer list while `cdr'ing down a list, as in the procedure `append',
+which takes two lists as arguments and combines their elements to make
+a new list:
+
+ (append squares odds)
+ (1 4 9 16 25 1 3 5 7)
+
+ (append odds squares)
+ (1 3 5 7 1 4 9 16 25)
+
+ `Append' is also implemented using a recursive plan. To `append'
+lists `list1' and `list2', do the following:
+
+ * If `list1' is the empty list, then the result is just `list2'.
+
+ * Otherwise, `append' the `cdr' of `list1' and `list2', and `cons'
+ the `car' of `list1' onto the result:
+
+
+ (define (append list1 list2)
+ (if (null? list1)
+ list2
+ (cons (car list1) (append (cdr list1) list2))))
+
+ *Exercise 2.17:* Define a procedure `last-pair' that returns the
+ list that contains only the last element of a given (nonempty)
+ list:
+
+ (last-pair (list 23 72 149 34))
+ (34)
+
+ *Exercise 2.18:* Define a procedure `reverse' that takes a list as
+ argument and returns a list of the same elements in reverse order:
+
+ (reverse (list 1 4 9 16 25))
+ (25 16 9 4 1)
+
+ *Exercise 2.19:* Consider the change-counting program of section
+ *Note 1-2-2::. It would be nice to be able to easily change the
+ currency used by the program, so that we could compute the number
+ of ways to change a British pound, for example. As the program is
+ written, the knowledge of the currency is distributed partly into
+ the procedure `first-denomination' and partly into the procedure
+ `count-change' (which knows that there are five kinds of U.S.
+ coins). It would be nicer to be able to supply a list of coins to
+ be used for making change.
+
+ We want to rewrite the procedure `cc' so that its second argument
+ is a list of the values of the coins to use rather than an integer
+ specifying which coins to use. We could then have lists that
+ defined each kind of currency:
+
+ (define us-coins (list 50 25 10 5 1))
+
+ (define uk-coins (list 100 50 20 10 5 2 1 0.5))
+
+ We could then call `cc' as follows:
+
+ (cc 100 us-coins)
+ 292
+
+ To do this will require changing the program `cc' somewhat. It
+ will still have the same form, but it will access its second
+ argument differently, as follows:
+
+ (define (cc amount coin-values)
+ (cond ((= amount 0) 1)
+ ((or (< amount 0) (no-more? coin-values)) 0)
+ (else
+ (+ (cc amount
+ (except-first-denomination coin-values))
+ (cc (- amount
+ (first-denomination coin-values))
+ coin-values)))))
+
+ Define the procedures `first-denomination',
+ `except-first-denomination', and `no-more?' in terms of primitive
+ operations on list structures. Does the order of the list
+ `coin-values' affect the answer produced by `cc'? Why or why not?
+
+ *Exercise 2.20:* The procedures `+', `*', and `list' take
+ arbitrary numbers of arguments. One way to define such procedures
+ is to use `define' with notation "dotted-tail notation". In a
+ procedure definition, a parameter list that has a dot before the
+ last parameter name indicates that, when the procedure is called,
+ the initial parameters (if any) will have as values the initial
+ arguments, as usual, but the final parameter's value will be a "list"
+ of any remaining arguments. For instance, given the definition
+
+ (define (f x y . z) <BODY>)
+
+ the procedure `f' can be called with two or more arguments. If we
+ evaluate
+
+ (f 1 2 3 4 5 6)
+
+ then in the body of `f', `x' will be 1, `y' will be 2, and `z'
+ will be the list `(3 4 5 6)'. Given the definition
+
+ (define (g . w) <BODY>)
+
+ the procedure `g' can be called with zero or more arguments. If we
+ evaluate
+
+ (g 1 2 3 4 5 6)
+
+ then in the body of `g', `w' will be the list `(1 2 3 4 5 6)'.(4)
+
+ Use this notation to write a procedure `same-parity' that takes
+ one or more integers and returns a list of all the arguments that
+ have the same even-odd parity as the first argument. For example,
+
+ (same-parity 1 2 3 4 5 6 7)
+ (1 3 5 7)
+
+ (same-parity 2 3 4 5 6 7)
+ (2 4 6)
+
+Mapping over lists
+..................
+
+One extremely useful operation is to apply some transformation to each
+element in a list and generate the list of results. For instance, the
+following procedure scales each number in a list by a given factor:
+
+ (define (scale-list items factor)
+ (if (null? items)
+ nil
+ (cons (* (car items) factor)
+ (scale-list (cdr items) factor))))
+
+ (scale-list (list 1 2 3 4 5) 10)
+ (10 20 30 40 50)
+
+ We can abstract this general idea and capture it as a common pattern
+expressed as a higher-order procedure, just as in section *Note 1-3::.
+The higher-order procedure here is called `map'. `Map' takes as
+arguments a procedure of one argument and a list, and returns a list of
+the results produced by applying the procedure to each element in the
+list:(5)
+
+ (define (map proc items)
+ (if (null? items)
+ nil
+ (cons (proc (car items))
+ (map proc (cdr items)))))
+
+ (map abs (list -10 2.5 -11.6 17))
+ (10 2.5 11.6 17)
+
+ (map (lambda (x) (* x x))
+ (list 1 2 3 4))
+ (1 4 9 16)
+
+ Now we can give a new definition of `scale-list' in terms of `map':
+
+ (define (scale-list items factor)
+ (map (lambda (x) (* x factor))
+ items))
+
+ `Map' is an important construct, not only because it captures a
+common pattern, but because it establishes a higher level of
+abstraction in dealing with lists. In the original definition of
+`scale-list', the recursive structure of the program draws attention to
+the element-by-element processing of the list. Defining `scale-list'
+in terms of `map' suppresses that level of detail and emphasizes that
+scaling transforms a list of elements to a list of results. The
+difference between the two definitions is not that the computer is
+performing a different process (it isn't) but that we think about the
+process differently. In effect, `map' helps establish an abstraction
+barrier that isolates the implementation of procedures that transform
+lists from the details of how the elements of the list are extracted
+and combined. Like the barriers shown in *Note Figure 2-1::, this
+abstraction gives us the flexibility to change the low-level details of
+how sequences are implemented, while preserving the conceptual
+framework of operations that transform sequences to sequences. Section
+*Note 2-2-3:: expands on this use of sequences as a framework for
+organizing programs.
+
+ *Exercise 2.21:* The procedure `square-list' takes a list of
+ numbers as argument and returns a list of the squares of those
+ numbers.
+
+ (square-list (list 1 2 3 4))
+ (1 4 9 16)
+
+ Here are two different definitions of `square-list'. Complete
+ both of them by filling in the missing expressions:
+
+ (define (square-list items)
+ (if (null? items)
+ nil
+ (cons <??> <??>)))
+
+ (define (square-list items)
+ (map <??> <??>))
+
+ *Exercise 2.22:* Louis Reasoner tries to rewrite the first
+ `square-list' procedure of *Note Exercise 2-21:: so that it
+ evolves an iterative process:
+
+ (define (square-list items)
+ (define (iter things answer)
+ (if (null? things)
+ answer
+ (iter (cdr things)
+ (cons (square (car things))
+ answer))))
+ (iter items nil))
+
+ Unfortunately, defining `square-list' this way produces the answer
+ list in the reverse order of the one desired. Why?
+
+ Louis then tries to fix his bug by interchanging the arguments to
+ `cons':
+
+ (define (square-list items)
+ (define (iter things answer)
+ (if (null? things)
+ answer
+ (iter (cdr things)
+ (cons answer
+ (square (car things))))))
+ (iter items nil))
+
+ This doesn't work either. Explain.
+
+ *Exercise 2.23:* The procedure `for-each' is similar to `map'. It
+ takes as arguments a procedure and a list of elements. However,
+ rather than forming a list of the results, `for-each' just applies
+ the procedure to each of the elements in turn, from left to right.
+ The values returned by applying the procedure to the elements are
+ not used at all--`for-each' is used with procedures that perform
+ an action, such as printing. For example,
+
+ (for-each (lambda (x) (newline) (display x))
+ (list 57 321 88))
+ 57
+ 321
+ 88
+
+ The value returned by the call to `for-each' (not illustrated
+ above) can be something arbitrary, such as true. Give an
+ implementation of `for-each'.
+
+ ---------- Footnotes ----------
+
+ (1) In this book, we use "list" to mean a chain of pairs terminated
+by the end-of-list marker. In contrast, the term "list structure"
+refers to any data structure made out of pairs, not just to lists.
+
+ (2) Since nested applications of `car' and `cdr' are cumbersome to
+write, Lisp dialects provide abbreviations for them--for instance,
+
+ (cadr (ARG)) = (car (cdr (ARG)))
+
+ The names of all such procedures start with `c' and end with `r'.
+Each `a' between them stands for a `car' operation and each `d' for a
+`cdr' operation, to be applied in the same order in which they appear
+in the name. The names `car' and `cdr' persist because simple
+combinations like `cadr' are pronounceable.
+
+ (3) It's remarkable how much energy in the standardization of Lisp
+dialects has been dissipated in arguments that are literally over
+nothing: Should `nil' be an ordinary name? Should the value of `nil'
+be a symbol? Should it be a list? Should it be a pair? In Scheme,
+`nil' is an ordinary name, which we use in this section as a variable
+whose value is the end-of-list marker (just as `true' is an ordinary
+variable that has a true value). Other dialects of Lisp, including
+Common Lisp, treat `nil' as a special symbol. The authors of this
+book, who have endured too many language standardization brawls, would
+like to avoid the entire issue. Once we have introduced quotation in
+section *Note 2-3::, we will denote the empty list as `'()' and
+dispense with the variable `nil' entirely.
+
+ (4) To define `f' and `g' using `lambda' we would write
+
+ (define f (lambda (x y . z) <BODY>))
+ (define g (lambda w <BODY>))
+
+ (5) [Footnote 12] Scheme standardly provides a `map' procedure that
+is more general than the one described here. This more general `map'
+takes a procedure of n arguments, together with n lists, and applies
+the procedure to all the first elements of the lists, all the second
+elements of the lists, and so on, returning a list of the results. For
+example:
+
+ (map + (list 1 2 3) (list 40 50 60) (list 700 800 900))
+ (741 852 963)
+
+ (map (lambda (x y) (+ x (* 2 y)))
+ (list 1 2 3)
+ (list 4 5 6))
+ (9 12 15)
+
+\1f
+File: sicp.info, Node: 2-2-2, Next: 2-2-3, Prev: 2-2-1, Up: 2-2
+
+2.2.2 Hierarchical Structures
+-----------------------------
+
+The representation of sequences in terms of lists generalizes naturally
+to represent sequences whose elements may themselves be sequences. For
+example, we can regard the object `((1 2) 3 4)' constructed by
+
+ (cons (list 1 2) (list 3 4))
+
+as a list of three items, the first of which is itself a list, `(1 2)'.
+Indeed, this is suggested by the form in which the result is printed by
+the interpreter. *Note Figure 2-5:: shows the representation of this
+structure in terms of pairs.
+
+ *Figure 2.5:* Structure formed by `(cons (list 1 2) (list 3 4))'.
+
+ (3 4)
+ |
+ V
+ ((1 2) 3 4) +---+---+ +---+---+ +---+---+
+ ---->| * | *-+----------------->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ | | |
+ V V V
+ (1 2) +---+---+ +---+---+ +---+ +---+
+ ---->| * | *-+---->| * | / | | 3 | | 4 |
+ +-|-+---+ +-|-+---+ +---+ +---+
+ | |
+ V V
+ +---+ +---+
+ | 1 | | 2 |
+ +---+ +---+
+
+ Another way to think of sequences whose elements are sequences is as "trees".
+The elements of the sequence are the branches of the tree, and
+elements that are themselves sequences are subtrees. *Note Figure 2-6::
+shows the structure in *Note Figure 2-5:: viewed as a tree.
+
+ *Figure 2.6:* The list structure in *Note Figure 2-5:: viewed as a
+ tree.
+
+ ((1 2) 3 4)
+ /\\
+ / | \
+ (1 2) 3 4
+ / \
+ 1 2
+
+ Recursion is a natural tool for dealing with tree structures, since
+we can often reduce operations on trees to operations on their
+branches, which reduce in turn to operations on the branches of the
+branches, and so on, until we reach the leaves of the tree. As an
+example, compare the `length' procedure of section *Note 2-2-1:: with
+the `count-leaves' procedure, which returns the total number of leaves
+of a tree:
+
+ (define x (cons (list 1 2) (list 3 4)))
+
+ (length x)
+ 3
+
+ (count-leaves x)
+ 4
+
+ (list x x)
+ (((1 2) 3 4) ((1 2) 3 4))
+
+ (length (list x x))
+ 2
+
+ (count-leaves (list x x))
+ 8
+
+ To implement `count-leaves', recall the recursive plan for computing
+`length':
+
+ * `Length' of a list `x' is 1 plus `length' of the `cdr' of `x'.
+
+ * `Length' of the empty list is 0.
+
+
+ `Count-leaves' is similar. The value for the empty list is the same:
+
+ * `Count-leaves' of the empty list is 0.
+
+
+ But in the reduction step, where we strip off the `car' of the list,
+we must take into account that the `car' may itself be a tree whose
+leaves we need to count. Thus, the appropriate reduction step is
+
+ * `Count-leaves' of a tree `x' is `count-leaves' of the `car' of `x'
+ plus `count-leaves' of the `cdr' of `x'.
+
+
+ Finally, by taking `car's we reach actual leaves, so we need another
+base case:
+
+ * `Count-leaves' of a leaf is 1.
+
+
+ To aid in writing recursive procedures on trees, Scheme provides the
+primitive predicate `pair?', which tests whether its argument is a
+pair. Here is the complete procedure:(1)
+
+ (define (count-leaves x)
+ (cond ((null? x) 0)
+ ((not (pair? x)) 1)
+ (else (+ (count-leaves (car x))
+ (count-leaves (cdr x))))))
+
+ *Exercise 2.24:* Suppose we evaluate the expression `(list 1 (list
+ 2 (list 3 4)))'. Give the result printed by the interpreter, the
+ corresponding box-and-pointer structure, and the interpretation of
+ this as a tree (as in *Note Figure 2-6::).
+
+ *Exercise 2.25:* Give combinations of `car's and `cdr's that will
+ pick 7 from each of the following lists:
+
+ (1 3 (5 7) 9)
+
+ ((7))
+
+ (1 (2 (3 (4 (5 (6 7))))))
+
+ *Exercise 2.26:* Suppose we define `x' and `y' to be two lists:
+
+ (define x (list 1 2 3))
+
+ (define y (list 4 5 6))
+
+ What result is printed by the interpreter in response to
+ evaluating each of the following expressions:
+
+ (append x y)
+
+ (cons x y)
+
+ (list x y)
+
+ *Exercise 2.27:* Modify your `reverse' procedure of *Note Exercise
+ 2-18:: to produce a `deep-reverse' procedure that takes a list as
+ argument and returns as its value the list with its elements
+ reversed and with all sublists deep-reversed as well. For example,
+
+ (define x (list (list 1 2) (list 3 4)))
+
+ x
+ ((1 2) (3 4))
+
+ (reverse x)
+ ((3 4) (1 2))
+
+ (deep-reverse x)
+ ((4 3) (2 1))
+
+ *Exercise 2.28:* Write a procedure `fringe' that takes as argument
+ a tree (represented as a list) and returns a list whose elements
+ are all the leaves of the tree arranged in left-to-right order.
+ For example,
+
+ (define x (list (list 1 2) (list 3 4)))
+
+ (fringe x)
+ (1 2 3 4)
+
+ (fringe (list x x))
+ (1 2 3 4 1 2 3 4)
+
+ *Exercise 2.29:* A binary mobile consists of two branches, a left
+ branch and a right branch. Each branch is a rod of a certain
+ length, from which hangs either a weight or another binary mobile.
+ We can represent a binary mobile using compound data by
+ constructing it from two branches (for example, using `list'):
+
+ (define (make-mobile left right)
+ (list left right))
+
+ A branch is constructed from a `length' (which must be a number)
+ together with a `structure', which may be either a number
+ (representing a simple weight) or another mobile:
+
+ (define (make-branch length structure)
+ (list length structure))
+
+ a. Write the corresponding selectors `left-branch' and
+ `right-branch', which return the branches of a mobile, and
+ `branch-length' and `branch-structure', which return the
+ components of a branch.
+
+ b. Using your selectors, define a procedure `total-weight' that
+ returns the total weight of a mobile.
+
+ c. A mobile is said to be "balanced" if the torque applied by
+ its top-left branch is equal to that applied by its top-right
+ branch (that is, if the length of the left rod multiplied by
+ the weight hanging from that rod is equal to the
+ corresponding product for the right side) and if each of the
+ submobiles hanging off its branches is balanced. Design a
+ predicate that tests whether a binary mobile is balanced.
+
+ d. Suppose we change the representation of mobiles so that the
+ constructors are
+
+ (define (make-mobile left right)
+ (cons left right))
+
+ (define (make-branch length structure)
+ (cons length structure))
+
+ How much do you need to change your programs to convert to
+ the new representation?
+
+
+Mapping over trees
+..................
+
+Just as `map' is a powerful abstraction for dealing with sequences,
+`map' together with recursion is a powerful abstraction for dealing with
+trees. For instance, the `scale-tree' procedure, analogous to
+`scale-list' of section *Note 2-2-1::, takes as arguments a numeric
+factor and a tree whose leaves are numbers. It returns a tree of the
+same shape, where each number is multiplied by the factor. The
+recursive plan for `scale-tree' is similar to the one for
+`count-leaves':
+
+ (define (scale-tree tree factor)
+ (cond ((null? tree) nil)
+ ((not (pair? tree)) (* tree factor))
+ (else (cons (scale-tree (car tree) factor)
+ (scale-tree (cdr tree) factor)))))
+
+ (scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7))
+ 10)
+ (10 (20 (30 40) 50) (60 70))
+
+ Another way to implement `scale-tree' is to regard the tree as a
+sequence of sub-trees and use `map'. We map over the sequence, scaling
+each sub-tree in turn, and return the list of results. In the base
+case, where the tree is a leaf, we simply multiply by the factor:
+
+ (define (scale-tree tree factor)
+ (map (lambda (sub-tree)
+ (if (pair? sub-tree)
+ (scale-tree sub-tree factor)
+ (* sub-tree factor)))
+ tree))
+
+ Many tree operations can be implemented by similar combinations of
+sequence operations and recursion.
+
+ *Exercise 2.30:* Define a procedure `square-tree' analogous to the
+ `square-list' procedure of *Note Exercise 2-21::. That is,
+ `square-list' should behave as follows:
+
+ (square-tree
+ (list 1
+ (list 2 (list 3 4) 5)
+ (list 6 7)))
+ (1 (4 (9 16) 25) (36 49))
+
+ Define `square-tree' both directly (i.e., without using any
+ higher-order procedures) and also by using `map' and recursion.
+
+ *Exercise 2.31:* Abstract your answer to *Note Exercise 2-30:: to
+ produce a procedure `tree-map' with the property that
+ `square-tree' could be defined as
+
+ (define (square-tree tree) (tree-map square tree))
+
+ *Exercise 2.32:* We can represent a set as a list of distinct
+ elements, and we can represent the set of all subsets of the set as
+ a list of lists. For example, if the set is `(1 2 3)', then the
+ set of all subsets is `(() (3) (2) (2 3) (1) (1 3) (1 2) (1 2
+ 3))'. Complete the following definition of a procedure that
+ generates the set of subsets of a set and give a clear explanation
+ of why it works:
+
+ (define (subsets s)
+ (if (null? s)
+ (list nil)
+ (let ((rest (subsets (cdr s))))
+ (append rest (map <??> rest)))))
+
+ ---------- Footnotes ----------
+
+ (1) The order of the first two clauses in the `cond' matters, since
+the empty list satisfies `null?' and also is not a pair.
+
+\1f
+File: sicp.info, Node: 2-2-3, Next: 2-2-4, Prev: 2-2-2, Up: 2-2
+
+2.2.3 Sequences as Conventional Interfaces
+------------------------------------------
+
+In working with compound data, we've stressed how data abstraction
+permits us to design programs without becoming enmeshed in the details
+of data representations, and how abstraction preserves for us the
+flexibility to experiment with alternative representations. In this
+section, we introduce another powerful design principle for working
+with data structures--the use of "conventional interfaces".
+
+ In section *Note 1-3:: we saw how program abstractions, implemented
+as higher-order procedures, can capture common patterns in programs
+that deal with numerical data. Our ability to formulate analogous
+operations for working with compound data depends crucially on the
+style in which we manipulate our data structures. Consider, for
+example, the following procedure, analogous to the `count-leaves'
+procedure of section *Note 2-2-2::, which takes a tree as argument and
+computes the sum of the squares of the leaves that are odd:
+
+ (define (sum-odd-squares tree)
+ (cond ((null? tree) 0)
+ ((not (pair? tree))
+ (if (odd? tree) (square tree) 0))
+ (else (+ (sum-odd-squares (car tree))
+ (sum-odd-squares (cdr tree))))))
+
+ On the surface, this procedure is very different from the following
+one, which constructs a list of all the even Fibonacci numbers
+_Fib_(k), where k is less than or equal to a given integer n:
+
+ (define (even-fibs n)
+ (define (next k)
+ (if (> k n)
+ nil
+ (let ((f (fib k)))
+ (if (even? f)
+ (cons f (next (+ k 1)))
+ (next (+ k 1))))))
+ (next 0))
+
+ Despite the fact that these two procedures are structurally very
+different, a more abstract description of the two computations reveals
+a great deal of similarity. The first program
+
+ * enumerates the leaves of a tree;
+
+ * filters them, selecting the odd ones;
+
+ * squares each of the selected ones; and
+
+ * accumulates the results using `+', starting with 0.
+
+
+ The second program
+
+ * enumerates the integers from 0 to n;
+
+ * computes the Fibonacci number for each integer;
+
+ * filters them, selecting the even ones; and
+
+ * accumulates the results using `cons', starting with the empty
+ list.
+
+
+ A signal-processing engineer would find it natural to conceptualize
+these processes in terms of signals flowing through a cascade of
+stages, each of which implements part of the program plan, as shown in
+*Note Figure 2-7::. In `sum-odd-squares', we begin with an "enumerator",
+which generates a "signal" consisting of the leaves of a given tree.
+This signal is passed through a "filter", which eliminates all but the
+odd elements. The resulting signal is in turn passed through a "map",
+which is a "transducer" that applies the `square' procedure to each
+element. The output of the map is then fed to an "accumulator", which
+combines the elements using `+', starting from an initial 0. The plan
+for `even-fibs' is analogous.
+
+ *Figure 2.7:* The signal-flow plans for the procedures
+ `sum-odd-squares' (top) and `even-fibs' (bottom) reveal the
+ commonality between the two programs.
+
+ +-------------+ +-------------+ +-------------+ +-------------+
+ | enumerate: |-->| filter: |-->| map: |-->| accumulate: |
+ | tree leaves | | odd? | | square | | +, 0 |
+ +-------------+ +-------------+ +-------------+ +-------------+
+
+ +-------------+ +-------------+ +-------------+ +-------------+
+ | enumerate: |-->| map: |-->| filter: |-->| accumulate: |
+ | integers | | fib | | even? | | cons, () |
+ +-------------+ +-------------+ +-------------+ +-------------+
+
+ Unfortunately, the two procedure definitions above fail to exhibit
+this signal-flow structure. For instance, if we examine the
+`sum-odd-squares' procedure, we find that the enumeration is
+implemented partly by the `null?' and `pair?' tests and partly by the
+tree-recursive structure of the procedure. Similarly, the accumulation
+is found partly in the tests and partly in the addition used in the
+recursion. In general, there are no distinct parts of either procedure
+that correspond to the elements in the signal-flow description. Our
+two procedures decompose the computations in a different way, spreading
+the enumeration over the program and mingling it with the map, the
+filter, and the accumulation. If we could organize our programs to
+make the signal-flow structure manifest in the procedures we write, this
+would increase the conceptual clarity of the resulting code.
+
+Sequence Operations
+...................
+
+The key to organizing programs so as to more clearly reflect the
+signal-flow structure is to concentrate on the "signals" that flow from
+one stage in the process to the next. If we represent these signals as
+lists, then we can use list operations to implement the processing at
+each of the stages. For instance, we can implement the mapping stages
+of the signal-flow diagrams using the `map' procedure from section
+*Note 2-2-1:::
+
+ (map square (list 1 2 3 4 5))
+ (1 4 9 16 25)
+
+ Filtering a sequence to select only those elements that satisfy a
+given predicate is accomplished by
+
+ (define (filter predicate sequence)
+ (cond ((null? sequence) nil)
+ ((predicate (car sequence))
+ (cons (car sequence)
+ (filter predicate (cdr sequence))))
+ (else (filter predicate (cdr sequence)))))
+
+ For example,
+
+ (filter odd? (list 1 2 3 4 5))
+ (1 3 5)
+
+ Accumulations can be implemented by
+
+ (define (accumulate op initial sequence)
+ (if (null? sequence)
+ initial
+ (op (car sequence)
+ (accumulate op initial (cdr sequence)))))
+
+ (accumulate + 0 (list 1 2 3 4 5))
+ 15
+
+ (accumulate * 1 (list 1 2 3 4 5))
+ 120
+
+ (accumulate cons nil (list 1 2 3 4 5))
+ (1 2 3 4 5)
+
+ All that remains to implement signal-flow diagrams is to enumerate
+the sequence of elements to be processed. For `even-fibs', we need to
+generate the sequence of integers in a given range, which we can do as
+follows:
+
+ (define (enumerate-interval low high)
+ (if (> low high)
+ nil
+ (cons low (enumerate-interval (+ low 1) high))))
+
+ (enumerate-interval 2 7)
+ (2 3 4 5 6 7)
+
+ To enumerate the leaves of a tree, we can use(1)
+
+ (define (enumerate-tree tree)
+ (cond ((null? tree) nil)
+ ((not (pair? tree)) (list tree))
+ (else (append (enumerate-tree (car tree))
+ (enumerate-tree (cdr tree))))))
+
+ (enumerate-tree (list 1 (list 2 (list 3 4)) 5))
+ (1 2 3 4 5)
+
+ Now we can reformulate `sum-odd-squares' and `even-fibs' as in the
+signal-flow diagrams. For `sum-odd-squares', we enumerate the sequence
+of leaves of the tree, filter this to keep only the odd numbers in the
+sequence, square each element, and sum the results:
+
+ (define (sum-odd-squares tree)
+ (accumulate +
+ 0
+ (map square
+ (filter odd?
+ (enumerate-tree tree)))))
+
+ For `even-fibs', we enumerate the integers from 0 to n, generate the
+Fibonacci number for each of these integers, filter the resulting
+sequence to keep only the even elements, and accumulate the results
+into a list:
+
+ (define (even-fibs n)
+ (accumulate cons
+ nil
+ (filter even?
+ (map fib
+ (enumerate-interval 0 n)))))
+
+ The value of expressing programs as sequence operations is that this
+helps us make program designs that are modular, that is, designs that
+are constructed by combining relatively independent pieces. We can
+encourage modular design by providing a library of standard components
+together with a conventional interface for connecting the components in
+flexible ways.
+
+ Modular construction is a powerful strategy for controlling
+complexity in engineering design. In real signal-processing
+applications, for example, designers regularly build systems by
+cascading elements selected from standardized families of filters and
+transducers. Similarly, sequence operations provide a library of
+standard program elements that we can mix and match. For instance, we
+can reuse pieces from the `sum-odd-squares' and `even-fibs' procedures
+in a program that constructs a list of the squares of the first n + 1
+Fibonacci numbers:
+
+ (define (list-fib-squares n)
+ (accumulate cons
+ nil
+ (map square
+ (map fib
+ (enumerate-interval 0 n)))))
+
+ (list-fib-squares 10)
+ (0 1 1 4 9 25 64 169 441 1156 3025)
+
+ We can rearrange the pieces and use them in computing the product of
+the odd integers in a sequence:
+
+ (define (product-of-squares-of-odd-elements sequence)
+ (accumulate *
+ 1
+ (map square
+ (filter odd? sequence))))
+
+ (product-of-squares-of-odd-elements (list 1 2 3 4 5))
+ 225
+
+ We can also formulate conventional data-processing applications in
+terms of sequence operations. Suppose we have a sequence of personnel
+records and we want to find the salary of the highest-paid programmer.
+Assume that we have a selector `salary' that returns the salary of a
+record, and a predicate `programmer?' that tests if a record is for a
+programmer. Then we can write
+
+ (define (salary-of-highest-paid-programmer records)
+ (accumulate max
+ 0
+ (map salary
+ (filter programmer? records))))
+
+ These examples give just a hint of the vast range of operations that
+can be expressed as sequence operations.(2)
+
+ Sequences, implemented here as lists, serve as a conventional
+interface that permits us to combine processing modules. Additionally,
+when we uniformly represent structures as sequences, we have localized
+the data-structure dependencies in our programs to a small number of
+sequence operations. By changing these, we can experiment with
+alternative representations of sequences, while leaving the overall
+design of our programs intact. We will exploit this capability in
+section *Note 3-5::, when we generalize the sequence-processing
+paradigm to admit infinite sequences.
+
+ *Exercise 2.33:* Fill in the missing expressions to complete the
+ following definitions of some basic list-manipulation operations
+ as accumulations:
+
+ (define (map p sequence)
+ (accumulate (lambda (x y) <??>) nil sequence))
+
+ (define (append seq1 seq2)
+ (accumulate cons <??> <??>))
+
+ (define (length sequence)
+ (accumulate <??> 0 sequence))
+
+ *Exercise 2.34:* Evaluating a polynomial in x at a given value of
+ x can be formulated as an accumulation. We evaluate the polynomial
+
+ a_n r^n | a_(n-1) r^(n-1) + ... + a_1 r + a_0
+
+ using a well-known algorithm called "Horner's rule", which
+ structures the computation as
+
+ (... (a_n r + a_(n-1)) r + ... + a_1) r + a_0
+
+ In other words, we start with a_n, multiply by x, add a_(n-1),
+ multiply by x, and so on, until we reach a_0.(3)
+
+ Fill in the following template to produce a procedure that
+ evaluates a polynomial using Horner's rule. Assume that the
+ coefficients of the polynomial are arranged in a sequence, from
+ a_0 through a_n.
+
+ (define (horner-eval x coefficient-sequence)
+ (accumulate (lambda (this-coeff higher-terms) <??>)
+ 0
+ coefficient-sequence))
+
+ For example, to compute 1 + 3x + 5x^3 + x^(5) at x = 2 you would
+ evaluate
+
+ (horner-eval 2 (list 1 3 0 5 0 1))
+
+ *Exercise 2.35:* Redefine `count-leaves' from section *Note
+ 2-2-2:: as an accumulation:
+
+ (define (count-leaves t)
+ (accumulate <??> <??> (map <??> <??>)))
+
+ *Exercise 2.36:* The procedure `accumulate-n' is similar to
+ `accumulate' except that it takes as its third argument a sequence
+ of sequences, which are all assumed to have the same number of
+ elements. It applies the designated accumulation procedure to
+ combine all the first elements of the sequences, all the second
+ elements of the sequences, and so on, and returns a sequence of
+ the results. For instance, if `s' is a sequence containing four
+ sequences, `((1 2 3) (4 5 6) (7 8 9) (10 11 12)),' then the value
+ of `(accumulate-n + 0 s)' should be the sequence `(22 26 30)'.
+ Fill in the missing expressions in the following definition of
+ `accumulate-n':
+
+ (define (accumulate-n op init seqs)
+ (if (null? (car seqs))
+ nil
+ (cons (accumulate op init <??>)
+ (accumulate-n op init <??>))))
+
+Exercise 2.37
+.............
+
+Suppose we represent vectors v = (v_i) as sequences of numbers, and
+matrices m = (m_(ij)) as sequences of vectors (the rows of the matrix).
+For example, the matrix
+
+ +- -+
+ | 1 2 3 4 |
+ | 4 5 6 6 |
+ | 6 7 8 9 |
+ +- -+
+
+is represented as the sequence `((1 2 3 4) (4 5 6 6) (6 7 8 9))'. With
+this representation, we can use sequence operations to concisely
+express the basic matrix and vector operations. These operations
+(which are described in any book on matrix algebra) are the following:
+
+ __
+ (dot-product v w) returns the sum >_i v_i w_i
+
+ (matrix-*-vector m v) returns the vector t,
+ __
+ where t_i = >_j m_(ij) v_j
+
+ (matrix-*-matrix m n) returns the matrix p,
+ __
+ where p_(ij) = >_k m_(ik) n_(kj)
+
+ (transpose m) returns the matrix n,
+ where n_(ij) = m_(ji)
+
+ We can define the dot product as(4)
+
+ (define (dot-product v w)
+ (accumulate + 0 (map * v w)))
+
+ Fill in the missing expressions in the following procedures for
+computing the other matrix operations. (The procedure `accumulate-n'
+is defined in *Note Exercise 2-36::.)
+
+ (define (matrix-*-vector m v)
+ (map <??> m))
+
+ (define (transpose mat)
+ (accumulate-n <??> <??> mat))
+
+ (define (matrix-*-matrix m n)
+ (let ((cols (transpose n)))
+ (map <??> m)))
+
+ *Exercise 2.38:* The `accumulate' procedure is also known as
+ `fold-right', because it combines the first element of the
+ sequence with the result of combining all the elements to the
+ right. There is also a `fold-left', which is similar to
+ `fold-right', except that it combines elements working in the
+ opposite direction:
+
+ (define (fold-left op initial sequence)
+ (define (iter result rest)
+ (if (null? rest)
+ result
+ (iter (op result (car rest))
+ (cdr rest))))
+ (iter initial sequence))
+
+ What are the values of
+
+ (fold-right / 1 (list 1 2 3))
+
+ (fold-left / 1 (list 1 2 3))
+
+ (fold-right list nil (list 1 2 3))
+
+ (fold-left list nil (list 1 2 3))
+
+ Give a property that `op' should satisfy to guarantee that
+ `fold-right' and `fold-left' will produce the same values for any
+ sequence.
+
+ *Exercise 2.39:* Complete the following definitions of `reverse'
+ (*Note Exercise 2-18::) in terms of `fold-right' and `fold-left'
+ from *Note Exercise 2-38:::
+
+ (define (reverse sequence)
+ (fold-right (lambda (x y) <??>) nil sequence))
+
+ (define (reverse sequence)
+ (fold-left (lambda (x y) <??>) nil sequence))
+
+Nested Mappings
+...............
+
+We can extend the sequence paradigm to include many computations that
+are commonly expressed using nested loops.(5) Consider this problem:
+Given a positive integer n, find all ordered pairs of distinct positive
+integers i and j, where 1 <= j< i<= n, such that i + j is prime. For
+example, if n is 6, then the pairs are the following:
+
+ i | 2 3 4 4 5 6 6
+ j | 1 2 1 3 2 1 5
+ ------+---------------
+ i + j | 3 5 5 7 7 7 11
+
+ A natural way to organize this computation is to generate the
+sequence of all ordered pairs of positive integers less than or equal
+to n, filter to select those pairs whose sum is prime, and then, for
+each pair (i, j) that passes through the filter, produce the triple
+(i,j,i + j).
+
+ Here is a way to generate the sequence of pairs: For each integer i
+<= n, enumerate the integers j<i, and for each such i and j generate
+the pair (i,j). In terms of sequence operations, we map along the
+sequence `(enumerate-interval 1 n)'. For each i in this sequence, we
+map along the sequence `(enumerate-interval 1 (- i 1))'. For each j in
+this latter sequence, we generate the pair `(list i j)'. This gives us
+a sequence of pairs for each i. Combining all the sequences for all
+the i (by accumulating with `append') produces the required sequence of
+pairs:(6)
+
+ (accumulate append
+ nil
+ (map (lambda (i)
+ (map (lambda (j) (list i j))
+ (enumerate-interval 1 (- i 1))))
+ (enumerate-interval 1 n)))
+
+ The combination of mapping and accumulating with `append' is so
+common in this sort of program that we will isolate it as a separate
+procedure:
+
+ (define (flatmap proc seq)
+ (accumulate append nil (map proc seq)))
+
+ Now filter this sequence of pairs to find those whose sum is prime.
+The filter predicate is called for each element of the sequence; its
+argument is a pair and it must extract the integers from the pair.
+Thus, the predicate to apply to each element in the sequence is
+
+ (define (prime-sum? pair)
+ (prime? (+ (car pair) (cadr pair))))
+
+ Finally, generate the sequence of results by mapping over the
+filtered pairs using the following procedure, which constructs a triple
+consisting of the two elements of the pair along with their sum:
+
+ (define (make-pair-sum pair)
+ (list (car pair) (cadr pair) (+ (car pair) (cadr pair))))
+
+ Combining all these steps yields the complete procedure:
+
+ (define (prime-sum-pairs n)
+ (map make-pair-sum
+ (filter prime-sum?
+ (flatmap
+ (lambda (i)
+ (map (lambda (j) (list i j))
+ (enumerate-interval 1 (- i 1))))
+ (enumerate-interval 1 n)))))
+
+ Nested mappings are also useful for sequences other than those that
+enumerate intervals. Suppose we wish to generate all the permutations
+of a set S; that is, all the ways of ordering the items in the set.
+For instance, the permutations of {1,2,3} are {1,2,3}, {1,3,2},
+{2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. Here is a plan for generating
+the permutations of S: For each item x in S, recursively generate the
+sequence of permutations of S - x,(7) and adjoin x to the front of each
+one. This yields, for each x in S, the sequence of permutations of S
+that begin with x. Combining these sequences for all x gives all the
+permutations of S:(8)
+
+ (define (permutations s)
+ (if (null? s) ; empty set?
+ (list nil) ; sequence containing empty set
+ (flatmap (lambda (x)
+ (map (lambda (p) (cons x p))
+ (permutations (remove x s))))
+ s)))
+
+ Notice how this strategy reduces the problem of generating
+permutations of S to the problem of generating the permutations of sets
+with fewer elements than S. In the terminal case, we work our way down
+to the empty list, which represents a set of no elements. For this, we
+generate `(list nil)', which is a sequence with one item, namely the
+set with no elements. The `remove' procedure used in `permutations'
+returns all the items in a given sequence except for a given item.
+This can be expressed as a simple filter:
+
+ (define (remove item sequence)
+ (filter (lambda (x) (not (= x item)))
+ sequence))
+
+ *Exercise 2.40:* Define a procedure `unique-pairs' that, given an
+ integer n, generates the sequence of pairs (i,j) with 1 <= j< i <=
+ n. Use `unique-pairs' to simplify the definition of
+ `prime-sum-pairs' given above.
+
+ *Exercise 2.41:* Write a procedure to find all ordered triples of
+ distinct positive integers i, j, and k less than or equal to a
+ given integer n that sum to a given integer s.
+
+ *Figure 2.8:* A solution to the eight-queens puzzle.
+
+ +---+---+---+---+---+---+---+---+
+ | | | | | | Q | | |
+ +---+---+---+---+---+---+---+---+
+ | | | Q | | | | | |
+ +---+---+---+---+---+---+---+---+
+ | Q | | | | | | | |
+ +---+---+---+---+---+---+---+---+
+ | | | | | | | Q | |
+ +---+---+---+---+---+---+---+---+
+ | | | | | Q | | | |
+ +---+---+---+---+---+---+---+---+
+ | | | | | | | | Q |
+ +---+---+---+---+---+---+---+---+
+ | | Q | | | | | | |
+ +---+---+---+---+---+---+---+---+
+ | | | | Q | | | | |
+ +---+---+---+---+---+---+---+---+
+
+ *Exercise 2.42:* The "eight-queens puzzle" asks how to place eight
+ queens on a chessboard so that no queen is in check from any other
+ (i.e., no two queens are in the same row, column, or diagonal).
+ One possible solution is shown in *Note Figure 2-8::. One way to
+ solve the puzzle is to work across the board, placing a queen in
+ each column. Once we have placed k - 1 queens, we must place the
+ kth queen in a position where it does not check any of the queens
+ already on the board. We can formulate this approach recursively:
+ Assume that we have already generated the sequence of all possible
+ ways to place k - 1 queens in the first k - 1 columns of the
+ board. For each of these ways, generate an extended set of
+ positions by placing a queen in each row of the kth column. Now
+ filter these, keeping only the positions for which the queen in
+ the kth column is safe with respect to the other queens. This
+ produces the sequence of all ways to place k queens in the first k
+ columns. By continuing this process, we will produce not only one
+ solution, but all solutions to the puzzle.
+
+ We implement this solution as a procedure `queens', which returns a
+ sequence of all solutions to the problem of placing n queens on an
+ n*n chessboard. `Queens' has an internal procedure `queen-cols'
+ that returns the sequence of all ways to place queens in the first
+ k columns of the board.
+
+ (define (queens board-size)
+ (define (queen-cols k)
+ (if (= k 0)
+ (list empty-board)
+ (filter
+ (lambda (positions) (safe? k positions))
+ (flatmap
+ (lambda (rest-of-queens)
+ (map (lambda (new-row)
+ (adjoin-position new-row k rest-of-queens))
+ (enumerate-interval 1 board-size)))
+ (queen-cols (- k 1))))))
+ (queen-cols board-size))
+
+ In this procedure `rest-of-queens' is a way to place k - 1 queens
+ in the first k - 1 columns, and `new-row' is a proposed row in
+ which to place the queen for the kth column. Complete the program
+ by implementing the representation for sets of board positions,
+ including the procedure `adjoin-position', which adjoins a new
+ row-column position to a set of positions, and `empty-board',
+ which represents an empty set of positions. You must also write
+ the procedure `safe?', which determines for a set of positions,
+ whether the queen in the kth column is safe with respect to the
+ others. (Note that we need only check whether the new queen is
+ safe--the other queens are already guaranteed safe with respect to
+ each other.)
+
+ *Exercise 2.43:* Louis Reasoner is having a terrible time doing
+ *Note Exercise 2-42::. His `queens' procedure seems to work, but
+ it runs extremely slowly. (Louis never does manage to wait long
+ enough for it to solve even the 6*6 case.) When Louis asks Eva Lu
+ Ator for help, she points out that he has interchanged the order
+ of the nested mappings in the `flatmap', writing it as
+
+ (flatmap
+ (lambda (new-row)
+ (map (lambda (rest-of-queens)
+ (adjoin-position new-row k rest-of-queens))
+ (queen-cols (- k 1))))
+ (enumerate-interval 1 board-size))
+
+ Explain why this interchange makes the program run slowly.
+ Estimate how long it will take Louis's program to solve the
+ eight-queens puzzle, assuming that the program in *Note Exercise
+ 2-42:: solves the puzzle in time T.
+
+ ---------- Footnotes ----------
+
+ (1) This is, in fact, precisely the `fringe' procedure from *Note
+Exercise 2-28::. Here we've renamed it to emphasize that it is part of
+a family of general sequence-manipulation procedures.
+
+ (2) Richard Waters (1979) developed a program that automatically
+analyzes traditional Fortran programs, viewing them in terms of maps,
+filters, and accumulations. He found that fully 90 percent of the code
+in the Fortran Scientific Subroutine Package fits neatly into this
+paradigm. One of the reasons for the success of Lisp as a programming
+language is that lists provide a standard medium for expressing ordered
+collections so that they can be manipulated using higher-order
+operations. The programming language APL owes much of its power and
+appeal to a similar choice. In APL all data are represented as arrays,
+and there is a universal and convenient set of generic operators for
+all sorts of array operations.
+
+ (3) According to Knuth (1981), this rule was formulated by W. G.
+Horner early in the nineteenth century, but the method was actually used
+by Newton over a hundred years earlier. Horner's rule evaluates the
+polynomial using fewer additions and multiplications than does the
+straightforward method of first computing a_n x^n, then adding
+a_(n-1)x^(n-1), and so on. In fact, it is possible to prove that any
+algorithm for evaluating arbitrary polynomials must use at least as
+many additions and multiplications as does Horner's rule, and thus
+Horner's rule is an optimal algorithm for polynomial evaluation. This
+was proved (for the number of additions) by A. M. Ostrowski in a 1954
+paper that essentially founded the modern study of optimal algorithms.
+The analogous statement for multiplications was proved by V. Y. Pan in
+1966. The book by Borodin and Munro (1975) provides an overview of
+these and other results about optimal algorithms.
+
+ (4) This definition uses the extended version of `map' described in
+*Note Footnote 12::.
+
+ (5) This approach to nested mappings was shown to us by David
+Turner, whose languages KRC and Miranda provide elegant formalisms for
+dealing with these constructs. The examples in this section (see also
+*Note Exercise 2-42::) are adapted from Turner 1981. In section *Note
+3-5-3::, we'll see how this approach generalizes to infinite sequences.
+
+ (6) We're representing a pair here as a list of two elements rather
+than as a Lisp pair. Thus, the "pair" (i,j) is represented as `(list i
+j)', not `(cons i j)'.
+
+ (7) The set S - x is the set of all elements of S, excluding x.
+
+ (8) Semicolons in Scheme code are used to introduce "comments".
+Everything from the semicolon to the end of the line is ignored by the
+interpreter. In this book we don't use many comments; we try to make
+our programs self-documenting by using descriptive names.
+
+\1f
+File: sicp.info, Node: 2-2-4, Prev: 2-2-3, Up: 2-2
+
+2.2.4 Example: A Picture Language
+---------------------------------
+
+This section presents a simple language for drawing pictures that
+illustrates the power of data abstraction and closure, and also
+exploits higher-order procedures in an essential way. The language is
+designed to make it easy to experiment with patterns such as the ones
+in *Note Figure 2-9::, which are composed of repeated elements that are
+shifted and scaled.(1) In this language, the data objects being
+combined are represented as procedures rather than as list structure.
+Just as `cons', which satisfies the closure property, allowed us to
+easily build arbitrarily complicated list structure, the operations in
+this language, which also satisfy the closure property, allow us to
+easily build arbitrarily complicated patterns.
+
+ *Figure 2.9:* Designs generated with the picture language.
+
+ [two graphic images not included]
+
+The picture language
+....................
+
+When we began our study of programming in section *Note 1-1::, we
+emphasized the importance of describing a language by focusing on the
+language's primitives, its means of combination, and its means of
+abstraction. We'll follow that framework here.
+
+ Part of the elegance of this picture language is that there is only
+one kind of element, called a "painter". A painter draws an image that
+is shifted and scaled to fit within a designated parallelogram-shaped
+frame. For example, there's a primitive painter we'll call `wave' that
+makes a crude line drawing, as shown in *Note Figure 2-10::. The
+actual shape of the drawing depends on the frame--all four images in
+*Note Figure 2-10:: are produced by the same `wave' painter, but with
+respect to four different frames. Painters can be more elaborate than
+this: The primitive painter called `rogers' paints a picture of MIT's
+founder, William Barton Rogers, as shown in *Note Figure 2-11::.(2) The
+four images in *Note Figure 2-11:: are drawn with respect to the same
+four frames as the `wave' images in *Note Figure 2-10::.
+
+ To combine images, we use various operations that construct new
+painters from given painters. For example, the `beside' operation
+takes two painters and produces a new, compound painter that draws the
+first painter's image in the left half of the frame and the second
+painter's image in the right half of the frame. Similarly, `below'
+takes two painters and produces a compound painter that draws the first
+painter's image below the second painter's image. Some operations
+transform a single painter to produce a new painter. For example,
+`flip-vert' takes a painter and produces a painter that draws its image
+upside-down, and `flip-horiz' produces a painter that draws the
+original painter's image left-to-right reversed.
+
+ *Figure 2.10:* Images produced by the `wave' painter, with respect
+ to four different frames. The frames, shown with dotted lines,
+ are not part of the images.
+
+ [four graphic images not included]
+
+ *Figure 2.11:* Images of William Barton Rogers, founder and first
+ president of MIT, painted with respect to the same four frames as
+ in *Note Figure 2-10:: (original image reprinted with the
+ permission of the MIT Museum).
+
+ [four graphic images not included]
+
+ *Note Figure 2-12:: shows the drawing of a painter called `wave4'
+that is built up in two stages starting from `wave':
+
+ (define wave2 (beside wave (flip-vert wave)))
+ (define wave4 (below wave2 wave2))
+
+ *Figure 2.12:* Creating a complex figure, starting from the `wave'
+ painter of *Note Figure 2-10::.
+
+ [two graphic images not included]
+
+ (define wave2 (define wave4
+ (beside wave (flip-vert wave))) (below wave2 wave2))
+
+ In building up a complex image in this manner we are exploiting the
+fact that painters are closed under the language's means of
+combination. The `beside' or `below' of two painters is itself a
+painter; therefore, we can use it as an element in making more complex
+painters. As with building up list structure using `cons', the closure
+of our data under the means of combination is crucial to the ability to
+create complex structures while using only a few operations.
+
+ Once we can combine painters, we would like to be able to abstract
+typical patterns of combining painters. We will implement the painter
+operations as Scheme procedures. This means that we don't need a
+special abstraction mechanism in the picture language: Since the means
+of combination are ordinary Scheme procedures, we automatically have
+the capability to do anything with painter operations that we can do
+with procedures. For example, we can abstract the pattern in `wave4' as
+
+ (define (flipped-pairs painter)
+ (let ((painter2 (beside painter (flip-vert painter))))
+ (below painter2 painter2)))
+
+and define `wave4' as an instance of this pattern:
+
+ (define wave4 (flipped-pairs wave))
+
+ We can also define recursive operations. Here's one that makes
+painters split and branch towards the right as shown in figures *Note
+Figure 2-13:: and *Note Figure 2-14:::
+
+ (define (right-split painter n)
+ (if (= n 0)
+ painter
+ (let ((smaller (right-split painter (- n 1))))
+ (beside painter (below smaller smaller)))))
+
+ *Figure 2.13:* Recursive plans for `right-split' and
+ `corner-split'.
+
+ +-------------+-------------+ +------+------+-------------+
+ | | | | up- | up- | |
+ | | right-split | | split| split| corner-split|
+ | | | | | | |
+ | | n-1 | | n-1 | n-1 | n-1 |
+ | | | | | | |
+ | identity +-------------+ +------+------+-------------+
+ | | | | | right-split |
+ | | right-split | | | n-1 |
+ | | | | identity +-------------+
+ | | n-1 | | | right-split |
+ | | | | | n-1 |
+ +-------------+-------------+ +-------------+-------------+
+
+ right-split n corner-split n
+
+ We can produce balanced patterns by branching upwards as well as
+towards the right (see *Note Exercise 2-44:: and figures *Note Figure
+2-13:: and *Note Figure 2-14::):
+
+ (define (corner-split painter n)
+ (if (= n 0)
+ painter
+ (let ((up (up-split painter (- n 1)))
+ (right (right-split painter (- n 1))))
+ (let ((top-left (beside up up))
+ (bottom-right (below right right))
+ (corner (corner-split painter (- n 1))))
+ (beside (below painter top-left)
+ (below bottom-right corner))))))
+
+ *Figure 2.14:* The recursive operations `right-split' and
+ `corner-split' applied to the painters `wave' and `rogers'.
+ Combining four `corner-split' figures produces symmetric
+ `square-limit' designs as shown in *Note Figure 2-9::.
+
+ [two graphic images not included]
+
+ (right-split wave 4) (right-split rogers 4)
+
+ [two graphic images not included]
+
+ (corner-split wave 4) (corner-split rogers 4)
+
+ By placing four copies of a `corner-split' appropriately, we obtain a
+pattern called `square-limit', whose application to `wave' and `rogers'
+is shown in *Note Figure 2-9:::
+
+ (define (square-limit painter n)
+ (let ((quarter (corner-split painter n)))
+ (let ((half (beside (flip-horiz quarter) quarter)))
+ (below (flip-vert half) half))))
+
+ *Exercise 2.44:* Define the procedure `up-split' used by
+ `corner-split'. It is similar to `right-split', except that it
+ switches the roles of `below' and `beside'.
+
+Higher-order operations
+.......................
+
+In addition to abstracting patterns of combining painters, we can work
+at a higher level, abstracting patterns of combining painter
+operations. That is, we can view the painter operations as elements to
+manipulate and can write means of combination for these
+elements--procedures that take painter operations as arguments and
+create new painter operations.
+
+ For example, `flipped-pairs' and `square-limit' each arrange four
+copies of a painter's image in a square pattern; they differ only in
+how they orient the copies. One way to abstract this pattern of
+painter combination is with the following procedure, which takes four
+one-argument painter operations and produces a painter operation that
+transforms a given painter with those four operations and arranges the
+results in a square. `Tl', `tr', `bl', and `br' are the
+transformations to apply to the top left copy, the top right copy, the
+bottom left copy, and the bottom right copy, respectively.
+
+ (define (square-of-four tl tr bl br)
+ (lambda (painter)
+ (let ((top (beside (tl painter) (tr painter)))
+ (bottom (beside (bl painter) (br painter))))
+ (below bottom top))))
+
+ Then `flipped-pairs' can be defined in terms of `square-of-four' as
+follows:(3)
+
+ (define (flipped-pairs painter)
+ (let ((combine4 (square-of-four identity flip-vert
+ identity flip-vert)))
+ (combine4 painter)))
+
+and `square-limit' can be expressed as(4)
+
+ (define (square-limit painter n)
+ (let ((combine4 (square-of-four flip-horiz identity
+ rotate180 flip-vert)))
+ (combine4 (corner-split painter n))))
+
+ *Exercise 2.45:* `Right-split' and `up-split' can be expressed as
+ instances of a general splitting operation. Define a procedure
+ `split' with the property that evaluating
+
+ (define right-split (split beside below))
+ (define up-split (split below beside))
+
+ produces procedures `right-split' and `up-split' with the same
+ behaviors as the ones already defined.
+
+Frames
+......
+
+Before we can show how to implement painters and their means of
+combination, we must first consider frames. A frame can be described
+by three vectors--an origin vector and two edge vectors. The origin
+vector specifies the offset of the frame's origin from some absolute
+origin in the plane, and the edge vectors specify the offsets of the
+frame's corners from its origin. If the edges are perpendicular, the
+frame will be rectangular. Otherwise the frame will be a more general
+parallelogram.
+
+ *Note Figure 2-15:: shows a frame and its associated vectors. In
+accordance with data abstraction, we need not be specific yet about how
+frames are represented, other than to say that there is a constructor
+`make-frame', which takes three vectors and produces a frame, and three
+corresponding selectors `origin-frame', `edge1-frame', and
+`edge2-frame' (see *Note Exercise 2-47::).
+
+ *Figure 2.15:* A frame is described by three vectors - an origin
+ and two edges.
+
+ __
+ __-- \
+ __-- \
+ __ __-- \ __
+ |\ __-- \__-|
+ \- __--
+ frame \ __--
+ edge2 \ __-- frame
+ vector \ __-- edge1
+ \_-- vector
+ - <--+
+ frame |
+ origin +-- (0,0) point
+ vector on display screen
+
+ We will use coordinates in the unit square (0<= x,y<= 1) to specify
+images. With each frame, we associate a "frame coordinate map", which
+will be used to shift and scale images to fit the frame. The map
+transforms the unit square into the frame by mapping the vector v =
+(x,y) to the vector sum
+
+ Origin(Frame) + r * Edge_1(Frame) + y * Edge_2(Frame)
+
+For example, (0,0) is mapped to the origin of the frame, (1,1) to the
+vertex diagonally opposite the origin, and (0.5,0.5) to the center of
+the frame. We can create a frame's coordinate map with the following
+procedure:(5)
+
+ (define (frame-coord-map frame)
+ (lambda (v)
+ (add-vect
+ (origin-frame frame)
+ (add-vect (scale-vect (xcor-vect v)
+ (edge1-frame frame))
+ (scale-vect (ycor-vect v)
+ (edge2-frame frame))))))
+
+ Observe that applying `frame-coord-map' to a frame returns a
+procedure that, given a vector, returns a vector. If the argument
+vector is in the unit square, the result vector will be in the frame.
+For example,
+
+ ((frame-coord-map a-frame) (make-vect 0 0))
+
+returns the same vector as
+
+ (origin-frame a-frame)
+
+ *Exercise 2.46:* A two-dimensional vector v running from the
+ origin to a point can be represented as a pair consisting of an
+ x-coordinate and a y-coordinate. Implement a data abstraction for
+ vectors by giving a constructor `make-vect' and corresponding
+ selectors `xcor-vect' and `ycor-vect'. In terms of your selectors
+ and constructor, implement procedures `add-vect', `sub-vect', and
+ `scale-vect' that perform the operations vector addition, vector
+ subtraction, and multiplying a vector by a scalar:
+
+ (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)
+ (x_1, y_1) - (x_2, y_2) = (x_1 - x_2, y_1 - y_2)
+ s * (x, y) = (sx, sy)
+
+ *Exercise 2.47:* Here are two possible constructors for frames:
+
+ (define (make-frame origin edge1 edge2)
+ (list origin edge1 edge2))
+
+ (define (make-frame origin edge1 edge2)
+ (cons origin (cons edge1 edge2)))
+
+ For each constructor supply the appropriate selectors to produce an
+ implementation for frames.
+
+Painters
+........
+
+A painter is represented as a procedure that, given a frame as
+argument, draws a particular image shifted and scaled to fit the frame.
+That is to say, if `p' is a painter and `f' is a frame, then we
+produce `p''s image in `f' by calling `p' with `f' as argument.
+
+ The details of how primitive painters are implemented depend on the
+particular characteristics of the graphics system and the type of image
+to be drawn. For instance, suppose we have a procedure `draw-line'
+that draws a line on the screen between two specified points. Then we
+can create painters for line drawings, such as the `wave' painter in
+*Note Figure 2-10::, from lists of line segments as follows:(6)
+
+ (define (segments->painter segment-list)
+ (lambda (frame)
+ (for-each
+ (lambda (segment)
+ (draw-line
+ ((frame-coord-map frame) (start-segment segment))
+ ((frame-coord-map frame) (end-segment segment))))
+ segment-list)))
+
+ The segments are given using coordinates with respect to the unit
+square. For each segment in the list, the painter transforms the
+segment endpoints with the frame coordinate map and draws a line
+between the transformed points.
+
+ Representing painters as procedures erects a powerful abstraction
+barrier in the picture language. We can create and intermix all sorts
+of primitive painters, based on a variety of graphics capabilities. The
+details of their implementation do not matter. Any procedure can serve
+as a painter, provided that it takes a frame as argument and draws
+something scaled to fit the frame.(7)
+
+ *Exercise 2.48:* A directed line segment in the plane can be
+ represented as a pair of vectors--the vector running from the
+ origin to the start-point of the segment, and the vector running
+ from the origin to the end-point of the segment. Use your vector
+ representation from *Note Exercise 2-46:: to define a
+ representation for segments with a constructor `make-segment' and
+ selectors `start-segment' and `end-segment'.
+
+ *Exercise 2.49:* Use `segments->painter' to define the following
+ primitive painters:
+
+ a. The painter that draws the outline of the designated frame.
+
+ b. The painter that draws an "X" by connecting opposite corners
+ of the frame.
+
+ c. The painter that draws a diamond shape by connecting the
+ midpoints of the sides of the frame.
+
+ d. The `wave' painter.
+
+
+Transforming and combining painters
+...................................
+
+An operation on painters (such as `flip-vert' or `beside') works by
+creating a painter that invokes the original painters with respect to
+frames derived from the argument frame. Thus, for example, `flip-vert'
+doesn't have to know how a painter works in order to flip it--it just
+has to know how to turn a frame upside down: The flipped painter just
+uses the original painter, but in the inverted frame.
+
+ Painter operations are based on the procedure `transform-painter',
+which takes as arguments a painter and information on how to transform
+a frame and produces a new painter. The transformed painter, when
+called on a frame, transforms the frame and calls the original painter
+on the transformed frame. The arguments to `transform-painter' are
+points (represented as vectors) that specify the corners of the new
+frame: When mapped into the frame, the first point specifies the new
+frame's origin and the other two specify the ends of its edge vectors.
+Thus, arguments within the unit square specify a frame contained within
+the original frame.
+
+ (define (transform-painter painter origin corner1 corner2)
+ (lambda (frame)
+ (let ((m (frame-coord-map frame)))
+ (let ((new-origin (m origin)))
+ (painter
+ (make-frame new-origin
+ (sub-vect (m corner1) new-origin)
+ (sub-vect (m corner2) new-origin)))))))
+
+ Here's how to flip painter images vertically:
+
+ (define (flip-vert painter)
+ (transform-painter painter
+ (make-vect 0.0 1.0) ; new `origin'
+ (make-vect 1.0 1.0) ; new end of `edge1'
+ (make-vect 0.0 0.0))) ; new end of `edge2'
+
+ Using `transform-painter', we can easily define new transformations.
+For example, we can define a painter that shrinks its image to the
+upper-right quarter of the frame it is given:
+
+ (define (shrink-to-upper-right painter)
+ (transform-painter painter
+ (make-vect 0.5 0.5)
+ (make-vect 1.0 0.5)
+ (make-vect 0.5 1.0)))
+
+ Other transformations rotate images counterclockwise by 90 degrees(8)
+
+ (define (rotate90 painter)
+ (transform-painter painter
+ (make-vect 1.0 0.0)
+ (make-vect 1.0 1.0)
+ (make-vect 0.0 0.0)))
+
+or squash images towards the center of the frame:(9)
+
+ (define (squash-inwards painter)
+ (transform-painter painter
+ (make-vect 0.0 0.0)
+ (make-vect 0.65 0.35)
+ (make-vect 0.35 0.65)))
+
+ Frame transformation is also the key to defining means of combining
+two or more painters. The `beside' procedure, for example, takes two
+painters, transforms them to paint in the left and right halves of an
+argument frame respectively, and produces a new, compound painter.
+When the compound painter is given a frame, it calls the first
+transformed painter to paint in the left half of the frame and calls
+the second transformed painter to paint in the right half of the frame:
+
+ (define (beside painter1 painter2)
+ (let ((split-point (make-vect 0.5 0.0)))
+ (let ((paint-left
+ (transform-painter painter1
+ (make-vect 0.0 0.0)
+ split-point
+ (make-vect 0.0 1.0)))
+ (paint-right
+ (transform-painter painter2
+ split-point
+ (make-vect 1.0 0.0)
+ (make-vect 0.5 1.0))))
+ (lambda (frame)
+ (paint-left frame)
+ (paint-right frame)))))
+
+ Observe how the painter data abstraction, and in particular the
+representation of painters as procedures, makes `beside' easy to
+implement. The `beside' procedure need not know anything about the
+details of the component painters other than that each painter will
+draw something in its designated frame.
+
+ *Exercise 2.50:* Define the transformation `flip-horiz', which
+ flips painters horizontally, and transformations that rotate
+ painters counterclockwise by 180 degrees and 270 degrees.
+
+ *Exercise 2.51:* Define the `below' operation for painters.
+ `Below' takes two painters as arguments. The resulting painter,
+ given a frame, draws with the first painter in the bottom of the
+ frame and with the second painter in the top. Define `below' in
+ two different ways--first by writing a procedure that is analogous
+ to the `beside' procedure given above, and again in terms of
+ `beside' and suitable rotation operations (from *Note Exercise
+ 2-50::).
+
+Levels of language for robust design
+....................................
+
+The picture language exercises some of the critical ideas we've
+introduced about abstraction with procedures and data. The fundamental
+data abstractions, painters, are implemented using procedural
+representations, which enables the language to handle different basic
+drawing capabilities in a uniform way. The means of combination
+satisfy the closure property, which permits us to easily build up
+complex designs. Finally, all the tools for abstracting procedures are
+available to us for abstracting means of combination for painters.
+
+ We have also obtained a glimpse of another crucial idea about
+languages and program design. This is the approach of "stratified
+design", the notion that a complex system should be structured as a
+sequence of levels that are described using a sequence of languages.
+Each level is constructed by combining parts that are regarded as
+primitive at that level, and the parts constructed at each level are
+used as primitives at the next level. The language used at each level
+of a stratified design has primitives, means of combination, and means
+of abstraction appropriate to that level of detail.
+
+ Stratified design pervades the engineering of complex systems. For
+example, in computer engineering, resistors and transistors are
+combined (and described using a language of analog circuits) to produce
+parts such as and-gates and or-gates, which form the primitives of a
+language for digital-circuit design.(10) These parts are combined to
+build processors, bus structures, and memory systems, which are in turn
+combined to form computers, using languages appropriate to computer
+architecture. Computers are combined to form distributed systems, using
+languages appropriate for describing network interconnections, and so
+on.
+
+ As a tiny example of stratification, our picture language uses
+primitive elements (primitive painters) that are created using a
+language that specifies points and lines to provide the lists of line
+segments for `segments->painter', or the shading details for a painter
+like `rogers'. The bulk of our description of the picture language
+focused on combining these primitives, using geometric combiners such
+as `beside' and `below'. We also worked at a higher level, regarding
+`beside' and `below' as primitives to be manipulated in a language
+whose operations, such as `square-of-four', capture common patterns of
+combining geometric combiners.
+
+ Stratified design helps make programs "robust", that is, it makes it
+likely that small changes in a specification will require
+correspondingly small changes in the program. For instance, suppose we
+wanted to change the image based on `wave' shown in *Note Figure 2-9::.
+We could work at the lowest level to change the detailed appearance of
+the `wave' element; we could work at the middle level to change the way
+`corner-split' replicates the `wave'; we could work at the highest
+level to change how `square-limit' arranges the four copies of the
+corner. In general, each level of a stratified design provides a
+different vocabulary for expressing the characteristics of the system,
+and a different kind of ability to change it.
+
+ *Exercise 2.52:* Make changes to the square limit of `wave' shown
+ in *Note Figure 2-9:: by working at each of the levels described
+ above. In particular:
+
+ a. Add some segments to the primitive `wave' painter of *Note
+ Exercise 2-49:: (to add a smile, for example).
+
+ b. Change the pattern constructed by `corner-split' (for
+ example, by using only one copy of the `up-split' and
+ `right-split' images instead of two).
+
+ c. Modify the version of `square-limit' that uses
+ `square-of-four' so as to assemble the corners in a different
+ pattern. (For example, you might make the big Mr. Rogers
+ look outward from each corner of the square.)
+
+
+ ---------- Footnotes ----------
+
+ (1) The picture language is based on the language Peter Henderson
+created to construct images like M.C. Escher's "Square Limit" woodcut
+(see Henderson 1982). The woodcut incorporates a repeated scaled
+pattern, similar to the arrangements drawn using the `square-limit'
+procedure in this section.
+
+ (2) William Barton Rogers (1804-1882) was the founder and first
+president of MIT. A geologist and talented teacher, he taught at
+William and Mary College and at the University of Virginia. In 1859 he
+moved to Boston, where he had more time for research, worked on a plan
+for establishing a "polytechnic institute," and served as
+Massachusetts's first State Inspector of Gas Meters.
+
+ When MIT was established in 1861, Rogers was elected its first
+president. Rogers espoused an ideal of "useful learning" that was
+different from the university education of the time, with its
+overemphasis on the classics, which, as he wrote, "stand in the way of
+the broader, higher and more practical instruction and discipline of
+the natural and social sciences." This education was likewise to be
+different from narrow trade-school education. In Rogers's words:
+
+ The world-enforced distinction between the practical and the
+ scientific worker is utterly futile, and the whole experience of
+ modern times has demonstrated its utter worthlessness.
+
+ Rogers served as president of MIT until 1870, when he resigned due to
+ill health. In 1878 the second president of MIT, John Runkle, resigned
+under the pressure of a financial crisis brought on by the Panic of
+1873 and strain of fighting off attempts by Harvard to take over MIT.
+Rogers returned to hold the office of president until 1881.
+
+ Rogers collapsed and died while addressing MIT's graduating class at
+the commencement exercises of 1882. Runkle quoted Rogers's last words
+in a memorial address delivered that same year:
+
+ "As I stand here today and see what the Institute is, ... I call
+ to mind the beginnings of science. I remember one hundred and
+ fifty years ago Stephen Hales published a pamphlet on the subject
+ of illuminating gas, in which he stated that his researches had
+ demonstrated that 128 grains of bituminous coal - " "Bituminous
+ coal," these were his last words on earth. Here he bent forward,
+ as if consulting some notes on the table before him, then slowly
+ regaining an erect position, threw up his hands, and was
+ translated from the scene of his earthly labors and triumphs to
+ "the tomorrow of death," where the mysteries of life are solved,
+ and the disembodied spirit finds unending satisfaction in
+ contemplating the new and still unfathomable mysteries of the
+ infinite future.
+
+ In the words of Francis A. Walker (MIT's third president):
+
+ All his life he had borne himself most faithfully and heroically,
+ and he died as so good a knight would surely have wished, in
+ harness, at his post, and in the very part and act of public duty.
+
+ (3) Equivalently, we could write
+
+ (define flipped-pairs
+ (square-of-four identity flip-vert identity flip-vert))
+
+ (4) `Rotate180' rotates a painter by 180 degrees (see *Note Exercise
+2-50::). Instead of `rotate180' we could say `(compose flip-vert
+flip-horiz)', using the `compose' procedure from *Note Exercise 1-42::.
+
+ (5) `Frame-coord-map' uses the vector operations described in *Note
+Exercise 2-46:: below, which we assume have been implemented using some
+representation for vectors. Because of data abstraction, it doesn't
+matter what this vector representation is, so long as the vector
+operations behave correctly.
+
+ (6) `Segments->painter' uses the representation for line segments
+described in *Note Exercise 2-48:: below. It also uses the `for-each'
+procedure described in *Note Exercise 2-23::.
+
+ (7) For example, the `rogers' painter of *Note Figure 2-11:: was
+constructed from a gray-level image. For each point in a given frame,
+the `rogers' painter determines the point in the image that is mapped
+to it under the frame coordinate map, and shades it accordingly. By
+allowing different types of painters, we are capitalizing on the
+abstract data idea discussed in section *Note 2-1-3::, where we argued
+that a rational-number representation could be anything at all that
+satisfies an appropriate condition. Here we're using the fact that a
+painter can be implemented in any way at all, so long as it draws
+something in the designated frame. Section *Note 2-1-3:: also showed
+how pairs could be implemented as procedures. Painters are our second
+example of a procedural representation for data.
+
+ (8) `Rotate90' is a pure rotation only for square frames, because it
+also stretches and shrinks the image to fit into the rotated frame.
+
+ (9) The diamond-shaped images in figures *Note Figure 2-10:: and
+*Note Figure 2-11:: were created with `squash-inwards' applied to
+`wave' and `rogers'.
+
+ (10) Section *Note 3-3-4:: describes one such language.
+
+\1f
+File: sicp.info, Node: 2-3, Next: 2-4, Prev: 2-2, Up: Chapter 2
+
+2.3 Symbolic Data
+=================
+
+All the compound data objects we have used so far were constructed
+ultimately from numbers. In this section we extend the
+representational capability of our language by introducing the ability
+to work with arbitrary symbols as data.
+
+* Menu:
+
+* 2-3-1:: Quotation
+* 2-3-2:: Example: Symbolic Differentiation
+* 2-3-3:: Example: Representing Sets
+* 2-3-4:: Example: Huffman Encoding Trees
+
+\1f
+File: sicp.info, Node: 2-3-1, Next: 2-3-2, Prev: 2-3, Up: 2-3
+
+2.3.1 Quotation
+---------------
+
+If we can form compound data using symbols, we can have lists such as
+
+ (a b c d)
+ (23 45 17)
+ ((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4))
+
+ Lists containing symbols can look just like the expressions of our
+language:
+
+ (* (+ 23 45) (+ x 9))
+
+ (define (fact n) (if (= n 1) 1 (* n (fact (- n 1)))))
+
+ In order to manipulate symbols we need a new element in our
+language: the ability to "quote" a data object. Suppose we want to
+construct the list `(a b)'. We can't accomplish this with `(list a
+b)', because this expression constructs a list of the "values" of `a'
+and `b' rather than the symbols themselves. This issue is well known
+in the context of natural languages, where words and sentences may be
+regarded either as semantic entities or as character strings (syntactic
+entities). The common practice in natural languages is to use
+quotation marks to indicate that a word or a sentence is to be treated
+literally as a string of characters. For instance, the first letter of
+"John" is clearly "J." If we tell somebody "say your name aloud," we
+expect to hear that person's name. However, if we tell somebody "say
+`your name' aloud," we expect to hear the words "your name." Note that
+we are forced to nest quotation marks to describe what somebody else
+might say.(1)
+
+ We can follow this same practice to identify lists and symbols that
+are to be treated as data objects rather than as expressions to be
+evaluated. However, our format for quoting differs from that of
+natural languages in that we place a quotation mark (traditionally, the
+single quote symbol `'') only at the beginning of the object to be
+quoted. We can get away with this in Scheme syntax because we rely on
+blanks and parentheses to delimit objects. Thus, the meaning of the
+single quote character is to quote the next object.(2)
+
+ Now we can distinguish between symbols and their values:
+
+ (define a 1)
+
+ (define b 2)
+
+ (list a b)
+ (1 2)
+
+ (list 'a 'b)
+ (a b)
+
+ (list 'a b)
+ (a 2)
+
+ Quotation also allows us to type in compound objects, using the
+conventional printed representation for lists:(3)
+
+ (car '(a b c))
+ a
+
+ (cdr '(a b c))
+ (b c)
+
+ In keeping with this, we can obtain the empty list by evaluating
+`'()', and thus dispense with the variable `nil'.
+
+ One additional primitive used in manipulating symbols is `eq?', which
+takes two symbols as arguments and tests whether they are the same.(4)
+Using `eq?', we can implement a useful procedure called `memq'. This
+takes two arguments, a symbol and a list. If the symbol is not
+contained in the list (i.e., is not `eq?' to any item in the list),
+then `memq' returns false. Otherwise, it returns the sublist of the
+list beginning with the first occurrence of the symbol:
+
+ (define (memq item x)
+ (cond ((null? x) false)
+ ((eq? item (car x)) x)
+ (else (memq item (cdr x)))))
+
+ For example, the value of
+
+ (memq 'apple '(pear banana prune))
+
+is false, whereas the value of
+
+ (memq 'apple '(x (apple sauce) y apple pear))
+
+is `(apple pear)'.
+
+ *Exercise 2.53:* What would the interpreter print in response to
+ evaluating each of the following expressions?
+
+ (list 'a 'b 'c)
+
+ (list (list 'george))
+
+ (cdr '((x1 x2) (y1 y2)))
+
+ (cadr '((x1 x2) (y1 y2)))
+
+ (pair? (car '(a short list)))
+
+ (memq 'red '((red shoes) (blue socks)))
+
+ (memq 'red '(red shoes blue socks))
+
+ *Exercise 2.54:* Two lists are said to be `equal?' if they contain
+ equal elements arranged in the same order. For example,
+
+ (equal? '(this is a list) '(this is a list))
+
+ is true, but
+
+ (equal? '(this is a list) '(this (is a) list))
+
+ is false. To be more precise, we can define `equal?' recursively
+ in terms of the basic `eq?' equality of symbols by saying that `a'
+ and `b' are `equal?' if they are both symbols and the symbols are
+ `eq?', or if they are both lists such that `(car a)' is `equal?'
+ to `(car b)' and `(cdr a)' is `equal?' to `(cdr b)'. Using this
+ idea, implement `equal?' as a procedure.(5)
+
+ *Exercise 2.55:* Eva Lu Ator types to the interpreter the
+ expression
+
+ (car ''abracadabra)
+
+ To her surprise, the interpreter prints back `quote'. Explain.
+
+ ---------- Footnotes ----------
+
+ (1) Allowing quotation in a language wreaks havoc with the ability
+to reason about the language in simple terms, because it destroys the
+notion that equals can be substituted for equals. For example, three
+is one plus two, but the word "three" is not the phrase "one plus two."
+Quotation is powerful because it gives us a way to build expressions
+that manipulate other expressions (as we will see when we write an
+interpreter in *Note Chapter 4::). But allowing statements in a
+language that talk about other statements in that language makes it
+very difficult to maintain any coherent principle of what "equals can
+be substituted for equals" should mean. For example, if we know that
+the evening star is the morning star, then from the statement "the
+evening star is Venus" we can deduce "the morning star is Venus."
+However, given that "John knows that the evening star is Venus" we
+cannot infer that "John knows that the morning star is Venus."
+
+ (2) The single quote is different from the double quote we have been
+using to enclose character strings to be printed. Whereas the single
+quote can be used to denote lists or symbols, the double quote is used
+only with character strings. In this book, the only use for character
+strings is as items to be printed.
+
+ (3) Strictly, our use of the quotation mark violates the general
+rule that all compound expressions in our language should be delimited
+by parentheses and look like lists. We can recover this consistency by
+introducing a special form `quote', which serves the same purpose as
+the quotation mark. Thus, we would type `(quote a)' instead of `'a',
+and we would type `(quote (a b c))' instead of `'(a b c)'. This is
+precisely how the interpreter works. The quotation mark is just a
+single-character abbreviation for wrapping the next complete expression
+with `quote' to form `(quote <EXPRESSION>)'. This is important because
+it maintains the principle that any expression seen by the interpreter
+can be manipulated as a data object. For instance, we could construct
+the expression `(car '(a b c))', which is the same as `(car (quote (a b
+c)))', by evaluating `(list 'car (list 'quote '(a b c)))'.
+
+ (4) We can consider two symbols to be "the same" if they consist of
+the same characters in the same order. Such a definition skirts a deep
+issue that we are not yet ready to address: the meaning of "sameness"
+in a programming language. We will return to this in *Note Chapter 3::
+(section *Note 3-1-3::).
+
+ (5) In practice, programmers use `equal?' to compare lists that
+contain numbers as well as symbols. Numbers are not considered to be
+symbols. The question of whether two numerically equal numbers (as
+tested by `=') are also `eq?' is highly implementation-dependent. A
+better definition of `equal?' (such as the one that comes as a
+primitive in Scheme) would also stipulate that if `a' and `b' are both
+numbers, then `a' and `b' are `equal?' if they are numerically equal.
+
+\1f
+File: sicp.info, Node: 2-3-2, Next: 2-3-3, Prev: 2-3-1, Up: 2-3
+
+2.3.2 Example: Symbolic Differentiation
+---------------------------------------
+
+As an illustration of symbol manipulation and a further illustration of
+data abstraction, consider the design of a procedure that performs
+symbolic differentiation of algebraic expressions. We would like the
+procedure to take as arguments an algebraic expression and a variable
+and to return the derivative of the expression with respect to the
+variable. For example, if the arguments to the procedure are ax^2 + bx
++ c and x, the procedure should return 2ax + b. Symbolic
+differentiation is of special historical significance in Lisp. It was
+one of the motivating examples behind the development of a computer
+language for symbol manipulation. Furthermore, it marked the beginning
+of the line of research that led to the development of powerful systems
+for symbolic mathematical work, which are currently being used by a
+growing number of applied mathematicians and physicists.
+
+ In developing the symbolic-differentiation program, we will follow
+the same strategy of data abstraction that we followed in developing
+the rational-number system of section *Note 2-1-1::. That is, we will
+first define a differentiation algorithm that operates on abstract
+objects such as "sums," "products," and "variables" without worrying
+about how these are to be represented. Only afterward will we address
+the representation problem.
+
+The differentiation program with abstract data
+..............................................
+
+In order to keep things simple, we will consider a very simple
+symbolic-differentiation program that handles expressions that are
+built up using only the operations of addition and multiplication with
+two arguments. Differentiation of any such expression can be carried
+out by applying the following reduction rules:
+
+ dc
+ -- = 0 for c a constant, or a variable different from x
+ dx
+
+ dx
+ -- = 1
+ dx
+
+ d(u + v) du dv
+ -------- = -- + --
+ dx dx dx
+
+ d(uv) / dv \ / du \
+ ----- = u | -- | + v | -- |
+ dx \ dx / \ dx /
+
+ Observe that the latter two rules are recursive in nature. That is,
+to obtain the derivative of a sum we first find the derivatives of the
+terms and add them. Each of the terms may in turn be an expression
+that needs to be decomposed. Decomposing into smaller and smaller
+pieces will eventually produce pieces that are either constants or
+variables, whose derivatives will be either 0 or 1.
+
+ To embody these rules in a procedure we indulge in a little wishful
+thinking, as we did in designing the rational-number implementation.
+If we had a means for representing algebraic expressions, we should be
+able to tell whether an expression is a sum, a product, a constant, or
+a variable. We should be able to extract the parts of an expression.
+For a sum, for example we want to be able to extract the addend (first
+term) and the augend (second term). We should also be able to
+construct expressions from parts. Let us assume that we already have
+procedures to implement the following selectors, constructors, and
+predicates:
+
+ (variable? e) Is `e' a variable?
+ (same-variable? v1 v2) Are `v1' and `v2' the same variable?
+ (sum? e) Is `e' a sum?
+ (addend e) Addend of the sum `e'.
+ (augend e) Augend of the sum `e'.
+ (make-sum a1 a2) Construct the sum of `a1' and `a2'.
+ (product? e) Is `e' a product?
+ (multiplier e) Multiplier of the product `e'.
+ (multiplicand e) Multiplicand of the product `e'.
+ (make-product m1 m2) Construct the product of `m1' and `m2'.
+
+ Using these, and the primitive predicate `number?', which identifies
+numbers, we can express the differentiation rules as the following
+procedure:
+
+ (define (deriv exp var)
+ (cond ((number? exp) 0)
+ ((variable? exp)
+ (if (same-variable? exp var) 1 0))
+ ((sum? exp)
+ (make-sum (deriv (addend exp) var)
+ (deriv (augend exp) var)))
+ ((product? exp)
+ (make-sum
+ (make-product (multiplier exp)
+ (deriv (multiplicand exp) var))
+ (make-product (deriv (multiplier exp) var)
+ (multiplicand exp))))
+ (else
+ (error "unknown expression type -- DERIV" exp))))
+
+ This `deriv' procedure incorporates the complete differentiation
+algorithm. Since it is expressed in terms of abstract data, it will
+work no matter how we choose to represent algebraic expressions, as
+long as we design a proper set of selectors and constructors. This is
+the issue we must address next.
+
+Representing algebraic expressions
+..................................
+
+We can imagine many ways to use list structure to represent algebraic
+expressions. For example, we could use lists of symbols that mirror
+the usual algebraic notation, representing ax + b as the list `(a * x +
+b)' . However, one especially straightforward choice is to use the same
+parenthesized prefix notation that Lisp uses for combinations; that is,
+to represent ax + b as `(+ (* a x) b)'. Then our data representation
+for the differentiation problem is as follows:
+
+ * The variables are symbols. They are identified by the primitive
+ predicate `symbol?':
+
+ (define (variable? x) (symbol? x))
+
+ * Two variables are the same if the symbols representing them are
+ `eq?':
+
+ (define (same-variable? v1 v2)
+ (and (variable? v1) (variable? v2) (eq? v1 v2)))
+
+ * Sums and products are constructed as lists:
+
+ (define (make-sum a1 a2) (list '+ a1 a2))
+
+ (define (make-product m1 m2) (list '* m1 m2))
+
+ * A sum is a list whose first element is the symbol `+':
+
+ (define (sum? x)
+ (and (pair? x) (eq? (car x) '+)))
+
+ * The addend is the second item of the sum list:
+
+ (define (addend s) (cadr s))
+
+ * The augend is the third item of the sum list:
+
+ (define (augend s) (caddr s))
+
+ * A product is a list whose first element is the symbol `*':
+
+ (define (product? x)
+ (and (pair? x) (eq? (car x) '*)))
+
+ * The multiplier is the second item of the product list:
+
+ (define (multiplier p) (cadr p))
+
+ * The multiplicand is the third item of the product list:
+
+ (define (multiplicand p) (caddr p))
+
+
+ Thus, we need only combine these with the algorithm as embodied by
+`deriv' in order to have a working symbolic-differentiation program.
+Let us look at some examples of its behavior:
+
+ (deriv '(+ x 3) 'x)
+ (+ 1 0)
+
+ (deriv '(* x y) 'x)
+ (+ (* x 0) (* 1 y))
+
+ (deriv '(* (* x y) (+ x 3)) 'x)
+ (+ (* (* x y) (+ 1 0))
+ (* (+ (* x 0) (* 1 y))
+ (+ x 3)))
+
+ The program produces answers that are correct; however, they are
+unsimplified. It is true that
+
+ d(xy)
+ ----- = x * 0 + 1 * y
+ dx
+
+but we would like the program to know that x * 0 = 0, 1 * y = y, and 0
++ y = y. The answer for the second example should have been simply
+`y'. As the third example shows, this becomes a serious issue when the
+expressions are complex.
+
+ Our difficulty is much like the one we encountered with the
+rational-number implementation: we haven't reduced answers to simplest
+form. To accomplish the rational-number reduction, we needed to change
+only the constructors and the selectors of the implementation. We can
+adopt a similar strategy here. We won't change `deriv' at all.
+Instead, we will change `make-sum' so that if both summands are
+numbers, `make-sum' will add them and return their sum. Also, if one
+of the summands is 0, then `make-sum' will return the other summand.
+
+ (define (make-sum a1 a2)
+ (cond ((=number? a1 0) a2)
+ ((=number? a2 0) a1)
+ ((and (number? a1) (number? a2)) (+ a1 a2))
+ (else (list '+ a1 a2))))
+
+ This uses the procedure `=number?', which checks whether an
+expression is equal to a given number:
+
+ (define (=number? exp num)
+ (and (number? exp) (= exp num)))
+
+ Similarly, we will change `make-product' to build in the rules that 0
+times anything is 0 and 1 times anything is the thing itself:
+
+ (define (make-product m1 m2)
+ (cond ((or (=number? m1 0) (=number? m2 0)) 0)
+ ((=number? m1 1) m2)
+ ((=number? m2 1) m1)
+ ((and (number? m1) (number? m2)) (* m1 m2))
+ (else (list '* m1 m2))))
+
+ Here is how this version works on our three examples:
+
+ (deriv '(+ x 3) 'x)
+ 1
+
+ (deriv '(* x y) 'x)
+ y
+
+ (deriv '(* (* x y) (+ x 3)) 'x)
+ (+ (* x y) (* y (+ x 3)))
+
+ Although this is quite an improvement, the third example shows that
+there is still a long way to go before we get a program that puts
+expressions into a form that we might agree is "simplest." The problem
+of algebraic simplification is complex because, among other reasons, a
+form that may be simplest for one purpose may not be for another.
+
+ *Exercise 2.56:* Show how to extend the basic differentiator to
+ handle more kinds of expressions. For instance, implement the
+ differentiation rule
+
+ n_1 n_2
+ --- = --- if and only if n_1 d_2 = n_2 d_1
+ d_1 d_2
+
+ by adding a new clause to the `deriv' program and defining
+ appropriate procedures `exponentiation?', `base', `exponent', and
+ `make-exponentiation'. (You may use the symbol `**' to denote
+ exponentiation.) Build in the rules that anything raised to the
+ power 0 is 1 and anything raised to the power 1 is the thing
+ itself.
+
+ *Exercise 2.57:* Extend the differentiation program to handle sums
+ and products of arbitrary numbers of (two or more) terms. Then
+ the last example above could be expressed as
+
+ (deriv '(* x y (+ x 3)) 'x)
+
+ Try to do this by changing only the representation for sums and
+ products, without changing the `deriv' procedure at all. For
+ example, the `addend' of a sum would be the first term, and the
+ `augend' would be the sum of the rest of the terms.
+
+ *Exercise 2.58:* Suppose we want to modify the differentiation
+ program so that it works with ordinary mathematical notation, in
+ which `+' and `*' are infix rather than prefix operators. Since
+ the differentiation program is defined in terms of abstract data,
+ we can modify it to work with different representations of
+ expressions solely by changing the predicates, selectors, and
+ constructors that define the representation of the algebraic
+ expressions on which the differentiator is to operate.
+
+ a. Show how to do this in order to differentiate algebraic
+ expressions presented in infix form, such as `(x + (3 * (x +
+ (y + 2))))'. To simplify the task, assume that `+' and `*'
+ always take two arguments and that expressions are fully
+ parenthesized.
+
+ b. The problem becomes substantially harder if we allow standard
+ algebraic notation, such as `(x + 3 * (x + y + 2))', which
+ drops unnecessary parentheses and assumes that multiplication
+ is done before addition. Can you design appropriate
+ predicates, selectors, and constructors for this notation
+ such that our derivative program still works?
+
+
+\1f
+File: sicp.info, Node: 2-3-3, Next: 2-3-4, Prev: 2-3-2, Up: 2-3
+
+2.3.3 Example: Representing Sets
+--------------------------------
+
+In the previous examples we built representations for two kinds of
+compound data objects: rational numbers and algebraic expressions. In
+one of these examples we had the choice of simplifying (reducing) the
+expressions at either construction time or selection time, but other
+than that the choice of a representation for these structures in terms
+of lists was straightforward. When we turn to the representation of
+sets, the choice of a representation is not so obvious. Indeed, there
+are a number of possible representations, and they differ significantly
+from one another in several ways.
+
+ Informally, a set is simply a collection of distinct objects. To
+give a more precise definition we can employ the method of data
+abstraction. That is, we define "set" by specifying the operations
+that are to be used on sets. These are `union-set',
+`intersection-set', `element-of-set?', and `adjoin-set'.
+`Element-of-set?' is a predicate that determines whether a given
+element is a member of a set. `Adjoin-set' takes an object and a set
+as arguments and returns a set that contains the elements of the
+original set and also the adjoined element. `Union-set' computes the
+union of two sets, which is the set containing each element that
+appears in either argument. `Intersection-set' computes the
+intersection of two sets, which is the set containing only elements
+that appear in both arguments. From the viewpoint of data abstraction,
+we are free to design any representation that implements these
+operations in a way consistent with the interpretations given above.(1)
+
+Sets as unordered lists
+.......................
+
+One way to represent a set is as a list of its elements in which no
+element appears more than once. The empty set is represented by the
+empty list. In this representation, `element-of-set?' is similar to
+the procedure `memq' of section *Note 2-3-1::. It uses `equal?'
+instead of `eq?' so that the set elements need not be symbols:
+
+ (define (element-of-set? x set)
+ (cond ((null? set) false)
+ ((equal? x (car set)) true)
+ (else (element-of-set? x (cdr set)))))
+
+ Using this, we can write `adjoin-set'. If the object to be adjoined
+is already in the set, we just return the set. Otherwise, we use
+`cons' to add the object to the list that represents the set:
+
+ (define (adjoin-set x set)
+ (if (element-of-set? x set)
+ set
+ (cons x set)))
+
+ For `intersection-set' we can use a recursive strategy. If we know
+how to form the intersection of `set2' and the `cdr' of `set1', we only
+need to decide whether to include the `car' of `set1' in this. But
+this depends on whether `(car set1)' is also in `set2'. Here is the
+resulting procedure:
+
+ (define (intersection-set set1 set2)
+ (cond ((or (null? set1) (null? set2)) '())
+ ((element-of-set? (car set1) set2)
+ (cons (car set1)
+ (intersection-set (cdr set1) set2)))
+ (else (intersection-set (cdr set1) set2))))
+
+ In designing a representation, one of the issues we should be
+concerned with is efficiency. Consider the number of steps required by
+our set operations. Since they all use `element-of-set?', the speed of
+this operation has a major impact on the efficiency of the set
+implementation as a whole. Now, in order to check whether an object is
+a member of a set, `element-of-set?' may have to scan the entire set.
+(In the worst case, the object turns out not to be in the set.) Hence,
+if the set has n elements, `element-of-set?' might take up to n steps.
+Thus, the number of steps required grows as [theta](n). The number of
+steps required by `adjoin-set', which uses this operation, also grows
+as [theta](n). For `intersection-set', which does an `element-of-set?'
+check for each element of `set1', the number of steps required grows as
+the product of the sizes of the sets involved, or [theta](n^2) for two
+sets of size n. The same will be true of `union-set'.
+
+ *Exercise 2.59:* Implement the `union-set' operation for the
+ unordered-list representation of sets.
+
+ *Exercise 2.60:* We specified that a set would be represented as a
+ list with no duplicates. Now suppose we allow duplicates. For
+ instance, the set {1,2,3} could be represented as the list `(2 3 2
+ 1 3 2 2)'. Design procedures `element-of-set?', `adjoin-set',
+ `union-set', and `intersection-set' that operate on this
+ representation. How does the efficiency of each compare with the
+ corresponding procedure for the non-duplicate representation? Are
+ there applications for which you would use this representation in
+ preference to the non-duplicate one?
+
+Sets as ordered lists
+.....................
+
+One way to speed up our set operations is to change the representation
+so that the set elements are listed in increasing order. To do this,
+we need some way to compare two objects so that we can say which is
+bigger. For example, we could compare symbols lexicographically, or we
+could agree on some method for assigning a unique number to an object
+and then compare the elements by comparing the corresponding numbers.
+To keep our discussion simple, we will consider only the case where the
+set elements are numbers, so that we can compare elements using `>' and
+`<'. We will represent a set of numbers by listing its elements in
+increasing order. Whereas our first representation above allowed us to
+represent the set {1,3,6,10} by listing the elements in any order, our
+new representation allows only the list `(1 3 6 10)'.
+
+ One advantage of ordering shows up in `element-of-set?': In checking
+for the presence of an item, we no longer have to scan the entire set.
+If we reach a set element that is larger than the item we are looking
+for, then we know that the item is not in the set:
+
+ (define (element-of-set? x set)
+ (cond ((null? set) false)
+ ((= x (car set)) true)
+ ((< x (car set)) false)
+ (else (element-of-set? x (cdr set)))))
+
+ How many steps does this save? In the worst case, the item we are
+looking for may be the largest one in the set, so the number of steps
+is the same as for the unordered representation. On the other hand, if
+we search for items of many different sizes we can expect that
+sometimes we will be able to stop searching at a point near the
+beginning of the list and that other times we will still need to
+examine most of the list. On the average we should expect to have to
+examine about half of the items in the set. Thus, the average number
+of steps required will be about n/2. This is still [theta](n) growth,
+but it does save us, on the average, a factor of 2 in number of steps
+over the previous implementation.
+
+ We obtain a more impressive speedup with `intersection-set'. In the
+unordered representation this operation required [theta](n^2) steps,
+because we performed a complete scan of `set2' for each element of
+`set1'. But with the ordered representation, we can use a more clever
+method. Begin by comparing the initial elements, `x1' and `x2', of the
+two sets. If `x1' equals `x2', then that gives an element of the
+intersection, and the rest of the intersection is the intersection of
+the `cdr's of the two sets. Suppose, however, that `x1' is less than
+`x2'. Since `x2' is the smallest element in `set2', we can immediately
+conclude that `x1' cannot appear anywhere in `set2' and hence is not in
+the intersection. Hence, the intersection is equal to the intersection
+of `set2' with the `cdr' of `set1'. Similarly, if `x2' is less than
+`x1', then the intersection is given by the intersection of `set1' with
+the `cdr' of `set2'. Here is the procedure:
+
+ (define (intersection-set set1 set2)
+ (if (or (null? set1) (null? set2))
+ '()
+ (let ((x1 (car set1)) (x2 (car set2)))
+ (cond ((= x1 x2)
+ (cons x1
+ (intersection-set (cdr set1)
+ (cdr set2))))
+ ((< x1 x2)
+ (intersection-set (cdr set1) set2))
+ ((< x2 x1)
+ (intersection-set set1 (cdr set2)))))))
+
+ To estimate the number of steps required by this process, observe
+that at each step we reduce the intersection problem to computing
+intersections of smaller sets--removing the first element from `set1'
+or `set2' or both. Thus, the number of steps required is at most the
+sum of the sizes of `set1' and `set2', rather than the product of the
+sizes as with the unordered representation. This is [theta](n) growth
+rather than [theta](n^2)--a considerable speedup, even for sets of
+moderate size.
+
+ *Exercise 2.61:* Give an implementation of `adjoin-set' using the
+ ordered representation. By analogy with `element-of-set?' show
+ how to take advantage of the ordering to produce a procedure that
+ requires on the average about half as many steps as with the
+ unordered representation.
+
+ *Exercise 2.62:* Give a [theta](n) implementation of `union-set'
+ for sets represented as ordered lists.
+
+Sets as binary trees
+....................
+
+We can do better than the ordered-list representation by arranging the
+set elements in the form of a tree. Each node of the tree holds one
+element of the set, called the "entry" at that node, and a link to each
+of two other (possibly empty) nodes. The "left" link points to
+elements smaller than the one at the node, and the "right" link to
+elements greater than the one at the node. *Note Figure 2-16:: shows
+some trees that represent the set {1,3,5,7,9,11}. The same set may be
+represented by a tree in a number of different ways. The only thing we
+require for a valid representation is that all elements in the left
+subtree be smaller than the node entry and that all elements in the
+right subtree be larger.
+
+ *Figure 2.16:* Various binary trees that represent the set
+ {1,3,5,7,9,11}.
+
+ 7 3 5
+ /\ /\ /\
+ 3 9 1 7 3 9
+ /\ \ /\ / /\
+ 1 5 11 5 9 1 7 11
+ \
+ 11
+
+ The advantage of the tree representation is this: Suppose we want to
+check whether a number x is contained in a set. We begin by comparing
+x with the entry in the top node. If x is less than this, we know that
+we need only search the left subtree; if x is greater, we need only
+search the right subtree. Now, if the tree is "balanced," each of
+these subtrees will be about half the size of the original. Thus, in
+one step we have reduced the problem of searching a tree of size n to
+searching a tree of size n/2. Since the size of the tree is halved at
+each step, we should expect that the number of steps needed to search a
+tree of size n grows as [theta](`log' n).(2) For large sets, this will
+be a significant speedup over the previous representations.
+
+ We can represent trees by using lists. Each node will be a list of
+three items: the entry at the node, the left subtree, and the right
+subtree. A left or a right subtree of the empty list will indicate
+that there is no subtree connected there. We can describe this
+representation by the following procedures:(3)
+
+ (define (entry tree) (car tree))
+
+ (define (left-branch tree) (cadr tree))
+
+ (define (right-branch tree) (caddr tree))
+
+ (define (make-tree entry left right)
+ (list entry left right))
+
+ Now we can write the `element-of-set?' procedure using the strategy
+described above:
+
+ (define (element-of-set? x set)
+ (cond ((null? set) false)
+ ((= x (entry set)) true)
+ ((< x (entry set))
+ (element-of-set? x (left-branch set)))
+ ((> x (entry set))
+ (element-of-set? x (right-branch set)))))
+
+ Adjoining an item to a set is implemented similarly and also requires
+[theta](`log' n) steps. To adjoin an item `x', we compare `x' with the
+node entry to determine whether `x' should be added to the right or to
+the left branch, and having adjoined `x' to the appropriate branch we
+piece this newly constructed branch together with the original entry
+and the other branch. If `x' is equal to the entry, we just return the
+node. If we are asked to adjoin `x' to an empty tree, we generate a
+tree that has `x' as the entry and empty right and left branches. Here
+is the procedure:
+
+ (define (adjoin-set x set)
+ (cond ((null? set) (make-tree x '() '()))
+ ((= x (entry set)) set)
+ ((< x (entry set))
+ (make-tree (entry set)
+ (adjoin-set x (left-branch set))
+ (right-branch set)))
+ ((> x (entry set))
+ (make-tree (entry set)
+ (left-branch set)
+ (adjoin-set x (right-branch set))))))
+
+ The above claim that searching the tree can be performed in a
+logarithmic number of steps rests on the assumption that the tree is
+"balanced," i.e., that the left and the right subtree of every tree
+have approximately the same number of elements, so that each subtree
+contains about half the elements of its parent. But how can we be
+certain that the trees we construct will be balanced? Even if we start
+with a balanced tree, adding elements with `adjoin-set' may produce an
+unbalanced result. Since the position of a newly adjoined element
+depends on how the element compares with the items already in the set,
+we can expect that if we add elements "randomly" the tree will tend to
+be balanced on the average. But this is not a guarantee. For example,
+if we start with an empty set and adjoin the numbers 1 through 7 in
+sequence we end up with the highly unbalanced tree shown in *Note
+Figure 2-17::. In this tree all the left subtrees are empty, so it has
+no advantage over a simple ordered list. One way to solve this problem
+is to define an operation that transforms an arbitrary tree into a
+balanced tree with the same elements. Then we can perform this
+transformation after every few `adjoin-set' operations to keep our set
+in balance. There are also other ways to solve this problem, most of
+which involve designing new data structures for which searching and
+insertion both can be done in [theta](`log' n) steps.(4)
+
+ *Figure 2.17:* Unbalanced tree produced by adjoining 1 through 7
+ in sequence.
+
+ 1
+ \
+ 2
+ \
+ 4
+ \
+ 5
+ \
+ 6
+ \
+ 7
+
+ *Exercise 2.63:* Each of the following two procedures converts a
+ binary tree to a list.
+
+ (define (tree->list-1 tree)
+ (if (null? tree)
+ '()
+ (append (tree->list-1 (left-branch tree))
+ (cons (entry tree)
+ (tree->list-1 (right-branch tree))))))
+
+ (define (tree->list-2 tree)
+ (define (copy-to-list tree result-list)
+ (if (null? tree)
+ result-list
+ (copy-to-list (left-branch tree)
+ (cons (entry tree)
+ (copy-to-list (right-branch tree)
+ result-list)))))
+ (copy-to-list tree '()))
+
+ a. Do the two procedures produce the same result for every tree?
+ If not, how do the results differ? What lists do the two
+ procedures produce for the trees in *Note Figure 2-16::?
+
+ b. Do the two procedures have the same order of growth in the
+ number of steps required to convert a balanced tree with n
+ elements to a list? If not, which one grows more slowly?
+
+
+ *Exercise 2.64:* The following procedure `list->tree' converts an
+ ordered list to a balanced binary tree. The helper procedure
+ `partial-tree' takes as arguments an integer n and list of at
+ least n elements and constructs a balanced tree containing the
+ first n elements of the list. The result returned by
+ `partial-tree' is a pair (formed with `cons') whose `car' is the
+ constructed tree and whose `cdr' is the list of elements not
+ included in the tree.
+
+ (define (list->tree elements)
+ (car (partial-tree elements (length elements))))
+
+ (define (partial-tree elts n)
+ (if (= n 0)
+ (cons '() elts)
+ (let ((left-size (quotient (- n 1) 2)))
+ (let ((left-result (partial-tree elts left-size)))
+ (let ((left-tree (car left-result))
+ (non-left-elts (cdr left-result))
+ (right-size (- n (+ left-size 1))))
+ (let ((this-entry (car non-left-elts))
+ (right-result (partial-tree (cdr non-left-elts)
+ right-size)))
+ (let ((right-tree (car right-result))
+ (remaining-elts (cdr right-result)))
+ (cons (make-tree this-entry left-tree right-tree)
+ remaining-elts))))))))
+
+ a. Write a short paragraph explaining as clearly as you can how
+ `partial-tree' works. Draw the tree produced by `list->tree'
+ for the list `(1 3 5 7 9 11)'.
+
+ b. What is the order of growth in the number of steps required by
+ `list->tree' to convert a list of n elements?
+
+
+ *Exercise 2.65:* Use the results of *Note Exercise 2-63:: and
+ *Note Exercise 2-64:: to give [theta](n) implementations of
+ `union-set' and `intersection-set' for sets implemented as
+ (balanced) binary trees.(5)
+
+Sets and information retrieval
+..............................
+
+We have examined options for using lists to represent sets and have
+seen how the choice of representation for a data object can have a
+large impact on the performance of the programs that use the data.
+Another reason for concentrating on sets is that the techniques
+discussed here appear again and again in applications involving
+information retrieval.
+
+ Consider a data base containing a large number of individual
+records, such as the personnel files for a company or the transactions
+in an accounting system. A typical data-management system spends a
+large amount of time accessing or modifying the data in the records and
+therefore requires an efficient method for accessing records. This is
+done by identifying a part of each record to serve as an identifying "key".
+A key can be anything that uniquely identifies the record. For a
+personnel file, it might be an employee's ID number. For an accounting
+system, it might be a transaction number. Whatever the key is, when we
+define the record as a data structure we should include a `key'
+selector procedure that retrieves the key associated with a given
+record.
+
+ Now we represent the data base as a set of records. To locate the
+record with a given key we use a procedure `lookup', which takes as
+arguments a key and a data base and which returns the record that has
+that key, or false if there is no such record. `Lookup' is implemented
+in almost the same way as `element-of-set?'. For example, if the set
+of records is implemented as an unordered list, we could use
+
+ (define (lookup given-key set-of-records)
+ (cond ((null? set-of-records) false)
+ ((equal? given-key (key (car set-of-records)))
+ (car set-of-records))
+ (else (lookup given-key (cdr set-of-records)))))
+
+ Of course, there are better ways to represent large sets than as
+unordered lists. Information-retrieval systems in which records have
+to be "randomly accessed" are typically implemented by a tree-based
+method, such as the binary-tree representation discussed previously.
+In designing such a system the methodology of data abstraction can be a
+great help. The designer can create an initial implementation using a
+simple, straightforward representation such as unordered lists. This
+will be unsuitable for the eventual system, but it can be useful in
+providing a "quick and dirty" data base with which to test the rest of
+the system. Later on, the data representation can be modified to be
+more sophisticated. If the data base is accessed in terms of abstract
+selectors and constructors, this change in representation will not
+require any changes to the rest of the system.
+
+ *Exercise 2.66:* Implement the `lookup' procedure for the case
+ where the set of records is structured as a binary tree, ordered
+ by the numerical values of the keys.
+
+ ---------- Footnotes ----------
+
+ (1) If we want to be more formal, we can specify "consistent with
+the interpretations given above" to mean that the operations satisfy a
+collection of rules such as these:
+
+ * For any set `S' and any object `x', `(element-of-set? x
+ (adjoin-set x S))' is true (informally: "Adjoining an object to a
+ set produces a set that contains the object").
+
+ * For any sets `S' and `T' and any object `x', `(element-of-set? x
+ (union-set S T))' is equal to `(or (element-of-set? x S)
+ (element-of-set? x T))' (informally: "The elements of `(union S
+ T)' are the elements that are in `S' or in `T'").
+
+ * For any object `x', `(element-of-set? x '())' is false
+ (informally: "No object is an element of the empty set").
+
+
+ (2) Halving the size of the problem at each step is the
+distinguishing characteristic of logarithmic growth, as we saw with the
+fast-exponentiation algorithm of section *Note 1-2-4:: and the
+half-interval search method of section *Note 1-3-3::.
+
+ (3) We are representing sets in terms of trees, and trees in terms
+of lists--in effect, a data abstraction built upon a data abstraction.
+We can regard the procedures `entry', `left-branch', `right-branch',
+and `make-tree' as a way of isolating the abstraction of a "binary
+tree" from the particular way we might wish to represent such a tree in
+terms of list structure.
+
+ (4) Examples of such structures include "B-trees" and "red-black
+trees". There is a large literature on data structures devoted to this
+problem. See Cormen, Leiserson, and Rivest 1990.
+
+ (5) *Note Exercise 2-63:: through *Note Exercise 2-65:: are due to
+Paul Hilfinger.
+
+\1f
+File: sicp.info, Node: 2-3-4, Prev: 2-3-3, Up: 2-3
+
+2.3.4 Example: Huffman Encoding Trees
+-------------------------------------
+
+This section provides practice in the use of list structure and data
+abstraction to manipulate sets and trees. The application is to
+methods for representing data as sequences of ones and zeros (bits).
+For example, the ASCII standard code used to represent text in
+computers encodes each character as a sequence of seven bits. Using
+seven bits allows us to distinguish 2^(7), or 128, possible different
+characters. In general, if we want to distinguish n different symbols,
+we will need to use `log'_2 n bits per symbol. If all our messages are
+made up of the eight symbols A, B, C, D, E, F, G, and H, we can choose
+a code with three bits per character, for example
+
+ A 000 C 010 E 100 G 110
+ B 001 D 011 F 101 H 111
+
+With this code, the message
+
+ BACADAEAFABBAAAGAH
+
+is encoded as the string of 54 bits
+
+ 001000010000011000100000101000001001000000000110000111
+
+ Codes such as ASCII and the A-through-H code above are known as "fixed-length"
+codes, because they represent each symbol in the message with the same
+number of bits. It is sometimes advantageous to use "variable-length"
+codes, in which different symbols may be represented by different
+numbers of bits. For example, Morse code does not use the same number
+of dots and dashes for each letter of the alphabet. In particular, E,
+the most frequent letter, is represented by a single dot. In general,
+if our messages are such that some symbols appear very frequently and
+some very rarely, we can encode data more efficiently (i.e., using
+fewer bits per message) if we assign shorter codes to the frequent
+symbols. Consider the following alternative code for the letters A
+through H:
+
+ A 0 C 1010 E 1100 G 1110
+ B 100 D 1011 F 1101 H 1111
+
+With this code, the same message as above is encoded as the string
+
+ 100010100101101100011010100100000111001111
+
+ This string contains 42 bits, so it saves more than 20% in space in
+comparison with the fixed-length code shown above.
+
+ One of the difficulties of using a variable-length code is knowing
+when you have reached the end of a symbol in reading a sequence of
+zeros and ones. Morse code solves this problem by using a special "separator
+code" (in this case, a pause) after the sequence of dots and dashes for
+each letter. Another solution is to design the code in such a way that
+no complete code for any symbol is the beginning (or "prefix") of the
+code for another symbol. Such a code is called a "prefix code". In
+the example above, A is encoded by 0 and B is encoded by 100, so no
+other symbol can have a code that begins with 0 or with 100.
+
+ In general, we can attain significant savings if we use
+variable-length prefix codes that take advantage of the relative
+frequencies of the symbols in the messages to be encoded. One
+particular scheme for doing this is called the Huffman encoding method,
+after its discoverer, David Huffman. A Huffman code can be represented
+as a binary tree whose leaves are the symbols that are encoded. At
+each non-leaf node of the tree there is a set containing all the
+symbols in the leaves that lie below the node. In addition, each
+symbol at a leaf is assigned a weight (which is its relative
+frequency), and each non-leaf node contains a weight that is the sum of
+all the weights of the leaves lying below it. The weights are not used
+in the encoding or the decoding process. We will see below how they
+are used to help construct the tree.
+
+ *Figure 2.18:* A Huffman encoding tree.
+
+ {A B C D E F G H} 17
+ *
+ / \
+ / \
+ A 8 * {B C D E F G H} 9
+ __________/ \_____________
+ / \
+ {B C D} 5 * * {E F G H} 4
+ / \ ___/ \___
+ / \ / \
+ B 3 * {C D} 2 {E F} 2 * * {G H} 2
+ / \ / \ / \
+ / \ / \ / \
+ C 1 D 1 E 1 F 1 G 1 H 1
+
+ *Note Figure 2-18:: shows the Huffman tree for the A-through-H code
+given above. The weights at the leaves indicate that the tree was
+designed for messages in which A appears with relative frequency 8, B
+with relative frequency 3, and the other letters each with relative
+frequency 1.
+
+ Given a Huffman tree, we can find the encoding of any symbol by
+starting at the root and moving down until we reach the leaf that holds
+the symbol. Each time we move down a left branch we add a 0 to the
+code, and each time we move down a right branch we add a 1. (We decide
+which branch to follow by testing to see which branch either is the
+leaf node for the symbol or contains the symbol in its set.) For
+example, starting from the root of the tree in *Note Figure 2-18::, we
+arrive at the leaf for D by following a right branch, then a left
+branch, then a right branch, then a right branch; hence, the code for D
+is 1011.
+
+ To decode a bit sequence using a Huffman tree, we begin at the root
+and use the successive zeros and ones of the bit sequence to determine
+whether to move down the left or the right branch. Each time we come
+to a leaf, we have generated a new symbol in the message, at which
+point we start over from the root of the tree to find the next symbol.
+For example, suppose we are given the tree above and the sequence
+10001010. Starting at the root, we move down the right branch, (since
+the first bit of the string is 1), then down the left branch (since the
+second bit is 0), then down the left branch (since the third bit is
+also 0). This brings us to the leaf for B, so the first symbol of the
+decoded message is B. Now we start again at the root, and we make a
+left move because the next bit in the string is 0. This brings us to
+the leaf for A. Then we start again at the root with the rest of the
+string 1010, so we move right, left, right, left and reach C. Thus,
+the entire message is BAC.
+
+Generating Huffman trees
+........................
+
+Given an "alphabet" of symbols and their relative frequencies, how do we
+construct the "best" code? (In other words, which tree will encode
+messages with the fewest bits?) Huffman gave an algorithm for doing
+this and showed that the resulting code is indeed the best
+variable-length code for messages where the relative frequency of the
+symbols matches the frequencies with which the code was constructed.
+We will not prove this optimality of Huffman codes here, but we will
+show how Huffman trees are constructed.(1)
+
+ The algorithm for generating a Huffman tree is very simple. The idea
+is to arrange the tree so that the symbols with the lowest frequency
+appear farthest away from the root. Begin with the set of leaf nodes,
+containing symbols and their frequencies, as determined by the initial
+data from which the code is to be constructed. Now find two leaves with
+the lowest weights and merge them to produce a node that has these two
+nodes as its left and right branches. The weight of the new node is the
+sum of the two weights. Remove the two leaves from the original set and
+replace them by this new node. Now continue this process. At each step,
+merge two nodes with the smallest weights, removing them from the set
+and replacing them with a node that has these two as its left and right
+branches. The process stops when there is only one node left, which is
+the root of the entire tree. Here is how the Huffman tree of *Note
+Figure 2-18:: was generated:
+
+ Initial leaves {(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)}
+ Merge {(A 8) (B 3) ({C D} 2) (E 1) (F 1) (G 1) (H 1)}
+ Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) (G 1) (H 1)}
+ Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) ({G H} 2)}
+ Merge {(A 8) (B 3) ({C D} 2) ({E F G H} 4)}
+ Merge {(A 8) ({B C D} 5) ({E F G H} 4)}
+ Merge {(A 8) ({B C D E F G H} 9)}
+ Final merge {({A B C D E F G H} 17)}
+
+ The algorithm does not always specify a unique tree, because there
+may not be unique smallest-weight nodes at each step. Also, the choice
+of the order in which the two nodes are merged (i.e., which will be the
+right branch and which will be the left branch) is arbitrary.
+
+Representing Huffman trees
+..........................
+
+In the exercises below we will work with a system that uses Huffman
+trees to encode and decode messages and generates Huffman trees
+according to the algorithm outlined above. We will begin by discussing
+how trees are represented.
+
+ Leaves of the tree are represented by a list consisting of the symbol
+`leaf', the symbol at the leaf, and the weight:
+
+ (define (make-leaf symbol weight)
+ (list 'leaf symbol weight))
+
+ (define (leaf? object)
+ (eq? (car object) 'leaf))
+
+ (define (symbol-leaf x) (cadr x))
+
+ (define (weight-leaf x) (caddr x))
+
+ A general tree will be a list of a left branch, a right branch, a
+set of symbols, and a weight. The set of symbols will be simply a list
+of the symbols, rather than some more sophisticated set representation.
+When we make a tree by merging two nodes, we obtain the weight of the
+tree as the sum of the weights of the nodes, and the set of symbols as
+the union of the sets of symbols for the nodes. Since our symbol sets
+are represented as lists, we can form the union by using the `append'
+procedure we defined in section *Note 2-2-1:::
+
+ (define (make-code-tree left right)
+ (list left
+ right
+ (append (symbols left) (symbols right))
+ (+ (weight left) (weight right))))
+
+ If we make a tree in this way, we have the following selectors:
+
+ (define (left-branch tree) (car tree))
+
+ (define (right-branch tree) (cadr tree))
+
+ (define (symbols tree)
+ (if (leaf? tree)
+ (list (symbol-leaf tree))
+ (caddr tree)))
+
+ (define (weight tree)
+ (if (leaf? tree)
+ (weight-leaf tree)
+ (cadddr tree)))
+
+ The procedures `symbols' and `weight' must do something slightly
+different depending on whether they are called with a leaf or a general
+tree. These are simple examples of "generic procedures" (procedures
+that can handle more than one kind of data), which we will have much
+more to say about in sections *Note 2-4:: and *Note 2-5::.
+
+The decoding procedure
+......................
+
+The following procedure implements the decoding algorithm. It takes as
+arguments a list of zeros and ones, together with a Huffman tree.
+
+ (define (decode bits tree)
+ (define (decode-1 bits current-branch)
+ (if (null? bits)
+ '()
+ (let ((next-branch
+ (choose-branch (car bits) current-branch)))
+ (if (leaf? next-branch)
+ (cons (symbol-leaf next-branch)
+ (decode-1 (cdr bits) tree))
+ (decode-1 (cdr bits) next-branch)))))
+ (decode-1 bits tree))
+
+ (define (choose-branch bit branch)
+ (cond ((= bit 0) (left-branch branch))
+ ((= bit 1) (right-branch branch))
+ (else (error "bad bit -- CHOOSE-BRANCH" bit))))
+
+ The procedure `decode-1' takes two arguments: the list of remaining
+bits and the current position in the tree. It keeps moving "down" the
+tree, choosing a left or a right branch according to whether the next
+bit in the list is a zero or a one. (This is done with the procedure
+`choose-branch'.) When it reaches a leaf, it returns the symbol at
+that leaf as the next symbol in the message by `cons'ing it onto the
+result of decoding the rest of the message, starting at the root of the
+tree. Note the error check in the final clause of `choose-branch',
+which complains if the procedure finds something other than a zero or a
+one in the input data.
+
+Sets of weighted elements
+.........................
+
+In our representation of trees, each non-leaf node contains a set of
+symbols, which we have represented as a simple list. However, the
+tree-generating algorithm discussed above requires that we also work
+with sets of leaves and trees, successively merging the two smallest
+items. Since we will be required to repeatedly find the smallest item
+in a set, it is convenient to use an ordered representation for this
+kind of set.
+
+ We will represent a set of leaves and trees as a list of elements,
+arranged in increasing order of weight. The following `adjoin-set'
+procedure for constructing sets is similar to the one described in
+*Note Exercise 2-61::; however, items are compared by their weights,
+and the element being added to the set is never already in it.
+
+ (define (adjoin-set x set)
+ (cond ((null? set) (list x))
+ ((< (weight x) (weight (car set))) (cons x set))
+ (else (cons (car set)
+ (adjoin-set x (cdr set))))))
+
+ The following procedure takes a list of symbol-frequency pairs such
+as `((A 4) (B 2) (C 1) (D 1))' and constructs an initial ordered set of
+leaves, ready to be merged according to the Huffman algorithm:
+
+ (define (make-leaf-set pairs)
+ (if (null? pairs)
+ '()
+ (let ((pair (car pairs)))
+ (adjoin-set (make-leaf (car pair) ; symbol
+ (cadr pair)) ; frequency
+ (make-leaf-set (cdr pairs))))))
+
+ *Exercise 2.67:* Define an encoding tree and a sample message:
+
+ (define sample-tree
+ (make-code-tree (make-leaf 'A 4)
+ (make-code-tree
+ (make-leaf 'B 2)
+ (make-code-tree (make-leaf 'D 1)
+ (make-leaf 'C 1)))))
+
+ (define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))
+
+ Use the `decode' procedure to decode the message, and give the
+ result.
+
+ *Exercise 2.68:* The `encode' procedure takes as arguments a
+ message and a tree and produces the list of bits that gives the
+ encoded message.
+
+ (define (encode message tree)
+ (if (null? message)
+ '()
+ (append (encode-symbol (car message) tree)
+ (encode (cdr message) tree))))
+
+ `Encode-symbol' is a procedure, which you must write, that returns
+ the list of bits that encodes a given symbol according to a given
+ tree. You should design `encode-symbol' so that it signals an
+ error if the symbol is not in the tree at all. Test your
+ procedure by encoding the result you obtained in *Note Exercise
+ 2-67:: with the sample tree and seeing whether it is the same as
+ the original sample message.
+
+ *Exercise 2.69:* The following procedure takes as its argument a
+ list of symbol-frequency pairs (where no symbol appears in more
+ than one pair) and generates a Huffman encoding tree according to
+ the Huffman algorithm.
+
+ (define (generate-huffman-tree pairs)
+ (successive-merge (make-leaf-set pairs)))
+
+ `Make-leaf-set' is the procedure given above that transforms the
+ list of pairs into an ordered set of leaves. `Successive-merge'
+ is the procedure you must write, using `make-code-tree' to
+ successively merge the smallest-weight elements of the set until
+ there is only one element left, which is the desired Huffman tree.
+ (This procedure is slightly tricky, but not really complicated.
+ If you find yourself designing a complex procedure, then you are
+ almost certainly doing something wrong. You can take significant
+ advantage of the fact that we are using an ordered set
+ representation.)
+
+ *Exercise 2.70:* The following eight-symbol alphabet with
+ associated relative frequencies was designed to efficiently encode
+ the lyrics of 1950s rock songs. (Note that the "symbols" of an
+ "alphabet" need not be individual letters.)
+
+ A 2 NA 16
+ BOOM 1 SHA 3
+ GET 2 YIP 9
+ JOB 2 WAH 1
+
+ Use `generate-huffman-tree' (*Note Exercise 2-69::) to generate a
+ corresponding Huffman tree, and use `encode' (*Note Exercise
+ 2-68::) to encode the following message:
+
+ Get a job
+
+ Sha na na na na na na na na
+
+ Get a job
+
+ Sha na na na na na na na na
+
+ Wah yip yip yip yip yip yip yip yip yip
+
+ Sha boom
+
+ How many bits are required for the encoding? What is the smallest
+ number of bits that would be needed to encode this song if we used
+ a fixed-length code for the eight-symbol alphabet?
+
+ *Exercise 2.71:* Suppose we have a Huffman tree for an alphabet of
+ n symbols, and that the relative frequencies of the symbols are 1,
+ 2, 4, ..., 2^(n-1). Sketch the tree for n=5; for n=10. In such a
+ tree (for general n) how may bits are required to encode the most
+ frequent symbol? the least frequent symbol?
+
+ *Exercise 2.72:* Consider the encoding procedure that you designed
+ in *Note Exercise 2-68::. What is the order of growth in the
+ number of steps needed to encode a symbol? Be sure to include the
+ number of steps needed to search the symbol list at each node
+ encountered. To answer this question in general is difficult.
+ Consider the special case where the relative frequencies of the n
+ symbols are as described in *Note Exercise 2-71::, and give the
+ order of growth (as a function of n) of the number of steps needed
+ to encode the most frequent and least frequent symbols in the
+ alphabet.
+
+ ---------- Footnotes ----------
+
+ (1) See Hamming 1980 for a discussion of the mathematical properties
+of Huffman codes.
+
+\1f
+File: sicp.info, Node: 2-4, Next: 2-5, Prev: 2-3, Up: Chapter 2
+
+2.4 Multiple Representations for Abstract Data
+==============================================
+
+We have introduced data abstraction, a methodology for structuring
+systems in such a way that much of a program can be specified
+independent of the choices involved in implementing the data objects
+that the program manipulates. For example, we saw in section *Note
+2-1-1:: how to separate the task of designing a program that uses
+rational numbers from the task of implementing rational numbers in
+terms of the computer language's primitive mechanisms for constructing
+compound data. The key idea was to erect an abstraction barrier - in
+this case, the selectors and constructors for rational numbers
+(`make-rat', `numer', `denom')--that isolates the way rational numbers
+are used from their underlying representation in terms of list
+structure. A similar abstraction barrier isolates the details of the
+procedures that perform rational arithmetic (`add-rat', `sub-rat',
+`mul-rat', and `div-rat') from the "higher-level" procedures that use
+rational numbers. The resulting program has the structure shown in
+*Note Figure 2-1::.
+
+ These data-abstraction barriers are powerful tools for controlling
+complexity. By isolating the underlying representations of data
+objects, we can divide the task of designing a large program into
+smaller tasks that can be performed separately. But this kind of data
+abstraction is not yet powerful enough, because it may not always make
+sense to speak of "the underlying representation" for a data object.
+
+ For one thing, there might be more than one useful representation
+for a data object, and we might like to design systems that can deal
+with multiple representations. To take a simple example, complex
+numbers may be represented in two almost equivalent ways: in
+rectangular form (real and imaginary parts) and in polar form
+(magnitude and angle). Sometimes rectangular form is more appropriate
+and sometimes polar form is more appropriate. Indeed, it is perfectly
+plausible to imagine a system in which complex numbers are represented
+in both ways, and in which the procedures for manipulating complex
+numbers work with either representation.
+
+ More importantly, programming systems are often designed by many
+people working over extended periods of time, subject to requirements
+that change over time. In such an environment, it is simply not
+possible for everyone to agree in advance on choices of data
+representation. So in addition to the data-abstraction barriers that
+isolate representation from use, we need abstraction barriers that
+isolate different design choices from each other and permit different
+choices to coexist in a single program. Furthermore, since large
+programs are often created by combining pre-existing modules that were
+designed in isolation, we need conventions that permit programmers to
+incorporate modules into larger systems "additively", that is, without
+having to redesign or reimplement these modules.
+
+ In this section, we will learn how to cope with data that may be
+represented in different ways by different parts of a program. This
+requires constructing "generic procedures"--procedures that can operate
+on data that may be represented in more than one way. Our main
+technique for building generic procedures will be to work in terms of
+data objects that have tags "type tags", that is, data objects that
+include explicit information about how they are to be processed. We
+will also discuss "data-directed" programming, a powerful and
+convenient implementation strategy for additively assembling systems
+with generic operations.
+
+ We begin with the simple complex-number example. We will see how
+type tags and data-directed style enable us to design separate
+rectangular and polar representations for complex numbers while
+maintaining the notion of an abstract "complex-number" data object. We
+will accomplish this by defining arithmetic procedures for complex
+numbers (`add-complex', `sub-complex', `mul-complex', and
+`div-complex') in terms of generic selectors that access parts of a
+complex number independent of how the number is represented. The
+resulting complex-number system, as shown in *Note Figure 2-19::,
+contains two different kinds of abstraction barriers. The "horizontal"
+abstraction barriers play the same role as the ones in *Note Figure
+2-1::. They isolate "higher-level" operations from "lower-level"
+representations. In addition, there is a "vertical" barrier that gives
+us the ability to separately design and install alternative
+representations.
+
+ *Figure 2.19:* Data-abstraction barriers in the complex-number
+ system.
+
+ Programs that use complex numbers
+ +-------------------------------------------------+
+ --| add-complex sub-complex mul-complex div-complex |--
+ +-------------------------------------------------+
+ Complex arithmetic package
+ ---------------------------+---------------------------
+ Rectangular | Polar
+ representation | representation
+ ---------------------------+---------------------------
+ List structure and primitive machine arithmetic
+
+ In section *Note 2-5:: we will show how to use type tags and
+data-directed style to develop a generic arithmetic package. This
+provides procedures (`add', `mul', and so on) that can be used to
+manipulate all sorts of "numbers" and can be easily extended when a new
+kind of number is needed. In section *Note 2-5-3::, we'll show how to
+use generic arithmetic in a system that performs symbolic algebra.
+
+* Menu:
+
+* 2-4-1:: Representations for Complex Numbers
+* 2-4-2:: Tagged data
+* 2-4-3:: Data-Directed Programming and Additivity
+
+\1f
+File: sicp.info, Node: 2-4-1, Next: 2-4-2, Prev: 2-4, Up: 2-4
+
+2.4.1 Representations for Complex Numbers
+-----------------------------------------
+
+We will develop a system that performs arithmetic operations on complex
+numbers as a simple but unrealistic example of a program that uses
+generic operations. We begin by discussing two plausible
+representations for complex numbers as ordered pairs: rectangular form
+(real part and imaginary part) and polar form (magnitude and angle).(1)
+Section *Note 2-4-2:: will show how both representations can be made
+to coexist in a single system through the use of type tags and generic
+operations.
+
+ Like rational numbers, complex numbers are naturally represented as
+ordered pairs. The set of complex numbers can be thought of as a
+two-dimensional space with two orthogonal axes, the "real" axis and the
+"imaginary" axis. (See *Note Figure 2-20::.) From this point of view,
+the complex number z = x + iy (where i^2 = - 1) can be thought of as
+the point in the plane whose real coordinate is x and whose imaginary
+coordinate is y. Addition of complex numbers reduces in this
+representation to addition of coordinates:
+
+ Real-part(z_1 + z_2) = Real-part(z_1) + Real-part(z_2)
+
+ Imaginary-part(z_1 + z_2) = Imaginary-part(z_1) + Imaginary-part(z_2)
+
+ When multiplying complex numbers, it is more natural to think in
+terms of representing a complex number in polar form, as a magnitude
+and an angle (r and A in *Note Figure 2-20::). The product of two
+complex numbers is the vector obtained by stretching one complex number
+by the length of the other and then rotating it through the angle of
+the other:
+
+ Magnitude(z_1 * z_2) = Magnitude(z_1) * Magnitude(z_2)
+
+ Angle(z_1 * z_2) = Angle(z_1) + Angle(z_2)
+
+ *Figure 2.20:* Complex numbers as points in the plane.
+
+ Imaginary
+ ^
+ |
+ y |.........................* z = x + ?y = r e^(?A)
+ | __-- .
+ | __-- .
+ | r __-- .
+ | __-- .
+ | __-- \ .
+ |__-- A | .
+ ----+----------+-------------------> Real
+ x
+
+ Thus, there are two different representations for complex numbers,
+which are appropriate for different operations. Yet, from the
+viewpoint of someone writing a program that uses complex numbers, the
+principle of data abstraction suggests that all the operations for
+manipulating complex numbers should be available regardless of which
+representation is used by the computer. For example, it is often
+useful to be able to find the magnitude of a complex number that is
+specified by rectangular coordinates. Similarly, it is often useful to
+be able to determine the real part of a complex number that is
+specified by polar coordinates.
+
+ To design such a system, we can follow the same data-abstraction
+strategy we followed in designing the rational-number package in
+section *Note 2-1-1::. Assume that the operations on complex numbers
+are implemented in terms of four selectors: `real-part', `imag-part',
+`magnitude', and `angle'. Also assume that we have two procedures for
+constructing complex numbers: `make-from-real-imag' returns a complex
+number with specified real and imaginary parts, and `make-from-mag-ang'
+returns a complex number with specified magnitude and angle. These
+procedures have the property that, for any complex number `z', both
+
+ (make-from-real-imag (real-part z) (imag-part z))
+
+and
+
+ (make-from-mag-ang (magnitude z) (angle z))
+
+produce complex numbers that are equal to `z'.
+
+ Using these constructors and selectors, we can implement arithmetic
+on complex numbers using the "abstract data" specified by the
+constructors and selectors, just as we did for rational numbers in
+section *Note 2-1-1::. As shown in the formulas above, we can add and
+subtract complex numbers in terms of real and imaginary parts while
+multiplying and dividing complex numbers in terms of magnitudes and
+angles:
+
+ (define (add-complex z1 z2)
+ (make-from-real-imag (+ (real-part z1) (real-part z2))
+ (+ (imag-part z1) (imag-part z2))))
+
+ (define (sub-complex z1 z2)
+ (make-from-real-imag (- (real-part z1) (real-part z2))
+ (- (imag-part z1) (imag-part z2))))
+
+ (define (mul-complex z1 z2)
+ (make-from-mag-ang (* (magnitude z1) (magnitude z2))
+ (+ (angle z1) (angle z2))))
+
+ (define (div-complex z1 z2)
+ (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
+ (- (angle z1) (angle z2))))
+
+ To complete the complex-number package, we must choose a
+representation and we must implement the constructors and selectors in
+terms of primitive numbers and primitive list structure. There are two
+obvious ways to do this: We can represent a complex number in
+"rectangular form" as a pair (real part, imaginary part) or in "polar
+form" as a pair (magnitude, angle). Which shall we choose?
+
+ In order to make the different choices concrete, imagine that there
+are two programmers, Ben Bitdiddle and Alyssa P. Hacker, who are
+independently designing representations for the complex-number system.
+Ben chooses to represent complex numbers in rectangular form. With
+this choice, selecting the real and imaginary parts of a complex number
+is straightforward, as is constructing a complex number with given real
+and imaginary parts. To find the magnitude and the angle, or to
+construct a complex number with a given magnitude and angle, he uses
+the trigonometric relations
+
+ __________
+ x = r cos A r = ./ x^2 + y^2
+
+ y = r sin A A = arctan(y,x)
+
+which relate the real and imaginary parts (x, y) to the magnitude and
+the angle (r, A).(2) Ben's representation is therefore given by the
+following selectors and constructors:
+
+ (define (real-part z) (car z))
+
+ (define (imag-part z) (cdr z))
+
+ (define (magnitude z)
+ (sqrt (+ (square (real-part z)) (square (imag-part z)))))
+
+ (define (angle z)
+ (atan (imag-part z) (real-part z)))
+
+ (define (make-from-real-imag x y) (cons x y))
+
+ (define (make-from-mag-ang r a)
+ (cons (* r (cos a)) (* r (sin a))))
+
+ Alyssa, in contrast, chooses to represent complex numbers in polar
+form. For her, selecting the magnitude and angle is straightforward,
+but she has to use the trigonometric relations to obtain the real and
+imaginary parts. Alyssa's representation is:
+
+ (define (real-part z)
+ (* (magnitude z) (cos (angle z))))
+
+ (define (imag-part z)
+ (* (magnitude z) (sin (angle z))))
+
+ (define (magnitude z) (car z))
+
+ (define (angle z) (cdr z))
+
+ (define (make-from-real-imag x y)
+ (cons (sqrt (+ (square x) (square y)))
+ (atan y x)))
+
+ (define (make-from-mag-ang r a) (cons r a))
+
+ The discipline of data abstraction ensures that the same
+implementation of `add-complex', `sub-complex', `mul-complex', and
+`div-complex' will work with either Ben's representation or Alyssa's
+representation.
+
+ ---------- Footnotes ----------
+
+ (1) In actual computational systems, rectangular form is preferable
+to polar form most of the time because of roundoff errors in conversion
+between rectangular and polar form. This is why the complex-number
+example is unrealistic. Nevertheless, it provides a clear illustration
+of the design of a system using generic operations and a good
+introduction to the more substantial systems to be developed later in
+this chapter.
+
+ (2) The arctangent function referred to here, computed by Scheme's
+`atan' procedure, is defined so as to take two arguments y and x and to
+return the angle whose tangent is y/x. The signs of the arguments
+determine the quadrant of the angle.
+
+\1f
+File: sicp.info, Node: 2-4-2, Next: 2-4-3, Prev: 2-4-1, Up: 2-4
+
+2.4.2 Tagged data
+-----------------
+
+One way to view data abstraction is as an application of the "principle
+of least commitment." In implementing the complex-number system in
+section *Note 2-4-1::, we can use either Ben's rectangular
+representation or Alyssa's polar representation. The abstraction
+barrier formed by the selectors and constructors permits us to defer to
+the last possible moment the choice of a concrete representation for
+our data objects and thus retain maximum flexibility in our system
+design.
+
+ The principle of least commitment can be carried to even further
+extremes. If we desire, we can maintain the ambiguity of
+representation even _after_ we have designed the selectors and
+constructors, and elect to use both Ben's representation _and_ Alyssa's
+representation. If both representations are included in a single
+system, however, we will need some way to distinguish data in polar
+form from data in rectangular form. Otherwise, if we were asked, for
+instance, to find the `magnitude' of the pair (3,4), we wouldn't know
+whether to answer 5 (interpreting the number in rectangular form) or 3
+(interpreting the number in polar form). A straightforward way to
+accomplish this distinction is to include a "type tag"--the symbol
+`rectangular' or `polar'--as part of each complex number. Then when we
+need to manipulate a complex number we can use the tag to decide which
+selector to apply.
+
+ In order to manipulate tagged data, we will assume that we have
+procedures `type-tag' and `contents' that extract from a data object
+the tag and the actual contents (the polar or rectangular coordinates,
+in the case of a complex number). We will also postulate a procedure
+`attach-tag' that takes a tag and contents and produces a tagged data
+object. A straightforward way to implement this is to use ordinary
+list structure:
+
+ (define (attach-tag type-tag contents)
+ (cons type-tag contents))
+
+ (define (type-tag datum)
+ (if (pair? datum)
+ (car datum)
+ (error "Bad tagged datum -- TYPE-TAG" datum)))
+
+ (define (contents datum)
+ (if (pair? datum)
+ (cdr datum)
+ (error "Bad tagged datum -- CONTENTS" datum)))
+
+ Using these procedures, we can define predicates `rectangular?' and
+`polar?', which recognize polar and rectangular numbers, respectively:
+
+ (define (rectangular? z)
+ (eq? (type-tag z) 'rectangular))
+
+ (define (polar? z)
+ (eq? (type-tag z) 'polar))
+
+ With type tags, Ben and Alyssa can now modify their code so that
+their two different representations can coexist in the same system.
+Whenever Ben constructs a complex number, he tags it as rectangular.
+Whenever Alyssa constructs a complex number, she tags it as polar. In
+addition, Ben and Alyssa must make sure that the names of their
+procedures do not conflict. One way to do this is for Ben to append
+the suffix `rectangular' to the name of each of his representation
+procedures and for Alyssa to append `polar' to the names of hers. Here
+is Ben's revised rectangular representation from section *Note 2-4-1:::
+
+ (define (real-part-rectangular z) (car z))
+
+ (define (imag-part-rectangular z) (cdr z))
+
+ (define (magnitude-rectangular z)
+ (sqrt (+ (square (real-part-rectangular z))
+ (square (imag-part-rectangular z)))))
+
+ (define (angle-rectangular z)
+ (atan (imag-part-rectangular z)
+ (real-part-rectangular z)))
+
+ (define (make-from-real-imag-rectangular x y)
+ (attach-tag 'rectangular (cons x y)))
+
+ (define (make-from-mag-ang-rectangular r a)
+ (attach-tag 'rectangular
+ (cons (* r (cos a)) (* r (sin a)))))
+
+and here is Alyssa's revised polar representation:
+
+ (define (real-part-polar z)
+ (* (magnitude-polar z) (cos (angle-polar z))))
+
+ (define (imag-part-polar z)
+ (* (magnitude-polar z) (sin (angle-polar z))))
+
+ (define (magnitude-polar z) (car z))
+
+ (define (angle-polar z) (cdr z))
+
+ (define (make-from-real-imag-polar x y)
+ (attach-tag 'polar
+ (cons (sqrt (+ (square x) (square y)))
+ (atan y x))))
+
+ (define (make-from-mag-ang-polar r a)
+ (attach-tag 'polar (cons r a)))
+
+ Each generic selector is implemented as a procedure that checks the
+tag of its argument and calls the appropriate procedure for handling
+data of that type. For example, to obtain the real part of a complex
+number, `real-part' examines the tag to determine whether to use Ben's
+`real-part-rectangular' or Alyssa's `real-part-polar'. In either case,
+we use `contents' to extract the bare, untagged datum and send this to
+the rectangular or polar procedure as required:
+
+ (define (real-part z)
+ (cond ((rectangular? z)
+ (real-part-rectangular (contents z)))
+ ((polar? z)
+ (real-part-polar (contents z)))
+ (else (error "Unknown type -- REAL-PART" z))))
+
+ (define (imag-part z)
+ (cond ((rectangular? z)
+ (imag-part-rectangular (contents z)))
+ ((polar? z)
+ (imag-part-polar (contents z)))
+ (else (error "Unknown type -- IMAG-PART" z))))
+
+ (define (magnitude z)
+ (cond ((rectangular? z)
+ (magnitude-rectangular (contents z)))
+ ((polar? z)
+ (magnitude-polar (contents z)))
+ (else (error "Unknown type -- MAGNITUDE" z))))
+
+ (define (angle z)
+ (cond ((rectangular? z)
+ (angle-rectangular (contents z)))
+ ((polar? z)
+ (angle-polar (contents z)))
+ (else (error "Unknown type -- ANGLE" z))))
+
+ To implement the complex-number arithmetic operations, we can use
+the same procedures `add-complex', `sub-complex', `mul-complex', and
+`div-complex' from section *Note 2-4-1::, because the selectors they
+call are generic, and so will work with either representation. For
+example, the procedure `add-complex' is still
+
+ (define (add-complex z1 z2)
+ (make-from-real-imag (+ (real-part z1) (real-part z2))
+ (+ (imag-part z1) (imag-part z2))))
+
+ Finally, we must choose whether to construct complex numbers using
+Ben's representation or Alyssa's representation. One reasonable choice
+is to construct rectangular numbers whenever we have real and imaginary
+parts and to construct polar numbers whenever we have magnitudes and
+angles:
+
+ (define (make-from-real-imag x y)
+ (make-from-real-imag-rectangular x y))
+
+ (define (make-from-mag-ang r a)
+ (make-from-mag-ang-polar r a))
+
+ *Figure 2.21:* Structure of the generic complex-arithmetic system.
+
+ +--------------------------------------------------+
+ ----| add-complex sub-complex mul-complex- div-complex |----
+ +--------------------------------------------------+
+ Complex arithmetic package
+ +-----------------------+
+ | real-part imag-part |
+ -----------------| |------------------
+ | magnitude angle |
+ +-----------+-----------+
+ Rectangular | Polar
+ representation | representation
+ -----------------------------+------------------------------
+ List structure and primitive machine arithmetic
+
+ The resulting complex-number system has the structure shown in *Note
+Figure 2-21::. The system has been decomposed into three relatively
+independent parts: the complex-number-arithmetic operations, Alyssa's
+polar implementation, and Ben's rectangular implementation. The polar
+and rectangular implementations could have been written by Ben and
+Alyssa working separately, and both of these can be used as underlying
+representations by a third programmer implementing the
+complex-arithmetic procedures in terms of the abstract
+constructor/selector interface.
+
+ Since each data object is tagged with its type, the selectors
+operate on the data in a generic manner. That is, each selector is
+defined to have a behavior that depends upon the particular type of
+data it is applied to. Notice the general mechanism for interfacing
+the separate representations: Within a given representation
+implementation (say, Alyssa's polar package) a complex number is an
+untyped pair (magnitude, angle). When a generic selector operates on a
+number of `polar' type, it strips off the tag and passes the contents on
+to Alyssa's code. Conversely, when Alyssa constructs a number for
+general use, she tags it with a type so that it can be appropriately
+recognized by the higher-level procedures. This discipline of
+stripping off and attaching tags as data objects are passed from level
+to level can be an important organizational strategy, as we shall see
+in section *Note 2-5::.
+
+\1f
+File: sicp.info, Node: 2-4-3, Prev: 2-4-2, Up: 2-4
+
+2.4.3 Data-Directed Programming and Additivity
+----------------------------------------------
+
+The general strategy of checking the type of a datum and calling an
+appropriate procedure is called "dispatching on type". This is a
+powerful strategy for obtaining modularity in system design. Oh the
+other hand, implementing the dispatch as in section *Note 2-4-2:: has
+two significant weaknesses. One weakness is that the generic interface
+procedures (`real-part', `imag-part', `magnitude', and `angle') must
+know about all the different representations. For instance, suppose we
+wanted to incorporate a new representation for complex numbers into our
+complex-number system. We would need to identify this new
+representation with a type, and then add a clause to each of the
+generic interface procedures to check for the new type and apply the
+appropriate selector for that representation.
+
+ Another weakness of the technique is that even though the individual
+representations can be designed separately, we must guarantee that no
+two procedures in the entire system have the same name. This is why
+Ben and Alyssa had to change the names of their original procedures
+from section *Note 2-4-1::.
+
+ The issue underlying both of these weaknesses is that the technique
+for implementing generic interfaces is not "additive". The person
+implementing the generic selector procedures must modify those
+procedures each time a new representation is installed, and the people
+interfacing the individual representations must modify their code to
+avoid name conflicts. In each of these cases, the changes that must be
+made to the code are straightforward, but they must be made
+nonetheless, and this is a source of inconvenience and error. This is
+not much of a problem for the complex-number system as it stands, but
+suppose there were not two but hundreds of different representations
+for complex numbers. And suppose that there were many generic
+selectors to be maintained in the abstract-data interface. Suppose, in
+fact, that no one programmer knew all the interface procedures or all
+the representations. The problem is real and must be addressed in such
+programs as large-scale data-base-management systems.
+
+ What we need is a means for modularizing the system design even
+further. This is provided by the programming technique known as programming
+"data-directed programming". To understand how data-directed
+programming works, begin with the observation that whenever we deal
+with a set of generic operations that are common to a set of different
+types we are, in effect, dealing with a two-dimensional table that
+contains the possible operations on one axis and the possible types on
+the other axis. The entries in the table are the procedures that
+implement each operation for each type of argument presented. In the
+complex-number system developed in the previous section, the
+correspondence between operation name, data type, and actual procedure
+was spread out among the various conditional clauses in the generic
+interface procedures. But the same information could have been
+organized in a table, as shown in *Note Figure 2-22::.
+
+ Data-directed programming is the technique of designing programs to
+work with such a table directly. Previously, we implemented the
+mechanism that interfaces the complex-arithmetic code with the two
+representation packages as a set of procedures that each perform an
+explicit dispatch on type. Here we will implement the interface as a
+single procedure that looks up the combination of the operation name
+and argument type in the table to find the correct procedure to apply,
+and then applies it to the contents of the argument. If we do this,
+then to add a new representation package to the system we need not
+change any existing procedures; we need only add new entries to the
+table.
+
+ *Figure 2.22:* Table of operations for the complex-number system.
+
+ | Types
+ Operations | Polar | Rectangular
+ ===========+=================+======================
+ real-part | real-part-polar | real-part-rectangular
+ imag-part | imag-part-polar | imag-part-rectangular
+ magnitude | magnitude-polar | magnitude-rectangular
+ angle | angle-polar | angle-rectangular
+
+ To implement this plan, assume that we have two procedures, `put' and
+`get', for manipulating the operation-and-type table:
+
+ * `(put <OP> <TYPE> <ITEM>)' installs the `<ITEM>' in the table,
+ indexed by the `<OP>' and the `<TYPE>'.
+
+ * `(get <OP> <TYPE>)' looks up the `<OP>', `<TYPE>' entry in the
+ table and returns the item found there. If no item is found,
+ `get' returns false.
+
+
+ For now, we can assume that `put' and `get' are included in our
+language. In *Note Chapter 3:: (section *Note 3-3-3::, *Note Exercise
+3-24::) we will see how to implement these and other operations for
+manipulating tables.
+
+ Here is how data-directed programming can be used in the
+complex-number system. Ben, who developed the rectangular
+representation, implements his code just as he did originally. He
+defines a collection of procedures, or a "package", and interfaces
+these to the rest of the system by adding entries to the table that
+tell the system how to operate on rectangular numbers. This is
+accomplished by calling the following procedure:
+
+ (define (install-rectangular-package)
+ ;; internal procedures
+ (define (real-part z) (car z))
+ (define (imag-part z) (cdr z))
+ (define (make-from-real-imag x y) (cons x y))
+ (define (magnitude z)
+ (sqrt (+ (square (real-part z))
+ (square (imag-part z)))))
+ (define (angle z)
+ (atan (imag-part z) (real-part z)))
+ (define (make-from-mag-ang r a)
+ (cons (* r (cos a)) (* r (sin a))))
+
+ ;; interface to the rest of the system
+ (define (tag x) (attach-tag 'rectangular x))
+ (put 'real-part '(rectangular) real-part)
+ (put 'imag-part '(rectangular) imag-part)
+ (put 'magnitude '(rectangular) magnitude)
+ (put 'angle '(rectangular) angle)
+ (put 'make-from-real-imag 'rectangular
+ (lambda (x y) (tag (make-from-real-imag x y))))
+ (put 'make-from-mag-ang 'rectangular
+ (lambda (r a) (tag (make-from-mag-ang r a))))
+ 'done)
+
+ Notice that the internal procedures here are the same procedures
+from section *Note 2-4-1:: that Ben wrote when he was working in
+isolation. No changes are necessary in order to interface them to the
+rest of the system. Moreover, since these procedure definitions are
+internal to the installation procedure, Ben needn't worry about name
+conflicts with other procedures outside the rectangular package. To
+interface these to the rest of the system, Ben installs his `real-part'
+procedure under the operation name `real-part' and the type
+`(rectangular)', and similarly for the other selectors.(1) The
+interface also defines the constructors to be used by the external
+system.(2) These are identical to Ben's internally defined
+constructors, except that they attach the tag.
+
+ Alyssa's polar package is analogous:
+
+ (define (install-polar-package)
+ ;; internal procedures
+ (define (magnitude z) (car z))
+ (define (angle z) (cdr z))
+ (define (make-from-mag-ang r a) (cons r a))
+ (define (real-part z)
+ (* (magnitude z) (cos (angle z))))
+ (define (imag-part z)
+ (* (magnitude z) (sin (angle z))))
+ (define (make-from-real-imag x y)
+ (cons (sqrt (+ (square x) (square y)))
+ (atan y x)))
+
+ ;; interface to the rest of the system
+ (define (tag x) (attach-tag 'polar x))
+ (put 'real-part '(polar) real-part)
+ (put 'imag-part '(polar) imag-part)
+ (put 'magnitude '(polar) magnitude)
+ (put 'angle '(polar) angle)
+ (put 'make-from-real-imag 'polar
+ (lambda (x y) (tag (make-from-real-imag x y))))
+ (put 'make-from-mag-ang 'polar
+ (lambda (r a) (tag (make-from-mag-ang r a))))
+ 'done)
+
+ Even though Ben and Alyssa both still use their original procedures
+defined with the same names as each other's (e.g., `real-part'), these
+definitions are now internal to different procedures (see section *Note
+1-1-8::), so there is no name conflict.
+
+ The complex-arithmetic selectors access the table by means of a
+general "operation" procedure called `apply-generic', which applies a
+generic operation to some arguments. `Apply-generic' looks in the
+table under the name of the operation and the types of the arguments
+and applies the resulting procedure if one is present:(3)
+
+ (define (apply-generic op . args)
+ (let ((type-tags (map type-tag args)))
+ (let ((proc (get op type-tags)))
+ (if proc
+ (apply proc (map contents args))
+ (error
+ "No method for these types -- APPLY-GENERIC"
+ (list op type-tags))))))
+
+ Using `apply-generic', we can define our generic selectors as
+follows:
+
+ (define (real-part z) (apply-generic 'real-part z))
+ (define (imag-part z) (apply-generic 'imag-part z))
+ (define (magnitude z) (apply-generic 'magnitude z))
+ (define (angle z) (apply-generic 'angle z))
+
+ Observe that these do not change at all if a new representation is
+added to the system.
+
+ We can also extract from the table the constructors to be used by
+the programs external to the packages in making complex numbers from
+real and imaginary parts and from magnitudes and angles. As in section
+*Note 2-4-2::, we construct rectangular numbers whenever we have real
+and imaginary parts, and polar numbers whenever we have magnitudes and
+angles:
+
+ (define (make-from-real-imag x y)
+ ((get 'make-from-real-imag 'rectangular) x y))
+
+ (define (make-from-mag-ang r a)
+ ((get 'make-from-mag-ang 'polar) r a))
+
+ *Exercise 2.73:* Section *Note 2-3-2:: described a program that
+ performs symbolic differentiation:
+
+ (define (deriv exp var)
+ (cond ((number? exp) 0)
+ ((variable? exp) (if (same-variable? exp var) 1 0))
+ ((sum? exp)
+ (make-sum (deriv (addend exp) var)
+ (deriv (augend exp) var)))
+ ((product? exp)
+ (make-sum
+ (make-product (multiplier exp)
+ (deriv (multiplicand exp) var))
+ (make-product (deriv (multiplier exp) var)
+ (multiplicand exp))))
+ <MORE RULES CAN BE ADDED HERE>
+ (else (error "unknown expression type -- DERIV" exp))))
+
+ We can regard this program as performing a dispatch on the type of
+ the expression to be differentiated. In this situation the "type
+ tag" of the datum is the algebraic operator symbol (such as `+')
+ and the operation being performed is `deriv'. We can transform
+ this program into data-directed style by rewriting the basic
+ derivative procedure as
+
+ (define (deriv exp var)
+ (cond ((number? exp) 0)
+ ((variable? exp) (if (same-variable? exp var) 1 0))
+ (else ((get 'deriv (operator exp)) (operands exp)
+ var))))
+
+ (define (operator exp) (car exp))
+
+ (define (operands exp) (cdr exp))
+
+ a. Explain what was done above. Why can't we assimilate the
+ predicates `number?' and `same-variable?' into the
+ data-directed dispatch?
+
+ b. Write the procedures for derivatives of sums and products,
+ and the auxiliary code required to install them in the table
+ used by the program above.
+
+ c. Choose any additional differentiation rule that you like,
+ such as the one for exponents (*Note Exercise 2-56::), and
+ install it in this data-directed system.
+
+ d. In this simple algebraic manipulator the type of an
+ expression is the algebraic operator that binds it together.
+ Suppose, however, we indexed the procedures in the opposite
+ way, so that the dispatch line in `deriv' looked like
+
+ ((get (operator exp) 'deriv) (operands exp) var)
+
+ What corresponding changes to the derivative system are
+ required?
+
+
+ *Exercise 2.74:* Insatiable Enterprises, Inc., is a highly
+ decentralized conglomerate company consisting of a large number of
+ independent divisions located all over the world. The company's
+ computer facilities have just been interconnected by means of a
+ clever network-interfacing scheme that makes the entire network
+ appear to any user to be a single computer. Insatiable's
+ president, in her first attempt to exploit the ability of the
+ network to extract administrative information from division files,
+ is dismayed to discover that, although all the division files have
+ been implemented as data structures in Scheme, the particular data
+ structure used varies from division to division. A meeting of
+ division managers is hastily called to search for a strategy to
+ integrate the files that will satisfy headquarters' needs while
+ preserving the existing autonomy of the divisions.
+
+ Show how such a strategy can be implemented with data-directed
+ programming. As an example, suppose that each division's
+ personnel records consist of a single file, which contains a set
+ of records keyed on employees' names. The structure of the set
+ varies from division to division. Furthermore, each employee's
+ record is itself a set (structured differently from division to
+ division) that contains information keyed under identifiers such
+ as `address' and `salary'. In particular:
+
+ a. Implement for headquarters a `get-record' procedure that
+ retrieves a specified employee's record from a specified
+ personnel file. The procedure should be applicable to any
+ division's file. Explain how the individual divisions' files
+ should be structured. In particular, what type information
+ must be supplied?
+
+ b. Implement for headquarters a `get-salary' procedure that
+ returns the salary information from a given employee's record
+ from any division's personnel file. How should the record be
+ structured in order to make this operation work?
+
+ c. Implement for headquarters a `find-employee-record'
+ procedure. This should search all the divisions' files for
+ the record of a given employee and return the record. Assume
+ that this procedure takes as arguments an employee's name and
+ a list of all the divisions' files.
+
+ d. When Insatiable takes over a new company, what changes must
+ be made in order to incorporate the new personnel information
+ into the central system?
+
+
+Message passing
+...............
+
+The key idea of data-directed programming is to handle generic
+operations in programs by dealing explicitly with operation-and-type
+tables, such as the table in *Note Figure 2-22::. The style of
+programming we used in section *Note 2-4-2:: organized the required
+dispatching on type by having each operation take care of its own
+dispatching. In effect, this decomposes the operation-and-type table
+into rows, with each generic operation procedure representing a row of
+the table.
+
+ An alternative implementation strategy is to decompose the table
+into columns and, instead of using "intelligent operations" that
+dispatch on data types, to work with "intelligent data objects" that
+dispatch on operation names. We can do this by arranging things so
+that a data object, such as a rectangular number, is represented as a
+procedure that takes as input the required operation name and performs
+the operation indicated. In such a discipline, `make-from-real-imag'
+could be written as
+
+ (define (make-from-real-imag x y)
+ (define (dispatch op)
+ (cond ((eq? op 'real-part) x)
+ ((eq? op 'imag-part) y)
+ ((eq? op 'magnitude)
+ (sqrt (+ (square x) (square y))))
+ ((eq? op 'angle) (atan y x))
+ (else
+ (error "Unknown op -- MAKE-FROM-REAL-IMAG" op))))
+ dispatch)
+
+ The corresponding `apply-generic' procedure, which applies a generic
+operation to an argument, now simply feeds the operation's name to the
+data object and lets the object do the work:(4)
+
+ (define (apply-generic op arg) (arg op))
+
+ Note that the value returned by `make-from-real-imag' is a
+procedure--the internal `dispatch' procedure. This is the procedure
+that is invoked when `apply-generic' requests an operation to be
+performed.
+
+ This style of programming is called "message passing". The name
+comes from the image that a data object is an entity that receives the
+requested operation name as a "message." We have already seen an
+example of message passing in section *Note 2-1-3::, where we saw how
+`cons', `car', and `cdr' could be defined with no data objects but only
+procedures. Here we see that message passing is not a mathematical
+trick but a useful technique for organizing systems with generic
+operations. In the remainder of this chapter we will continue to use
+data-directed programming, rather than message passing, to discuss
+generic arithmetic operations. In *Note Chapter 3:: we will return to
+message passing, and we will see that it can be a powerful tool for
+structuring simulation programs.
+
+ *Exercise 2.75:* Implement the constructor `make-from-mag-ang' in
+ message-passing style. This procedure should be analogous to the
+ `make-from-real-imag' procedure given above.
+
+ *Exercise 2.76:* As a large system with generic operations
+ evolves, new types of data objects or new operations may be needed.
+ For each of the three strategies--generic operations with explicit
+ dispatch, data-directed style, and message-passing-style--describe
+ the changes that must be made to a system in order to add new
+ types or new operations. Which organization would be most
+ appropriate for a system in which new types must often be added?
+ Which would be most appropriate for a system in which new
+ operations must often be added?
+
+ ---------- Footnotes ----------
+
+ (1) We use the list `(rectangular)' rather than the symbol
+`rectangular' to allow for the possibility of operations with multiple
+arguments, not all of the same type.
+
+ (2) The type the constructors are installed under needn't be a list
+because a constructor is always used to make an object of one
+particular type.
+
+ (3) `Apply-generic' uses the dotted-tail notation described in *Note
+Exercise 2-20::, because different generic operations may take
+different numbers of arguments. In `apply-generic', `op' has as its
+value the first argument to `apply-generic' and `args' has as its value
+a list of the remaining arguments.
+
+ `Apply-generic' also uses the primitive procedure `apply', which
+takes two arguments, a procedure and a list. `Apply' applies the
+procedure, using the elements in the list as arguments. For example,
+
+ (apply + (list 1 2 3 4))
+
+returns 10.
+
+ (4) One limitation of this organization is it permits only generic
+procedures of one argument.
+
+\1f
+File: sicp.info, Node: 2-5, Prev: 2-4, Up: Chapter 2
+
+2.5 Systems with Generic Operations
+===================================
+
+In the previous section, we saw how to design systems in which data
+objects can be represented in more than one way. The key idea is to
+link the code that specifies the data operations to the several
+representations by means of generic interface procedures. Now we will
+see how to use this same idea not only to define operations that are
+generic over different representations but also to define operations
+that are generic over different kinds of arguments. We have already
+seen several different packages of arithmetic operations: the primitive
+arithmetic (`+', `-', `*', `/') built into our language, the
+rational-number arithmetic (`add-rat', `sub-rat', `mul-rat', `div-rat')
+of section *Note 2-1-1::, and the complex-number arithmetic that we
+implemented in section *Note 2-4-3::. We will now use data-directed
+techniques to construct a package of arithmetic operations that
+incorporates all the arithmetic packages we have already constructed.
+
+ *Note Figure 2-23:: shows the structure of the system we shall
+build. Notice the abstraction barriers. From the perspective of
+someone using "numbers," there is a single procedure `add' that
+operates on whatever numbers are supplied. `Add' is part of a generic
+interface that allows the separate ordinary-arithmetic,
+rational-arithmetic, and complex-arithmetic packages to be accessed
+uniformly by programs that use numbers. Any individual arithmetic
+package (such as the complex package) may itself be accessed through
+generic procedures (such as `add-complex') that combine packages
+designed for different representations (such as rectangular and polar).
+Moreover, the structure of the system is additive, so that one can
+design the individual arithmetic packages separately and combine them
+to produce a generic arithmetic system.
+
+ *Figure 2.23:* Generic arithmetic system.
+
+ Programs that use numbers
+ +-----------------+
+ ---------------------------| add sub mul div |-------------------
+ +-----------------+
+ Generic arithmetic package
+ +-----------------+ +-------------------------+
+ | add-rat sub-rat | | add-complex sub-complex | +---------+
+ -| |-+-| |-+-| + - * / |-
+ | mul-rat div-rat | | | mul-complex div-complex | | +---------+
+ +-----------------+ | +-------------------------+ |
+ Rational | Complex artithmetic | Ordinary
+ arithmetic +--------------+--------------+ arithmetic
+ | Rectangular | Polar |
+ ---------------------+--------------+--------------+-------------
+
+* Menu:
+
+* 2-5-1:: Generic Arithmetic Operations
+* 2-5-2:: Combining Data of Different Types
+* 2-5-3:: Example: Symbolic Algebra
+
+\1f
+File: sicp.info, Node: 2-5-1, Next: 2-5-2, Prev: 2-5, Up: 2-5
+
+2.5.1 Generic Arithmetic Operations
+-----------------------------------
+
+The task of designing generic arithmetic operations is analogous to
+that of designing the generic complex-number operations. We would
+like, for instance, to have a generic addition procedure `add' that
+acts like ordinary primitive addition `+' on ordinary numbers, like
+`add-rat' on rational numbers, and like `add-complex' on complex
+numbers. We can implement `add', and the other generic arithmetic
+operations, by following the same strategy we used in section *Note
+2-4-3:: to implement the generic selectors for complex numbers. We
+will attach a type tag to each kind of number and cause the generic
+procedure to dispatch to an appropriate package according to the data
+type of its arguments.
+
+ The generic arithmetic procedures are defined as follows:
+
+ (define (add x y) (apply-generic 'add x y))
+ (define (sub x y) (apply-generic 'sub x y))
+ (define (mul x y) (apply-generic 'mul x y))
+ (define (div x y) (apply-generic 'div x y))
+
+ We begin by installing a package for handling "ordinary" numbers,
+that is, the primitive numbers of our language. We will tag these with
+the symbol `scheme-number'. The arithmetic operations in this package
+are the primitive arithmetic procedures (so there is no need to define
+extra procedures to handle the untagged numbers). Since these
+operations each take two arguments, they are installed in the table
+keyed by the list `(scheme-number scheme-number)':
+
+ (define (install-scheme-number-package)
+ (define (tag x)
+ (attach-tag 'scheme-number x))
+ (put 'add '(scheme-number scheme-number)
+ (lambda (x y) (tag (+ x y))))
+ (put 'sub '(scheme-number scheme-number)
+ (lambda (x y) (tag (- x y))))
+ (put 'mul '(scheme-number scheme-number)
+ (lambda (x y) (tag (* x y))))
+ (put 'div '(scheme-number scheme-number)
+ (lambda (x y) (tag (/ x y))))
+ (put 'make 'scheme-number
+ (lambda (x) (tag x)))
+ 'done)
+
+ Users of the Scheme-number package will create (tagged) ordinary
+numbers by means of the procedure:
+
+ (define (make-scheme-number n)
+ ((get 'make 'scheme-number) n))
+
+ Now that the framework of the generic arithmetic system is in place,
+we can readily include new kinds of numbers. Here is a package that
+performs rational arithmetic. Notice that, as a benefit of additivity,
+we can use without modification the rational-number code from section
+*Note 2-1-1:: as the internal procedures in the package:
+
+ (define (install-rational-package)
+ ;; internal procedures
+ (define (numer x) (car x))
+ (define (denom x) (cdr x))
+ (define (make-rat n d)
+ (let ((g (gcd n d)))
+ (cons (/ n g) (/ d g))))
+ (define (add-rat x y)
+ (make-rat (+ (* (numer x) (denom y))
+ (* (numer y) (denom x)))
+ (* (denom x) (denom y))))
+ (define (sub-rat x y)
+ (make-rat (- (* (numer x) (denom y))
+ (* (numer y) (denom x)))
+ (* (denom x) (denom y))))
+ (define (mul-rat x y)
+ (make-rat (* (numer x) (numer y))
+ (* (denom x) (denom y))))
+ (define (div-rat x y)
+ (make-rat (* (numer x) (denom y))
+ (* (denom x) (numer y))))
+
+ ;; interface to rest of the system
+ (define (tag x) (attach-tag 'rational x))
+ (put 'add '(rational rational)
+ (lambda (x y) (tag (add-rat x y))))
+ (put 'sub '(rational rational)
+ (lambda (x y) (tag (sub-rat x y))))
+ (put 'mul '(rational rational)
+ (lambda (x y) (tag (mul-rat x y))))
+ (put 'div '(rational rational)
+ (lambda (x y) (tag (div-rat x y))))
+
+ (put 'make 'rational
+ (lambda (n d) (tag (make-rat n d))))
+ 'done)
+
+ (define (make-rational n d)
+ ((get 'make 'rational) n d))
+
+ We can install a similar package to handle complex numbers, using
+the tag `complex'. In creating the package, we extract from the table
+the operations `make-from-real-imag' and `make-from-mag-ang' that were
+defined by the rectangular and polar packages. Additivity permits us
+to use, as the internal operations, the same `add-complex',
+`sub-complex', `mul-complex', and `div-complex' procedures from section
+*Note 2-4-1::.
+
+ (define (install-complex-package)
+ ;; imported procedures from rectangular and polar packages
+ (define (make-from-real-imag x y)
+ ((get 'make-from-real-imag 'rectangular) x y))
+ (define (make-from-mag-ang r a)
+ ((get 'make-from-mag-ang 'polar) r a))
+
+ ;; internal procedures
+ (define (add-complex z1 z2)
+ (make-from-real-imag (+ (real-part z1) (real-part z2))
+ (+ (imag-part z1) (imag-part z2))))
+ (define (sub-complex z1 z2)
+ (make-from-real-imag (- (real-part z1) (real-part z2))
+ (- (imag-part z1) (imag-part z2))))
+ (define (mul-complex z1 z2)
+ (make-from-mag-ang (* (magnitude z1) (magnitude z2))
+ (+ (angle z1) (angle z2))))
+ (define (div-complex z1 z2)
+ (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
+ (- (angle z1) (angle z2))))
+
+ ;; interface to rest of the system
+ (define (tag z) (attach-tag 'complex z))
+ (put 'add '(complex complex)
+ (lambda (z1 z2) (tag (add-complex z1 z2))))
+ (put 'sub '(complex complex)
+ (lambda (z1 z2) (tag (sub-complex z1 z2))))
+ (put 'mul '(complex complex)
+ (lambda (z1 z2) (tag (mul-complex z1 z2))))
+ (put 'div '(complex complex)
+ (lambda (z1 z2) (tag (div-complex z1 z2))))
+ (put 'make-from-real-imag 'complex
+ (lambda (x y) (tag (make-from-real-imag x y))))
+ (put 'make-from-mag-ang 'complex
+ (lambda (r a) (tag (make-from-mag-ang r a))))
+ 'done)
+
+ Programs outside the complex-number package can construct complex
+numbers either from real and imaginary parts or from magnitudes and
+angles. Notice how the underlying procedures, originally defined in
+the rectangular and polar packages, are exported to the complex
+package, and exported from there to the outside world.
+
+ (define (make-complex-from-real-imag x y)
+ ((get 'make-from-real-imag 'complex) x y))
+
+ (define (make-complex-from-mag-ang r a)
+ ((get 'make-from-mag-ang 'complex) r a))
+
+ What we have here is a two-level tag system. A typical complex
+number, such as 3 + 4i in rectangular form, would be represented as
+shown in *Note Figure 2-24::. The outer tag (`complex') is used to
+direct the number to the complex package. Once within the complex
+package, the next tag (`rectangular') is used to direct the number to
+the rectangular package. In a large and complicated system there might
+be many levels, each interfaced with the next by means of generic
+operations. As a data object is passed "downward," the outer tag that
+is used to direct it to the appropriate package is stripped off (by
+applying `contents') and the next level of tag (if any) becomes visible
+to be used for further dispatching.
+
+ *Figure 2.24:* Representation of 3 + 4i in rectangular form.
+
+ +---+---+ +---+---+ +---+---+
+ ---->| * | *-+---->| * | *-+---->| * | * |
+ +-|-+---+ +-|-+---+ +-|-+-|-+
+ | | | |
+ V V V V
+ +---------+ +-------------+ +---+ +---+
+ | complex | | rectangular | | 3 | | 4 |
+ +---------+ +-------------+ +---+ +---+
+
+ In the above packages, we used `add-rat', `add-complex', and the
+other arithmetic procedures exactly as originally written. Once these
+definitions are internal to different installation procedures, however,
+they no longer need names that are distinct from each other: we could
+simply name them `add', `sub', `mul', and `div' in both packages.
+
+ *Exercise 2.77:* Louis Reasoner tries to evaluate the expression
+ `(magnitude z)' where `z' is the object shown in *Note Figure
+ 2-24::. To his surprise, instead of the answer 5 he gets an error
+ message from `apply-generic', saying there is no method for the
+ operation `magnitude' on the types `(complex)'. He shows this
+ interaction to Alyssa P. Hacker, who says "The problem is that the
+ complex-number selectors were never defined for `complex' numbers,
+ just for `polar' and `rectangular' numbers. All you have to do to
+ make this work is add the following to the `complex' package:"
+
+ (put 'real-part '(complex) real-part)
+ (put 'imag-part '(complex) imag-part)
+ (put 'magnitude '(complex) magnitude)
+ (put 'angle '(complex) angle)
+
+ Describe in detail why this works. As an example, trace through
+ all the procedures called in evaluating the expression `(magnitude
+ z)' where `z' is the object shown in *Note Figure 2-24::. In
+ particular, how many times is `apply-generic' invoked? What
+ procedure is dispatched to in each case?
+
+ *Exercise 2.78:* The internal procedures in the `scheme-number'
+ package are essentially nothing more than calls to the primitive
+ procedures `+', `-', etc. It was not possible to use the
+ primitives of the language directly because our type-tag system
+ requires that each data object have a type attached to it. In
+ fact, however, all Lisp implementations do have a type system,
+ which they use internally. Primitive predicates such as `symbol?'
+ and `number?' determine whether data objects have particular
+ types. Modify the definitions of `type-tag', `contents', and
+ `attach-tag' from section *Note 2-4-2:: so that our generic system
+ takes advantage of Scheme's internal type system. That is to say,
+ the system should work as before except that ordinary numbers
+ should be represented simply as Scheme numbers rather than as
+ pairs whose `car' is the symbol `scheme-number'.
+
+ *Exercise 2.79:* Define a generic equality predicate `equ?' that
+ tests the equality of two numbers, and install it in the generic
+ arithmetic package. This operation should work for ordinary
+ numbers, rational numbers, and complex numbers.
+
+ *Exercise 2.80:* Define a generic predicate `=zero?' that tests if
+ its argument is zero, and install it in the generic arithmetic
+ package. This operation should work for ordinary numbers, rational
+ numbers, and complex numbers.
+
+\1f
+File: sicp.info, Node: 2-5-2, Next: 2-5-3, Prev: 2-5-1, Up: 2-5
+
+2.5.2 Combining Data of Different Types
+---------------------------------------
+
+We have seen how to define a unified arithmetic system that encompasses
+ordinary numbers, complex numbers, rational numbers, and any other type
+of number we might decide to invent, but we have ignored an important
+issue. The operations we have defined so far treat the different data
+types as being completely independent. Thus, there are separate
+packages for adding, say, two ordinary numbers, or two complex numbers.
+What we have not yet considered is the fact that it is meaningful to
+define operations that cross the type boundaries, such as the addition
+of a complex number to an ordinary number. We have gone to great pains
+to introduce barriers between parts of our programs so that they can be
+developed and understood separately. We would like to introduce the
+cross-type operations in some carefully controlled way, so that we can
+support them without seriously violating our module boundaries.
+
+ One way to handle cross-type operations is to design a different
+procedure for each possible combination of types for which the
+operation is valid. For example, we could extend the complex-number
+package so that it provides a procedure for adding complex numbers to
+ordinary numbers and installs this in the table using the tag `(complex
+scheme-number)':(1)
+
+ ;; to be included in the complex package
+ (define (add-complex-to-schemenum z x)
+ (make-from-real-imag (+ (real-part z) x)
+ (imag-part z)))
+
+ (put 'add '(complex scheme-number)
+ (lambda (z x) (tag (add-complex-to-schemenum z x))))
+
+ This technique works, but it is cumbersome. With such a system, the
+cost of introducing a new type is not just the construction of the
+package of procedures for that type but also the construction and
+installation of the procedures that implement the cross-type
+operations. This can easily be much more code than is needed to define
+the operations on the type itself. The method also undermines our
+ability to combine separate packages additively, or least to limit the
+extent to which the implementors of the individual packages need to
+take account of other packages. For instance, in the example above, it
+seems reasonable that handling mixed operations on complex numbers and
+ordinary numbers should be the responsibility of the complex-number
+package. Combining rational numbers and complex numbers, however,
+might be done by the complex package, by the rational package, or by
+some third package that uses operations extracted from these two
+packages. Formulating coherent policies on the division of
+responsibility among packages can be an overwhelming task in designing
+systems with many packages and many cross-type operations.
+
+Coercion
+........
+
+In the general situation of completely unrelated operations acting on
+completely unrelated types, implementing explicit cross-type operations,
+cumbersome though it may be, is the best that one can hope for.
+Fortunately, we can usually do better by taking advantage of additional
+structure that may be latent in our type system. Often the different
+data types are not completely independent, and there may be ways by
+which objects of one type may be viewed as being of another type. This
+process is called "coercion". For example, if we are asked to
+arithmetically combine an ordinary number with a complex number, we can
+view the ordinary number as a complex number whose imaginary part is
+zero. This transforms the problem to that of combining two complex
+numbers, which can be handled in the ordinary way by the
+complex-arithmetic package.
+
+ In general, we can implement this idea by designing coercion
+procedures that transform an object of one type into an equivalent
+object of another type. Here is a typical coercion procedure, which
+transforms a given ordinary number to a complex number with that real
+part and zero imaginary part:
+
+ (define (scheme-number->complex n)
+ (make-complex-from-real-imag (contents n) 0))
+
+ We install these coercion procedures in a special coercion table,
+indexed under the names of the two types:
+
+ (put-coercion 'scheme-number 'complex scheme-number->complex)
+
+ (We assume that there are `put-coercion' and `get-coercion'
+procedures available for manipulating this table.) Generally some of
+the slots in the table will be empty, because it is not generally
+possible to coerce an arbitrary data object of each type into all other
+types. For example, there is no way to coerce an arbitrary complex
+number to an ordinary number, so there will be no general
+`complex->scheme-number' procedure included in the table.
+
+ Once the coercion table has been set up, we can handle coercion in a
+uniform manner by modifying the `apply-generic' procedure of section
+*Note 2-4-3::. When asked to apply an operation, we first check
+whether the operation is defined for the arguments' types, just as
+before. If so, we dispatch to the procedure found in the
+operation-and-type table. Otherwise, we try coercion. For simplicity,
+we consider only the case where there are two arguments.(2) We check
+the coercion table to see if objects of the first type can be coerced
+to the second type. If so, we coerce the first argument and try the
+operation again. If objects of the first type cannot in general be
+coerced to the second type, we try the coercion the other way around to
+see if there is a way to coerce the second argument to the type of the
+first argument. Finally, if there is no known way to coerce either
+type to the other type, we give up. Here is the procedure:
+
+ (define (apply-generic op . args)
+ (let ((type-tags (map type-tag args)))
+ (let ((proc (get op type-tags)))
+ (if proc
+ (apply proc (map contents args))
+ (if (= (length args) 2)
+ (let ((type1 (car type-tags))
+ (type2 (cadr type-tags))
+ (a1 (car args))
+ (a2 (cadr args)))
+ (let ((t1->t2 (get-coercion type1 type2))
+ (t2->t1 (get-coercion type2 type1)))
+ (cond (t1->t2
+ (apply-generic op (t1->t2 a1) a2))
+ (t2->t1
+ (apply-generic op a1 (t2->t1 a2)))
+ (else
+ (error "No method for these types"
+ (list op type-tags))))))
+ (error "No method for these types"
+ (list op type-tags)))))))
+
+ This coercion scheme has many advantages over the method of defining
+explicit cross-type operations, as outlined above. Although we still
+need to write coercion procedures to relate the types (possibly n^2
+procedures for a system with n types), we need to write only one
+procedure for each pair of types rather than a different procedure for
+each collection of types and each generic operation.(3) What we are
+counting on here is the fact that the appropriate transformation
+between types depends only on the types themselves, not on the
+operation to be applied.
+
+ On the other hand, there may be applications for which our coercion
+scheme is not general enough. Even when neither of the objects to be
+combined can be converted to the type of the other it may still be
+possible to perform the operation by converting both objects to a third
+type. In order to deal with such complexity and still preserve
+modularity in our programs, it is usually necessary to build systems
+that take advantage of still further structure in the relations among
+types, as we discuss next.
+
+Hierarchies of types
+....................
+
+The coercion scheme presented above relied on the existence of natural
+relations between pairs of types. Often there is more "global"
+structure in how the different types relate to each other. For
+instance, suppose we are building a generic arithmetic system to handle
+integers, rational numbers, real numbers, and complex numbers. In such
+a system, it is quite natural to regard an integer as a special kind of
+rational number, which is in turn a special kind of real number, which
+is in turn a special kind of complex number. What we actually have is
+a so-called "hierarchy of types", in which, for example, integers are a "subtype"
+of rational numbers (i.e., any operation that can be applied to a
+rational number can automatically be applied to an integer).
+Conversely, we say that rational numbers form a "supertype" of
+integers. The particular hierarchy we have here is of a very simple
+kind, in which each type has at most one supertype and at most one
+subtype. Such a structure, called a "tower", is illustrated in *Note
+Figure 2-25::.
+
+ *Figure 2.25:* A tower of types.
+
+ complex
+ ^
+ |
+ real
+ ^
+ |
+ rational
+ ^
+ |
+ integer
+
+ If we have a tower structure, then we can greatly simplify the
+problem of adding a new type to the hierarchy, for we need only specify
+how the new type is embedded in the next supertype above it and how it
+is the supertype of the type below it. For example, if we want to add
+an integer to a complex number, we need not explicitly define a special
+coercion procedure `integer->complex'. Instead, we define how an
+integer can be transformed into a rational number, how a rational
+number is transformed into a real number, and how a real number is
+transformed into a complex number. We then allow the system to
+transform the integer into a complex number through these steps and
+then add the two complex numbers.
+
+ We can redesign our `apply-generic' procedure in the following way:
+For each type, we need to supply a `raise' procedure, which "raises"
+objects of that type one level in the tower. Then when the system is
+required to operate on objects of different types it can successively
+raise the lower types until all the objects are at the same level in
+the tower. (*Note Exercise 2-83:: and *Note Exercise 2-84:: concern
+the details of implementing such a strategy.)
+
+ Another advantage of a tower is that we can easily implement the
+notion that every type "inherits" all operations defined on a
+supertype. For instance, if we do not supply a special procedure for
+finding the real part of an integer, we should nevertheless expect that
+`real-part' will be defined for integers by virtue of the fact that
+integers are a subtype of complex numbers. In a tower, we can arrange
+for this to happen in a uniform way by modifying `apply-generic'. If
+the required operation is not directly defined for the type of the
+object given, we raise the object to its supertype and try again. We
+thus crawl up the tower, transforming our argument as we go, until we
+either find a level at which the desired operation can be performed or
+hit the top (in which case we give up).
+
+ Yet another advantage of a tower over a more general hierarchy is
+that it gives us a simple way to "lower" a data object to the simplest
+representation. For example, if we add 2 + 3i to 4 - 3i, it would be
+nice to obtain the answer as the integer 6 rather than as the complex
+number 6 + 0i. *Note Exercise 2-85:: discusses a way to implement such
+a lowering operation. (The trick is that we need a general way to
+distinguish those objects that can be lowered, such as 6 + 0i, from
+those that cannot, such as 6 + 2i.)
+
+ *Figure 2.26:* Relations among types of geometric figures.
+
+ polygon
+ / \
+ / \
+ triangle quadrilateral
+ / \ / \
+ / \ / \
+ isosceles right trapezoid kite
+ triangle triangle | |
+ | \ | | |
+ | \ | | |
+ equilateral isosceles parallelogram |
+ triangle right | \ |
+ triangle | \ |
+ rectangle rhombus
+ \ /
+ \ /
+ square
+
+Inadequacies of hierarchies
+...........................
+
+If the data types in our system can be naturally arranged in a tower,
+this greatly simplifies the problems of dealing with generic operations
+on different types, as we have seen. Unfortunately, this is usually
+not the case. *Note Figure 2-26:: illustrates a more complex
+arrangement of mixed types, this one showing relations among different
+types of geometric figures. We see that, in general, a type may have
+more than one subtype. Triangles and quadrilaterals, for instance, are
+both subtypes of polygons. In addition, a type may have more than one
+supertype. For example, an isosceles right triangle may be regarded
+either as an isosceles triangle or as a right triangle. This
+multiple-supertypes issue is particularly thorny, since it means that
+there is no unique way to "raise" a type in the hierarchy. Finding the
+"correct" supertype in which to apply an operation to an object may
+involve considerable searching through the entire type network on the
+part of a procedure such as `apply-generic'. Since there generally are
+multiple subtypes for a type, there is a similar problem in coercing a
+value "down" the type hierarchy. Dealing with large numbers of
+interrelated types while still preserving modularity in the design of
+large systems is very difficult, and is an area of much current
+research.(4)
+
+ *Exercise 2.81:* Louis Reasoner has noticed that `apply-generic'
+ may try to coerce the arguments to each other's type even if they
+ already have the same type. Therefore, he reasons, we need to put
+ procedures in the coercion table to "coerce" arguments of each
+ type to their own type. For example, in addition to the
+ `scheme-number->complex' coercion shown above, he would do:
+
+ (define (scheme-number->scheme-number n) n)
+ (define (complex->complex z) z)
+ (put-coercion 'scheme-number 'scheme-number
+ scheme-number->scheme-number)
+ (put-coercion 'complex 'complex complex->complex)
+
+ a. With Louis's coercion procedures installed, what happens if
+ `apply-generic' is called with two arguments of type
+ `scheme-number' or two arguments of type `complex' for an
+ operation that is not found in the table for those types?
+ For example, assume that we've defined a generic
+ exponentiation operation:
+
+ (define (exp x y) (apply-generic 'exp x y))
+
+ and have put a procedure for exponentiation in the
+ Scheme-number package but not in any other package:
+
+ ;; following added to Scheme-number package
+ (put 'exp '(scheme-number scheme-number)
+ (lambda (x y) (tag (expt x y)))) ; using primitive `expt'
+
+ What happens if we call `exp' with two complex numbers as
+ arguments?
+
+ b. Is Louis correct that something had to be done about coercion
+ with arguments of the same type, or does `apply-generic' work
+ correctly as is?
+
+ c. Modify `apply-generic' so that it doesn't try coercion if the
+ two arguments have the same type.
+
+
+ *Exercise 2.82:* Show how to generalize `apply-generic' to handle
+ coercion in the general case of multiple arguments. One strategy
+ is to attempt to coerce all the arguments to the type of the first
+ argument, then to the type of the second argument, and so on.
+ Give an example of a situation where this strategy (and likewise
+ the two-argument version given above) is not sufficiently general.
+ (Hint: Consider the case where there are some suitable mixed-type
+ operations present in the table that will not be tried.)
+
+ *Exercise 2.83:* Suppose you are designing a generic arithmetic
+ system for dealing with the tower of types shown in *Note Figure
+ 2-25::: integer, rational, real, complex. For each type (except
+ complex), design a procedure that raises objects of that type one
+ level in the tower. Show how to install a generic `raise'
+ operation that will work for each type (except complex).
+
+ *Exercise 2.84:* Using the `raise' operation of *Note Exercise
+ 2-83::, modify the `apply-generic' procedure so that it coerces
+ its arguments to have the same type by the method of successive
+ raising, as discussed in this section. You will need to devise a
+ way to test which of two types is higher in the tower. Do this in
+ a manner that is "compatible" with the rest of the system and will
+ not lead to problems in adding new levels to the tower.
+
+ *Exercise 2.85:* This section mentioned a method for "simplifying"
+ a data object by lowering it in the tower of types as far as
+ possible. Design a procedure `drop' that accomplishes this for the
+ tower described in *Note Exercise 2-83::. The key is to decide,
+ in some general way, whether an object can be lowered. For
+ example, the complex number 1.5 + 0i can be lowered as far as
+ `real', the complex number 1 + 0i can be lowered as far as
+ `integer', and the complex number 2 + 3i cannot be lowered at all.
+ Here is a plan for determining whether an object can be lowered:
+ Begin by defining a generic operation `project' that "pushes" an
+ object down in the tower. For example, projecting a complex
+ number would involve throwing away the imaginary part. Then a
+ number can be dropped if, when we `project' it and `raise' the
+ result back to the type we started with, we end up with something
+ equal to what we started with. Show how to implement this idea in
+ detail, by writing a `drop' procedure that drops an object as far
+ as possible. You will need to design the various projection
+ operations(5) and install `project' as a generic operation in the
+ system. You will also need to make use of a generic equality
+ predicate, such as described in *Note Exercise 2-79::. Finally,
+ use `drop' to rewrite `apply-generic' from *Note Exercise 2-84::
+ so that it "simplifies" its answers.
+
+ *Exercise 2.86:* Suppose we want to handle complex numbers whose
+ real parts, imaginary parts, magnitudes, and angles can be either
+ ordinary numbers, rational numbers, or other numbers we might wish
+ to add to the system. Describe and implement the changes to the
+ system needed to accommodate this. You will have to define
+ operations such as `sine' and `cosine' that are generic over
+ ordinary numbers and rational numbers.
+
+ ---------- Footnotes ----------
+
+ (1) We also have to supply an almost identical procedure to handle
+the types `(scheme-number complex)'.
+
+ (2) See *Note Exercise 2-82:: for generalizations.
+
+ (3) If we are clever, we can usually get by with fewer than n^2
+coercion procedures. For instance, if we know how to convert from type
+1 to type 2 and from type 2 to type 3, then we can use this knowledge to
+convert from type 1 to type 3. This can greatly decrease the number of
+coercion procedures we need to supply explicitly when we add a new type
+to the system. If we are willing to build the required amount of
+sophistication into our system, we can have it search the "graph" of
+relations among types and automatically generate those coercion
+procedures that can be inferred from the ones that are supplied
+explicitly.
+
+ (4) This statement, which also appears in the first edition of this
+book, is just as true now as it was when we wrote it twelve years ago.
+Developing a useful, general framework for expressing the relations
+among different types of entities (what philosophers call "ontology")
+seems intractably difficult. The main difference between the confusion
+that existed ten years ago and the confusion that exists now is that
+now a variety of inadequate ontological theories have been embodied in
+a plethora of correspondingly inadequate programming languages. For
+example, much of the complexity of object-oriented programming
+languages--and the subtle and confusing differences among contemporary
+object-oriented languages--centers on the treatment of generic
+operations on interrelated types. Our own discussion of computational
+objects in *Note Chapter 3:: avoids these issues entirely. Readers
+familiar with object-oriented programming will notice that we have much
+to say in *Note Chapter 3:: about local state, but we do not even
+mention "classes" or "inheritance." In fact, we suspect that these
+problems cannot be adequately addressed in terms of computer-language
+design alone, without also drawing on work in knowledge representation
+and automated reasoning.
+
+ (5) A real number can be projected to an integer using the `round'
+primitive, which returns the closest integer to its argument.
+
+\1f
+File: sicp.info, Node: 2-5-3, Prev: 2-5-2, Up: 2-5
+
+2.5.3 Example: Symbolic Algebra
+-------------------------------
+
+The manipulation of symbolic algebraic expressions is a complex process
+that illustrates many of the hardest problems that occur in the design
+of large-scale systems. An algebraic expression, in general, can be
+viewed as a hierarchical structure, a tree of operators applied to
+operands. We can construct algebraic expressions by starting with a
+set of primitive objects, such as constants and variables, and
+combining these by means of algebraic operators, such as addition and
+multiplication. As in other languages, we form abstractions that
+enable us to refer to compound objects in simple terms. Typical
+abstractions in symbolic algebra are ideas such as linear combination,
+polynomial, rational function, or trigonometric function. We can
+regard these as compound "types," which are often useful for directing
+the processing of expressions. For example, we could describe the
+expression
+
+ x^2 sin (y^2 + 1) + r cos 2y + cos(y^3 - 2y^2)
+
+as a polynomial in x with coefficients that are trigonometric functions
+of polynomials in y whose coefficients are integers.
+
+ We will not attempt to develop a complete algebraic-manipulation
+system here. Such systems are exceedingly complex programs, embodying
+deep algebraic knowledge and elegant algorithms. What we will do is
+look at a simple but important part of algebraic manipulation: the
+arithmetic of polynomials. We will illustrate the kinds of decisions
+the designer of such a system faces, and how to apply the ideas of
+abstract data and generic operations to help organize this effort.
+
+Arithmetic on polynomials
+.........................
+
+Our first task in designing a system for performing arithmetic on
+polynomials is to decide just what a polynomial is. Polynomials are
+normally defined relative to certain variables (the "indeterminates" of
+the polynomial). For simplicity, we will restrict ourselves to
+polynomials having just one indeterminate ("univariate
+polynomials").(1) We will define a polynomial to be a sum of terms,
+each of which is either a coefficient, a power of the indeterminate, or
+a product of a coefficient and a power of the indeterminate. A
+coefficient is defined as an algebraic expression that is not dependent
+upon the indeterminate of the polynomial. For example,
+
+ 5x^2 + 3r + 7
+
+is a simple polynomial in x, and
+
+ (y^2 + 1)r^3 + (2y)x + 1
+
+is a polynomial in x whose coefficients are polynomials in y.
+
+ Already we are skirting some thorny issues. Is the first of these
+polynomials the same as the polynomial 5y^2 + 3y + 7, or not? A
+reasonable answer might be "yes, if we are considering a polynomial
+purely as a mathematical function, but no, if we are considering a
+polynomial to be a syntactic form." The second polynomial is
+algebraically equivalent to a polynomial in y whose coefficients are
+polynomials in x. Should our system recognize this, or not?
+Furthermore, there are other ways to represent a polynomial--for
+example, as a product of factors, or (for a univariate polynomial) as
+the set of roots, or as a listing of the values of the polynomial at a
+specified set of points.(2) We can finesse these questions by deciding
+that in our algebraic-manipulation system a "polynomial" will be a
+particular syntactic form, not its underlying mathematical meaning.
+
+ Now we must consider how to go about doing arithmetic on
+polynomials. In this simple system, we will consider only addition and
+multiplication. Moreover, we will insist that two polynomials to be
+combined must have the same indeterminate.
+
+ We will approach the design of our system by following the familiar
+discipline of data abstraction. We will represent polynomials using a
+data structure called a "poly", which consists of a variable and a
+collection of terms. We assume that we have selectors `variable' and
+`term-list' that extract those parts from a poly and a constructor
+`make-poly' that assembles a poly from a given variable and a term
+list. A variable will be just a symbol, so we can use the
+`same-variable?' procedure of section *Note 2-3-2:: to compare
+variables. The following procedures define addition and multiplication
+of polys:
+
+ (define (add-poly p1 p2)
+ (if (same-variable? (variable p1) (variable p2))
+ (make-poly (variable p1)
+ (add-terms (term-list p1)
+ (term-list p2)))
+ (error "Polys not in same var -- ADD-POLY"
+ (list p1 p2))))
+
+ (define (mul-poly p1 p2)
+ (if (same-variable? (variable p1) (variable p2))
+ (make-poly (variable p1)
+ (mul-terms (term-list p1)
+ (term-list p2)))
+ (error "Polys not in same var -- MUL-POLY"
+ (list p1 p2))))
+
+ To incorporate polynomials into our generic arithmetic system, we
+need to supply them with type tags. We'll use the tag `polynomial',
+and install appropriate operations on tagged polynomials in the
+operation table. We'll embed all our code in an installation procedure
+for the polynomial package, similar to the ones in section *Note
+2-5-1:::
+
+ (define (install-polynomial-package)
+ ;; internal procedures
+ ;; representation of poly
+ (define (make-poly variable term-list)
+ (cons variable term-list))
+ (define (variable p) (car p))
+ (define (term-list p) (cdr p))
+ <_procedures `same-variable?' and `variable?' from section 2.3.2_>
+
+ ;; representation of terms and term lists
+ <_procedures `adjoin-term' ... `coeff' from text below_>
+
+ ;; continued on next page
+
+ (define (add-poly p1 p2) ...)
+ <_procedures used by `add-poly'_>
+ (define (mul-poly p1 p2) ...)
+ <_procedures used by `mul-poly'_>
+
+ ;; interface to rest of the system
+ (define (tag p) (attach-tag 'polynomial p))
+ (put 'add '(polynomial polynomial)
+ (lambda (p1 p2) (tag (add-poly p1 p2))))
+ (put 'mul '(polynomial polynomial)
+ (lambda (p1 p2) (tag (mul-poly p1 p2))))
+ (put 'make 'polynomial
+ (lambda (var terms) (tag (make-poly var terms))))
+ 'done)
+
+ Polynomial addition is performed termwise. Terms of the same order
+(i.e., with the same power of the indeterminate) must be combined.
+This is done by forming a new term of the same order whose coefficient
+is the sum of the coefficients of the addends. Terms in one addend for
+which there are no terms of the same order in the other addend are
+simply accumulated into the sum polynomial being constructed.
+
+ In order to manipulate term lists, we will assume that we have a
+constructor `the-empty-termlist' that returns an empty term list and a
+constructor `adjoin-term' that adjoins a new term to a term list. We
+will also assume that we have a predicate `empty-termlist?' that tells
+if a given term list is empty, a selector `first-term' that extracts
+the highest-order term from a term list, and a selector `rest-terms'
+that returns all but the highest-order term. To manipulate terms, we
+will suppose that we have a constructor `make-term' that constructs a
+term with given order and coefficient, and selectors `order' and
+`coeff' that return, respectively, the order and the coefficient of the
+term. These operations allow us to consider both terms and term lists
+as data abstractions, whose concrete representations we can worry about
+separately.
+
+ Here is the procedure that constructs the term list for the sum of
+two polynomials:(3)
+
+ (define (add-terms L1 L2)
+ (cond ((empty-termlist? L1) L2)
+ ((empty-termlist? L2) L1)
+ (else
+ (let ((t1 (first-term L1)) (t2 (first-term L2)))
+ (cond ((> (order t1) (order t2))
+ (adjoin-term
+ t1 (add-terms (rest-terms L1) L2)))
+ ((< (order t1) (order t2))
+ (adjoin-term
+ t2 (add-terms L1 (rest-terms L2))))
+ (else
+ (adjoin-term
+ (make-term (order t1)
+ (add (coeff t1) (coeff t2)))
+ (add-terms (rest-terms L1)
+ (rest-terms L2)))))))))
+
+ The most important point to note here is that we used the generic
+addition procedure `add' to add together the coefficients of the terms
+being combined. This has powerful consequences, as we will see below.
+
+ In order to multiply two term lists, we multiply each term of the
+first list by all the terms of the other list, repeatedly using
+`mul-term-by-all-terms', which multiplies a given term by all terms in
+a given term list. The resulting term lists (one for each term of the
+first list) are accumulated into a sum. Multiplying two terms forms a
+term whose order is the sum of the orders of the factors and whose
+coefficient is the product of the coefficients of the factors:
+
+ (define (mul-terms L1 L2)
+ (if (empty-termlist? L1)
+ (the-empty-termlist)
+ (add-terms (mul-term-by-all-terms (first-term L1) L2)
+ (mul-terms (rest-terms L1) L2))))
+
+ (define (mul-term-by-all-terms t1 L)
+ (if (empty-termlist? L)
+ (the-empty-termlist)
+ (let ((t2 (first-term L)))
+ (adjoin-term
+ (make-term (+ (order t1) (order t2))
+ (mul (coeff t1) (coeff t2)))
+ (mul-term-by-all-terms t1 (rest-terms L))))))
+
+ This is really all there is to polynomial addition and
+multiplication. Notice that, since we operate on terms using the
+generic procedures `add' and `mul', our polynomial package is
+automatically able to handle any type of coefficient that is known
+about by the generic arithmetic package. If we include a coercion
+mechanism such as one of those discussed in section *Note 2-5-2::, then
+we also are automatically able to handle operations on polynomials of
+different coefficient types, such as
+
+ / 2 \
+ [3x^2 + (2 + 3i)x + 7] * | x^4 + --- x^2 + (5 + 3i) |
+ \ 3 /
+
+ Because we installed the polynomial addition and multiplication
+procedures `add-poly' and `mul-poly' in the generic arithmetic system
+as the `add' and `mul' operations for type `polynomial', our system is
+also automatically able to handle polynomial operations such as
+
+ [(y + 1)x^2 + (y^2 + 1)x + (y - 1)] * [(y - 2)x + (y^3 + 7)]
+
+ The reason is that when the system tries to combine coefficients, it
+will dispatch through `add' and `mul'. Since the coefficients are
+themselves polynomials (in y), these will be combined using `add-poly'
+and `mul-poly'. The result is a kind of "data-directed recursion" in
+which, for example, a call to `mul-poly' will result in recursive calls
+to `mul-poly' in order to multiply the coefficients. If the
+coefficients of the coefficients were themselves polynomials (as might
+be used to represent polynomials in three variables), the data
+direction would ensure that the system would follow through another
+level of recursive calls, and so on through as many levels as the
+structure of the data dictates.(4)
+
+Representing term lists
+.......................
+
+Finally, we must confront the job of implementing a good representation
+for term lists. A term list is, in effect, a set of coefficients keyed
+by the order of the term. Hence, any of the methods for representing
+sets, as discussed in section *Note 2-3-3::, can be applied to this
+task. On the other hand, our procedures `add-terms' and `mul-terms'
+always access term lists sequentially from highest to lowest order.
+Thus, we will use some kind of ordered list representation.
+
+ How should we structure the list that represents a term list? One
+consideration is the "density" of the polynomials we intend to
+manipulate. A polynomial is said to be "dense" if it has nonzero
+coefficients in terms of most orders. If it has many zero terms it is
+said to be "sparse". For example,
+
+ A : x^5 + 2x^4 + 3x^2 - 2x - 5
+
+is a dense polynomial, whereas
+
+ B : x^100 + 2x^2 + 1
+
+is sparse.
+
+ The term lists of dense polynomials are most efficiently represented
+as lists of the coefficients. For example, A above would be nicely
+represented as `(1 2 0 3 -2 -5)'. The order of a term in this
+representation is the length of the sublist beginning with that term's
+coefficient, decremented by 1.(5) This would be a terrible
+representation for a sparse polynomial such as B: There would be a
+giant list of zeros punctuated by a few lonely nonzero terms. A more
+reasonable representation of the term list of a sparse polynomial is as
+a list of the nonzero terms, where each term is a list containing the
+order of the term and the coefficient for that order. In such a
+scheme, polynomial B is efficiently represented as `((100 1) (2 2) (0
+1))'. As most polynomial manipulations are performed on sparse
+polynomials, we will use this method. We will assume that term lists
+are represented as lists of terms, arranged from highest-order to
+lowest-order term. Once we have made this decision, implementing the
+selectors and constructors for terms and term lists is
+straightforward:(6)
+
+ (define (adjoin-term term term-list)
+ (if (=zero? (coeff term))
+ term-list
+ (cons term term-list)))
+
+ (define (the-empty-termlist) '())
+ (define (first-term term-list) (car term-list))
+ (define (rest-terms term-list) (cdr term-list))
+ (define (empty-termlist? term-list) (null? term-list))
+
+ (define (make-term order coeff) (list order coeff))
+ (define (order term) (car term))
+ (define (coeff term) (cadr term))
+
+where `=zero?' is as defined in *Note Exercise 2-80::. (See also *Note
+Exercise 2-87:: below.)
+
+ Users of the polynomial package will create (tagged) polynomials by
+means of the procedure:
+
+ (define (make-polynomial var terms)
+ ((get 'make 'polynomial) var terms))
+
+ *Exercise 2.87:* Install `=zero?' for polynomials in the generic
+ arithmetic package. This will allow `adjoin-term' to work for
+ polynomials with coefficients that are themselves polynomials.
+
+ *Exercise 2.88:* Extend the polynomial system to include
+ subtraction of polynomials. (Hint: You may find it helpful to
+ define a generic negation operation.)
+
+ *Exercise 2.89:* Define procedures that implement the term-list
+ representation described above as appropriate for dense
+ polynomials.
+
+ *Exercise 2.90:* Suppose we want to have a polynomial system that
+ is efficient for both sparse and dense polynomials. One way to do
+ this is to allow both kinds of term-list representations in our
+ system. The situation is analogous to the complex-number example
+ of section *Note 2-4::, where we allowed both rectangular and
+ polar representations. To do this we must distinguish different
+ types of term lists and make the operations on term lists generic.
+ Redesign the polynomial system to implement this generalization.
+ This is a major effort, not a local change.
+
+ *Exercise 2.91:* A univariate polynomial can be divided by another
+ one to produce a polynomial quotient and a polynomial remainder.
+ For example,
+
+ x^5 - 1
+ ------- = x^3 + x, remainder x - 1
+ x^2 - 1
+
+ Division can be performed via long division. That is, divide the
+ highest-order term of the dividend by the highest-order term of
+ the divisor. The result is the first term of the quotient. Next,
+ multiply the result by the divisor, subtract that from the
+ dividend, and produce the rest of the answer by recursively
+ dividing the difference by the divisor. Stop when the order of the
+ divisor exceeds the order of the dividend and declare the dividend
+ to be the remainder. Also, if the dividend ever becomes zero,
+ return zero as both quotient and remainder.
+
+ We can design a `div-poly' procedure on the model of `add-poly' and
+ `mul-poly'. The procedure checks to see if the two polys have the
+ same variable. If so, `div-poly' strips off the variable and
+ passes the problem to `div-terms', which performs the division
+ operation on term lists. `Div-poly' finally reattaches the
+ variable to the result supplied by `div-terms'. It is convenient
+ to design `div-terms' to compute both the quotient and the
+ remainder of a division. `Div-terms' can take two term lists as
+ arguments and return a list of the quotient term list and the
+ remainder term list.
+
+ Complete the following definition of `div-terms' by filling in the
+ missing expressions. Use this to implement `div-poly', which
+ takes two polys as arguments and returns a list of the quotient
+ and remainder polys.
+
+ (define (div-terms L1 L2)
+ (if (empty-termlist? L1)
+ (list (the-empty-termlist) (the-empty-termlist))
+ (let ((t1 (first-term L1))
+ (t2 (first-term L2)))
+ (if (> (order t2) (order t1))
+ (list (the-empty-termlist) L1)
+ (let ((new-c (div (coeff t1) (coeff t2)))
+ (new-o (- (order t1) (order t2))))
+ (let ((rest-of-result
+ <COMPUTE REST OF RESULT RECURSIVELY>
+ ))
+ <FORM COMPLETE RESULT>
+ ))))))
+
+Hierarchies of types in symbolic algebra
+........................................
+
+Our polynomial system illustrates how objects of one type (polynomials)
+may in fact be complex objects that have objects of many different
+types as parts. This poses no real difficulty in defining generic
+operations. We need only install appropriate generic operations for
+performing the necessary manipulations of the parts of the compound
+types. In fact, we saw that polynomials form a kind of "recursive data
+abstraction," in that parts of a polynomial may themselves be
+polynomials. Our generic operations and our data-directed programming
+style can handle this complication without much trouble.
+
+ On the other hand, polynomial algebra is a system for which the data
+types cannot be naturally arranged in a tower. For instance, it is
+possible to have polynomials in x whose coefficients are polynomials in
+y. It is also possible to have polynomials in y whose coefficients are
+polynomials in x. Neither of these types is "above" the other in any
+natural way, yet it is often necessary to add together elements from
+each set. There are several ways to do this. One possibility is to
+convert one polynomial to the type of the other by expanding and
+rearranging terms so that both polynomials have the same principal
+variable. One can impose a towerlike structure on this by ordering the
+variables and thus always converting any polynomial to a "canonical
+form" with the highest-priority variable dominant and the
+lower-priority variables buried in the coefficients. This strategy
+works fairly well, except that the conversion may expand a polynomial
+unnecessarily, making it hard to read and perhaps less efficient to
+work with. The tower strategy is certainly not natural for this domain
+or for any domain where the user can invent new types dynamically using
+old types in various combining forms, such as trigonometric functions,
+power series, and integrals.
+
+ It should not be surprising that controlling coercion is a serious
+problem in the design of large-scale algebraic-manipulation systems.
+Much of the complexity of such systems is concerned with relationships
+among diverse types. Indeed, it is fair to say that we do not yet
+completely understand coercion. In fact, we do not yet completely
+understand the concept of a data type. Nevertheless, what we know
+provides us with powerful structuring and modularity principles to
+support the design of large systems.
+
+ *Exercise 2.92:* By imposing an ordering on variables, extend the
+ polynomial package so that addition and multiplication of
+ polynomials works for polynomials in different variables. (This
+ is not easy!)
+
+Extended exercise: Rational functions
+.....................................
+
+We can extend our generic arithmetic system to include functions
+"rational functions". These are "fractions" whose numerator and
+denominator are polynomials, such as
+
+ x + 1
+ -------
+ x^3 - 1
+
+ The system should be able to add, subtract, multiply, and divide
+rational functions, and to perform such computations as
+
+ x + 1 x x^3 + 2x^2 + 3x + 1
+ ------- + ------- = -------------------
+ x^3 - 1 x^2 - 1 x^4 + x^3 - x - 1
+
+(Here the sum has been simplified by removing common factors. Ordinary
+"cross multiplication" would have produced a fourth-degree polynomial
+over a fifth-degree polynomial.)
+
+ If we modify our rational-arithmetic package so that it uses generic
+operations, then it will do what we want, except for the problem of
+reducing fractions to lowest terms.
+
+ *Exercise 2.93:* Modify the rational-arithmetic package to use
+ generic operations, but change `make-rat' so that it does not
+ attempt to reduce fractions to lowest terms. Test your system by
+ calling `make-rational' on two polynomials to produce a rational
+ function
+
+ (define p1 (make-polynomial 'x '((2 1)(0 1))))
+ (define p2 (make-polynomial 'x '((3 1)(0 1))))
+ (define rf (make-rational p2 p1))
+
+ Now add `rf' to itself, using `add'. You will observe that this
+ addition procedure does not reduce fractions to lowest terms.
+
+ We can reduce polynomial fractions to lowest terms using the same
+ idea we used with integers: modifying `make-rat' to divide both
+ the numerator and the denominator by their greatest common
+ divisor. The notion of "greatest common divisor" makes sense for
+ polynomials. In fact, we can compute the GCD of two polynomials
+ using essentially the same Euclid's Algorithm that works for
+ integers.(7) The integer version is
+
+ (define (gcd a b)
+ (if (= b 0)
+ a
+ (gcd b (remainder a b))))
+
+ Using this, we could make the obvious modification to define a GCD
+ operation that works on term lists:
+
+ (define (gcd-terms a b)
+ (if (empty-termlist? b)
+ a
+ (gcd-terms b (remainder-terms a b))))
+
+ where `remainder-terms' picks out the remainder component of the
+ list returned by the term-list division operation `div-terms' that
+ was implemented in *Note Exercise 2-91::.
+
+ *Exercise 2.94:* Using `div-terms', implement the procedure
+ `remainder-terms' and use this to define `gcd-terms' as above.
+ Now write a procedure `gcd-poly' that computes the polynomial GCD
+ of two polys. (The procedure should signal an error if the two
+ polys are not in the same variable.) Install in the system a
+ generic operation `greatest-common-divisor' that reduces to
+ `gcd-poly' for polynomials and to ordinary `gcd' for ordinary
+ numbers. As a test, try
+
+ (define p1 (make-polynomial 'x '((4 1) (3 -1) (2 -2) (1 2))))
+ (define p2 (make-polynomial 'x '((3 1) (1 -1))))
+ (greatest-common-divisor p1 p2)
+
+ and check your result by hand.
+
+ *Exercise 2.95:* Define P_1, P_2, and P_3 to be the polynomials
+
+ P_1 : x^2 - 2x + 1
+
+ P_2 : 11x^2 + 7
+
+ P_3 : 13x + 5
+
+ Now define Q_1 to be the product of P_1 and P_2 and Q_2 to be the
+ product of P_1 and P_3, and use `greatest-common-divisor' (*Note
+ Exercise 2-94::) to compute the GCD of Q_1 and Q_2. Note that the
+ answer is not the same as P_1. This example introduces noninteger
+ operations into the computation, causing difficulties with the GCD
+ algorithm.(8) To understand what is happening, try tracing
+ `gcd-terms' while computing the GCD or try performing the division
+ by hand.
+
+ We can solve the problem exhibited in *Note Exercise 2-95:: if we
+ use the following modification of the GCD algorithm (which really
+ works only in the case of polynomials with integer coefficients).
+ Before performing any polynomial division in the GCD computation,
+ we multiply the dividend by an integer constant factor, chosen to
+ guarantee that no fractions will arise during the division
+ process. Our answer will thus differ from the actual GCD by an
+ integer constant factor, but this does not matter in the case of
+ reducing rational functions to lowest terms; the GCD will be used
+ to divide both the numerator and denominator, so the integer
+ constant factor will cancel out.
+
+ More precisely, if P and Q are polynomials, let O_1 be the order of
+ P (i.e., the order of the largest term of P) and let O_2 be the
+ order of Q. Let c be the leading coefficient of Q. Then it can be
+ shown that, if we multiply P by the "integerizing factor" c^(1+O_1
+ -O_2), the resulting polynomial can be divided by Q by using the
+ `div-terms' algorithm without introducing any fractions. The
+ operation of multiplying the dividend by this constant and then
+ dividing is sometimes called the "pseudodivision" of P by Q. The
+ remainder of the division is called the "pseudoremainder".
+
+ *Exercise 2.96:*
+ a. Implement the procedure `pseudoremainder-terms', which is
+ just like `remainder-terms' except that it multiplies the
+ dividend by the integerizing factor described above before
+ calling `div-terms'. Modify `gcd-terms' to use
+ `pseudoremainder-terms', and verify that
+ `greatest-common-divisor' now produces an answer with integer
+ coefficients on the example in *Note Exercise 2-95::.
+
+ b. The GCD now has integer coefficients, but they are larger
+ than those of P_1. Modify `gcd-terms' so that it removes
+ common factors from the coefficients of the answer by
+ dividing all the coefficients by their (integer) greatest
+ common divisor.
+
+
+ Thus, here is how to reduce a rational function to lowest terms:
+
+ * Compute the GCD of the numerator and denominator, using the
+ version of `gcd-terms' from *Note Exercise 2-96::.
+
+ * When you obtain the GCD, multiply both numerator and
+ denominator by the same integerizing factor before dividing
+ through by the GCD, so that division by the GCD will not
+ introduce any noninteger coefficients. As the factor you can
+ use the leading coefficient of the GCD raised to the power 1
+ + O_1 - O_2, where O_2 is the order of the GCD and O_1 is the
+ maximum of the orders of the numerator and denominator. This
+ will ensure that dividing the numerator and denominator by
+ the GCD will not introduce any fractions.
+
+ * The result of this operation will be a numerator and
+ denominator with integer coefficients. The coefficients will
+ normally be very large because of all of the integerizing
+ factors, so the last step is to remove the redundant factors
+ by computing the (integer) greatest common divisor of all the
+ coefficients of the numerator and the denominator and
+ dividing through by this factor.
+
+
+ *Exercise 2.97:*
+ a. Implement this algorithm as a procedure `reduce-terms' that
+ takes two term lists `n' and `d' as arguments and returns a
+ list `nn', `dd', which are `n' and `d' reduced to lowest
+ terms via the algorithm given above. Also write a procedure
+ `reduce-poly', analogous to `add-poly', that checks to see if
+ the two polys have the same variable. If so, `reduce-poly'
+ strips off the variable and passes the problem to
+ `reduce-terms', then reattaches the variable to the two term
+ lists supplied by `reduce-terms'.
+
+ b. Define a procedure analogous to `reduce-terms' that does what
+ the original `make-rat' did for integers:
+
+ (define (reduce-integers n d)
+ (let ((g (gcd n d)))
+ (list (/ n g) (/ d g))))
+
+ and define `reduce' as a generic operation that calls
+ `apply-generic' to dispatch to either `reduce-poly' (for
+ `polynomial' arguments) or `reduce-integers' (for
+ `scheme-number' arguments). You can now easily make the
+ rational-arithmetic package reduce fractions to lowest terms
+ by having `make-rat' call `reduce' before combining the given
+ numerator and denominator to form a rational number. The
+ system now handles rational expressions in either integers or
+ polynomials. To test your program, try the example at the
+ beginning of this extended exercise:
+
+ (define p1 (make-polynomial 'x '((1 1)(0 1))))
+ (define p2 (make-polynomial 'x '((3 1)(0 -1))))
+ (define p3 (make-polynomial 'x '((1 1))))
+ (define p4 (make-polynomial 'x '((2 1)(0 -1))))
+
+ (define rf1 (make-rational p1 p2))
+ (define rf2 (make-rational p3 p4))
+
+ (add rf1 rf2)
+
+ See if you get the correct answer, correctly reduced to
+ lowest terms.
+
+ The GCD computation is at the heart of any system that does
+ operations on rational functions. The algorithm used above,
+ although mathematically straightforward, is extremely slow.
+ The slowness is due partly to the large number of division
+ operations and partly to the enormous size of the
+ intermediate coefficients generated by the pseudodivisions.
+ One of the active areas in the development of
+ algebraic-manipulation systems is the design of better
+ algorithms for computing polynomial GCDs.(9)
+
+
+ ---------- Footnotes ----------
+
+ (1) On the other hand, we will allow polynomials whose coefficients
+are themselves polynomials in other variables. This will give us
+essentially the same representational power as a full multivariate
+system, although it does lead to coercion problems, as discussed below.
+
+ (2) For univariate polynomials, giving the value of a polynomial at
+a given set of points can be a particularly good representation. This
+makes polynomial arithmetic extremely simple. To obtain, for example,
+the sum of two polynomials represented in this way, we need only add
+the values of the polynomials at corresponding points. To transform
+back to a more familiar representation, we can use the Lagrange
+interpolation formula, which shows how to recover the coefficients of a
+polynomial of degree n given the values of the polynomial at n + 1
+points.
+
+ (3) This operation is very much like the ordered `union-set'
+operation we developed in exercise *Note Exercise 2-62::. In fact, if
+we think of the terms of the polynomial as a set ordered according to
+the power of the indeterminate, then the program that produces the term
+list for a sum is almost identical to `union-set'.
+
+ (4) To make this work completely smoothly, we should also add to our
+generic arithmetic system the ability to coerce a "number" to a
+polynomial by regarding it as a polynomial of degree zero whose
+coefficient is the number. This is necessary if we are going to
+perform operations such as
+
+ [x^2 + (y + 1)x + 5] + [x^2 + 2x + 1]
+
+which requires adding the coefficient y + 1 to the coefficient 2.
+
+ (5) In these polynomial examples, we assume that we have implemented
+the generic arithmetic system using the type mechanism suggested in
+*Note Exercise 2-78::. Thus, coefficients that are ordinary numbers
+will be represented as the numbers themselves rather than as pairs
+whose `car' is the symbol `scheme-number'.
+
+ (6) Although we are assuming that term lists are ordered, we have
+implemented `adjoin-term' to simply `cons' the new term onto the
+existing term list. We can get away with this so long as we guarantee
+that the procedures (such as `add-terms') that use `adjoin-term' always
+call it with a higher-order term than appears in the list. If we did
+not want to make such a guarantee, we could have implemented
+`adjoin-term' to be similar to the `adjoin-set' constructor for the
+ordered-list representation of sets (*Note Exercise 2-61::).
+
+ (7) The fact that Euclid's Algorithm works for polynomials is
+formalized in algebra by saying that polynomials form a kind of
+algebraic domain called a "Euclidean ring". A Euclidean ring is a
+domain that admits addition, subtraction, and commutative
+multiplication, together with a way of assigning to each element x of
+the ring a positive integer "measure" m(x) with the properties that
+m(xy)>= m(x) for any nonzero x and y and that, given any x and y, there
+exists a q such that y = qx + r and either r = 0 or m(r)< m(x). From
+an abstract point of view, this is what is needed to prove that
+Euclid's Algorithm works. For the domain of integers, the measure m of
+an integer is the absolute value of the integer itself. For the domain
+of polynomials, the measure of a polynomial is its degree.
+
+ (8) In an implementation like MIT Scheme, this produces a polynomial
+that is indeed a divisor of Q_1 and Q_2, but with rational
+coefficients. In many other Scheme systems, in which division of
+integers can produce limited-precision decimal numbers, we may fail to
+get a valid divisor.
+
+ (9) One extremely efficient and elegant method for computing
+polynomial GCDs was discovered by Richard Zippel (1979). The method is
+a probabilistic algorithm, as is the fast test for primality that we
+discussed in *Note Chapter 1::. Zippel's book (1993) describes this
+method, together with other ways to compute polynomial GCDs.
+
+\1f
+File: sicp.info, Node: Chapter 3, Next: Chapter 4, Prev: Chapter 2, Up: Top
+
+3 Modularity, Objects, and State
+********************************
+
+ [greek not included here]
+
+ (Even while it changes, it stands still.)
+
+ --Heraclitus
+
+ Plus c,a change, plus c'est la me*me chose.
+
+ --Alphonse Karr
+
+ The preceding chapters introduced the basic elements from which
+programs are made. We saw how primitive procedures and primitive data
+are combined to construct compound entities, and we learned that
+abstraction is vital in helping us to cope with the complexity of large
+systems. But these tools are not sufficient for designing programs.
+Effective program synthesis also requires organizational principles
+that can guide us in formulating the overall design of a program. In
+particular, we need strategies to help us structure large systems so
+that they will be "modular", that is, so that they can be divided
+"naturally" into coherent parts that can be separately developed and
+maintained.
+
+ One powerful design strategy, which is particularly appropriate to
+the construction of programs for modeling physical systems, is to base
+the structure of our programs on the structure of the system being
+modeled. For each object in the system, we construct a corresponding
+computational object. For each system action, we define a symbolic
+operation in our computational model. Our hope in using this strategy
+is that extending the model to accommodate new objects or new actions
+will require no strategic changes to the program, only the addition of
+the new symbolic analogs of those objects or actions. If we have been
+successful in our system organization, then to add a new feature or
+debug an old one we will have to work on only a localized part of the
+system.
+
+ To a large extent, then, the way we organize a large program is
+dictated by our perception of the system to be modeled. In this
+chapter we will investigate two prominent organizational strategies
+arising from two rather different "world views" of the structure of
+systems. The first organizational strategy concentrates on "objects",
+viewing a large system as a collection of distinct objects whose
+behaviors may change over time. An alternative organizational strategy
+concentrates on the "streams" of information that flow in the system,
+much as an electrical engineer views a signal-processing system.
+
+ Both the object-based approach and the stream-processing approach
+raise significant linguistic issues in programming. With objects, we
+must be concerned with how a computational object can change and yet
+maintain its identity. This will force us to abandon our old
+substitution model of computation (section *Note 1-1-5::) in favor of a
+more mechanistic but less theoretically tractable "environment model"
+of computation. The difficulties of dealing with objects, change, and
+identity are a fundamental consequence of the need to grapple with time
+in our computational models. These difficulties become even greater
+when we allow the possibility of concurrent execution of programs. The
+stream approach can be most fully exploited when we decouple simulated
+time in our model from the order of the events that take place in the
+computer during evaluation. We will accomplish this using a technique
+known as "delayed evaluation".
+
+* Menu:
+
+* 3-1:: Assignment and Local State
+* 3-2:: The Environment Model of Evaluation
+* 3-3:: Modeling with Mutable Data
+* 3-4:: Concurrency: Time Is of the Essence
+* 3-5:: Streams
+
+\1f
+File: sicp.info, Node: 3-1, Next: 3-2, Prev: Chapter 3, Up: Chapter 3
+
+3.1 Assignment and Local State
+==============================
+
+We ordinarily view the world as populated by independent objects, each
+of which has a state that changes over time. An object is said to
+"have state" if its behavior is influenced by its history. A bank
+account, for example, has state in that the answer to the question "Can
+I withdraw $100?" depends upon the history of deposit and withdrawal
+transactions. We can characterize an object's state by one or more "state
+variables", which among them maintain enough information about history
+to determine the object's current behavior. In a simple banking
+system, we could characterize the state of an account by a current
+balance rather than by remembering the entire history of account
+transactions.
+
+ In a system composed of many objects, the objects are rarely
+completely independent. Each may influence the states of others
+through interactions, which serve to couple the state variables of one
+object to those of other objects. Indeed, the view that a system is
+composed of separate objects is most useful when the state variables of
+the system can be grouped into closely coupled subsystems that are only
+loosely coupled to other subsystems.
+
+ This view of a system can be a powerful framework for organizing
+computational models of the system. For such a model to be modular, it
+should be decomposed into computational objects that model the actual
+objects in the system. Each computational object must have its own "local
+state variables" describing the actual object's state. Since the
+states of objects in the system being modeled change over time, the
+state variables of the corresponding computational objects must also
+change. If we choose to model the flow of time in the system by the
+elapsed time in the computer, then we must have a way to construct
+computational objects whose behaviors change as our programs run. In
+particular, if we wish to model state variables by ordinary symbolic
+names in the programming language, then the language must provide an operator
+"assignment operator" to enable us to change the value associated with
+a name.
+
+* Menu:
+
+* 3-1-1:: Local State Variables
+* 3-1-2:: The Benefits of Introducing Assignment
+* 3-1-3:: The Costs of Introducing Assignment
+
+\1f
+File: sicp.info, Node: 3-1-1, Next: 3-1-2, Prev: 3-1, Up: 3-1
+
+3.1.1 Local State Variables
+---------------------------
+
+To illustrate what we mean by having a computational object with
+time-varying state, let us model the situation of withdrawing money
+from a bank account. We will do this using a procedure `withdraw',
+which takes as argument an `amount' to be withdrawn. If there is
+enough money in the account to accommodate the withdrawal, then
+`withdraw' should return the balance remaining after the withdrawal.
+Otherwise, `withdraw' should return the message _Insufficient funds_.
+For example, if we begin with $100 in the account, we should obtain the
+following sequence of responses using `withdraw':
+
+ (withdraw 25)
+ 75
+
+ (withdraw 25)
+ 50
+
+ (withdraw 60)
+ "Insufficient funds"
+
+ (withdraw 15)
+ 35
+
+ Observe that the expression `(withdraw 25)', evaluated twice, yields
+different values. This is a new kind of behavior for a procedure.
+Until now, all our procedures could be viewed as specifications for
+computing mathematical functions. A call to a procedure computed the
+value of the function applied to the given arguments, and two calls to
+the same procedure with the same arguments always produced the same
+result.(1)
+
+ To implement `withdraw', we can use a variable `balance' to indicate
+the balance of money in the account and define `withdraw' as a procedure
+that accesses `balance'. The `withdraw' procedure checks to see if
+`balance' is at least as large as the requested `amount'. If so,
+`withdraw' decrements `balance' by `amount' and returns the new value
+of `balance'. Otherwise, `withdraw' returns the _Insufficient funds_
+message. Here are the definitions of `balance' and `withdraw':
+
+ (define balance 100)
+
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+
+ Decrementing `balance' is accomplished by the expression
+
+ (set! balance (- balance amount))
+
+ This uses the `set!' special form, whose syntax is
+
+ (set! <NAME> <NEW-VALUE>)
+
+ Here <NAME> is a symbol and <NEW-VALUE> is any expression. `Set!'
+changes <NAME> so that its value is the result obtained by evaluating
+<NEW-VALUE>. In the case at hand, we are changing `balance' so that
+its new value will be the result of subtracting `amount' from the
+previous value of `balance'.(2)
+
+ `Withdraw' also uses the `begin' special form to cause two
+expressions to be evaluated in the case where the `if' test is true:
+first decrementing `balance' and then returning the value of `balance'.
+In general, evaluating the expression
+
+ (begin <EXP_1> <EXP_2> ... <EXP_K>)
+
+causes the expressions <EXP_1> through <EXP_K> to be evaluated in
+sequence and the value of the final expression <EXP_K> to be returned
+as the value of the entire `begin' form.(3)
+
+ Although `withdraw' works as desired, the variable `balance' presents
+a problem. As specified above, `balance' is a name defined in the
+global environment and is freely accessible to be examined or modified
+by any procedure. It would be much better if we could somehow make
+`balance' internal to `withdraw', so that `withdraw' would be the only
+procedure that could access `balance' directly and any other procedure
+could access `balance' only indirectly (through calls to `withdraw').
+This would more accurately model the notion that `balance' is a local
+state variable used by `withdraw' to keep track of the state of the
+account.
+
+ We can make `balance' internal to `withdraw' by rewriting the
+definition as follows:
+
+ (define new-withdraw
+ (let ((balance 100))
+ (lambda (amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))))
+
+ What we have done here is use `let' to establish an environment with
+a local variable `balance', bound to the initial value 100. Within this
+local environment, we use `lambda' to create a procedure that takes
+`amount' as an argument and behaves like our previous `withdraw'
+procedure. This procedure--returned as the result of evaluating the
+`let' expression--is `new-withdraw', which behaves in precisely the
+same way as `withdraw' but whose variable `balance' is not accessible
+by any other procedure.(4)
+
+ Combining `set!' with local variables is the general programming
+technique we will use for constructing computational objects with local
+state. Unfortunately, using this technique raises a serious problem:
+When we first introduced procedures, we also introduced the
+substitution model of evaluation (section *Note 1-1-5::) to provide an
+interpretation of what procedure application means. We said that
+applying a procedure should be interpreted as evaluating the body of
+the procedure with the formal parameters replaced by their values. The
+trouble is that, as soon as we introduce assignment into our language,
+substitution is no longer an adequate model of procedure application.
+(We will see why this is so in section *Note 3-1-3::.) As a
+consequence, we technically have at this point no way to understand why
+the `new-withdraw' procedure behaves as claimed above. In order to
+really understand a procedure such as `new-withdraw', we will need to
+develop a new model of procedure application. In section *Note 3-2::
+we will introduce such a model, together with an explanation of `set!'
+and local variables. First, however, we examine some variations on the
+theme established by `new-withdraw'.
+
+ The following procedure, `make-withdraw', creates "withdrawal
+processors." The formal parameter `balance' in `make-withdraw'
+specifies the initial amount of money in the account.(5)
+
+ (define (make-withdraw balance)
+ (lambda (amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds")))
+
+ `Make-withdraw' can be used as follows to create two objects `W1' and
+`W2':
+
+ (define W1 (make-withdraw 100))
+ (define W2 (make-withdraw 100))
+
+ (W1 50)
+ 50
+
+ (W2 70)
+ 30
+
+ (W2 40)
+ "Insufficient funds"
+
+ (W1 40)
+ 10
+
+ Observe that `W1' and `W2' are completely independent objects, each
+with its own local state variable `balance'. Withdrawals from one do
+not affect the other.
+
+ We can also create objects that handle deposits as well as
+withdrawals, and thus we can represent simple bank accounts. Here is a
+procedure that returns a "bank-account object" with a specified initial
+balance:
+
+ (define (make-account balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) withdraw)
+ ((eq? m 'deposit) deposit)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch)
+
+ Each call to `make-account' sets up an environment with a local state
+variable `balance'. Within this environment, `make-account' defines
+procedures `deposit' and `withdraw' that access `balance' and an
+additional procedure `dispatch' that takes a "message" as input and
+returns one of the two local procedures. The `dispatch' procedure
+itself is returned as the value that represents the bank-account
+object. This is precisely the "message-passing" style of programming
+that we saw in section *Note 2-4-3::, although here we are using it in
+conjunction with the ability to modify local variables.
+
+ `Make-account' can be used as follows:
+
+ (define acc (make-account 100))
+
+ ((acc 'withdraw) 50)
+ 50
+
+ ((acc 'withdraw) 60)
+ "Insufficient funds"
+
+ ((acc 'deposit) 40)
+ 90
+
+ ((acc 'withdraw) 60)
+ 30
+
+ Each call to `acc' returns the locally defined `deposit' or
+`withdraw' procedure, which is then applied to the specified `amount'.
+As was the case with `make-withdraw', another call to `make-account'
+
+ (define acc2 (make-account 100))
+
+will produce a completely separate account object, which maintains its
+own local `balance'.
+
+ *Exercise 3.1:* An "accumulator" is a procedure that is called
+ repeatedly with a single numeric argument and accumulates its
+ arguments into a sum. Each time it is called, it returns the
+ currently accumulated sum. Write a procedure `make-accumulator'
+ that generates accumulators, each maintaining an independent sum.
+ The input to `make-accumulator' should specify the initial value
+ of the sum; for example
+
+ (define A (make-accumulator 5))
+
+ (A 10)
+ 15
+
+ (A 10)
+ 25
+
+ *Exercise 3.2:* In software-testing applications, it is useful to
+ be able to count the number of times a given procedure is called
+ during the course of a computation. Write a procedure
+ `make-monitored' that takes as input a procedure, `f', that itself
+ takes one input. The result returned by `make-monitored' is a
+ third procedure, say `mf', that keeps track of the number of times
+ it has been called by maintaining an internal counter. If the
+ input to `mf' is the special symbol `how-many-calls?', then `mf'
+ returns the value of the counter. If the input is the special
+ symbol `reset-count', then `mf' resets the counter to zero. For
+ any other input, `mf' returns the result of calling `f' on that
+ input and increments the counter. For instance, we could make a
+ monitored version of the `sqrt' procedure:
+
+ (define s (make-monitored sqrt))
+
+ (s 100)
+ 10
+
+ (s 'how-many-calls?)
+ 1
+
+ *Exercise 3.3:* Modify the `make-account' procedure so that it
+ creates password-protected accounts. That is, `make-account'
+ should take a symbol as an additional argument, as in
+
+ (define acc (make-account 100 'secret-password))
+
+ The resulting account object should process a request only if it
+ is accompanied by the password with which the account was created,
+ and should otherwise return a complaint:
+
+ ((acc 'secret-password 'withdraw) 40)
+ 60
+
+ ((acc 'some-other-password 'deposit) 50)
+ "Incorrect password"
+
+ *Exercise 3.4:* Modify the `make-account' procedure of *Note
+ Exercise 3-3:: by adding another local state variable so that, if
+ an account is accessed more than seven consecutive times with an
+ incorrect password, it invokes the procedure `call-the-cops'.
+
+ ---------- Footnotes ----------
+
+ (1) Actually, this is not quite true. One exception was the
+random-number generator in section *Note 1-2-6::. Another exception
+involved the operation/type tables we introduced in section *Note
+2-4-3::, where the values of two calls to `get' with the same arguments
+depended on intervening calls to `put'. On the other hand, until we
+introduce assignment, we have no way to create such procedures
+ourselves.
+
+ (2) The value of a `set!' expression is implementation-dependent.
+`Set!' should be used only for its effect, not for its value.
+
+ The name `set!' reflects a naming convention used in Scheme:
+Operations that change the values of variables (or that change data
+structures, as we will see in section *Note 3-3::) are given names that
+end with an exclamation point. This is similar to the convention of
+designating predicates by names that end with a question mark.
+
+ (3) We have already used `begin' implicitly in our programs, because
+in Scheme the body of a procedure can be a sequence of expressions.
+Also, the <CONSEQUENT> part of each clause in a `cond' expression can
+be a sequence of expressions rather than a single expression.
+
+ (4) In programming-language jargon, the variable `balance' is said
+to be "encapsulated" within the `new-withdraw' procedure.
+Encapsulation reflects the general system-design principle known as the "hiding
+principle": One can make a system more modular and robust by protecting
+parts of the system from each other; that is, by providing information
+access only to those parts of the system that have a "need to know."
+
+ (5) In contrast with `new-withdraw' above, we do not have to use
+`let' to make `balance' a local variable, since formal parameters are
+already local. This will be clearer after the discussion of the
+environment model of evaluation in section *Note 3-2::. (See also
+*Note Exercise 3-10::.)
+
+\1f
+File: sicp.info, Node: 3-1-2, Next: 3-1-3, Prev: 3-1-1, Up: 3-1
+
+3.1.2 The Benefits of Introducing Assignment
+--------------------------------------------
+
+As we shall see, introducing assignment into our programming language
+leads us into a thicket of difficult conceptual issues. Nevertheless,
+viewing systems as collections of objects with local state is a
+powerful technique for maintaining a modular design. As a simple
+example, consider the design of a procedure `rand' that, whenever it is
+called, returns an integer chosen at random.
+
+ It is not at all clear what is meant by "chosen at random." What we
+presumably want is for successive calls to `rand' to produce a sequence
+of numbers that has statistical properties of uniform distribution. We
+will not discuss methods for generating suitable sequences here.
+Rather, let us assume that we have a procedure `rand-update' that has
+the property that if we start with a given number x_1 and form
+
+ x_2 = (rand-update x_1)
+ x_3 = (rand-update x_2)
+
+then the sequence of values x_1, x_2, x_3, ..., will have the desired
+statistical properties.(1)
+
+ We can implement `rand' as a procedure with a local state variable
+`x' that is initialized to some fixed value `random-init'. Each call
+to `rand' computes `rand-update' of the current value of `x', returns
+this as the random number, and also stores this as the new value of `x'.
+
+ (define rand
+ (let ((x random-init))
+ (lambda ()
+ (set! x (rand-update x))
+ x)))
+
+ Of course, we could generate the same sequence of random numbers
+without using assignment by simply calling `rand-update' directly.
+However, this would mean that any part of our program that used random
+numbers would have to explicitly remember the current value of `x' to
+be passed as an argument to `rand-update'. To realize what an
+annoyance this would be, consider using random numbers to implement a
+technique called simulation "Monte Carlo simulation".
+
+ The Monte Carlo method consists of choosing sample experiments at
+random from a large set and then making deductions on the basis of the
+probabilities estimated from tabulating the results of those
+experiments. For example, we can approximate [pi] using the fact that
+6/[pi]^2 is the probability that two integers chosen at random will
+have no factors in common; that is, that their greatest common divisor
+will be 1.(2) To obtain the approximation to [pi], we perform a large
+number of experiments. In each experiment we choose two integers at
+random and perform a test to see if their GCD is 1. The fraction of
+times that the test is passed gives us our estimate of 6/[pi]^2, and
+from this we obtain our approximation to [pi].
+
+ The heart of our program is a procedure `monte-carlo', which takes as
+arguments the number of times to try an experiment, together with the
+experiment, represented as a no-argument procedure that will return
+either true or false each time it is run. `Monte-carlo' runs the
+experiment for the designated number of trials and returns a number
+telling the fraction of the trials in which the experiment was found to
+be true.
+
+ (define (estimate-pi trials)
+ (sqrt (/ 6 (monte-carlo trials cesaro-test))))
+
+ (define (cesaro-test)
+ (= (gcd (rand) (rand)) 1))
+
+ (define (monte-carlo trials experiment)
+ (define (iter trials-remaining trials-passed)
+ (cond ((= trials-remaining 0)
+ (/ trials-passed trials))
+ ((experiment)
+ (iter (- trials-remaining 1) (+ trials-passed 1)))
+ (else
+ (iter (- trials-remaining 1) trials-passed))))
+ (iter trials 0))
+
+ Now let us try the same computation using `rand-update' directly
+rather than `rand', the way we would be forced to proceed if we did not
+use assignment to model local state:
+
+ (define (estimate-pi trials)
+ (sqrt (/ 6 (random-gcd-test trials random-init))))
+
+ (define (random-gcd-test trials initial-x)
+ (define (iter trials-remaining trials-passed x)
+ (let ((x1 (rand-update x)))
+ (let ((x2 (rand-update x1)))
+ (cond ((= trials-remaining 0)
+ (/ trials-passed trials))
+ ((= (gcd x1 x2) 1)
+ (iter (- trials-remaining 1)
+ (+ trials-passed 1)
+ x2))
+ (else
+ (iter (- trials-remaining 1)
+ trials-passed
+ x2))))))
+ (iter trials 0 initial-x))
+
+ While the program is still simple, it betrays some painful breaches
+of modularity. In our first version of the program, using `rand', we
+can express the Monte Carlo method directly as a general `monte-carlo'
+procedure that takes as an argument an arbitrary `experiment' procedure.
+In our second version of the program, with no local state for the
+random-number generator, `random-gcd-test' must explicitly manipulate
+the random numbers `x1' and `x2' and recycle `x2' through the iterative
+loop as the new input to `rand-update'. This explicit handling of the
+random numbers intertwines the structure of accumulating test results
+with the fact that our particular experiment uses two random numbers,
+whereas other Monte Carlo experiments might use one random number or
+three. Even the top-level procedure `estimate-pi' has to be concerned
+with supplying an initial random number. The fact that the
+random-number generator's insides are leaking out into other parts of
+the program makes it difficult for us to isolate the Monte Carlo idea
+so that it can be applied to other tasks. In the first version of the
+program, assignment encapsulates the state of the random-number
+generator within the `rand' procedure, so that the details of
+random-number generation remain independent of the rest of the program.
+
+ The general phenomenon illustrated by the Monte Carlo example is
+this: From the point of view of one part of a complex process, the
+other parts appear to change with time. They have hidden time-varying
+local state. If we wish to write computer programs whose structure
+reflects this decomposition, we make computational objects (such as
+bank accounts and random-number generators) whose behavior changes with
+time. We model state with local state variables, and we model the
+changes of state with assignments to those variables.
+
+ It is tempting to conclude this discussion by saying that, by
+introducing assignment and the technique of hiding state in local
+variables, we are able to structure systems in a more modular fashion
+than if all state had to be manipulated explicitly, by passing
+additional parameters. Unfortunately, as we shall see, the story is
+not so simple.
+
+ *Exercise 3.5:* "Monte Carlo integration" is a method of
+ estimating definite integrals by means of Monte Carlo simulation.
+ Consider computing the area of a region of space described by a
+ predicate P(x, y) that is true for points (x, y) in the region and
+ false for points not in the region. For example, the region
+ contained within a circle of radius 3 centered at (5, 7) is
+ described by the predicate that tests whether (x - 5)^2 + (y -
+ 7)^2 <= 3^2. To estimate the area of the region described by such
+ a predicate, begin by choosing a rectangle that contains the
+ region. For example, a rectangle with diagonally opposite corners
+ at (2, 4) and (8, 10) contains the circle above. The desired
+ integral is the area of that portion of the rectangle that lies in
+ the region. We can estimate the integral by picking, at random,
+ points (x,y) that lie in the rectangle, and testing P(x, y) for
+ each point to determine whether the point lies in the region. If
+ we try this with many points, then the fraction of points that
+ fall in the region should give an estimate of the proportion of
+ the rectangle that lies in the region. Hence, multiplying this
+ fraction by the area of the entire rectangle should produce an
+ estimate of the integral.
+
+ Implement Monte Carlo integration as a procedure
+ `estimate-integral' that takes as arguments a predicate `P', upper
+ and lower bounds `x1', `x2', `y1', and `y2' for the rectangle, and
+ the number of trials to perform in order to produce the estimate.
+ Your procedure should use the same `monte-carlo' procedure that
+ was used above to estimate [pi]. Use your `estimate-integral' to
+ produce an estimate of [pi] by measuring the area of a unit circle.
+
+ You will find it useful to have a procedure that returns a number
+ chosen at random from a given range. The following
+ `random-in-range' procedure implements this in terms of the
+ `random' procedure used in section *Note 1-2-6::, which returns a
+ nonnegative number less than its input.(3)
+
+ (define (random-in-range low high)
+ (let ((range (- high low)))
+ (+ low (random range))))
+
+ *Exercise 3.6:* It is useful to be able to reset a random-number
+ generator to produce a sequence starting from a given value.
+ Design a new `rand' procedure that is called with an argument that
+ is either the symbol `generate' or the symbol `reset' and behaves
+ as follows: `(rand 'generate)' produces a new random number;
+ `((rand 'reset) <NEW-VALUE>)' resets the internal state variable
+ to the designated <NEW-VALUE>. Thus, by resetting the state, one
+ can generate repeatable sequences. These are very handy to have
+ when testing and debugging programs that use random numbers.
+
+ ---------- Footnotes ----------
+
+ (1) One common way to implement `rand-update' is to use the rule
+that x is updated to ax + b modulo m, where a, b, and m are
+appropriately chosen integers. Chapter 3 of Knuth 1981 includes an
+extensive discussion of techniques for generating sequences of random
+numbers and establishing their statistical properties. Notice that the
+`rand-update' procedure computes a mathematical function: Given the
+same input twice, it produces the same output. Therefore, the number
+sequence produced by `rand-update' certainly is not "random," if by
+"random" we insist that each number in the sequence is unrelated to the
+preceding number. The relation between "real randomness" and so-called "pseudo-random"
+sequences, which are produced by well-determined computations and yet
+have suitable statistical properties, is a complex question involving
+difficult issues in mathematics and philosophy. Kolmogorov,
+Solomonoff, and Chaitin have made great progress in clarifying these
+issues; a discussion can be found in Chaitin 1975.
+
+ (2) This theorem is due to E. Cesa`ro. See section 4.5.2 of Knuth
+1981 for a discussion and a proof.
+
+ (3) MIT Scheme provides such a procedure. If `random' is given an
+exact integer (as in section *Note 1-2-6::) it returns an exact
+integer, but if it is given a decimal value (as in this exercise) it
+returns a decimal value.
+
+\1f
+File: sicp.info, Node: 3-1-3, Prev: 3-1-2, Up: 3-1
+
+3.1.3 The Costs of Introducing Assignment
+-----------------------------------------
+
+As we have seen, the `set!' operation enables us to model objects that
+have local state. However, this advantage comes at a price. Our
+programming language can no longer be interpreted in terms of the
+substitution model of procedure application that we introduced in
+section *Note 1-1-5::. Moreover, no simple model with "nice"
+mathematical properties can be an adequate framework for dealing with
+objects and assignment in programming languages.
+
+ So long as we do not use assignments, two evaluations of the same
+procedure with the same arguments will produce the same result, so that
+procedures can be viewed as computing mathematical functions.
+Programming without any use of assignments, as we did throughout the
+first two chapters of this book, is accordingly known as "functional
+programming".
+
+ To understand how assignment complicates matters, consider a
+simplified version of the `make-withdraw' procedure of section *Note
+3-1-1:: that does not bother to check for an insufficient amount:
+
+ (define (make-simplified-withdraw balance)
+ (lambda (amount)
+ (set! balance (- balance amount))
+ balance))
+
+ (define W (make-simplified-withdraw 25))
+
+ (W 20)
+ 5
+
+ (W 10)
+ - 5
+
+ Compare this procedure with the following `make-decrementer'
+procedure, which does not use `set!':
+
+ (define (make-decrementer balance)
+ (lambda (amount)
+ (- balance amount)))
+
+ `Make-decrementer' returns a procedure that subtracts its input from
+a designated amount `balance', but there is no accumulated effect over
+successive calls, as with `make-simplified-withdraw':
+
+ (define D (make-decrementer 25))
+
+ (D 20)
+ 5
+
+ (D 10)
+ 15
+
+ We can use the substitution model to explain how `make-decrementer'
+works. For instance, let us analyze the evaluation of the expression
+
+ ((make-decrementer 25) 20)
+
+ We first simplify the operator of the combination by substituting 25
+for `balance' in the body of `make-decrementer'. This reduces the
+expression to
+
+ ((lambda (amount) (- 25 amount)) 20)
+
+ Now we apply the operator by substituting 20 for `amount' in the
+body of the `lambda' expression:
+
+ (- 25 20)
+
+ The final answer is 5.
+
+ Observe, however, what happens if we attempt a similar substitution
+analysis with `make-simplified-withdraw':
+
+ ((make-simplified-withdraw 25) 20)
+
+ We first simplify the operator by substituting 25 for `balance' in
+the body of `make-simplified-withdraw'. This reduces the expression
+to(1)
+
+ ((lambda (amount) (set! balance (- 25 amount)) 25) 20)
+
+ Now we apply the operator by substituting 20 for `amount' in the
+body of the `lambda' expression:
+
+ (set! balance (- 25 20)) 25
+
+ If we adhered to the substitution model, we would have to say that
+the meaning of the procedure application is to first set `balance' to 5
+and then return 25 as the value of the expression. This gets the wrong
+answer. In order to get the correct answer, we would have to somehow
+distinguish the first occurrence of `balance' (before the effect of the
+`set!') from the second occurrence of `balance' (after the effect of
+the `set!'), and the substitution model cannot do this.
+
+ The trouble here is that substitution is based ultimately on the
+notion that the symbols in our language are essentially names for
+values. But as soon as we introduce `set!' and the idea that the value
+of a variable can change, a variable can no longer be simply a name.
+Now a variable somehow refers to a place where a value can be stored,
+and the value stored at this place can change. In section *Note 3-2::
+we will see how environments play this role of "place" in our
+computational model.
+
+Sameness and change
+...................
+
+The issue surfacing here is more profound than the mere breakdown of a
+particular model of computation. As soon as we introduce change into
+our computational models, many notions that were previously
+straightforward become problematical. Consider the concept of two
+things being "the same."
+
+ Suppose we call `make-decrementer' twice with the same argument to
+create two procedures:
+
+ (define D1 (make-decrementer 25))
+
+ (define D2 (make-decrementer 25))
+
+ Are `D1' and `D2' the same? An acceptable answer is yes, because
+`D1' and `D2' have the same computational behavior--each is a procedure
+that subtracts its input from 25. In fact, `D1' could be substituted
+for `D2' in any computation without changing the result.
+
+ Contrast this with making two calls to `make-simplified-withdraw':
+
+ (define W1 (make-simplified-withdraw 25))
+
+ (define W2 (make-simplified-withdraw 25))
+
+ Are `W1' and `W2' the same? Surely not, because calls to `W1' and
+`W2' have distinct effects, as shown by the following sequence of
+interactions:
+
+ (W1 20)
+ 5
+
+ (W1 20)
+ - 15
+
+ (W2 20)
+ 5
+
+ Even though `W1' and `W2' are "equal" in the sense that they are
+both created by evaluating the same expression,
+`(make-simplified-withdraw 25)', it is not true that `W1' could be
+substituted for `W2' in any expression without changing the result of
+evaluating the expression.
+
+ A language that supports the concept that "equals can be substituted
+for equals" in an expresssion without changing the value of the
+expression is said to be "referentially transparent". Referential
+transparency is violated when we include `set!' in our computer
+language. This makes it tricky to determine when we can simplify
+expressions by substituting equivalent expressions. Consequently,
+reasoning about programs that use assignment becomes drastically more
+difficult.
+
+ Once we forgo referential transparency, the notion of what it means
+for computational objects to be "the same" becomes difficult to capture
+in a formal way. Indeed, the meaning of "same" in the real world that
+our programs model is hardly clear in itself. In general, we can
+determine that two apparently identical objects are indeed "the same
+one" only by modifying one object and then observing whether the other
+object has changed in the same way. But how can we tell if an object
+has "changed" other than by observing the "same" object twice and
+seeing whether some property of the object differs from one observation
+to the next? Thus, we cannot determine "change" without some _a
+priori_ notion of "sameness," and we cannot determine sameness without
+observing the effects of change.
+
+ As an example of how this issue arises in programming, consider the
+situation where Peter and Paul have a bank account with $100 in it.
+There is a substantial difference between modeling this as
+
+ (define peter-acc (make-account 100))
+ (define paul-acc (make-account 100))
+
+and modeling it as
+
+ (define peter-acc (make-account 100))
+ (define paul-acc peter-acc)
+
+ In the first situation, the two bank accounts are distinct.
+Transactions made by Peter will not affect Paul's account, and vice
+versa. In the second situation, however, we have defined `paul-acc' to
+be _the same thing_ as `peter-acc'. In effect, Peter and Paul now have
+a joint bank account, and if Peter makes a withdrawal from `peter-acc'
+Paul will observe less money in `paul-acc'. These two similar but
+distinct situations can cause confusion in building computational
+models. With the shared account, in particular, it can be especially
+confusing that there is one object (the bank account) that has two
+different names (`peter-acc' and `paul-acc'); if we are searching for
+all the places in our program where `paul-acc' can be changed, we must
+remember to look also at things that change `peter-acc'.(2)
+
+ With reference to the above remarks on "sameness" and "change,"
+observe that if Peter and Paul could only examine their bank balances,
+and could not perform operations that changed the balance, then the
+issue of whether the two accounts are distinct would be moot. In
+general, so long as we never modify data objects, we can regard a
+compound data object to be precisely the totality of its pieces. For
+example, a rational number is determined by giving its numerator and
+its denominator. But this view is no longer valid in the presence of
+change, where a compound data object has an "identity" that is
+something different from the pieces of which it is composed. A bank
+account is still "the same" bank account even if we change the balance
+by making a withdrawal; conversely, we could have two different bank
+accounts with the same state information. This complication is a
+consequence, not of our programming language, but of our perception of
+a bank account as an object. We do not, for example, ordinarily regard
+a rational number as a changeable object with identity, such that we
+could change the numerator and still have "the same" rational number.
+
+Pitfalls of imperative programming
+..................................
+
+In contrast to functional programming, programming that makes extensive
+use of assignment is known as "imperative programming". In addition to
+raising complications about computational models, programs written in
+imperative style are susceptible to bugs that cannot occur in functional
+programs. For example, recall the iterative factorial program from
+section *Note 1-2-1:::
+
+ (define (factorial n)
+ (define (iter product counter)
+ (if (> counter n)
+ product
+ (iter (* counter product)
+ (+ counter 1))))
+ (iter 1 1))
+
+ Instead of passing arguments in the internal iterative loop, we
+could adopt a more imperative style by using explicit assignment to
+update the values of the variables `product' and `counter':
+
+ (define (factorial n)
+ (let ((product 1)
+ (counter 1))
+ (define (iter)
+ (if (> counter n)
+ product
+ (begin (set! product (* counter product))
+ (set! counter (+ counter 1))
+ (iter))))
+ (iter)))
+
+ This does not change the results produced by the program, but it
+does introduce a subtle trap. How do we decide the order of the
+assignments? As it happens, the program is correct as written. But
+writing the assignments in the opposite order
+
+ (set! counter (+ counter 1))
+ (set! product (* counter product))
+
+would have produced a different, incorrect result. In general,
+programming with assignment forces us to carefully consider the
+relative orders of the assignments to make sure that each statement is
+using the correct version of the variables that have been changed.
+This issue simply does not arise in functional programs.(3)
+
+ The complexity of imperative programs becomes even worse if we
+consider applications in which several processes execute concurrently.
+We will return to this in section *Note 3-4::. First, however, we will
+address the issue of providing a computational model for expressions
+that involve assignment, and explore the uses of objects with local
+state in designing simulations.
+
+ *Exercise 3.7:* Consider the bank account objects created by
+ `make-account', with the password modification described in *Note
+ Exercise 3-3::. Suppose that our banking system requires the
+ ability to make joint accounts. Define a procedure `make-joint'
+ that accomplishes this. `Make-joint' should take three arguments.
+ The first is a password-protected account. The second argument
+ must match the password with which the account was defined in
+ order for the `make-joint' operation to proceed. The third
+ argument is a new password. `Make-joint' is to create an
+ additional access to the original account using the new password.
+ For example, if `peter-acc' is a bank account with password
+ `open-sesame', then
+
+ (define paul-acc
+ (make-joint peter-acc 'open-sesame 'rosebud))
+
+ will allow one to make transactions on `peter-acc' using the name
+ `paul-acc' and the password `rosebud'. You may wish to modify your
+ solution to *Note Exercise 3-3:: to accommodate this new feature
+
+ *Exercise 3.8:* When we defined the evaluation model in section
+ *Note 1-1-3::, we said that the first step in evaluating an
+ expression is to evaluate its subexpressions. But we never
+ specified the order in which the subexpressions should be
+ evaluated (e.g., left to right or right to left). When we
+ introduce assignment, the order in which the arguments to a
+ procedure are evaluated can make a difference to the result.
+ Define a simple procedure `f' such that evaluating `(+ (f 0) (f
+ 1))' will return 0 if the arguments to `+' are evaluated from left
+ to right but will return 1 if the arguments are evaluated from
+ right to left.
+
+ ---------- Footnotes ----------
+
+ (1) We don't substitute for the occurrence of `balance' in the
+`set!' expression because the <NAME> in a `set!' is not evaluated. If
+we did substitute for it, we would get `(set! 25 (- 25 amount))', which
+makes no sense.
+
+ (2) The phenomenon of a single computational object being accessed
+by more than one name is known as "aliasing". The joint bank account
+situation illustrates a very simple example of an alias. In section
+*Note 3-3:: we will see much more complex examples, such as "distinct"
+compound data structures that share parts. Bugs can occur in our
+programs if we forget that a change to an object may also, as a "side
+effect," change a "different" object because the two "different"
+objects are actually a single object appearing under different aliases.
+These so-called "side-effect bugs" are so difficult to locate and to
+analyze that some people have proposed that programming languages be
+designed in such a way as to not allow side effects or aliasing
+(Lampson et al. 1981; Morris, Schmidt, and Wadler 1980).
+
+ (3) In view of this, it is ironic that introductory programming is
+most often taught in a highly imperative style. This may be a vestige
+of a belief, common throughout the 1960s and 1970s, that programs that
+call procedures must inherently be less efficient than programs that
+perform assignments. (Steele (1977) debunks this argument.)
+Alternatively it may reflect a view that step-by-step assignment is
+easier for beginners to visualize than procedure call. Whatever the
+reason, it often saddles beginning programmers with "should I set this
+variable before or after that one" concerns that can complicate
+programming and obscure the important ideas.
+
+\1f
+File: sicp.info, Node: 3-2, Next: 3-3, Prev: 3-1, Up: Chapter 3
+
+3.2 The Environment Model of Evaluation
+=======================================
+
+When we introduced compound procedures in *Note Chapter 1::, we used the
+substitution model of evaluation (section *Note 1-1-5::) to define what
+is meant by applying a procedure to arguments:
+
+ * To apply a compound procedure to arguments, evaluate the body of
+ the procedure with each formal parameter replaced by the
+ corresponding argument.
+
+
+ Once we admit assignment into our programming language, such a
+definition is no longer adequate. In particular, section *Note 3-1-3::
+argued that, in the presence of assignment, a variable can no longer be
+considered to be merely a name for a value. Rather, a variable must
+somehow designate a "place" in which values can be stored. In our new
+model of evaluation, these places will be maintained in structures
+called "environments".
+
+ An environment is a sequence of "frames". Each frame is a table
+(possibly empty) of "bindings", which associate variable names with
+their corresponding values. (A single frame may contain at most one
+binding for any variable.) Each frame also has a pointer to its environment
+"enclosing environment", unless, for the purposes of discussion, the
+frame is considered to be "global". The "value of a variable" with
+respect to an environment is the value given by the binding of the
+variable in the first frame in the environment that contains a binding
+for that variable. If no frame in the sequence specifies a binding for
+the variable, then the variable is said to be "unbound" in the
+environment.
+
+ *Figure 3.1:* A simple environment structure.
+
+ +--------+
+ | I |
+ | x: 3 |
+ | y: 5 |
+ +--------+
+ ^ ^
+ | |
+ C | | D
+ +---------+ | | +----------+
+ | II | | | | III |
+ | z: 6 +---+ +---+ m: 1 |
+ | x: 7 | | y: 2 |
+ +---------+ +----------+
+
+ *Note Figure 3-1:: shows a simple environment structure consisting
+of three frames, labeled I, II, and III. In the diagram, A, B, C, and
+D are pointers to environments. C and D point to the same environment.
+The variables `z' and `x' are bound in frame II, while `y' and `x' are
+bound in frame I. The value of `x' in environment D is 3. The value
+of `x' with respect to environment B is also 3. This is determined as
+follows: We examine the first frame in the sequence (frame III) and do
+not find a binding for `x', so we proceed to the enclosing environment
+D and find the binding in frame I. On the other hand, the value of `x'
+in environment A is 7, because the first frame in the sequence (frame
+II) contains a binding of `x' to 7. With respect to environment A, the
+binding of `x' to 7 in frame II is said to "shadow" the binding of `x'
+to 3 in frame I.
+
+ The environment is crucial to the evaluation process, because it
+determines the context in which an expression should be evaluated.
+Indeed, one could say that expressions in a programming language do
+not, in themselves, have any meaning. Rather, an expression acquires a
+meaning only with respect to some environment in which it is evaluated.
+Even the interpretation of an expression as straightforward as `(+ 1
+1)' depends on an understanding that one is operating in a context in
+which `+' is the symbol for addition. Thus, in our model of evaluation
+we will always speak of evaluating an expression with respect to some
+environment. To describe interactions with the interpreter, we will
+suppose that there is a global environment, consisting of a single frame
+(with no enclosing environment) that includes values for the symbols
+associated with the primitive procedures. For example, the idea that
+`+' is the symbol for addition is captured by saying that the symbol
+`+' is bound in the global environment to the primitive addition
+procedure.
+
+* Menu:
+
+* 3-2-1:: The Rules for Evaluation
+* 3-2-2:: Applying Simple Procedures
+* 3-2-3:: Frames as the Repository of Local State
+* 3-2-4:: Internal Definitions
+
+\1f
+File: sicp.info, Node: 3-2-1, Next: 3-2-2, Prev: 3-2, Up: 3-2
+
+3.2.1 The Rules for Evaluation
+------------------------------
+
+The overall specification of how the interpreter evaluates a combination
+remains the same as when we first introduced it in section *Note
+1-1-3:::
+
+ * To evaluate a combination:
+
+
+ 1. Evaluate the subexpressions of the combination.(1)
+
+ 2. Apply the value of the operator subexpression to the values of the
+ operand subexpressions.
+
+
+ The environment model of evaluation replaces the substitution model
+in specifying what it means to apply a compound procedure to arguments.
+
+ In the environment model of evaluation, a procedure is always a pair
+consisting of some code and a pointer to an environment. Procedures
+are created in one way only: by evaluating a `lambda' expression. This
+produces a procedure whose code is obtained from the text of the
+`lambda' expression and whose environment is the environment in which
+the `lambda' expression was evaluated to produce the procedure. For
+example, consider the procedure definition
+
+ (define (square x)
+ (* x x))
+
+evaluated in the global environment. The procedure definition syntax
+is just syntactic sugar for an underlying implicit `lambda' expression.
+It would have been equivalent to have used
+
+ (define square
+ (lambda (x) (* x x)))
+
+which evaluates `(lambda (x) (* x x))' and binds `square' to the
+resulting value, all in the global environment.
+
+ *Note Figure 3-2:: shows the result of evaluating this `define'
+expression. The procedure object is a pair whose code specifies that
+the procedure has one formal parameter, namely `x', and a procedure
+body `(* x x)'. The environment part of the procedure is a pointer to
+the global environment, since that is the environment in which the
+`lambda' expression was evaluated to produce the procedure. A new
+binding, which associates the procedure object with the symbol
+`square', has been added to the global frame. In general, `define'
+creates definitions by adding bindings to frames.
+
+ *Figure 3.2:* Environment structure produced by evaluating
+ `(define (square x) (* x x))' in the global environment.
+
+ +----------------------+
+ | other variables |
+ global --->| |
+ env | square: --+ |
+ +-----------|----------+
+ | ^
+ (define (square x) | |
+ (* x x)) V |
+ .---.---. |
+ | O | O-+---+
+ `-|-^---'
+ |
+ V
+ parameters: x
+ body: (* x x)
+
+ Now that we have seen how procedures are created, we can describe how
+procedures are applied. The environment model specifies: To apply a
+procedure to arguments, create a new environment containing a frame
+that binds the parameters to the values of the arguments. The
+enclosing environment of this frame is the environment specified by the
+procedure. Now, within this new environment, evaluate the procedure
+body.
+
+ To show how this rule is followed, *Note Figure 3-3:: illustrates
+the environment structure created by evaluating the expression `(square
+5)' in the global environment, where `square' is the procedure
+generated in *Note Figure 3-2::. Applying the procedure results in the
+creation of a new environment, labeled E1 in the figure, that begins
+with a frame in which `x', the formal parameter for the procedure, is
+bound to the argument 5. The pointer leading upward from this frame
+shows that the frame's enclosing environment is the global environment.
+The global environment is chosen here, because this is the environment
+that is indicated as part of the `square' procedure object. Within E1,
+we evaluate the body of the procedure, `(* x x)'. Since the value of
+`x' in E1 is 5, the result is `(* 5 5)', or 25.
+
+ *Figure 3.3:* Environment created by evaluating `(square 5)' in
+ the global environment.
+
+ +------------------------------------+
+ | other variables |
+ global -->| |
+ env | square: --+ |
+ +-----------|---------------------+--+
+ | ^ ^
+ (square 5) | | |
+ V | |
+ .---.---. | +---+--+
+ | O | O-+---+ E1 -->| x: 5 |
+ `-|-^---' +------+
+ |
+ V
+ parameters: x
+ body: (* x x)
+
+ The environment model of procedure application can be summarized by
+two rules:
+
+ * A procedure object is applied to a set of arguments by
+ constructing a frame, binding the formal parameters of the
+ procedure to the arguments of the call, and then evaluating the
+ body of the procedure in the context of the new environment
+ constructed. The new frame has as its enclosing environment the
+ environment part of the procedure object being applied.
+
+ * A procedure is created by evaluating a `lambda' expression
+ relative to a given environment. The resulting procedure object
+ is a pair consisting of the text of the `lambda' expression and a
+ pointer to the environment in which the procedure was created.
+
+
+ We also specify that defining a symbol using `define' creates a
+binding in the current environment frame and assigns to the symbol the
+indicated value.(2) Finally, we specify the behavior of `set!', the
+operation that forced us to introduce the environment model in the
+first place. Evaluating the expression `(set! <VARIABLE> <VALUE>)' in
+some environment locates the binding of the variable in the environment
+and changes that binding to indicate the new value. That is, one finds
+the first frame in the environment that contains a binding for the
+variable and modifies that frame. If the variable is unbound in the
+environment, then `set!' signals an error.
+
+ These evaluation rules, though considerably more complex than the
+substitution model, are still reasonably straightforward. Moreover,
+the evaluation model, though abstract, provides a correct description
+of how the interpreter evaluates expressions. In *Note Chapter 4:: we
+shall see how this model can serve as a blueprint for implementing a
+working interpreter. The following sections elaborate the details of
+the model by analyzing some illustrative programs.
+
+ ---------- Footnotes ----------
+
+ (1) ssignment introduces a subtlety into step 1 of the evaluation
+rule. As shown in *Note Exercise 3-8::, the presence of assignment
+allows us to write expressions that will produce different values
+depending on the order in which the subexpressions in a combination are
+evaluated. Thus, to be precise, we should specify an evaluation order
+in step 1 (e.g., left to right or right to left). However, this order
+should always be considered to be an implementation detail, and one
+should never write programs that depend on some particular order. For
+instance, a sophisticated compiler might optimize a program by varying
+the order in which subexpressions are evaluated.
+
+ (2) If there is already a binding for the variable in the current
+frame, then the binding is changed. This is convenient because it
+allows redefinition of symbols; however, it also means that `define'
+can be used to change values, and this brings up the issues of
+assignment without explicitly using `set!'. Because of this, some
+people prefer redefinitions of existing symbols to signal errors or
+warnings.
+
+\1f
+File: sicp.info, Node: 3-2-2, Next: 3-2-3, Prev: 3-2-1, Up: 3-2
+
+3.2.2 Applying Simple Procedures
+--------------------------------
+
+When we introduced the substitution model in section *Note 1-1-5:: we
+showed how the combination `(f 5)' evaluates to 136, given the
+following procedure definitions:
+
+ (define (square x)
+ (* x x))
+
+ (define (sum-of-squares x y)
+ (+ (square x) (square y)))
+
+ (define (f a)
+ (sum-of-squares (+ a 1) (* a 2)))
+
+ We can analyze the same example using the environment model. *Note
+Figure 3-4:: shows the three procedure objects created by evaluating
+the definitions of `f', `square', and `sum-of-squares' in the global
+environment. Each procedure object consists of some code, together
+with a pointer to the global environment.
+
+ *Figure 3.4:* Procedure objects in the global frame.
+
+ +--------------------------------------------+
+ | sum-of-squares: |
+ global -->| square: |
+ env | f: --+ |
+ +------|--------------+--------------+-------+
+ | ^ | ^ | ^
+ | | | | | |
+ V | V | V |
+ .---.---. | .---.---. | .---.---. |
+ | O | O-+-+ | O | O-+-+ | O | O-+-+
+ `-|-^---' `-|-^---' `-|-^---'
+ | | |
+ V V V
+ parameters: a parameters: x parameters: x, y
+ body: (sum-of-squares body: (* x x) body: (+ (square x)
+ (+ a 1) (square y))
+ (* a 2))
+
+ In *Note Figure 3-5:: we see the environment structure created by
+evaluating the expression `(f 5)'. The call to `f' creates a new
+environment E1 beginning with a frame in which `a', the formal
+parameter of `f', is bound to the argument 5. In E1, we evaluate the
+body of `f':
+
+ (sum-of-squares (+ a 1) (* a 2))
+
+ *Figure 3.5:* Environments created by evaluating `(f 5)' using the
+ procedures in *Note Figure 3-4::.
+
+ +-----------------------------------------------------+
+ global -->| |
+ env +-----------------------------------------------------+
+ ^ ^ ^ ^
+ (f 5) | | | |
+ +------+ +-------+ +------+ +-------+
+ E1 -->| a: 5 | E2 ->| x: 6 | E3 -->| x: 6 | E4 -->| x: 10 |
+ | | | y: 10 | | | | |
+ +------+ +-------+ +------+ +-------+
+ (sum-of-squares (+ (square x) (* x x) (* x x)
+ (+ a 1) (square u))
+ (+ a 2))
+
+ To evaluate this combination, we first evaluate the subexpressions.
+The first subexpression, `sum-of-squares', has a value that is a
+procedure object. (Notice how this value is found: We first look in
+the first frame of E1, which contains no binding for `sum-of-squares'.
+Then we proceed to the enclosing environment, i.e. the global
+environment, and find the binding shown in *Note Figure 3-4::.) The
+other two subexpressions are evaluated by applying the primitive
+operations `+' and `*' to evaluate the two combinations `(+ a 1)' and
+`(* a 2)' to obtain 6 and 10, respectively.
+
+ Now we apply the procedure object `sum-of-squares' to the arguments
+6 and 10. This results in a new environment E2 in which the formal
+parameters `x' and `y' are bound to the arguments. Within E2 we
+evaluate the combination `(+ (square x) (square y))'. This leads us to
+evaluate `(square x)', where `square' is found in the global frame and
+`x' is 6. Once again, we set up a new environment, E3, in which `x' is
+bound to 6, and within this we evaluate the body of `square', which is
+`(* x x)'. Also as part of applying `sum-of-squares', we must evaluate
+the subexpression `(square y)', where `y' is 10. This second call to
+`square' creates another environment, E4, in which `x', the formal
+parameter of `square', is bound to 10. And within E4 we must evaluate
+`(* x x)'.
+
+ The important point to observe is that each call to `square' creates
+a new environment containing a binding for `x'. We can see here how the
+different frames serve to keep separate the different local variables
+all named `x'. Notice that each frame created by `square' points to
+the global environment, since this is the environment indicated by the
+`square' procedure object.
+
+ After the subexpressions are evaluated, the results are returned.
+The values generated by the two calls to `square' are added by
+`sum-of-squares', and this result is returned by `f'. Since our focus
+here is on the environment structures, we will not dwell on how these
+returned values are passed from call to call; however, this is also an
+important aspect of the evaluation process, and we will return to it in
+detail in *Note Chapter 5::.
+
+ *Exercise 3.9:* In section *Note 1-2-1:: we used the substitution
+ model to analyze two procedures for computing factorials, a
+ recursive version
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* n (factorial (- n 1)))))
+
+ and an iterative version
+
+ (define (factorial n)
+ (fact-iter 1 1 n))
+
+ (define (fact-iter product counter max-count)
+ (if (> counter max-count)
+ product
+ (fact-iter (* counter product)
+ (+ counter 1)
+ max-count)))
+
+ Show the environment structures created by evaluating `(factorial
+ 6)' using each version of the `factorial' procedure.(1)
+
+ ---------- Footnotes ----------
+
+ (1) The environment model will not clarify our claim in section
+*Note 1-2-1:: that the interpreter can execute a procedure such as
+`fact-iter' in a constant amount of space using tail recursion. We
+will discuss tail recursion when we deal with the control structure of
+the interpreter in section *Note 5-4::.
+
+\1f
+File: sicp.info, Node: 3-2-3, Next: 3-2-4, Prev: 3-2-2, Up: 3-2
+
+3.2.3 Frames as the Repository of Local State
+---------------------------------------------
+
+We can turn to the environment model to see how procedures and
+assignment can be used to represent objects with local state. As an
+example, consider the "withdrawal processor" from section *Note 3-1-1::
+created by calling the procedure
+
+ (define (make-withdraw balance)
+ (lambda (amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds")))
+
+ Let us describe the evaluation of
+
+ (define W1 (make-withdraw 100))
+
+followed by
+
+ (W1 50)
+ 50
+
+ *Note Figure 3-6:: shows the result of defining the `make-withdraw'
+procedure in the global environment. This produces a procedure object
+that contains a pointer to the global environment. So far, this is no
+different from the examples we have already seen, except that the body
+of the procedure is itself a `lambda' expression.
+
+ *Figure 3.6:* Result of defining `make-withdraw' in the global
+ environment.
+
+ +---------------------------+
+ global -->| make-withdraw: --+ |
+ env +------------------|--------+
+ | ^
+ V |
+ .---.---. |
+ | O | O-+--+
+ `-|-^---'
+ |
+ V
+ parameters: balance
+ body: (lambda (amount)
+ (if (>= balance amount)
+ (begin (set! balance
+ (- balance amount))
+ balance)
+ "Insufficient funds"))
+
+ The interesting part of the computation happens when we apply the
+procedure `make-withdraw' to an argument:
+
+ (define W1 (make-withdraw 100))
+
+ We begin, as usual, by setting up an environment E1 in which the
+formal parameter `balance' is bound to the argument 100. Within this
+environment, we evaluate the body of `make-withdraw', namely the
+`lambda' expression. This constructs a new procedure object, whose code
+is as specified by the `lambda' and whose environment is E1, the
+environment in which the `lambda' was evaluated to produce the
+procedure. The resulting procedure object is the value returned by the
+call to `make-withdraw'. This is bound to `W1' in the global
+environment, since the `define' itself is being evaluated in the global
+environment. *Note Figure 3-7:: shows the resulting environment
+structure.
+
+ *Figure 3.7:* Result of evaluating `(define W1 (make-withdraw
+ 100))'.
+
+ +-----------------------------------------------+
+ | make-withdraw: -----------------------+ |
+ global -->| | |
+ | W1: --+ | |
+ +-------|-------------------------------|-------+
+ | ^ | ^
+ | | V |
+ | +-------+------+ .---.---. |
+ | E1 -->| balance: 100 | | O | O-+-+
+ | +--------------+ `-|-^---'
+ V ^ |
+ .---.---. | V
+ +-+-O | O-+------------+ parameters: balance
+ | `---^---' body: ...
+ V
+ parameters: amount
+ body: (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds")
+
+ Now we can analyze what happens when `W1' is applied to an argument:
+
+ (W1 50)
+ 50
+
+ We begin by constructing a frame in which `amount', the formal
+parameter of `W1', is bound to the argument 50. The crucial point to
+observe is that this frame has as its enclosing environment not the
+global environment, but rather the environment E1, because this is the
+environment that is specified by the `W1' procedure object. Within
+this new environment, we evaluate the body of the procedure:
+
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds")
+
+ The resulting environment structure is shown in *Note Figure 3-8::.
+The expression being evaluated references both `amount' and `balance'.
+`Amount' will be found in the first frame in the environment, while
+`balance' will be found by following the enclosing-environment pointer
+to E1.
+
+ *Figure 3.8:* Environments created by applying the procedure
+ object `W1'.
+
+ +---------------------------------------------------+
+ | make-withdraw: ... |
+ global -->| |
+ env | W1: --+ |
+ +-------|-------------------------------------------+
+ | ^
+ | |
+ | +-------+------+ Here is the balance
+ | E1 -->| balance: 100 | that will be changed
+ | +--------------+ by the set!.
+ V ^ ^
+ .---.---. | +----+
+ | O | O-+-----------+ |
+ `-|-^---' +------+-----+
+ | | amount: 50 |
+ V +------------+
+ parameters: amount (if (>= balance amount)
+ body: ... (begin (set! balance
+ (- balance amount))
+ balance)
+ "Insufficient funds")
+
+ When the `set!' is executed, the binding of `balance' in E1 is
+changed. At the completion of the call to `W1', `balance' is 50, and
+the frame that contains `balance' is still pointed to by the procedure
+object `W1'. The frame that binds `amount' (in which we executed the
+code that changed `balance') is no longer relevant, since the procedure
+call that constructed it has terminated, and there are no pointers to
+that frame from other parts of the environment. The next time `W1' is
+called, this will build a new frame that binds `amount' and whose
+enclosing environment is E1. We see that E1 serves as the "place" that
+holds the local state variable for the procedure object `W1'. *Note
+Figure 3-9:: shows the situation after the call to `W1'.
+
+ *Figure 3.9:* Environments after the call to `W1'.
+
+ +------------------------------------+
+ | make-withdraw: ... |
+ global --->| |
+ env | W1: --+ |
+ +-------|----------------------------+
+ | ^
+ | |
+ | +------+------+
+ | E1 --->| balance: 50 |
+ | +-------------+
+ V ^
+ .---.---. |
+ | O | O-+---------------+
+ `-|-^---'
+ |
+ V
+ parameters: amount
+ body: ...
+
+ Observe what happens when we create a second "withdraw" object by
+making another call to `make-withdraw':
+
+ (define W2 (make-withdraw 100))
+
+ This produces the environment structure of *Note Figure 3-10::,
+which shows that `W2' is a procedure object, that is, a pair with some
+code and an environment. The environment E2 for `W2' was created by
+the call to `make-withdraw'. It contains a frame with its own local
+binding for `balance'. On the other hand, `W1' and `W2' have the same
+code: the code specified by the `lambda' expression in the body of
+`make-withdraw'.(1) We see here why `W1' and `W2' behave as independent
+objects. Calls to `W1' reference the state variable `balance' stored
+in E1, whereas calls to `W2' reference the `balance' stored in E2.
+Thus, changes to the local state of one object do not affect the other
+object.
+
+ *Figure 3.10:* Using `(define W2 (make-withdraw 100))' to create a
+ second object.
+
+ +-------------------------------------------------+
+ | make-withdraw: ... |
+ global ->| W2: ---------------------------+ |
+ env | W1: --+ | |
+ +-------|------------------------|----------------+
+ | ^ | ^
+ | | | |
+ | +------+------+ | +------+-------+
+ | E1 ->| balance: 50 | | E2 ->| balance: 100 |
+ | +-------------+ | +--------------+
+ V ^ V ^
+ .---.---. | .---.---. |
+ | O | O-+----------+ | O | O-+----------+
+ `-|-^---' `-|-^---'
+ | +----------------------+
+ V V
+ parameters: amount
+ body: ...
+
+ *Exercise 3.10:* In the `make-withdraw' procedure, the local
+ variable `balance' is created as a parameter of `make-withdraw'.
+ We could also create the local state variable explicitly, using
+ `let', as follows:
+
+ (define (make-withdraw initial-amount)
+ (let ((balance initial-amount))
+ (lambda (amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))))
+
+ Recall from section *Note 1-3-2:: that `let' is simply syntactic
+ sugar for a procedure call:
+
+ (let ((<VAR> <EXP>)) <BODY>)
+
+ is interpreted as an alternate syntax for
+
+ ((lambda (<VAR>) <BODY>) <EXP>)
+
+ Use the environment model to analyze this alternate version of
+ `make-withdraw', drawing figures like the ones above to illustrate
+ the interactions
+
+ (define W1 (make-withdraw 100))
+
+ (W1 50)
+
+ (define W2 (make-withdraw 100))
+
+ Show that the two versions of `make-withdraw' create objects with
+ the same behavior. How do the environment structures differ for
+ the two versions?
+
+ ---------- Footnotes ----------
+
+ (1) Whether `W1' and `W2' share the same physical code stored in the
+computer, or whether they each keep a copy of the code, is a detail of
+the implementation. For the interpreter we implement in *Note Chapter
+4::, the code is in fact shared.
+
+\1f
+File: sicp.info, Node: 3-2-4, Prev: 3-2-3, Up: 3-2
+
+3.2.4 Internal Definitions
+--------------------------
+
+Section *Note 1-1-8:: introduced the idea that procedures can have
+internal definitions, thus leading to a block structure as in the
+following procedure to compute square roots:
+
+ (define (sqrt x)
+ (define (good-enough? guess)
+ (< (abs (- (square guess) x)) 0.001))
+ (define (improve guess)
+ (average guess (/ x guess)))
+ (define (sqrt-iter guess)
+ (if (good-enough? guess)
+ guess
+ (sqrt-iter (improve guess))))
+ (sqrt-iter 1.0))
+
+ Now we can use the environment model to see why these internal
+definitions behave as desired. *Note Figure 3-11:: shows the point in
+the evaluation of the expression `(sqrt 2)' where the internal
+procedure `good-enough?' has been called for the first time with
+`guess' equal to 1.
+
+ *Figure 3.11:* `Sqrt' procedure with internal definitions.
+
+ +--------------------------------------------------+
+ global -->| sqrt: --+ |
+ env | | |
+ +---------|----------------------------------------+
+ V ^ ^
+ .---.---. | |
+ +----------+-O | O-+---+ +----------+------------+
+ | `---^---' | x: 2 |
+ V E1 -->| good-enough?: -+ |
+ parameters: x | improve: ... | |
+ body: (define good-enough? ...) | sqrt-iter: ... | |
+ (define improve ...) +----------------|------+
+ (define sqrt-iter ...) ^ ^ | ^
+ (sqrt-iter 1.0) | | V |
+ +---------++ | .---.---. |
+ E2 -->| guess: 1 | | | O | O-+-+
+ +----------+ | `-|-^---'
+ call to sqrt-iter | |
+ | V
+ +---------++ parameters: guess
+ E3 -->| guess: 1 | body: (< (abs ...)
+ +----------+ ...)
+ call to good-enough?
+
+ Observe the structure of the environment. `Sqrt' is a symbol in the
+global environment that is bound to a procedure object whose associated
+environment is the global environment. When `sqrt' was called, a new
+environment E1 was formed, subordinate to the global environment, in
+which the parameter `x' is bound to 2. The body of `sqrt' was then
+evaluated in E1. Since the first expression in the body of `sqrt' is
+
+ (define (good-enough? guess)
+ (< (abs (- (square guess) x)) 0.001))
+
+evaluating this expression defined the procedure `good-enough?' in the
+environment E1. To be more precise, the symbol `good-enough?' was added
+to the first frame of E1, bound to a procedure object whose associated
+environment is E1. Similarly, `improve' and `sqrt-iter' were defined
+as procedures in E1. For conciseness, *Note Figure 3-11:: shows only
+the procedure object for `good-enough?'.
+
+ After the local procedures were defined, the expression `(sqrt-iter
+1.0)' was evaluated, still in environment E1. So the procedure object
+bound to `sqrt-iter' in E1 was called with 1 as an argument. This
+created an environment E2 in which `guess', the parameter of
+`sqrt-iter', is bound to 1. `Sqrt-iter' in turn called `good-enough?'
+with the value of `guess' (from E2) as the argument for `good-enough?'.
+This set up another environment, E3, in which `guess' (the parameter of
+`good-enough?') is bound to 1. Although `sqrt-iter' and `good-enough?'
+both have a parameter named `guess', these are two distinct local
+variables located in different frames. Also, E2 and E3 both have E1 as
+their enclosing environment, because the `sqrt-iter' and `good-enough?'
+procedures both have E1 as their environment part. One consequence of
+this is that the symbol `x' that appears in the body of `good-enough?'
+will reference the binding of `x' that appears in E1, namely the value
+of `x' with which the original `sqrt' procedure was called.
+
+ The environment model thus explains the two key properties that make
+local procedure definitions a useful technique for modularizing
+programs:
+
+ * The names of the local procedures do not interfere with names
+ external to the enclosing procedure, because the local procedure
+ names will be bound in the frame that the procedure creates when
+ it is run, rather than being bound in the global environment.
+
+ * The local procedures can access the arguments of the enclosing
+ procedure, simply by using parameter names as free variables.
+ This is because the body of the local procedure is evaluated in an
+ environment that is subordinate to the evaluation environment for
+ the enclosing procedure.
+
+
+ *Exercise 3.11:* In section *Note 3-2-3:: we saw how the
+ environment model described the behavior of procedures with local
+ state. Now we have seen how internal definitions work. A typical
+ message-passing procedure contains both of these aspects.
+ Consider the bank account procedure of section *Note 3-1-1:::
+
+ (define (make-account balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) withdraw)
+ ((eq? m 'deposit) deposit)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch)
+
+ Show the environment structure generated by the sequence of
+ interactions
+
+ (define acc (make-account 50))
+
+ ((acc 'deposit) 40)
+ 90
+
+ ((acc 'withdraw) 60)
+ 30
+
+ Where is the local state for `acc' kept? Suppose we define another
+ account
+
+ (define acc2 (make-account 100))
+
+ How are the local states for the two accounts kept distinct?
+ Which parts of the environment structure are shared between `acc'
+ and `acc2'?
+
+\1f
+File: sicp.info, Node: 3-3, Next: 3-4, Prev: 3-2, Up: Chapter 3
+
+3.3 Modeling with Mutable Data
+==============================
+
+Chapter 2 dealt with compound data as a means for constructing
+computational objects that have several parts, in order to model
+real-world objects that have several aspects. In that chapter we
+introduced the discipline of data abstraction, according to which data
+structures are specified in terms of constructors, which create data
+objects, and selectors, which access the parts of compound data
+objects. But we now know that there is another aspect of data that
+*Note Chapter 2:: did not address. The desire to model systems
+composed of objects that have changing state leads us to the need to
+modify compound data objects, as well as to construct and select from
+them. In order to model compound objects with changing state, we will
+design data abstractions to include, in addition to selectors and
+constructors, operations called "mutators", which modify data objects.
+For instance, modeling a banking system requires us to change account
+balances. Thus, a data structure for representing bank accounts might
+admit an operation
+
+ (set-balance! <ACCOUNT> <NEW-VALUE>)
+
+that changes the balance of the designated account to the designated
+new value. Data objects for which mutators are defined are known as objects
+"mutable data objects".
+
+ *Note Chapter 2:: introduced pairs as a general-purpose "glue" for
+synthesizing compound data. We begin this section by defining basic
+mutators for pairs, so that pairs can serve as building blocks for
+constructing mutable data objects. These mutators greatly enhance the
+representational power of pairs, enabling us to build data structures
+other than the sequences and trees that we worked with in section *Note
+2-2::. We also present some examples of simulations in which complex
+systems are modeled as collections of objects with local state.
+
+* Menu:
+
+* 3-3-1:: Mutable List Structure
+* 3-3-2:: Representing Queues
+* 3-3-3:: Representing Tables
+* 3-3-4:: A Simulator for Digital Circuits
+* 3-3-5:: Propagation of Constraints
+
+\1f
+File: sicp.info, Node: 3-3-1, Next: 3-3-2, Prev: 3-3, Up: 3-3
+
+3.3.1 Mutable List Structure
+----------------------------
+
+The basic operations on pairs--`cons', `car', and `cdr'--can be used to
+construct list structure and to select parts from list structure, but
+they are incapable of modifying list structure. The same is true of the
+list operations we have used so far, such as `append' and `list', since
+these can be defined in terms of `cons', `car', and `cdr'. To modify
+list structures we need new operations.
+
+ *Figure 3.12:* Lists `x': `((a b) c d)' and `y': `(e f)'.
+
+ +---+---+ +---+---+ +---+---+
+ x -->| * | *-+---->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ | V V
+ | +---+ +---+
+ | | c | | d |
+ | +---+ +---+
+ | +---+---+ +---+---+
+ +---------->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | a | | b |
+ +---+ +---+
+ +---+---+ +---+---+
+ y -->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | e | | f |
+ +---+ +---+
+
+ *Figure 3.13:* Effect of `(set-car! x y)' on the lists in *Note
+ Figure 3-12::.
+
+ +---+---+ +---+---+ +---+---+
+ x -->| * | *-+---->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ | V V
+ | +---+ +---+
+ | | c | | d |
+ | +---+ +---+
+ | +---+---+ +---+---+
+ | | * | *-+---->| * | / |
+ | +-|-+---+ +-|-+---+
+ | V V
+ | +---+ +---+
+ | | a | | b |
+ | +---+ +---+
+ +---------->+---+---+ +---+---+
+ | * | *-+---->| * | / |
+ y -->+-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | e | | f |
+ +---+ +---+
+
+ *Figure 3.14:* Effect of `(define z (cons y (cdr x)))' on the
+ lists in *Note Figure 3-12::.
+
+ +---+---+ +---+---+ +---+---+
+ x -->| * | *-+---->| * | *-+---->| * | / |
+ +-|-+---+ +-->+-|-+---+ +-|-+---+
+ | | V V
+ | | +---+ +---+
+ | | | c | | d |
+ | | +---+ +---+
+ | | +---+---+ +---+---+
+ +-------+-->| * | *-+---->| * | / |
+ | +-|-+---+ +-|-+---+
+ +---+---+ | V V
+ z -->| * | *-+-+ +---+ +---+
+ +-|-+---+ | a | | b |
+ | +---+ +---+
+ +---------->+---+---+ +---+---+
+ | * | *-+---->| * | / |
+ y -->+-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | e | | f |
+ +---+ +---+
+
+ *Figure 3.15:* Effect of `(set-cdr! x y)' on the lists in *Note
+ Figure 3-12::.
+
+ +---+---+ +---+---+ +---+---+
+ x -->| * | * | | * | *-+---->| * | / |
+ +-|-+-|-+ +-|-+---+ +-|-+---+
+ | | V V
+ | | +---+ +---+
+ | | | c | | d |
+ | | +---+ +---+
+ | | +---+---+ +---+---+
+ +---+------>| * | *-+---->| * | / |
+ | +-|-+---+ +-|-+---+
+ | V V
+ | +---+ +---+
+ | | a | | b |
+ | +---+ +---+
+ +------>+---+---+ +---+---+
+ | * | *-+---->| * | / |
+ y -->+-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | e | | f |
+ +---+ +---+
+
+ The primitive mutators for pairs are `set-car!' and `set-cdr!'.
+`Set-car!' takes two arguments, the first of which must be a pair. It
+modifies this pair, replacing the `car' pointer by a pointer to the
+second argument of `set-car!'.(1)
+
+ As an example, suppose that `x' is bound to the list `((a b) c d)'
+and `y' to the list `(e f)' as illustrated in *Note Figure 3-12::.
+Evaluating the expression ` (set-car! x y)' modifies the pair to which
+`x' is bound, replacing its `car' by the value of `y'. The result of
+the operation is shown in *Note Figure 3-13::. The structure `x' has
+been modified and would now be printed as `((e f) c d)'. The pairs
+representing the list `(a b)', identified by the pointer that was
+replaced, are now detached from the original structure.(2)
+
+ Compare *Note Figure 3-13:: with *Note Figure 3-14::, which
+illustrates the result of executing `(define z (cons y (cdr x)))' with
+`x' and `y' bound to the original lists of *Note Figure 3-12::. The
+variable `z' is now bound to a new pair created by the `cons'
+operation; the list to which `x' is bound is unchanged.
+
+ The `set-cdr!' operation is similar to `set-car!'. The only
+difference is that the `cdr' pointer of the pair, rather than the `car'
+pointer, is replaced. The effect of executing `(set-cdr! x y)' on the
+lists of *Note Figure 3-12:: is shown in *Note Figure 3-15::. Here the
+`cdr' pointer of `x' has been replaced by the pointer to `(e f)'.
+Also, the list `(c d)', which used to be the `cdr' of `x', is now
+detached from the structure.
+
+ `Cons' builds new list structure by creating new pairs, while
+`set-car!' and `set-cdr!' modify existing pairs. Indeed, we could
+implement `cons' in terms of the two mutators, together with a procedure
+`get-new-pair', which returns a new pair that is not part of any
+existing list structure. We obtain the new pair, set its `car' and
+`cdr' pointers to the designated objects, and return the new pair as
+the result of the `cons'.(3)
+
+ (define (cons x y)
+ (let ((new (get-new-pair)))
+ (set-car! new x)
+ (set-cdr! new y)
+ new))
+
+ *Exercise 3.12:* The following procedure for appending lists was
+ introduced in section *Note 2-2-1:::
+
+ (define (append x y)
+ (if (null? x)
+ y
+ (cons (car x) (append (cdr x) y))))
+
+ `Append' forms a new list by successively `cons'ing the elements of
+ `x' onto `y'. The procedure `append!' is similar to `append', but
+ it is a mutator rather than a constructor. It appends the lists
+ by splicing them together, modifying the final pair of `x' so that
+ its `cdr' is now `y'. (It is an error to call `append!' with an
+ empty `x'.)
+
+ (define (append! x y)
+ (set-cdr! (last-pair x) y)
+ x)
+
+ Here `last-pair' is a procedure that returns the last pair in its
+ argument:
+
+ (define (last-pair x)
+ (if (null? (cdr x))
+ x
+ (last-pair (cdr x))))
+
+ Consider the interaction
+
+ (define x (list 'a 'b))
+
+ (define y (list 'c 'd))
+
+ (define z (append x y))
+
+ z
+ (a b c d)
+
+ (cdr x)
+ <RESPONSE>
+
+ (define w (append! x y))
+
+ w
+ (a b c d)
+
+ (cdr x)
+ <RESPONSE>
+
+ What are the missing <RESPONSE>s? Draw box-and-pointer diagrams to
+ explain your answer.
+
+ *Exercise 3.13:* Consider the following `make-cycle' procedure,
+ which uses the `last-pair' procedure defined in *Note Exercise
+ 3-12:::
+
+ (define (make-cycle x)
+ (set-cdr! (last-pair x) x)
+ x)
+
+ Draw a box-and-pointer diagram that shows the structure `z'
+ created by
+
+ (define z (make-cycle (list 'a 'b 'c)))
+
+ What happens if we try to compute `(last-pair z)'?
+
+ *Exercise 3.14:* The following procedure is quite useful, although
+ obscure:
+
+ (define (mystery x)
+ (define (loop x y)
+ (if (null? x)
+ y
+ (let ((temp (cdr x)))
+ (set-cdr! x y)
+ (loop temp x))))
+ (loop x '()))
+
+ `Loop' uses the "temporary" variable `temp' to hold the old value
+ of the `cdr' of `x', since the `set-cdr!' on the next line
+ destroys the `cdr'. Explain what `mystery' does in general.
+ Suppose `v' is defined by `(define v (list 'a 'b 'c 'd))'. Draw the
+ box-and-pointer diagram that represents the list to which `v' is
+ bound. Suppose that we now evaluate `(define w (mystery v))'. Draw
+ box-and-pointer diagrams that show the structures `v' and `w' after
+ evaluating this expression. What would be printed as the values
+ of `v' and `w'?
+
+Sharing and identity
+....................
+
+We mentioned in section *Note 3-1-3:: the theoretical issues of
+"sameness" and "change" raised by the introduction of assignment.
+These issues arise in practice when individual pairs are "shared" among
+different data objects. For example, consider the structure formed by
+
+ (define x (list 'a 'b))
+ (define z1 (cons x x))
+
+ As shown in *Note Figure 3-16::, `z1' is a pair whose `car' and
+`cdr' both point to the same pair `x'. This sharing of `x' by the
+`car' and `cdr' of `z1' is a consequence of the straightforward way in
+which `cons' is implemented. In general, using `cons' to construct
+lists will result in an interlinked structure of pairs in which many
+individual pairs are shared by many different structures.
+
+ *Figure 3.16:* The list `z1' formed by `(cons x x)'.
+
+ +---+---+
+ z1 -->| * | * |
+ +-|-+-|-+
+ V V
+ +---+---+ +---+---+
+ x -->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+
+ V V
+ +---+ +---+
+ | a | | b |
+ +---+ +---+
+
+ *Figure 3.17:* The list `z2' formed by `(cons (list 'a 'b) (list
+ 'a 'b))'.
+
+ +---+---+ +---+---+ +---+---+
+ z2 -->| * | *-+---->| * | *-+---->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ | V V
+ | +---+ +---+
+ | | a | | b |
+ | +---+ +---+
+ | ^ ^
+ | | |
+ | +-|-+---+ +-|-+---+
+ +---------->| * | *-+---->| * | / |
+ +---+---+ +---+---+
+
+ In contrast to *Note Figure 3-16::, *Note Figure 3-17:: shows the
+structure created by
+
+ (define z2 (cons (list 'a 'b) (list 'a 'b)))
+
+ In this structure, the pairs in the two `(a b)' lists are distinct,
+although the actual symbols are shared.(4)
+
+ When thought of as a list, `z1' and `z2' both represent "the same"
+list, `((a b) a b)'. In general, sharing is completely undetectable if
+we operate on lists using only `cons', `car', and `cdr'. However, if
+we allow mutators on list structure, sharing becomes significant. As an
+example of the difference that sharing can make, consider the following
+procedure, which modifies the `car' of the structure to which it is
+applied:
+
+ (define (set-to-wow! x)
+ (set-car! (car x) 'wow)
+ x)
+
+ Even though `z1' and `z2' are "the same" structure, applying
+`set-to-wow!' to them yields different results. With `z1', altering
+the `car' also changes the `cdr', because in `z1' the `car' and the
+`cdr' are the same pair. With `z2', the `car' and `cdr' are distinct,
+so `set-to-wow!' modifies only the `car':
+
+ z1
+ ((a b) a b)
+
+ (set-to-wow! z1)
+ ((wow b) wow b)
+
+ z2
+ ((a b) a b)
+
+ (set-to-wow! z2)
+ ((wow b) a b)
+
+ One way to detect sharing in list structures is to use the predicate
+`eq?', which we introduced in section *Note 2-3-1:: as a way to test
+whether two symbols are equal. More generally, `(eq? x y)' tests
+whether `x' and `y' are the same object (that is, whether `x' and `y'
+are equal as pointers). Thus, with `z1' and `z2' as defined in figures
+*Note Figure 3-16:: and *Note Figure 3-17::, `(eq? (car z1) (cdr z1))'
+is true and `(eq? (car z2) (cdr z2))' is false.
+
+ As will be seen in the following sections, we can exploit sharing to
+greatly extend the repertoire of data structures that can be
+represented by pairs. On the other hand, sharing can also be
+dangerous, since modifications made to structures will also affect
+other structures that happen to share the modified parts. The mutation
+operations `set-car!' and `set-cdr!' should be used with care; unless
+we have a good understanding of how our data objects are shared,
+mutation can have unanticipated results.(5)
+
+ *Exercise 3.15:* Draw box-and-pointer diagrams to explain the
+ effect of `set-to-wow!' on the structures `z1' and `z2' above.
+
+ *Exercise 3.16:* Ben Bitdiddle decides to write a procedure to
+ count the number of pairs in any list structure. "It's easy," he
+ reasons. "The number of pairs in any structure is the number in
+ the `car' plus the number in the `cdr' plus one more to count the
+ current pair." So Ben writes the following procedure:
+
+ (define (count-pairs x)
+ (if (not (pair? x))
+ 0
+ (+ (count-pairs (car x))
+ (count-pairs (cdr x))
+ 1)))
+
+ Show that this procedure is not correct. In particular, draw
+ box-and-pointer diagrams representing list structures made up of
+ exactly three pairs for which Ben's procedure would return 3;
+ return 4; return 7; never return at all.
+
+ *Exercise 3.17:* Devise a correct version of the `count-pairs'
+ procedure of *Note Exercise 3-16:: that returns the number of
+ distinct pairs in any structure. (Hint: Traverse the structure,
+ maintaining an auxiliary data structure that is used to keep track
+ of which pairs have already been counted.)
+
+ *Exercise 3.18:* Write a procedure that examines a list and
+ determines whether it contains a cycle, that is, whether a program
+ that tried to find the end of the list by taking successive `cdr's
+ would go into an infinite loop. *Note Exercise 3-13:: constructed
+ such lists.
+
+ *Exercise 3.19:* Redo *Note Exercise 3-18:: using an algorithm
+ that takes only a constant amount of space. (This requires a very
+ clever idea.)
+
+Mutation is just assignment
+...........................
+
+When we introduced compound data, we observed in section *Note 2-1-3::
+that pairs can be represented purely in terms of procedures:
+
+ (define (cons x y)
+ (define (dispatch m)
+ (cond ((eq? m 'car) x)
+ ((eq? m 'cdr) y)
+ (else (error "Undefined operation -- CONS" m))))
+ dispatch)
+
+ (define (car z) (z 'car))
+
+ (define (cdr z) (z 'cdr))
+
+ The same observation is true for mutable data. We can implement
+mutable data objects as procedures using assignment and local state.
+For instance, we can extend the above pair implementation to handle
+`set-car!' and `set-cdr!' in a manner analogous to the way we
+implemented bank accounts using `make-account' in section *Note 3-1-1:::
+
+ (define (cons x y)
+ (define (set-x! v) (set! x v))
+ (define (set-y! v) (set! y v))
+ (define (dispatch m)
+ (cond ((eq? m 'car) x)
+ ((eq? m 'cdr) y)
+ ((eq? m 'set-car!) set-x!)
+ ((eq? m 'set-cdr!) set-y!)
+ (else (error "Undefined operation -- CONS" m))))
+ dispatch)
+
+ (define (car z) (z 'car))
+
+ (define (cdr z) (z 'cdr))
+
+ (define (set-car! z new-value)
+ ((z 'set-car!) new-value)
+ z)
+
+ (define (set-cdr! z new-value)
+ ((z 'set-cdr!) new-value)
+ z)
+
+ Assignment is all that is needed, theoretically, to account for the
+behavior of mutable data. As soon as we admit `set!' to our language,
+we raise all the issues, not only of assignment, but of mutable data in
+general.(6)
+
+ *Exercise 3.20:* Draw environment diagrams to illustrate the
+ evaluation of the sequence of expressions
+
+ (define x (cons 1 2))
+ (define z (cons x x))
+ (set-car! (cdr z) 17)
+
+ (car x)
+ 17
+
+ using the procedural implementation of pairs given above. (Compare
+ *Note Exercise 3-11::.)
+
+ ---------- Footnotes ----------
+
+ (1) `Set-car!' and `set-cdr!' return implementation-dependent
+values. Like `set!', they should be used only for their effect.
+
+ (2) We see from this that mutation operations on lists can create
+"garbage" that is not part of any accessible structure. We will see in
+section *Note 5-3-2:: that Lisp memory-management systems include a "garbage
+collector", which identifies and recycles the memory space used by
+unneeded pairs.
+
+ (3) `Get-new-pair' is one of the operations that must be implemented
+as part of the memory management required by a Lisp implementation. We
+will discuss this in section *Note 5-3-1::.
+
+ (4) The two pairs are distinct because each call to `cons' returns a
+new pair. The symbols are shared; in Scheme there is a unique symbol
+with any given name. Since Scheme provides no way to mutate a symbol,
+this sharing is undetectable. Note also that the sharing is what
+enables us to compare symbols using `eq?', which simply checks equality
+of pointers.
+
+ (5) The subtleties of dealing with sharing of mutable data objects
+reflect the underlying issues of "sameness" and "change" that were
+raised in section *Note 3-1-3::. We mentioned there that admitting
+change to our language requires that a compound object must have an
+"identity" that is something different from the pieces from which it is
+composed. In Lisp, we consider this "identity" to be the quality that
+is tested by `eq?', i.e., by equality of pointers. Since in most Lisp
+implementations a pointer is essentially a memory address, we are
+"solving the problem" of defining the identity of objects by
+stipulating that a data object "itself" is the information stored in
+some particular set of memory locations in the computer. This suffices
+for simple Lisp programs, but is hardly a general way to resolve the
+issue of "sameness" in computational models.
+
+ (6) On the other hand, from the viewpoint of implementation,
+assignment requires us to modify the environment, which is itself a
+mutable data structure. Thus, assignment and mutation are equipotent:
+Each can be implemented in terms of the other.
+
+\1f
+File: sicp.info, Node: 3-3-2, Next: 3-3-3, Prev: 3-3-1, Up: 3-3
+
+3.3.2 Representing Queues
+-------------------------
+
+The mutators `set-car!' and `set-cdr!' enable us to use pairs to
+construct data structures that cannot be built with `cons', `car', and
+`cdr' alone. This section shows how to use pairs to represent a data
+structure called a queue. Section *Note 3-3-3:: will show how to
+represent data structures called tables.
+
+ A "queue" is a sequence in which items are inserted at one end
+(called the "rear" of the queue) and deleted from the other end (the "front").
+*Note Figure 3-18:: shows an initially empty queue in which the items
+`a' and `b' are inserted. Then `a' is removed, `c' and `d' are
+inserted, and `b' is removed. Because items are always removed in the
+order in which they are inserted, a queue is sometimes called a "FIFO"
+(first in, first out) buffer.
+
+ *Figure 3.18:* Queue operations.
+
+ Operation Resulting Queue
+ (define q (make-queue))
+ (insert-queue! q 'a) a
+ (insert-queue! q 'b) a b
+ (delete-queue! q) b
+ (insert-queue! q 'c) b c
+ (insert-queue! q 'd) b c d
+ (delete-queue! q) c d
+
+ In terms of data abstraction, we can regard a queue as defined by
+the following set of operations:
+
+ * a constructor: `(make-queue)' returns an empty queue (a queue
+ containing no items).
+
+ * two selectors:
+
+ (empty-queue? <QUEUE>)
+
+ tests if the queue is empty.
+
+ (front-queue <QUEUE>)
+
+ returns the object at the front of the queue, signaling an error
+ if the queue is empty; it does not modify the queue.
+
+ * two mutators:
+
+ (insert-queue! <QUEUE> <ITEM>)
+
+ inserts the item at the rear of the queue and returns the modified
+ queue as its value.
+
+ (delete-queue! <QUEUE>)
+
+ removes the item at the front of the queue and returns the
+ modified queue as its value, signaling an error if the queue is
+ empty before the deletion.
+
+
+ Because a queue is a sequence of items, we could certainly represent
+it as an ordinary list; the front of the queue would be the `car' of
+the list, inserting an item in the queue would amount to appending a
+new element at the end of the list, and deleting an item from the queue
+would just be taking the `cdr' of the list. However, this
+representation is inefficient, because in order to insert an item we
+must scan the list until we reach the end. Since the only method we
+have for scanning a list is by successive `cdr' operations, this
+scanning requires [theta](n) steps for a list of n items. A simple
+modification to the list representation overcomes this disadvantage by
+allowing the queue operations to be implemented so that they require
+[theta](1) steps; that is, so that the number of steps needed is
+independent of the length of the queue.
+
+ The difficulty with the list representation arises from the need to
+scan to find the end of the list. The reason we need to scan is that,
+although the standard way of representing a list as a chain of pairs
+readily provides us with a pointer to the beginning of the list, it
+gives us no easily accessible pointer to the end. The modification
+that avoids the drawback is to represent the queue as a list, together
+with an additional pointer that indicates the final pair in the list.
+That way, when we go to insert an item, we can consult the rear pointer
+and so avoid scanning the list.
+
+ A queue is represented, then, as a pair of pointers, `front-ptr' and
+`rear-ptr', which indicate, respectively, the first and last pairs in an
+ordinary list. Since we would like the queue to be an identifiable
+object, we can use `cons' to combine the two pointers. Thus, the queue
+itself will be the `cons' of the two pointers. *Note Figure 3-19::
+illustrates this representation.
+
+ *Figure 3.19:* Implementation of a queue as a list with front and
+ rear pointers.
+
+ +---+---+
+ q -->| * | *-+-------------------+
+ +-|-+---+ |
+ | |
+ | front-ptr | rear-ptr
+ V V
+ +---+---+ +---+---+ +---+---+
+ | * | *-+--->| * | *-+--->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ V V V
+ +---+ +---+ +---+
+ | a | | b | | c |
+ +---+ +---+ +---+
+
+ To define the queue operations we use the following procedures,
+which enable us to select and to modify the front and rear pointers of
+a queue:
+
+ (define (front-ptr queue) (car queue))
+
+ (define (rear-ptr queue) (cdr queue))
+
+ (define (set-front-ptr! queue item) (set-car! queue item))
+
+ (define (set-rear-ptr! queue item) (set-cdr! queue item))
+
+ Now we can implement the actual queue operations. We will consider
+a queue to be empty if its front pointer is the empty list:
+
+ (define (empty-queue? queue) (null? (front-ptr queue)))
+
+ The `make-queue' constructor returns, as an initially empty queue, a
+pair whose `car' and `cdr' are both the empty list:
+
+ (define (make-queue) (cons '() '()))
+
+ To select the item at the front of the queue, we return the `car' of
+the pair indicated by the front pointer:
+
+ (define (front-queue queue)
+ (if (empty-queue? queue)
+ (error "FRONT called with an empty queue" queue)
+ (car (front-ptr queue))))
+
+ To insert an item in a queue, we follow the method whose result is
+indicated in *Note Figure 3-20::. We first create a new pair whose
+`car' is the item to be inserted and whose `cdr' is the empty list. If
+the queue was initially empty, we set the front and rear pointers of
+the queue to this new pair. Otherwise, we modify the final pair in the
+queue to point to the new pair, and also set the rear pointer to the
+new pair.
+
+ *Figure 3.20:* Result of using `(insert-queue! q 'd)' on the
+ queue of *Note Figure 3-19::.
+
+ +---+---+
+ q -->| * | *-+--------------------------------+
+ +-|-+---+ |
+ | |
+ | front-ptr | rear-ptr
+ V V
+ +---+---+ +---+---+ +---+---+ +---+---+
+ | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
+ V V V V
+ +---+ +---+ +---+ +---+
+ | a | | b | | c | | d |
+ +---+ +---+ +---+ +---+
+
+ (define (insert-queue! queue item)
+ (let ((new-pair (cons item '())))
+ (cond ((empty-queue? queue)
+ (set-front-ptr! queue new-pair)
+ (set-rear-ptr! queue new-pair)
+ queue)
+ (else
+ (set-cdr! (rear-ptr queue) new-pair)
+ (set-rear-ptr! queue new-pair)
+ queue))))
+
+ To delete the item at the front of the queue, we merely modify the
+front pointer so that it now points at the second item in the queue,
+which can be found by following the `cdr' pointer of the first item
+(see *Note Figure 3-21::):(1)
+
+ *Figure 3.21:* Result of using `(delete-queue! q)' on the queue
+ of *Note Figure 3-20::.
+
+ +---+---+
+ q -->| * | *-+--------------------------------+
+ +-|-+---+ |
+ +------------+ |
+ front-ptr | | rear-ptr
+ V V
+ +---+---+ +---+---+ +---+---+ +---+---+
+ | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
+ V V V V
+ +---+ +---+ +---+ +---+
+ | a | | b | | c | | d |
+ +---+ +---+ +---+ +---+
+
+ (define (delete-queue! queue)
+ (cond ((empty-queue? queue)
+ (error "DELETE! called with an empty queue" queue))
+ (else
+ (set-front-ptr! queue (cdr (front-ptr queue)))
+ queue)))
+
+ *Exercise 3.21:* Ben Bitdiddle decides to test the queue
+ implementation described above. He types in the procedures to the
+ Lisp interpreter and proceeds to try them out:
+
+ (define q1 (make-queue))
+
+ (insert-queue! q1 'a)
+ ((a) a)
+
+ (insert-queue! q1 'b)
+ ((a b) b)
+
+ (delete-queue! q1)
+ ((b) b)
+
+ (delete-queue! q1)
+ (() b)
+
+ "It's all wrong!" he complains. "The interpreter's response shows
+ that the last item is inserted into the queue twice. And when I
+ delete both items, the second `b' is still there, so the queue
+ isn't empty, even though it's supposed to be." Eva Lu Ator
+ suggests that Ben has misunderstood what is happening. "It's not
+ that the items are going into the queue twice," she explains.
+ "It's just that the standard Lisp printer doesn't know how to make
+ sense of the queue representation. If you want to see the queue
+ printed correctly, you'll have to define your own print procedure
+ for queues." Explain what Eva Lu is talking about. In particular,
+ show why Ben's examples produce the printed results that they do.
+ Define a procedure `print-queue' that takes a queue as input and
+ prints the sequence of items in the queue.
+
+ *Exercise 3.22:* Instead of representing a queue as a pair of
+ pointers, we can build a queue as a procedure with local state.
+ The local state will consist of pointers to the beginning and the
+ end of an ordinary list. Thus, the `make-queue' procedure will
+ have the form
+
+ (define (make-queue)
+ (let ((front-ptr ... )
+ (rear-ptr ... ))
+ <DEFINITIONS OF INTERNAL PROCEDURES>
+ (define (dispatch m) ...)
+ dispatch))
+
+ Complete the definition of `make-queue' and provide
+ implementations of the queue operations using this representation.
+
+ *Exercise 3.23:* A "deque" ("double-ended queue") is a sequence in
+ which items can be inserted and deleted at either the front or the
+ rear. Operations on deques are the constructor `make-deque', the
+ predicate `empty-deque?', selectors `front-deque' and
+ `rear-deque', and mutators `front-insert-deque!',
+ `rear-insert-deque!', `front-delete-deque!', and
+ `rear-delete-deque!'. Show how to represent deques using pairs,
+ and give implementations of the operations.(2) All operations
+ should be accomplished in [theta](1) steps.
+
+ ---------- Footnotes ----------
+
+ (1) If the first item is the final item in the queue, the front
+pointer will be the empty list after the deletion, which will mark the
+queue as empty; we needn't worry about updating the rear pointer, which
+will still point to the deleted item, because `empty-queue?' looks only
+at the front pointer.
+
+ (2) Be careful not to make the interpreter try to print a structure
+that contains cycles. (See *Note Exercise 3-13::.)
+
+\1f
+File: sicp.info, Node: 3-3-3, Next: 3-3-4, Prev: 3-3-2, Up: 3-3
+
+3.3.3 Representing Tables
+-------------------------
+
+When we studied various ways of representing sets in *Note Chapter 2::,
+we mentioned in section *Note 2-3-3:: the task of maintaining a table
+of records indexed by identifying keys. In the implementation of
+data-directed programming in section *Note 2-4-3::, we made extensive
+use of two-dimensional tables, in which information is stored and
+retrieved using two keys. Here we see how to build tables as mutable
+list structures.
+
+ We first consider a one-dimensional table, in which each value is
+stored under a single key. We implement the table as a list of
+records, each of which is implemented as a pair consisting of a key and
+the associated value. The records are glued together to form a list by
+pairs whose `car's point to successive records. These gluing pairs are
+called the "backbone" of the table. In order to have a place that we
+can change when we add a new record to the table, we build the table as
+a "headed list". A headed list has a special backbone pair at the
+beginning, which holds a dummy "record"--in this case the arbitrarily
+chosen symbol `*table*'. *Note Figure 3-22:: shows the box-and-pointer
+diagram for the table
+
+ a: 1
+ b: 2
+ c: 3
+
+ *Figure 3.22:* A table represented as a headed list.
+
+ +---+---+ +---+---+ +---+---+ +---+---+
+ | * | *-+--->| * | *-+--->| * | *-+--->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
+ | | | |
+ V V V V
+ +---------+ +---+---+ +---+---+ +---+---+
+ | *table* | | * | * | | * | * | | * | * |
+ +---------+ +-|-+-|-+ +-|-+-|-+ +-|-+-|-+
+ | | | | | |
+ V V V V V V
+ +---+ +---+ +---+ +---+ +---+ +---+
+ | a | | 1 | | b | | 2 | | c | | 3 |
+ +---+ +---+ +---+ +---+ +---+ +---+
+
+ To extract information from a table we use the `lookup' procedure,
+which takes a key as argument and returns the associated value (or
+false if there is no value stored under that key). `Lookup' is defined
+in terms of the `assoc' operation, which expects a key and a list of
+records as arguments. Note that `assoc' never sees the dummy record.
+`Assoc' returns the record that has the given key as its `car'.(1)
+`Lookup' then checks to see that the resulting record returned by
+`assoc' is not false, and returns the value (the `cdr') of the record.
+
+ (define (lookup key table)
+ (let ((record (assoc key (cdr table))))
+ (if record
+ (cdr record)
+ false)))
+
+ (define (assoc key records)
+ (cond ((null? records) false)
+ ((equal? key (caar records)) (car records))
+ (else (assoc key (cdr records)))))
+
+ To insert a value in a table under a specified key, we first use
+`assoc' to see if there is already a record in the table with this key.
+If not, we form a new record by `cons'ing the key with the value, and
+insert this at the head of the table's list of records, after the dummy
+record. If there already is a record with this key, we set the `cdr'
+of this record to the designated new value. The header of the table
+provides us with a fixed location to modify in order to insert the new
+record.(2)
+
+ (define (insert! key value table)
+ (let ((record (assoc key (cdr table))))
+ (if record
+ (set-cdr! record value)
+ (set-cdr! table
+ (cons (cons key value) (cdr table)))))
+ 'ok)
+
+ To construct a new table, we simply create a list containing the
+symbol `*table*':
+
+ (define (make-table)
+ (list '*table*))
+
+Two-dimensional tables
+......................
+
+In a two-dimensional table, each value is indexed by two keys. We can
+construct such a table as a one-dimensional table in which each key
+identifies a subtable. *Note Figure 3-23:: shows the box-and-pointer
+diagram for the table
+
+ math:
+ +: 43
+ -: 45
+ *: 42
+ letters:
+ a: 97
+ b: 98
+
+which has two subtables. (The subtables don't need a special header
+symbol, since the key that identifies the subtable serves this purpose.)
+
+ *Figure 3.23:* A two-dimensional table.
+
+ table
+ |
+ V
+ +---+---+ +---+---+ +---+---+
+ | * | *-+-->| * | *-+-->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ V | V
+ +-------+ | +---+---+ +---+---+ +---+---+
+ |*table*| | | * | *-+-->| * | *-+-->| * | / |
+ +-------+ | +-|-+---+ +-|-+---+ +-|-+---+
+ | V V V
+ | +-------+ +---+---+ +---+---+
+ | |letters| | * | * | | * | * |
+ | +-------+ +-|-+-|-+ +-|-+-|-+
+ | V V V V
+ | +---+ +---+ +---+ +---+
+ | | a | | 97| | b | | 98|
+ | +---+ +---+ +---+ +---+
+ V
+ +---+---+ +---+---+ +---+---+ +---+---+
+ | * | *-+-->| * | *-+-->| * | *-+-->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+ +-|-+---+
+ V V V V
+ +------+ +---+---+ +---+---+ +---+---+
+ | math | | * | * | | * | * | | * | * |
+ +------+ +-|-+-|-+ +-|-+-|-+ +-|-+-|-+
+ V V V V V V
+ +---+ +---+ +---+ +---+ +---+ +---+
+ | + | | 43| | - | | 45| | * | | 42|
+ +---+ +---+ +---+ +---+ +---+ +---+
+
+ When we look up an item, we use the first key to identify the
+correct subtable. Then we use the second key to identify the record
+within the subtable.
+
+ (define (lookup key-1 key-2 table)
+ (let ((subtable (assoc key-1 (cdr table))))
+ (if subtable
+ (let ((record (assoc key-2 (cdr subtable))))
+ (if record
+ (cdr record)
+ false))
+ false)))
+
+ To insert a new item under a pair of keys, we use `assoc' to see if
+there is a subtable stored under the first key. If not, we build a new
+subtable containing the single record (`key-2', `value') and insert it
+into the table under the first key. If a subtable already exists for
+the first key, we insert the new record into this subtable, using the
+insertion method for one-dimensional tables described above:
+
+ (define (insert! key-1 key-2 value table)
+ (let ((subtable (assoc key-1 (cdr table))))
+ (if subtable
+ (let ((record (assoc key-2 (cdr subtable))))
+ (if record
+ (set-cdr! record value)
+ (set-cdr! subtable
+ (cons (cons key-2 value)
+ (cdr subtable)))))
+ (set-cdr! table
+ (cons (list key-1
+ (cons key-2 value))
+ (cdr table)))))
+ 'ok)
+
+Creating local tables
+.....................
+
+The `lookup' and `insert!' operations defined above take the table as
+an argument. This enables us to use programs that access more than one
+table. Another way to deal with multiple tables is to have separate
+`lookup' and `insert!' procedures for each table. We can do this by
+representing a table procedurally, as an object that maintains an
+internal table as part of its local state. When sent an appropriate
+message, this "table object" supplies the procedure with which to
+operate on the internal table. Here is a generator for two-dimensional
+tables represented in this fashion:
+
+ (define (make-table)
+ (let ((local-table (list '*table*)))
+ (define (lookup key-1 key-2)
+ (let ((subtable (assoc key-1 (cdr local-table))))
+ (if subtable
+ (let ((record (assoc key-2 (cdr subtable))))
+ (if record
+ (cdr record)
+ false))
+ false)))
+ (define (insert! key-1 key-2 value)
+ (let ((subtable (assoc key-1 (cdr local-table))))
+ (if subtable
+ (let ((record (assoc key-2 (cdr subtable))))
+ (if record
+ (set-cdr! record value)
+ (set-cdr! subtable
+ (cons (cons key-2 value)
+ (cdr subtable)))))
+ (set-cdr! local-table
+ (cons (list key-1
+ (cons key-2 value))
+ (cdr local-table)))))
+ 'ok)
+ (define (dispatch m)
+ (cond ((eq? m 'lookup-proc) lookup)
+ ((eq? m 'insert-proc!) insert!)
+ (else (error "Unknown operation -- TABLE" m))))
+ dispatch))
+
+ Using `make-table', we could implement the `get' and `put'
+operations used in section *Note 2-4-3:: for data-directed programming,
+as follows:
+
+ (define operation-table (make-table))
+ (define get (operation-table 'lookup-proc))
+ (define put (operation-table 'insert-proc!))
+
+ `Get' takes as arguments two keys, and `put' takes as arguments two
+keys and a value. Both operations access the same local table, which is
+encapsulated within the object created by the call to `make-table'.
+
+ *Exercise 3.24:* In the table implementations above, the keys are
+ tested for equality using `equal?' (called by `assoc'). This is
+ not always the appropriate test. For instance, we might have a
+ table with numeric keys in which we don't need an exact match to
+ the number we're looking up, but only a number within some
+ tolerance of it. Design a table constructor `make-table' that
+ takes as an argument a `same-key?' procedure that will be used to
+ test "equality" of keys. `Make-table' should return a `dispatch'
+ procedure that can be used to access appropriate `lookup' and
+ `insert!' procedures for a local table.
+
+ *Exercise 3.25:* Generalizing one- and two-dimensional tables,
+ show how to implement a table in which values are stored under an
+ arbitrary number of keys and different values may be stored under
+ different numbers of keys. The `lookup' and `insert!' procedures
+ should take as input a list of keys used to access the table.
+
+ *Exercise 3.26:* To search a table as implemented above, one needs
+ to scan through the list of records. This is basically the
+ unordered list representation of section *Note 2-3-3::. For large
+ tables, it may be more efficient to structure the table in a
+ different manner. Describe a table implementation where the (key,
+ value) records are organized using a binary tree, assuming that
+ keys can be ordered in some way (e.g., numerically or
+ alphabetically). (Compare *Note Exercise 2-66:: of *Note Chapter
+ 2::.)
+
+ *Exercise 3.27:* "Memoization" (also called "tabulation") is a
+ technique that enables a procedure to record, in a local table,
+ values that have previously been computed. This technique can
+ make a vast difference in the performance of a program. A memoized
+ procedure maintains a table in which values of previous calls are
+ stored using as keys the arguments that produced the values. When
+ the memoized procedure is asked to compute a value, it first
+ checks the table to see if the value is already there and, if so,
+ just returns that value. Otherwise, it computes the new value in
+ the ordinary way and stores this in the table. As an example of
+ memoization, recall from section *Note 1-2-2:: the exponential
+ process for computing Fibonacci numbers:
+
+ (define (fib n)
+ (cond ((= n 0) 0)
+ ((= n 1) 1)
+ (else (+ (fib (- n 1))
+ (fib (- n 2))))))
+
+ The memoized version of the same procedure is
+
+ (define memo-fib
+ (memoize (lambda (n)
+ (cond ((= n 0) 0)
+ ((= n 1) 1)
+ (else (+ (memo-fib (- n 1))
+ (memo-fib (- n 2))))))))
+
+ where the memoizer is defined as
+
+ (define (memoize f)
+ (let ((table (make-table)))
+ (lambda (x)
+ (let ((previously-computed-result (lookup x table)))
+ (or previously-computed-result
+ (let ((result (f x)))
+ (insert! x result table)
+ result))))))
+
+ Draw an environment diagram to analyze the computation of
+ `(memo-fib 3)'. Explain why `memo-fib' computes the nth Fibonacci
+ number in a number of steps proportional to n. Would the scheme
+ still work if we had simply defined `memo-fib' to be `(memoize
+ fib)'?
+
+ ---------- Footnotes ----------
+
+ (1) Because `assoc' uses `equal?', it can recognize keys that are
+symbols, numbers, or list structure.
+
+ (2) Thus, the first backbone pair is the object that represents the
+table "itself"; that is, a pointer to the table is a pointer to this
+pair. This same backbone pair always starts the table. If we did not
+arrange things in this way, `insert!' would have to return a new value
+for the start of the table when it added a new record.
+
+\1f
+File: sicp.info, Node: 3-3-4, Next: 3-3-5, Prev: 3-3-3, Up: 3-3
+
+3.3.4 A Simulator for Digital Circuits
+--------------------------------------
+
+Designing complex digital systems, such as computers, is an important
+engineering activity. Digital systems are constructed by
+interconnecting simple elements. Although the behavior of these
+individual elements is simple, networks of them can have very complex
+behavior. Computer simulation of proposed circuit designs is an
+important tool used by digital systems engineers. In this section we
+design a system for performing digital logic simulations. This system
+typifies a kind of program called an "event-driven simulation", in
+which actions ("events") trigger further events that happen at a later
+time, which in turn trigger more events, and so so.
+
+ Our computational model of a circuit will be composed of objects that
+correspond to the elementary components from which the circuit is
+constructed. There are "wires", which carry "digital signals". A
+digital signal may at any moment have only one of two possible values,
+0 and 1. There are also various types of digital "function boxes",
+which connect wires carrying input signals to other output wires. Such
+boxes produce output signals computed from their input signals. The
+output signal is delayed by a time that depends on the type of the
+function box. For example, an "inverter" is a primitive function box
+that inverts its input. If the input signal to an inverter changes to
+0, then one inverter-delay later the inverter will change its output
+signal to 1. If the input signal to an inverter changes to 1, then one
+inverter-delay later the inverter will change its output signal to 0.
+We draw an inverter symbolically as in *Note Figure 3-24::. An "and-gate",
+also shown in *Note Figure 3-24::, is a primitive function box with two
+inputs and one output. It drives its output signal to a value that is
+the "logical and" of the inputs. That is, if both of its input signals
+become 1, then one and-gate-delay time later the and-gate will force
+its output signal to be 1; otherwise the output will be 0. An "or-gate"
+is a similar two-input primitive function box that drives its output
+signal to a value that is the "logical or" of the inputs. That is, the
+output will become 1 if at least one of the input signals is 1;
+otherwise the output will become 0.
+
+ *Figure 3.24:* Primitive functions in the digital logic simulator.
+
+ __ ___
+ |\ --| \ --\ \
+ --| >o-- | )-- ) >--
+ |/ --|__/ --/__/
+
+ Inverter And-gate Or-gate
+
+ We can connect primitive functions together to construct more complex
+functions. To accomplish this we wire the outputs of some function
+boxes to the inputs of other function boxes. For example, the "half-adder"
+circuit shown in *Note Figure 3-25:: consists of an or-gate, two
+and-gates, and an inverter. It takes two input signals, A and B, and
+has two output signals, S and C. S will become 1 whenever precisely
+one of A and B is 1, and C will become 1 whenever A and B are both 1.
+We can see from the figure that, because of the delays involved, the
+outputs may be generated at different times. Many of the difficulties
+in the design of digital circuits arise from this fact.
+
+ *Figure 3.25:* A half-adder circuit.
+
+ +--------------------------------------+
+ | ____ |
+ A --------*---\ \ D ___ |
+ | | > >---------------| \ |
+ | +--|---/___/ | )----- S
+ | | | |\ E +--|___/ |
+ | | | +--| >o---+ |
+ | | | ___ | |/ |
+ | | +---| \ | |
+ | | | )--*----------------------- C
+ B -----*------|___/ |
+ | |
+ +--------------------------------------+
+
+ We will now build a program for modeling the digital logic circuits
+we wish to study. The program will construct computational objects
+modeling the wires, which will "hold" the signals. Function boxes will
+be modeled by procedures that enforce the correct relationships among
+the signals.
+
+ One basic element of our simulation will be a procedure `make-wire',
+which constructs wires. For example, we can construct six wires as
+follows:
+
+ (define a (make-wire))
+ (define b (make-wire))
+ (define c (make-wire))
+
+ (define d (make-wire))
+ (define e (make-wire))
+ (define s (make-wire))
+
+ We attach a function box to a set of wires by calling a procedure
+that constructs that kind of box. The arguments to the constructor
+procedure are the wires to be attached to the box. For example, given
+that we can construct and-gates, or-gates, and inverters, we can wire
+together the half-adder shown in *Note Figure 3-25:::
+
+ (or-gate a b d)
+ ok
+
+ (and-gate a b c)
+ ok
+
+ (inverter c e)
+ ok
+
+ (and-gate d e s)
+ ok
+
+ Better yet, we can explicitly name this operation by defining a
+procedure `half-adder' that constructs this circuit, given the four
+external wires to be attached to the half-adder:
+
+ (define (half-adder a b s c)
+ (let ((d (make-wire)) (e (make-wire)))
+ (or-gate a b d)
+ (and-gate a b c)
+ (inverter c e)
+ (and-gate d e s)
+ 'ok))
+
+ The advantage of making this definition is that we can use
+`half-adder' itself as a building block in creating more complex
+circuits. *Note Figure 3-26::, for example, shows a "full-adder"
+composed of two half-adders and an or-gate.(1) We can construct a
+full-adder as follows:
+
+ (define (full-adder a b c-in sum c-out)
+ (let ((s (make-wire))
+ (c1 (make-wire))
+ (c2 (make-wire)))
+ (half-adder b c-in s c1)
+ (half-adder a s sum c2)
+ (or-gate c1 c2 c-out)
+ 'ok))
+
+ Having defined `full-adder' as a procedure, we can now use it as a
+building block for creating still more complex circuits. (For example,
+see *Note Exercise 3-30::.)
+
+ *Figure 3.26:* A full-adder circuit.
+
+ +----------------------------------+
+ | +-------+ |
+ A -----------------+ half +-------------- SUM
+ | +-------+ | adder | ____ |
+ B -----+ half +---+ +---\ \ |
+ | | adder | +-------+ >or >----- Cout
+ C -----+ +---------------/___/ |
+ | +-------+ |
+ +----------------------------------+
+
+ In essence, our simulator provides us with the tools to construct a
+language of circuits. If we adopt the general perspective on languages
+with which we approached the study of Lisp in section *Note 1-1::, we
+can say that the primitive function boxes form the primitive elements
+of the language, that wiring boxes together provides a means of
+combination, and that specifying wiring patterns as procedures serves
+as a means of abstraction.
+
+Primitive function boxes
+........................
+
+The primitive function boxes implement the "forces" by which a change
+in the signal on one wire influences the signals on other wires. To
+build function boxes, we use the following operations on wires:
+
+ * (get-signal <WIRE>)
+
+ returns the current value of the signal on the wire.
+
+ * (set-signal! <WIRE> <NEW VALUE>)
+
+ changes the value of the signal on the wire to the new value.
+
+ * (add-action! <WIRE> <PROCEDURE OF NO ARGUMENTS>)
+
+ asserts that the designated procedure should be run whenever the
+ signal on the wire changes value. Such procedures are the
+ vehicles by which changes in the signal value on the wire are
+ communicated to other wires.
+
+
+ In addition, we will make use of a procedure `after-delay' that
+takes a time delay and a procedure to be run and executes the given
+procedure after the given delay.
+
+ Using these procedures, we can define the primitive digital logic
+functions. To connect an input to an output through an inverter, we
+use `add-action!' to associate with the input wire a procedure that
+will be run whenever the signal on the input wire changes value. The
+procedure computes the `logical-not' of the input signal, and then,
+after one `inverter-delay', sets the output signal to be this new value:
+
+ (define (inverter input output)
+ (define (invert-input)
+ (let ((new-value (logical-not (get-signal input))))
+ (after-delay inverter-delay
+ (lambda ()
+ (set-signal! output new-value)))))
+ (add-action! input invert-input)
+ 'ok)
+
+ (define (logical-not s)
+ (cond ((= s 0) 1)
+ ((= s 1) 0)
+ (else (error "Invalid signal" s))))
+
+ An and-gate is a little more complex. The action procedure must be
+run if either of the inputs to the gate changes. It computes the
+`logical-and' (using a procedure analogous to `logical-not') of the
+values of the signals on the input wires and sets up a change to the
+new value to occur on the output wire after one `and-gate-delay'.
+
+ (define (and-gate a1 a2 output)
+ (define (and-action-procedure)
+ (let ((new-value
+ (logical-and (get-signal a1) (get-signal a2))))
+ (after-delay and-gate-delay
+ (lambda ()
+ (set-signal! output new-value)))))
+ (add-action! a1 and-action-procedure)
+ (add-action! a2 and-action-procedure)
+ 'ok)
+
+ *Exercise 3.28:* Define an or-gate as a primitive function box.
+ Your `or-gate' constructor should be similar to `and-gate'.
+
+ *Exercise 3.29:* Another way to construct an or-gate is as a
+ compound digital logic device, built from and-gates and inverters.
+ Define a procedure `or-gate' that accomplishes this. What is the
+ delay time of the or-gate in terms of `and-gate-delay' and
+ `inverter-delay'?
+
+ *Exercise 3.30:* *Note Figure 3-27:: shows a "ripple-carry adder"
+ formed by stringing together n full-adders. This is the simplest
+ form of parallel adder for adding two n-bit binary numbers. The
+ inputs A_1, A_2, A_3, ..., A_n and B_1, B_2, B_3, ..., B_n are the
+ two binary numbers to be added (each A_k and B_k is a 0 or a 1).
+ The circuit generates S_1, S_2, S_3, ..., S_n, the n bits of the
+ sum, and C, the carry from the addition. Write a procedure
+ `ripple-carry-adder' that generates this circuit. The procedure
+ should take as arguments three lists of n wires each--the A_k, the
+ B_k, and the S_k--and also another wire C. The major drawback of
+ the ripple-carry adder is the need to wait for the carry signals
+ to propagate. What is the delay needed to obtain the complete
+ output from an n-bit ripple-carry adder, expressed in terms of the
+ delays for and-gates, or-gates, and inverters?
+
+ *Figure 3.27:* A ripple-carry adder for n-bit numbers.
+
+ : : :
+ : A_1 B_1 C_1 A_2 B_2 C_2 A_3 B_3 C_3: : A_n B_n C_n=0
+ : | | +---+ | | +---+ | | +----- : | | +-
+ | | | | | | | | | | | | : : | | |
+ : ++---+---++ | ++---+---++ | ++---+---++ : : ++---+---++
+ : | FA | | | FA | | | FA | : : | FA |
+ : +--+---+--+ | +--+---+--+ | +--+---+--+ : : +--+---+--+
+ : | | | | | | | | : : | |
+ C ------+ | +-----+ | +-----+ | : ------+ |
+ : | C_1 | C_2 | : :C_(n-1) |
+ : | | | : : |
+ S_1 S_2 S_3 S_n
+
+Representing wires
+..................
+
+A wire in our simulation will be a computational object with two local
+state variables: a `signal-value' (initially taken to be 0) and a
+collection of `action-procedures' to be run when the signal changes
+value. We implement the wire, using message-passing style, as a
+collection of local procedures together with a `dispatch' procedure
+that selects the appropriate local operation, just as we did with the
+simple bank-account object in section *Note 3-1-1:::
+
+ (define (make-wire)
+ (let ((signal-value 0) (action-procedures '()))
+ (define (set-my-signal! new-value)
+ (if (not (= signal-value new-value))
+ (begin (set! signal-value new-value)
+ (call-each action-procedures))
+ 'done))
+
+ (define (accept-action-procedure! proc)
+ (set! action-procedures (cons proc action-procedures))
+ (proc))
+
+ (define (dispatch m)
+ (cond ((eq? m 'get-signal) signal-value)
+ ((eq? m 'set-signal!) set-my-signal!)
+ ((eq? m 'add-action!) accept-action-procedure!)
+ (else (error "Unknown operation -- WIRE" m))))
+ dispatch))
+
+ The local procedure `set-my-signal!' tests whether the new signal
+value changes the signal on the wire. If so, it runs each of the
+action procedures, using the following procedure `call-each', which
+calls each of the items in a list of no-argument procedures:
+
+ (define (call-each procedures)
+ (if (null? procedures)
+ 'done
+ (begin
+ ((car procedures))
+ (call-each (cdr procedures)))))
+
+ The local procedure `accept-action-procedure!' adds the given
+procedure to the list of procedures to be run, and then runs the new
+procedure once. (See *Note Exercise 3-31::.)
+
+ With the local `dispatch' procedure set up as specified, we can
+provide the following procedures to access the local operations on
+wires:(2)
+
+ (define (get-signal wire)
+ (wire 'get-signal))
+
+ (define (set-signal! wire new-value)
+ ((wire 'set-signal!) new-value))
+
+ (define (add-action! wire action-procedure)
+ ((wire 'add-action!) action-procedure))
+
+ Wires, which have time-varying signals and may be incrementally
+attached to devices, are typical of mutable objects. We have modeled
+them as procedures with local state variables that are modified by
+assignment. When a new wire is created, a new set of state variables
+is allocated (by the `let' expression in `make-wire') and a new
+`dispatch' procedure is constructed and returned, capturing the
+environment with the new state variables.
+
+ The wires are shared among the various devices that have been
+connected to them. Thus, a change made by an interaction with one
+device will affect all the other devices attached to the wire. The
+wire communicates the change to its neighbors by calling the action
+procedures provided to it when the connections were established.
+
+The agenda
+..........
+
+The only thing needed to complete the simulator is `after-delay'. The
+idea here is that we maintain a data structure, called an "agenda",
+that contains a schedule of things to do. The following operations are
+defined for agendas:
+
+ * `(make-agenda)' returns a new empty agenda.
+
+ * `(empty-agenda? <AGENDA>)' is true if the specified agenda is
+ empty.
+
+ * `(first-agenda-item <AGENDA>)' returns the first item on the
+ agenda.
+
+ * `(remove-first-agenda-item! <AGENDA>)' modifies the agenda by
+ removing the first item.
+
+ * `(add-to-agenda! <TIME> <ACTION> <AGENDA>)' modifies the agenda by
+ adding the given action procedure to be run at the specified time.
+
+ * `(current-time <AGENDA>)' returns the current simulation time.
+
+
+ The particular agenda that we use is denoted by `the-agenda'. The
+procedure `after-delay' adds new elements to `the-agenda':
+
+ (define (after-delay delay action)
+ (add-to-agenda! (+ delay (current-time the-agenda))
+ action
+ the-agenda))
+
+ The simulation is driven by the procedure `propagate', which
+operates on `the-agenda', executing each procedure on the agenda in
+sequence. In general, as the simulation runs, new items will be added
+to the agenda, and `propagate' will continue the simulation as long as
+there are items on the agenda:
+
+ (define (propagate)
+ (if (empty-agenda? the-agenda)
+ 'done
+ (let ((first-item (first-agenda-item the-agenda)))
+ (first-item)
+ (remove-first-agenda-item! the-agenda)
+ (propagate))))
+
+A sample simulation
+...................
+
+The following procedure, which places a "probe" on a wire, shows the
+simulator in action. The probe tells the wire that, whenever its signal
+changes value, it should print the new signal value, together with the
+current time and a name that identifies the wire:
+
+ (define (probe name wire)
+ (add-action! wire
+ (lambda ()
+ (newline)
+ (display name)
+ (display " ")
+ (display (current-time the-agenda))
+ (display " New-value = ")
+ (display (get-signal wire)))))
+
+ We begin by initializing the agenda and specifying delays for the
+primitive function boxes:
+
+ (define the-agenda (make-agenda))
+ (define inverter-delay 2)
+ (define and-gate-delay 3)
+ (define or-gate-delay 5)
+
+ Now we define four wires, placing probes on two of them:
+
+ (define input-1 (make-wire))
+ (define input-2 (make-wire))
+ (define sum (make-wire))
+ (define carry (make-wire))
+
+ (probe 'sum sum)
+ sum 0 New-value = 0
+
+ (probe 'carry carry)
+ carry 0 New-value = 0
+
+ Next we connect the wires in a half-adder circuit (as in *Note
+Figure 3-25::), set the signal on `input-1' to 1, and run the
+simulation:
+
+ (half-adder input-1 input-2 sum carry)
+ ok
+
+ (set-signal! input-1 1)
+ done
+
+ (propagate)
+ sum 8 New-value = 1
+ done
+
+ The `sum' signal changes to 1 at time 8. We are now eight time
+units from the beginning of the simulation. At this point, we can set
+the signal on `input-2' to 1 and allow the values to propagate:
+
+ (set-signal! input-2 1)
+ done
+
+ (propagate)
+ carry 11 New-value = 1
+ sum 16 New-value = 0
+ done
+
+ The `carry' changes to 1 at time 11 and the `sum' changes to 0 at
+time 16.
+
+ *Exercise 3.31:* The internal procedure `accept-action-procedure!'
+ defined in `make-wire' specifies that when a new action procedure
+ is added to a wire, the procedure is immediately run. Explain why
+ this initialization is necessary. In particular, trace through the
+ half-adder example in the paragraphs above and say how the
+ system's response would differ if we had defined
+ `accept-action-procedure!' as
+
+ (define (accept-action-procedure! proc)
+ (set! action-procedures (cons proc action-procedures)))
+
+Implementing the agenda
+.......................
+
+Finally, we give details of the agenda data structure, which holds the
+procedures that are scheduled for future execution.
+
+ The agenda is made up of "time segments". Each time segment is a
+pair consisting of a number (the time) and a queue (see *Note Exercise
+3-32::) that holds the procedures that are scheduled to be run during
+that time segment.
+
+ (define (make-time-segment time queue)
+ (cons time queue))
+
+ (define (segment-time s) (car s))
+
+ (define (segment-queue s) (cdr s))
+
+ We will operate on the time-segment queues using the queue
+operations described in section *Note 3-3-2::.
+
+ The agenda itself is a one-dimensional table of time segments. It
+differs from the tables described in section *Note 3-3-3:: in that the
+segments will be sorted in order of increasing time. In addition, we
+store the "current time" (i.e., the time of the last action that was
+processed) at the head of the agenda. A newly constructed agenda has
+no time segments and has a current time of 0:(3)
+
+ (define (make-agenda) (list 0))
+
+ (define (current-time agenda) (car agenda))
+
+ (define (set-current-time! agenda time)
+ (set-car! agenda time))
+
+ (define (segments agenda) (cdr agenda))
+
+ (define (set-segments! agenda segments)
+ (set-cdr! agenda segments))
+
+ (define (first-segment agenda) (car (segments agenda)))
+
+ (define (rest-segments agenda) (cdr (segments agenda)))
+
+ An agenda is empty if it has no time segments:
+
+ (define (empty-agenda? agenda)
+ (null? (segments agenda)))
+
+ To add an action to an agenda, we first check if the agenda is
+empty. If so, we create a time segment for the action and install this
+in the agenda. Otherwise, we scan the agenda, examining the time of
+each segment. If we find a segment for our appointed time, we add the
+action to the associated queue. If we reach a time later than the one
+to which we are appointed, we insert a new time segment into the agenda
+just before it. If we reach the end of the agenda, we must create a
+new time segment at the end.
+
+ (define (add-to-agenda! time action agenda)
+ (define (belongs-before? segments)
+ (or (null? segments)
+ (< time (segment-time (car segments)))))
+ (define (make-new-time-segment time action)
+ (let ((q (make-queue)))
+ (insert-queue! q action)
+ (make-time-segment time q)))
+ (define (add-to-segments! segments)
+ (if (= (segment-time (car segments)) time)
+ (insert-queue! (segment-queue (car segments))
+ action)
+ (let ((rest (cdr segments)))
+ (if (belongs-before? rest)
+ (set-cdr!
+ segments
+ (cons (make-new-time-segment time action)
+ (cdr segments)))
+ (add-to-segments! rest)))))
+ (let ((segments (segments agenda)))
+ (if (belongs-before? segments)
+ (set-segments!
+ agenda
+ (cons (make-new-time-segment time action)
+ segments))
+ (add-to-segments! segments))))
+
+ The procedure that removes the first item from the agenda deletes
+the item at the front of the queue in the first time segment. If this
+deletion makes the time segment empty, we remove it from the list of
+segments:(4)
+
+ (define (remove-first-agenda-item! agenda)
+ (let ((q (segment-queue (first-segment agenda))))
+ (delete-queue! q)
+ (if (empty-queue? q)
+ (set-segments! agenda (rest-segments agenda)))))
+
+ The first agenda item is found at the head of the queue in the first
+time segment. Whenever we extract an item, we also update the current
+time:(5)
+
+ (define (first-agenda-item agenda)
+ (if (empty-agenda? agenda)
+ (error "Agenda is empty -- FIRST-AGENDA-ITEM")
+ (let ((first-seg (first-segment agenda)))
+ (set-current-time! agenda (segment-time first-seg))
+ (front-queue (segment-queue first-seg)))))
+
+ *Exercise 3.32:* The procedures to be run during each time segment
+ of the agenda are kept in a queue. Thus, the procedures for each
+ segment are called in the order in which they were added to the
+ agenda (first in, first out). Explain why this order must be
+ used. In particular, trace the behavior of an and-gate whose
+ inputs change from 0,1 to 1,0 in the same segment and say how the
+ behavior would differ if we stored a segment's procedures in an
+ ordinary list, adding and removing procedures only at the front
+ (last in, first out).
+
+ ---------- Footnotes ----------
+
+ (1) A full-adder is a basic circuit element used in adding two
+binary numbers. Here A and B are the bits at corresponding positions in
+the two numbers to be added, and C_(in) is the carry bit from the
+addition one place to the right. The circuit generates SUM, which is
+the sum bit in the corresponding position, and C_(out), which is the
+carry bit to be propagated to the left.
+
+ (2) [Footnote 27] These procedures are simply syntactic sugar that
+allow us to use ordinary procedural syntax to access the local
+procedures of objects. It is striking that we can interchange the role
+of "procedures" and "data" in such a simple way. For example, if we
+write `(wire 'get-signal)' we think of `wire' as a procedure that is
+called with the message `get-signal' as input. Alternatively, writing
+`(get-signal wire)' encourages us to think of `wire' as a data object
+that is the input to a procedure `get-signal'. The truth of the matter
+is that, in a language in which we can deal with procedures as objects,
+there is no fundamental difference between "procedures" and "data," and
+we can choose our syntactic sugar to allow us to program in whatever
+style we choose.
+
+ (3) The agenda is a headed list, like the tables in section *Note
+3-3-3::, but since the list is headed by the time, we do not need an
+additional dummy header (such as the `*table*' symbol used with tables).
+
+ (4) Observe that the `if' expression in this procedure has no
+<ALTERNATIVE> expression. Such a "one-armed `if' statement" is used to
+decide whether to do something, rather than to select between two
+expressions. An `if' expression returns an unspecified value if the
+predicate is false and there is no <ALTERNATIVE>.
+
+ (5) In this way, the current time will always be the time of the
+action most recently processed. Storing this time at the head of the
+agenda ensures that it will still be available even if the associated
+time segment has been deleted.
+
+\1f
+File: sicp.info, Node: 3-3-5, Prev: 3-3-4, Up: 3-3
+
+3.3.5 Propagation of Constraints
+--------------------------------
+
+Computer programs are traditionally organized as one-directional
+computations, which perform operations on prespecified arguments to
+produce desired outputs. On the other hand, we often model systems in
+terms of relations among quantities. For example, a mathematical model
+of a mechanical structure might include the information that the
+deflection d of a metal rod is related to the force f on the rod, the
+length L of the rod, the cross-sectional area A, and the elastic
+modulus E via the equation
+
+ dAE = FL
+
+ Such an equation is not one-directional. Given any four of the
+quantities, we can use it to compute the fifth. Yet translating the
+equation into a traditional computer language would force us to choose
+one of the quantities to be computed in terms of the other four. Thus,
+a procedure for computing the area A could not be used to compute the
+deflection d, even though the computations of A and d arise from the
+same equation.(1)
+
+ In this section, we sketch the design of a language that enables us
+to work in terms of relations themselves. The primitive elements of
+the language are "primitive constraints", which state that certain
+relations hold between quantities. For example, `(adder a b c)'
+specifies that the quantities a, b, and c must be related by the
+equation a + b = c, `(multiplier x y z)' expresses the constraint xy =
+z, and `(constant 3.14 x)' says that the value of x must be 3.14.
+
+ Our language provides a means of combining primitive constraints in
+order to express more complex relations. We combine constraints by
+constructing "constraint networks", in which constraints are joined by "connectors".
+A connector is an object that "holds" a value that may participate in
+one or more constraints. For example, we know that the relationship
+between Fahrenheit and Celsius temperatures is
+
+ 9C = 5(F - 32)
+
+ Such a constraint can be thought of as a network consisting of
+primitive adder, multiplier, and constant constraints (*Note Figure
+3-28::). In the figure, we see on the left a multiplier box with three
+terminals, labeled m1, m2, and p. These connect the multiplier to the
+rest of the network as follows: The m1 terminal is linked to a
+connector C, which will hold the Celsius temperature. The m2 terminal
+is linked to a connector w, which is also linked to a constant box that
+holds 9. The p terminal, which the multiplier box constrains to be the
+product of m1 and m2, is linked to the p terminal of another multiplier
+box, whose m2 is connected to a constant 5 and whose m1 is connected to
+one of the terms in a sum.
+
+ *Figure 3.28:* The relation 9C = 5(F - 32) expressed as a
+ constraint network.
+
+ +---------+ +---------+ v +---------+
+ C -----+ m1 | u | m1 +-------+ a1 |
+ | * p +-----+ p * | | + s +---- F
+ +--+ m2 | | m2 +--+ +--+ a2 |
+ | +---------+ +---------+ | | +---------+
+ w | x| |y
+ | +-----+ +-----+ | | +-----+
+ +----+ 9 | | 5 +-----+ +-----+ 32 |
+ +-----+ +-----+ +-----+
+
+ Computation by such a network proceeds as follows: When a connector
+is given a value (by the user or by a constraint box to which it is
+linked), it awakens all of its associated constraints (except for the
+constraint that just awakened it) to inform them that it has a value.
+Each awakened constraint box then polls its connectors to see if there
+is enough information to determine a value for a connector. If so, the
+box sets that connector, which then awakens all of its associated
+constraints, and so on. For instance, in conversion between Celsius
+and Fahrenheit, w, x, and y are immediately set by the constant boxes
+to 9, 5, and 32, respectively. The connectors awaken the multipliers
+and the adder, which determine that there is not enough information to
+proceed. If the user (or some other part of the network) sets C to a
+value (say 25), the leftmost multiplier will be awakened, and it will
+set u to 25*9 = 225. Then u awakens the second multiplier, which sets
+v to 45, and v awakens the adder, which sets f to 77.
+
+Using the constraint system
+...........................
+
+To use the constraint system to carry out the temperature computation
+outlined above, we first create two connectors, `C' and `F', by calling
+the constructor `make-connector', and link `C' and `F' in an
+appropriate network:
+
+ (define C (make-connector))
+ (define F (make-connector))
+ (celsius-fahrenheit-converter C F)
+ ok
+
+ The procedure that creates the network is defined as follows:
+
+ (define (celsius-fahrenheit-converter c f)
+ (let ((u (make-connector))
+ (v (make-connector))
+ (w (make-connector))
+ (x (make-connector))
+ (y (make-connector)))
+ (multiplier c w u)
+ (multiplier v x u)
+ (adder v y f)
+ (constant 9 w)
+ (constant 5 x)
+ (constant 32 y)
+ 'ok))
+
+ This procedure creates the internal connectors `u', `v', `w', `x',
+and `y', and links them as shown in *Note Figure 3-28:: using the
+primitive constraint constructors `adder', `multiplier', and
+`constant'. Just as with the digital-circuit simulator of section
+*Note 3-3-4::, expressing these combinations of primitive elements in
+terms of procedures automatically provides our language with a means of
+abstraction for compound objects.
+
+ To watch the network in action, we can place probes on the
+connectors `C' and `F', using a `probe' procedure similar to the one we
+used to monitor wires in section *Note 3-3-4::. Placing a probe on a
+connector will cause a message to be printed whenever the connector is
+given a value:
+
+ (probe "Celsius temp" C)
+ (probe "Fahrenheit temp" F)
+
+ Next we set the value of `C' to 25. (The third argument to
+`set-value!' tells `C' that this directive comes from the `user'.)
+
+ (set-value! C 25 'user)
+ Probe: Celsius temp = 25
+ Probe: Fahrenheit temp = 77
+ done
+
+ The probe on `C' awakens and reports the value. `C' also propagates
+its value through the network as described above. This sets `F' to 77,
+which is reported by the probe on `F'.
+
+ Now we can try to set `F' to a new value, say 212:
+
+ (set-value! F 212 'user)
+ Error! Contradiction (77 212)
+
+ The connector complains that it has sensed a contradiction: Its
+value is 77, and someone is trying to set it to 212. If we really want
+to reuse the network with new values, we can tell `C' to forget its old
+value:
+
+ (forget-value! C 'user)
+ Probe: Celsius temp = ?
+ Probe: Fahrenheit temp = ?
+ done
+
+ `C' finds that the `user', who set its value originally, is now
+retracting that value, so `C' agrees to lose its value, as shown by the
+probe, and informs the rest of the network of this fact. This
+information eventually propagates to `F', which now finds that it has
+no reason for continuing to believe that its own value is 77. Thus,
+`F' also gives up its value, as shown by the probe.
+
+ Now that `F' has no value, we are free to set it to 212:
+
+ (set-value! F 212 'user)
+ Probe: Fahrenheit temp = 212
+ Probe: Celsius temp = 100
+ done
+
+ This new value, when propagated through the network, forces `C' to
+have a value of 100, and this is registered by the probe on `C'.
+Notice that the very same network is being used to compute `C' given
+`F' and to compute `F' given `C'. This nondirectionality of
+computation is the distinguishing feature of constraint-based systems.
+
+Implementing the constraint system
+..................................
+
+The constraint system is implemented via procedural objects with local
+state, in a manner very similar to the digital-circuit simulator of
+section *Note 3-3-4::. Although the primitive objects of the
+constraint system are somewhat more complex, the overall system is
+simpler, since there is no concern about agendas and logic delays.
+
+ The basic operations on connectors are the following:
+
+ * `(has-value? <CONNECTOR>)' tells whether the connector has a value.
+
+ * `(get-value <CONNECTOR>)' returns the connector's current value.
+
+ * `(set-value! <CONNECTOR> <NEW-VALUE> <INFORMANT>)' indicates that
+ the informant is requesting the connector to set its value to the
+ new value.
+
+ * `(forget-value! <CONNECTOR> <RETRACTOR>)' tells the connector that
+ the retractor is requesting it to forget its value.
+
+ * `(connect <CONNECTOR> <NEW-CONSTRAINT>)' tells the connector to
+ participate in the new constraint.
+
+
+ The connectors communicate with the constraints by means of the
+procedures `inform-about-value', which tells the given constraint that
+the connector has a value, and `inform-about-no-value', which tells the
+constraint that the connector has lost its value.
+
+ `Adder' constructs an adder constraint among summand connectors `a1'
+and `a2' and a `sum' connector. An adder is implemented as a procedure
+with local state (the procedure `me' below):
+
+ (define (adder a1 a2 sum)
+ (define (process-new-value)
+ (cond ((and (has-value? a1) (has-value? a2))
+ (set-value! sum
+ (+ (get-value a1) (get-value a2))
+ me))
+ ((and (has-value? a1) (has-value? sum))
+ (set-value! a2
+ (- (get-value sum) (get-value a1))
+ me))
+ ((and (has-value? a2) (has-value? sum))
+ (set-value! a1
+ (- (get-value sum) (get-value a2))
+ me))))
+ (define (process-forget-value)
+ (forget-value! sum me)
+ (forget-value! a1 me)
+ (forget-value! a2 me)
+ (process-new-value))
+ (define (me request)
+ (cond ((eq? request 'I-have-a-value)
+ (process-new-value))
+ ((eq? request 'I-lost-my-value)
+ (process-forget-value))
+ (else
+ (error "Unknown request -- ADDER" request))))
+ (connect a1 me)
+ (connect a2 me)
+ (connect sum me)
+ me)
+
+ `Adder' connects the new adder to the designated connectors and
+returns it as its value. The procedure `me', which represents the
+adder, acts as a dispatch to the local procedures. The following
+"syntax interfaces" (see footnote *Note Footnote 27:: in section *Note
+3-3-4::) are used in conjunction with the dispatch:
+
+ (define (inform-about-value constraint)
+ (constraint 'I-have-a-value))
+
+ (define (inform-about-no-value constraint)
+ (constraint 'I-lost-my-value))
+
+ The adder's local procedure `process-new-value' is called when the
+adder is informed that one of its connectors has a value. The adder
+first checks to see if both `a1' and `a2' have values. If so, it tells
+`sum' to set its value to the sum of the two addends. The `informant'
+argument to `set-value!' is `me', which is the adder object itself. If
+`a1' and `a2' do not both have values, then the adder checks to see if
+perhaps `a1' and `sum' have values. If so, it sets `a2' to the
+difference of these two. Finally, if `a2' and `sum' have values, this
+gives the adder enough information to set `a1'. If the adder is told
+that one of its connectors has lost a value, it requests that all of its
+connectors now lose their values. (Only those values that were set by
+this adder are actually lost.) Then it runs `process-new-value'. The
+reason for this last step is that one or more connectors may still have
+a value (that is, a connector may have had a value that was not
+originally set by the adder), and these values may need to be
+propagated back through the adder.
+
+ A multiplier is very similar to an adder. It will set its `product'
+to 0 if either of the factors is 0, even if the other factor is not
+known.
+
+ (define (multiplier m1 m2 product)
+ (define (process-new-value)
+ (cond ((or (and (has-value? m1) (= (get-value m1) 0))
+ (and (has-value? m2) (= (get-value m2) 0)))
+ (set-value! product 0 me))
+ ((and (has-value? m1) (has-value? m2))
+ (set-value! product
+ (* (get-value m1) (get-value m2))
+ me))
+ ((and (has-value? product) (has-value? m1))
+ (set-value! m2
+ (/ (get-value product) (get-value m1))
+ me))
+ ((and (has-value? product) (has-value? m2))
+ (set-value! m1
+ (/ (get-value product) (get-value m2))
+ me))))
+ (define (process-forget-value)
+ (forget-value! product me)
+ (forget-value! m1 me)
+ (forget-value! m2 me)
+ (process-new-value))
+ (define (me request)
+ (cond ((eq? request 'I-have-a-value)
+ (process-new-value))
+ ((eq? request 'I-lost-my-value)
+ (process-forget-value))
+ (else
+ (error "Unknown request -- MULTIPLIER" request))))
+ (connect m1 me)
+ (connect m2 me)
+ (connect product me)
+ me)
+
+ A `constant' constructor simply sets the value of the designated
+connector. Any `I-have-a-value' or `I-lost-my-value' message sent to
+the constant box will produce an error.
+
+ (define (constant value connector)
+ (define (me request)
+ (error "Unknown request -- CONSTANT" request))
+ (connect connector me)
+ (set-value! connector value me)
+ me)
+
+ Finally, a probe prints a message about the setting or unsetting of
+the designated connector:
+
+ (define (probe name connector)
+ (define (print-probe value)
+ (newline)
+ (display "Probe: ")
+ (display name)
+ (display " = ")
+ (display value))
+ (define (process-new-value)
+ (print-probe (get-value connector)))
+ (define (process-forget-value)
+ (print-probe "?"))
+ (define (me request)
+ (cond ((eq? request 'I-have-a-value)
+ (process-new-value))
+ ((eq? request 'I-lost-my-value)
+ (process-forget-value))
+ (else
+ (error "Unknown request -- PROBE" request))))
+ (connect connector me)
+ me)
+
+Representing connectors
+.......................
+
+A connector is represented as a procedural object with local state
+variables `value', the current value of the connector; `informant', the
+object that set the connector's value; and `constraints', a list of the
+constraints in which the connector participates.
+
+ (define (make-connector)
+ (let ((value false) (informant false) (constraints '()))
+ (define (set-my-value newval setter)
+ (cond ((not (has-value? me))
+ (set! value newval)
+ (set! informant setter)
+ (for-each-except setter
+ inform-about-value
+ constraints))
+ ((not (= value newval))
+ (error "Contradiction" (list value newval)))
+ (else 'ignored)))
+ (define (forget-my-value retractor)
+ (if (eq? retractor informant)
+ (begin (set! informant false)
+ (for-each-except retractor
+ inform-about-no-value
+ constraints))
+ 'ignored))
+ (define (connect new-constraint)
+ (if (not (memq new-constraint constraints))
+ (set! constraints
+ (cons new-constraint constraints)))
+ (if (has-value? me)
+ (inform-about-value new-constraint))
+ 'done)
+ (define (me request)
+ (cond ((eq? request 'has-value?)
+ (if informant true false))
+ ((eq? request 'value) value)
+ ((eq? request 'set-value!) set-my-value)
+ ((eq? request 'forget) forget-my-value)
+ ((eq? request 'connect) connect)
+ (else (error "Unknown operation -- CONNECTOR"
+ request))))
+ me))
+
+ The connector's local procedure `set-my-value' is called when there
+is a request to set the connector's value. If the connector does not
+currently have a value, it will set its value and remember as
+`informant' the constraint that requested the value to be set.(2) Then
+the connector will notify all of its participating constraints except
+the constraint that requested the value to be set. This is
+accomplished using the following iterator, which applies a designated
+procedure to all items in a list except a given one:
+
+ (define (for-each-except exception procedure list)
+ (define (loop items)
+ (cond ((null? items) 'done)
+ ((eq? (car items) exception) (loop (cdr items)))
+ (else (procedure (car items))
+ (loop (cdr items)))))
+ (loop list))
+
+ If a connector is asked to forget its value, it runs the local
+procedure `forget-my-value', which first checks to make sure that the
+request is coming from the same object that set the value originally.
+If so, the connector informs its associated constraints about the loss
+of the value.
+
+ The local procedure `connect' adds the designated new constraint to
+the list of constraints if it is not already in that list. Then, if
+the connector has a value, it informs the new constraint of this fact.
+
+ The connector's procedure `me' serves as a dispatch to the other
+internal procedures and also represents the connector as an object.
+The following procedures provide a syntax interface for the dispatch:
+
+ (define (has-value? connector)
+ (connector 'has-value?))
+
+ (define (get-value connector)
+ (connector 'value))
+
+ (define (set-value! connector new-value informant)
+ ((connector 'set-value!) new-value informant))
+
+ (define (forget-value! connector retractor)
+ ((connector 'forget) retractor))
+
+ (define (connect connector new-constraint)
+ ((connector 'connect) new-constraint))
+
+ *Exercise 3.33:* Using primitive multiplier, adder, and constant
+ constraints, define a procedure `averager' that takes three
+ connectors `a', `b', and `c' as inputs and establishes the
+ constraint that the value of `c' is the average of the values of
+ `a' and `b'.
+
+ *Exercise 3.34:* Louis Reasoner wants to build a squarer, a
+ constraint device with two terminals such that the value of
+ connector `b' on the second terminal will always be the square of
+ the value `a' on the first terminal. He proposes the following
+ simple device made from a multiplier:
+
+ (define (squarer a b)
+ (multiplier a a b))
+
+ There is a serious flaw in this idea. Explain.
+
+ *Exercise 3.35:* Ben Bitdiddle tells Louis that one way to avoid
+ the trouble in *Note Exercise 3-34:: is to define a squarer as a
+ new primitive constraint. Fill in the missing portions in Ben's
+ outline for a procedure to implement such a constraint:
+
+ (define (squarer a b)
+ (define (process-new-value)
+ (if (has-value? b)
+ (if (< (get-value b) 0)
+ (error "square less than 0 -- SQUARER" (get-value b))
+ <ALTERNATIVE1>)
+ <ALTERNATIVE2>))
+ (define (process-forget-value) <BODY1>)
+ (define (me request) <BODY2>)
+ <REST OF DEFINITION>
+ me)
+
+ *Exercise 3.36:* Suppose we evaluate the following sequence of
+ expressions in the global environment:
+
+ (define a (make-connector))
+ (define b (make-connector))
+ (set-value! a 10 'user)
+
+ At some time during evaluation of the `set-value!', the following
+ expression from the connector's local procedure is evaluated:
+
+ (for-each-except setter inform-about-value constraints)
+
+ Draw an environment diagram showing the environment in which the
+ above expression is evaluated.
+
+ *Exercise 3.37:* The `celsius-fahrenheit-converter' procedure is
+ cumbersome when compared with a more expression-oriented style of
+ definition, such as
+
+ (define (celsius-fahrenheit-converter x)
+ (c+ (c* (c/ (cv 9) (cv 5))
+ x)
+ (cv 32)))
+
+ (define C (make-connector))
+ (define F (celsius-fahrenheit-converter C))
+
+ Here `c+', `c*', etc. are the "constraint" versions of the
+ arithmetic operations. For example, `c+' takes two connectors as
+ arguments and returns a connector that is related to these by an
+ adder constraint:
+
+ (define (c+ x y)
+ (let ((z (make-connector)))
+ (adder x y z)
+ z))
+
+ Define analogous procedures `c-', `c*', `c/', and `cv' (constant
+ value) that enable us to define compound constraints as in the
+ converter example above.(3)
+
+ ---------- Footnotes ----------
+
+ (1) Constraint propagation first appeared in the incredibly
+forward-looking SKETCHPAD system of Ivan Sutherland (1963). A
+beautiful constraint-propagation system based on the Smalltalk language
+was developed by Alan Borning (1977) at Xerox Palo Alto Research
+Center. Sussman, Stallman, and Steele applied constraint propagation
+to electrical circuit analysis (Sussman and Stallman 1975; Sussman and
+Steele 1980). TK!Solver (Konopasek and Jayaraman 1984) is an extensive
+modeling environment based on constraints.
+
+ (2) The `setter' might not be a constraint. In our temperature
+example, we used `user' as the `setter'.
+
+ (3) The expression-oriented format is convenient because it avoids
+the need to name the intermediate expressions in a computation. Our
+original formulation of the constraint language is cumbersome in the
+same way that many languages are cumbersome when dealing with operations
+on compound data. For example, if we wanted to compute the product (a +
+b) * (c + d), where the variables represent vectors, we could work in
+"imperative style," using procedures that set the values of designated
+vector arguments but do not themselves return vectors as values:
+
+ (v-sum a b temp1)
+ (v-sum c d temp2)
+ (v-prod temp1 temp2 answer)
+
+ Alternatively, we could deal with expressions, using procedures that
+return vectors as values, and thus avoid explicitly mentioning `temp1'
+and `temp2':
+
+ (define answer (v-prod (v-sum a b) (v-sum c d)))
+
+ Since Lisp allows us to return compound objects as values of
+procedures, we can transform our imperative-style constraint language
+into an expression-oriented style as shown in this exercise. In
+languages that are impoverished in handling compound objects, such as
+Algol, Basic, and Pascal (unless one explicitly uses Pascal pointer
+variables), one is usually stuck with the imperative style when
+manipulating compound objects. Given the advantage of the
+expression-oriented format, one might ask if there is any reason to have
+implemented the system in imperative style, as we did in this section.
+One reason is that the non-expression-oriented constraint language
+provides a handle on constraint objects (e.g., the value of the `adder'
+procedure) as well as on connector objects. This is useful if we wish
+to extend the system with new operations that communicate with
+constraints directly rather than only indirectly via operations on
+connectors. Although it is easy to implement the expression-oriented
+style in terms of the imperative implementation, it is very difficult
+to do the converse.
+
+\1f
+File: sicp.info, Node: 3-4, Next: 3-5, Prev: 3-3, Up: Chapter 3
+
+3.4 Concurrency: Time Is of the Essence
+=======================================
+
+We've seen the power of computational objects with local state as tools
+for modeling. Yet, as section *Note 3-1-3:: warned, this power
+extracts a price: the loss of referential transparency, giving rise to
+a thicket of questions about sameness and change, and the need to
+abandon the substitution model of evaluation in favor of the more
+intricate environment model.
+
+ The central issue lurking beneath the complexity of state, sameness,
+and change is that by introducing assignment we are forced to admit "time"
+into our computational models. Before we introduced assignment, all
+our programs were timeless, in the sense that any expression that has a
+value always has the same value. In contrast, recall the example of
+modeling withdrawals from a bank account and returning the resulting
+balance, introduced at the beginning of section *Note 3-1-1:::
+
+ (withdraw 25)
+ 75
+
+ (withdraw 25)
+ 50
+
+ Here successive evaluations of the same expression yield different
+values. This behavior arises from the fact that the execution of
+assignment statements (in this case, assignments to the variable
+`balance') delineates "moments in time" when values change. The result
+of evaluating an expression depends not only on the expression itself,
+but also on whether the evaluation occurs before or after these
+moments. Building models in terms of computational objects with local
+state forces us to confront time as an essential concept in programming.
+
+ We can go further in structuring computational models to match our
+perception of the physical world. Objects in the world do not change
+one at a time in sequence. Rather we perceive them as acting "concurrently"--all
+at once. So it is often natural to model systems as collections of
+computational processes that execute concurrently. Just as we can make
+our programs modular by organizing models in terms of objects with
+separate local state, it is often appropriate to divide computational
+models into parts that evolve separately and concurrently. Even if the
+programs are to be executed on a sequential computer, the practice of
+writing programs as if they were to be executed concurrently forces the
+programmer to avoid inessential timing constraints and thus makes
+programs more modular.
+
+ In addition to making programs more modular, concurrent computation
+can provide a speed advantage over sequential computation. Sequential
+computers execute only one operation at a time, so the amount of time
+it takes to perform a task is proportional to the total number of
+operations performed.(1) However, if it is possible to decompose a
+problem into pieces that are relatively independent and need to
+communicate only rarely, it may be possible to allocate pieces to
+separate computing processors, producing a speed advantage proportional
+to the number of processors available.
+
+ Unfortunately, the complexities introduced by assignment become even
+more problematic in the presence of concurrency. The fact of
+concurrent execution, either because the world operates in parallel or
+because our computers do, entails additional complexity in our
+understanding of time.
+
+* Menu:
+
+* 3-4-1:: The Nature of Time in Concurrent Systems
+* 3-4-2:: Mechanisms for Controlling Concurrency
+
+ ---------- Footnotes ----------
+
+ (1) Most real processors actually execute a few operations at a
+time, following a strategy called "pipelining". Although this
+technique greatly improves the effective utilization of the hardware,
+it is used only to speed up the execution of a sequential instruction
+stream, while retaining the behavior of the sequential program.
+
+\1f
+File: sicp.info, Node: 3-4-1, Next: 3-4-2, Prev: 3-4, Up: 3-4
+
+3.4.1 The Nature of Time in Concurrent Systems
+----------------------------------------------
+
+On the surface, time seems straightforward. It is an ordering imposed
+on events.(1) For any events A and B, either A occurs before B, A and
+B are simultaneous, or A occurs after B. For instance, returning to
+the bank account example, suppose that Peter withdraws $10 and Paul
+withdraws $25 from a joint account that initially contains $100, leaving
+$65 in the account. Depending on the order of the two withdrawals, the
+sequence of balances in the account is either $100 -> $90 -> $65 or
+$100 -> $75 -> $65. In a computer implementation of the banking
+system, this changing sequence of balances could be modeled by
+successive assignments to a variable `balance'.
+
+ In complex situations, however, such a view can be problematic.
+Suppose that Peter and Paul, and other people besides, are accessing
+the same bank account through a network of banking machines distributed
+all over the world. The actual sequence of balances in the account
+will depend critically on the detailed timing of the accesses and the
+details of the communication among the machines.
+
+ This indeterminacy in the order of events can pose serious problems
+in the design of concurrent systems. For instance, suppose that the
+withdrawals made by Peter and Paul are implemented as two separate
+processes sharing a common variable `balance', each process specified
+by the procedure given in section *Note 3-1-1:::
+
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+
+ If the two processes operate independently, then Peter might test the
+balance and attempt to withdraw a legitimate amount. However, Paul
+might withdraw some funds in between the time that Peter checks the
+balance and the time Peter completes the withdrawal, thus invalidating
+Peter's test.
+
+ Things can be worse still. Consider the expression
+
+ (set! balance (- balance amount))
+
+executed as part of each withdrawal process. This consists of three
+steps: (1) accessing the value of the `balance' variable; (2) computing
+the new balance; (3) setting `balance' to this new value. If Peter and
+Paul's withdrawals execute this statement concurrently, then the two
+withdrawals might interleave the order in which they access `balance'
+and set it to the new value.
+
+ The timing diagram in *Note Figure 3-29:: depicts an order of events
+where `balance' starts at 100, Peter withdraws 10, Paul withdraws 25,
+and yet the final value of `balance' is 75. As shown in the diagram,
+the reason for this anomaly is that Paul's assignment of 75 to
+`balance' is made under the assumption that the value of `balance' to
+be decremented is 100. That assumption, however, became invalid when
+Peter changed `balance' to 90. This is a catastrophic failure for the
+banking system, because the total amount of money in the system is not
+conserved. Before the transactions, the total amount of money was
+$100. Afterwards, Peter has $10, Paul has $25, and the bank has $75.(2)
+
+ The general phenomenon illustrated here is that several processes
+may share a common state variable. What makes this complicated is that
+more than one process may be trying to manipulate the shared state at
+the same time. For the bank account example, during each transaction,
+each customer should be able to act as if the other customers did not
+exist. When a customer changes the balance in a way that depends on
+the balance, he must be able to assume that, just before the moment of
+change, the balance is still what he thought it was.
+
+Correct behavior of concurrent programs
+.......................................
+
+The above example typifies the subtle bugs that can creep into
+concurrent programs. The root of this complexity lies in the
+assignments to variables that are shared among the different processes.
+We already know that we must be careful in writing programs that use
+`set!', because the results of a computation depend on the order in
+which the assignments occur.(3) With concurrent processes we must be
+especially careful about assignments, because we may not be able to
+control the order of the assignments made by the different processes.
+If several such changes might be made concurrently (as with two
+depositors accessing a joint account) we need some way to ensure that
+our system behaves correctly. For example, in the case of withdrawals
+from a joint bank account, we must ensure that money is conserved. To
+make concurrent programs behave correctly, we may have to place some
+restrictions on concurrent execution.
+
+ *Figure 3.29:* Timing diagram showing how interleaving the order
+ of events in two banking withdrawals can lead to an incorrect
+ final balance.
+
+ | Peter Bank Paul
+ | ____
+ | / \
+ | .--------------| $100 |-------------.
+ | | \____/ |
+ | V V
+ | .----------------------. .----------------------.
+ | | Access balance: $100 | | Access balance: $100 |
+ | `----------+-----------' `----------+-----------'
+ | V V
+ | .----------------------. .----------------------.
+ | | new value: 100-10=90 | | new value: 100-25=75 |
+ | `----------+-----------' `----------+-----------'
+ | V |
+ | .----------------------. |
+ | | set! balance to $90 | |
+ | `----------+-----------' ____ |
+ | | / \ |
+ | `------------->| $ 90 | V
+ | \____/ .----------------------.
+ | | new value: 100-25=75 |
+ | ____ `----------+-----------'
+ | / \ |
+ | | $ 90 |<------------'
+ V \____/
+ time
+
+ One possible restriction on concurrency would stipulate that no two
+operations that change any shared state variables can occur at the same
+time. This is an extremely stringent requirement. For distributed
+banking, it would require the system designer to ensure that only one
+transaction could proceed at a time. This would be both inefficient
+and overly conservative. *Note Figure 3-30:: shows Peter and Paul
+sharing a bank account, where Paul has a private account as well. The
+diagram illustrates two withdrawals from the shared account (one by
+Peter and one by Paul) and a deposit to Paul's private account.(4) The
+two withdrawals from the shared account must not be concurrent (since
+both access and update the same account), and Paul's deposit and
+withdrawal must not be concurrent (since both access and update the
+amount in Paul's wallet). But there should be no problem permitting
+Paul's deposit to his private account to proceed concurrently with
+Peter's withdrawal from the shared account.
+
+ *Figure 3.30:* Concurrent deposits and withdrawals from a joint
+ account in Bank1 and a private account in Bank2.
+
+ | Peter Bank1 Paul Bank2
+ | ____ ____ ____ ____
+ | / \ / \ / \ / \
+ | | $7 |--. .--| $100 | | $5 |--. .--| $300 |
+ | \____/ V V \____/ \____/ V V \____/
+ | +---+ +---+
+ | | W | | D |
+ | ____ ++-++ ____ ____ ++-++ ____
+ | / \ | | / \ / \ | | / \
+ | | $17 |<-' `->| $90 |--. .--| $0 |<-' `->| $305 |
+ | \____/ \____/ V V \____/ \____/
+ | +---+
+ | | W |
+ | ____ ____ ++-++ ____ ____
+ | / \ / \ | | / \ / \
+ | | $17 | | $65 |<-' `->| $25 | | $305 |
+ | \____/ \____/ \____/ \____/
+ V
+ time
+
+ A less stringent restriction on concurrency would ensure that a
+concurrent system produces the same result as if the processes had run
+sequentially in some order. There are two important aspects to this
+requirement. First, it does not require the processes to actually run
+sequentially, but only to produce results that are the same _as if_
+they had run sequentially. For the example in *Note Figure 3-30::, the
+designer of the bank account system can safely allow Paul's deposit and
+Peter's withdrawal to happen concurrently, because the net result will
+be the same as if the two operations had happened sequentially.
+Second, there may be more than one possible "correct" result produced
+by a concurrent program, because we require only that the result be the
+same as for _some_ sequential order. For example, suppose that Peter
+and Paul's joint account starts out with $100, and Peter deposits $40
+while Paul concurrently withdraws half the money in the account. Then
+sequential execution could result in the account balance being either
+$70 or $90 (see *Note Exercise 3-38::).(5)
+
+ There are still weaker requirements for correct execution of
+concurrent programs. A program for simulating diffusion (say, the flow
+of heat in an object) might consist of a large number of processes,
+each one representing a small volume of space, that update their values
+concurrently. Each process repeatedly changes its value to the average
+of its own value and its neighbors' values. This algorithm converges
+to the right answer independent of the order in which the operations
+are done; there is no need for any restrictions on concurrent use of
+the shared values.
+
+ *Exercise 3.38:* Suppose that Peter, Paul, and Mary share a joint
+ bank account that initially contains $100. Concurrently, Peter
+ deposits $10, Paul withdraws $20, and Mary withdraws half the
+ money in the account, by executing the following commands:
+
+ Peter: (set! balance (+ balance 10))
+ Paul: (set! balance (- balance 20))
+ Mary: (set! balance (- balance (/ balance 2)))
+
+ a. List all the different possible values for `balance' after
+ these three transactions have been completed, assuming that
+ the banking system forces the three processes to run
+ sequentially in some order.
+
+ b. What are some other values that could be produced if the
+ system allows the processes to be interleaved? Draw timing
+ diagrams like the one in *Note Figure 3-29:: to explain how
+ these values can occur.
+
+
+ ---------- Footnotes ----------
+
+ (1) To quote some graffiti seen on a Cambridge building wall: "Time
+is a device that was invented to keep everything from happening at
+once."
+
+ (2) An even worse failure for this system could occur if the two
+`set!' operations attempt to change the balance simultaneously, in
+which case the actual data appearing in memory might end up being a
+random combination of the information being written by the two
+processes. Most computers have interlocks on the primitive
+memory-write operations, which protect against such simultaneous
+access. Even this seemingly simple kind of protection, however, raises
+implementation challenges in the design of multiprocessing computers,
+where elaborate "cache-coherence" protocols are required to ensure that
+the various processors will maintain a consistent view of memory
+contents, despite the fact that data may be replicated ("cached") among
+the different processors to increase the speed of memory access.
+
+ (3) The factorial program in section *Note 3-1-3:: illustrates this
+for a single sequential process.
+
+ (4) The columns show the contents of Peter's wallet, the joint
+account (in Bank1), Paul's wallet, and Paul's private account (in
+Bank2), before and after each withdrawal (W) and deposit (D). Peter
+withdraws $10 from Bank1; Paul deposits $5 in Bank2, then withdraws $25
+from Bank1.
+
+ (5) [Footnote 39] A more formal way to express this idea is to say
+that concurrent programs are inherently "nondeterministic". That is,
+they are described not by single-valued functions, but by functions
+whose results are sets of possible values. In section *Note 4-3:: we
+will study a language for expressing nondeterministic computations.
+
+\1f
+File: sicp.info, Node: 3-4-2, Prev: 3-4-1, Up: 3-4
+
+3.4.2 Mechanisms for Controlling Concurrency
+--------------------------------------------
+
+We've seen that the difficulty in dealing with concurrent processes is
+rooted in the need to consider the interleaving of the order of events
+in the different processes. For example, suppose we have two
+processes, one with three ordered events (a,b,c) and one with three
+ordered events (x,y,z). If the two processes run concurrently, with no
+constraints on how their execution is interleaved, then there are 20
+different possible orderings for the events that are consistent with
+the individual orderings for the two processes:
+
+ (a,b,c,x,y,z) (a,x,b,y,c,z) (x,a,b,c,y,z) (x,a,y,z,b,c)
+ (a,b,x,c,y,z) (a,x,b,y,z,c) (x,a,b,y,c,z) (x,y,a,b,c,z)
+ (a,b,x,y,c,z) (a,x,y,b,c,z) (x,a,b,y,z,c) (x,y,a,b,z,c)
+ (a,b,x,y,z,c) (a,x,y,b,z,c) (x,a,y,b,c,z) (x,y,a,z,b,c)
+ (a,x,b,c,y,z) (a,x,y,z,b,c) (x,a,y,b,z,c) (x,y,z,a,b,c)
+
+ As programmers designing this system, we would have to consider the
+effects of each of these 20 orderings and check that each behavior is
+acceptable. Such an approach rapidly becomes unwieldy as the numbers
+of processes and events increase.
+
+ A more practical approach to the design of concurrent systems is to
+devise general mechanisms that allow us to constrain the interleaving
+of concurrent processes so that we can be sure that the program
+behavior is correct. Many mechanisms have been developed for this
+purpose. In this section, we describe one of them, the "serializer".
+
+Serializing access to shared state
+..................................
+
+Serialization implements the following idea: Processes will execute
+concurrently, but there will be certain collections of procedures that
+cannot be executed concurrently. More precisely, serialization creates
+distinguished sets of procedures such that only one execution of a
+procedure in each serialized set is permitted to happen at a time. If
+some procedure in the set is being executed, then a process that
+attempts to execute any procedure in the set will be forced to wait
+until the first execution has finished.
+
+ We can use serialization to control access to shared variables. For
+example, if we want to update a shared variable based on the previous
+value of that variable, we put the access to the previous value of the
+variable and the assignment of the new value to the variable in the
+same procedure. We then ensure that no other procedure that assigns to
+the variable can run concurrently with this procedure by serializing
+all of these procedures with the same serializer. This guarantees that
+the value of the variable cannot be changed between an access and the
+corresponding assignment.
+
+Serializers in Scheme
+.....................
+
+To make the above mechanism more concrete, suppose that we have
+extended Scheme to include a procedure called `parallel-execute':
+
+ (parallel-execute <P_1> <P_2> ... <P_K>)
+
+ Each <P> must be a procedure of no arguments. `Parallel-execute'
+creates a separate process for each <P>, which applies <P> (to no
+arguments). These processes all run concurrently.(1)
+
+ As an example of how this is used, consider
+
+ (define x 10)
+
+ (parallel-execute (lambda () (set! x (* x x)))
+ (lambda () (set! x (+ x 1))))
+
+ This creates two concurrent processes--P_1, which sets `x' to `x'
+times `x', and P_2, which increments `x'. After execution is complete,
+`x' will be left with one of five possible values, depending on the
+interleaving of the events of P_1 and P_2:
+
+ 101: P_1 sets `x' to 100 and then P_2 increments
+ `x' to 101.
+ 121: P_2 increments `x' to 11 and then P_1 sets
+ `x' to `x' times `x'.
+ 110: P_2 changes `x' from 10 to 11 between the two
+ times that P_1 accesses the value of `x' during
+ the evaluation of `(* x x)'.
+ 11: P_2 accesses `x', then P_1 sets `x' to
+ 100, then P_2 sets `x'.
+ 100: P_1 accesses `x' (twice), then P_2 sets
+ `x' to 11, then P_1 sets `x'.
+
+ We can constrain the concurrency by using serialized procedures,
+which are created by "serializers". Serializers are constructed by
+`make-serializer', whose implementation is given below. A serializer
+takes a procedure as argument and returns a serialized procedure that
+behaves like the original procedure. All calls to a given serializer
+return serialized procedures in the same set.
+
+ Thus, in contrast to the example above, executing
+
+ (define x 10)
+
+ (define s (make-serializer))
+
+ (parallel-execute (s (lambda () (set! x (* x x))))
+ (s (lambda () (set! x (+ x 1)))))
+
+can produce only two possible values for `x', 101 or 121. The other
+possibilities are eliminated, because the execution of P_1 and P_2
+cannot be interleaved.
+
+ Here is a version of the `make-account' procedure from section *Note
+3-1-1::, where the deposits and withdrawals have been serialized:
+
+ (define (make-account balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (let ((protected (make-serializer)))
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) (protected withdraw))
+ ((eq? m 'deposit) (protected deposit))
+ ((eq? m 'balance) balance)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch))
+
+ With this implementation, two processes cannot be withdrawing from or
+depositing into a single account concurrently. This eliminates the
+source of the error illustrated in *Note Figure 3-29::, where Peter
+changes the account balance between the times when Paul accesses the
+balance to compute the new value and when Paul actually performs the
+assignment. On the other hand, each account has its own serializer, so
+that deposits and withdrawals for different accounts can proceed
+concurrently.
+
+ *Exercise 3.39:* Which of the five possibilities in the parallel
+ execution shown above remain if we instead serialize execution as
+ follows:
+
+ (define x 10)
+
+ (define s (make-serializer))
+
+ (parallel-execute (lambda () (set! x ((s (lambda () (* x x))))))
+ (s (lambda () (set! x (+ x 1)))))
+
+ *Exercise 3.40:* Give all possible values of `x' that can result
+ from executing
+
+ (define x 10)
+
+ (parallel-execute (lambda () (set! x (* x x)))
+ (lambda () (set! x (* x x x))))
+
+ Which of these possibilities remain if we instead use serialized
+ procedures:
+
+ (define x 10)
+
+ (define s (make-serializer))
+
+ (parallel-execute (s (lambda () (set! x (* x x))))
+ (s (lambda () (set! x (* x x x)))))
+
+ *Exercise 3.41:* Ben Bitdiddle worries that it would be better to
+ implement the bank account as follows (where the commented line
+ has been changed):
+
+ (define (make-account balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ ;; continued on next page
+
+ (let ((protected (make-serializer)))
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) (protected withdraw))
+ ((eq? m 'deposit) (protected deposit))
+ ((eq? m 'balance)
+ ((protected (lambda () balance)))) ; serialized
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch))
+
+ because allowing unserialized access to the bank balance can
+ result in anomalous behavior. Do you agree? Is there any
+ scenario that demonstrates Ben's concern?
+
+ *Exercise 3.42:* Ben Bitdiddle suggests that it's a waste of time
+ to create a new serialized procedure in response to every
+ `withdraw' and `deposit' message. He says that `make-account'
+ could be changed so that the calls to `protected' are done outside
+ the `dispatch' procedure. That is, an account would return the
+ same serialized procedure (which was created at the same time as
+ the account) each time it is asked for a withdrawal procedure.
+
+ (define (make-account balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (let ((protected (make-serializer)))
+ (let ((protected-withdraw (protected withdraw))
+ (protected-deposit (protected deposit)))
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) protected-withdraw)
+ ((eq? m 'deposit) protected-deposit)
+ ((eq? m 'balance) balance)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch)))
+
+ Is this a safe change to make? In particular, is there any
+ difference in what concurrency is allowed by these two versions of
+ `make-account' ?
+
+Complexity of using multiple shared resources
+.............................................
+
+Serializers provide a powerful abstraction that helps isolate the
+complexities of concurrent programs so that they can be dealt with
+carefully and (hopefully) correctly. However, while using serializers
+is relatively straightforward when there is only a single shared
+resource (such as a single bank account), concurrent programming can be
+treacherously difficult when there are multiple shared resources.
+
+ To illustrate one of the difficulties that can arise, suppose we
+wish to swap the balances in two bank accounts. We access each account
+to find the balance, compute the difference between the balances,
+withdraw this difference from one account, and deposit it in the other
+account. We could implement this as follows:(2)
+
+ (define (exchange account1 account2)
+ (let ((difference (- (account1 'balance)
+ (account2 'balance))))
+ ((account1 'withdraw) difference)
+ ((account2 'deposit) difference)))
+
+ This procedure works well when only a single process is trying to do
+the exchange. Suppose, however, that Peter and Paul both have access
+to accounts a1, a2, and a3, and that Peter exchanges a1 and a2 while
+Paul concurrently exchanges a1 and a3. Even with account deposits and
+withdrawals serialized for individual accounts (as in the `make-account'
+procedure shown above in this section), `exchange' can still produce
+incorrect results. For example, Peter might compute the difference in
+the balances for a1 and a2, but then Paul might change the balance in
+a1 before Peter is able to complete the exchange.(3) For correct
+behavior, we must arrange for the `exchange' procedure to lock out any
+other concurrent accesses to the accounts during the entire time of the
+exchange.
+
+ One way we can accomplish this is by using both accounts'
+serializers to serialize the entire `exchange' procedure. To do this,
+we will arrange for access to an account's serializer. Note that we
+are deliberately breaking the modularity of the bank-account object by
+exposing the serializer. The following version of `make-account' is
+identical to the original version given in section *Note 3-1-1::,
+except that a serializer is provided to protect the balance variable,
+and the serializer is exported via message passing:
+
+ (define (make-account-and-serializer balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (let ((balance-serializer (make-serializer)))
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) withdraw)
+ ((eq? m 'deposit) deposit)
+ ((eq? m 'balance) balance)
+ ((eq? m 'serializer) balance-serializer)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch))
+
+ We can use this to do serialized deposits and withdrawals. However,
+unlike our earlier serialized account, it is now the responsibility of
+each user of bank-account objects to explicitly manage the
+serialization, for example as follows:(4)
+
+ (define (deposit account amount)
+ (let ((s (account 'serializer))
+ (d (account 'deposit)))
+ ((s d) amount)))
+
+ Exporting the serializer in this way gives us enough flexibility to
+implement a serialized exchange program. We simply serialize the
+original `exchange' procedure with the serializers for both accounts:
+
+ (define (serialized-exchange account1 account2)
+ (let ((serializer1 (account1 'serializer))
+ (serializer2 (account2 'serializer)))
+ ((serializer1 (serializer2 exchange))
+ account1
+ account2)))
+
+ *Exercise 3.43:* Suppose that the balances in three accounts start
+ out as $10, $20, and $30, and that multiple processes run,
+ exchanging the balances in the accounts. Argue that if the
+ processes are run sequentially, after any number of concurrent
+ exchanges, the account balances should be $10, $20, and $30 in
+ some order. Draw a timing diagram like the one in *Note Figure
+ 3-29:: to show how this condition can be violated if the exchanges
+ are implemented using the first version of the account-exchange
+ program in this section. On the other hand, argue that even with
+ this `exchange' program, the sum of the balances in the accounts
+ will be preserved. Draw a timing diagram to show how even this
+ condition would be violated if we did not serialize the
+ transactions on individual accounts.
+
+ *Exercise 3.44:* Consider the problem of transferring an amount
+ from one account to another. Ben Bitdiddle claims that this can
+ be accomplished with the following procedure, even if there are
+ multiple people concurrently transferring money among multiple
+ accounts, using any account mechanism that serializes deposit and
+ withdrawal transactions, for example, the version of
+ `make-account' in the text above.
+
+ (define (transfer from-account to-account amount)
+ ((from-account 'withdraw) amount)
+ ((to-account 'deposit) amount))
+
+ Louis Reasoner claims that there is a problem here, and that we
+ need to use a more sophisticated method, such as the one required
+ for dealing with the exchange problem. Is Louis right? If not,
+ what is the essential difference between the transfer problem and
+ the exchange problem? (You should assume that the balance in
+ `from-account' is at least `amount'.)
+
+ *Exercise 3.45:* Louis Reasoner thinks our bank-account system is
+ unnecessarily complex and error-prone now that deposits and
+ withdrawals aren't automatically serialized. He suggests that
+ `make-account-and-serializer' should have exported the serializer
+ (for use by such procedures as `serialized-exchange') in addition
+ to (rather than instead of) using it to serialize accounts and
+ deposits as `make-account' did. He proposes to redefine accounts
+ as follows:
+
+ (define (make-account-and-serializer balance)
+ (define (withdraw amount)
+ (if (>= balance amount)
+ (begin (set! balance (- balance amount))
+ balance)
+ "Insufficient funds"))
+ (define (deposit amount)
+ (set! balance (+ balance amount))
+ balance)
+ (let ((balance-serializer (make-serializer)))
+ (define (dispatch m)
+ (cond ((eq? m 'withdraw) (balance-serializer withdraw))
+ ((eq? m 'deposit) (balance-serializer deposit))
+ ((eq? m 'balance) balance)
+ ((eq? m 'serializer) balance-serializer)
+ (else (error "Unknown request -- MAKE-ACCOUNT"
+ m))))
+ dispatch))
+
+ Then deposits are handled as with the original `make-account':
+
+ (define (deposit account amount)
+ ((account 'deposit) amount))
+
+ Explain what is wrong with Louis's reasoning. In particular,
+ consider what happens when `serialized-exchange' is called.
+
+Implementing serializers
+........................
+
+We implement serializers in terms of a more primitive synchronization
+mechanism called a "mutex". A mutex is an object that supports two
+operations--the mutex can be "acquired", and the mutex can be "released".
+Once a mutex has been acquired, no other acquire operations on that
+mutex may proceed until the mutex is released.(5) In our
+implementation, each serializer has an associated mutex. Given a
+procedure `p', the serializer returns a procedure that acquires the
+mutex, runs `p', and then releases the mutex. This ensures that only
+one of the procedures produced by the serializer can be running at
+once, which is precisely the serialization property that we need to
+guarantee.
+
+ (define (make-serializer)
+ (let ((mutex (make-mutex)))
+ (lambda (p)
+ (define (serialized-p . args)
+ (mutex 'acquire)
+ (let ((val (apply p args)))
+ (mutex 'release)
+ val))
+ serialized-p)))
+
+ The mutex is a mutable object (here we'll use a one-element list,
+which we'll refer to as a "cell") that can hold the value true or
+false. When the value is false, the mutex is available to be acquired.
+When the value is true, the mutex is unavailable, and any process that
+attempts to acquire the mutex must wait.
+
+ Our mutex constructor `make-mutex' begins by initializing the cell
+contents to false. To acquire the mutex, we test the cell. If the
+mutex is available, we set the cell contents to true and proceed.
+Otherwise, we wait in a loop, attempting to acquire over and over
+again, until we find that the mutex is available.(6) To release the
+mutex, we set the cell contents to false.
+
+ (define (make-mutex)
+ (let ((cell (list false)))
+ (define (the-mutex m)
+ (cond ((eq? m 'acquire)
+ (if (test-and-set! cell)
+ (the-mutex 'acquire))) ; retry
+ ((eq? m 'release) (clear! cell))))
+ the-mutex))
+
+ (define (clear! cell)
+ (set-car! cell false))
+
+ `Test-and-set!' tests the cell and returns the result of the test.
+In addition, if the test was false, `test-and-set!' sets the cell
+contents to true before returning false. We can express this behavior
+as the following procedure:
+
+ (define (test-and-set! cell)
+ (if (car cell)
+ true
+ (begin (set-car! cell true)
+ false)))
+
+ However, this implementation of `test-and-set!' does not suffice as
+it stands. There is a crucial subtlety here, which is the essential
+place where concurrency control enters the system: The `test-and-set!'
+operation must be performed "atomically". That is, we must guarantee
+that, once a process has tested the cell and found it to be false, the
+cell contents will actually be set to true before any other process can
+test the cell. If we do not make this guarantee, then the mutex can
+fail in a way similar to the bank-account failure in *Note Figure
+3-29::. (See *Note Exercise 3-46::.)
+
+ The actual implementation of `test-and-set!' depends on the details
+of how our system runs concurrent processes. For example, we might be
+executing concurrent processes on a sequential processor using a
+time-slicing mechanism that cycles through the processes, permitting
+each process to run for a short time before interrupting it and moving
+on to the next process. In that case, `test-and-set!' can work by
+disabling time slicing during the testing and setting.(7)
+Alternatively, multiprocessing computers provide instructions that
+support atomic operations directly in hardware.(8)
+
+ *Exercise 3.46:* Suppose that we implement `test-and-set!' using
+ an ordinary procedure as shown in the text, without attempting to
+ make the operation atomic. Draw a timing diagram like the one in
+ *Note Figure 3-29:: to demonstrate how the mutex implementation
+ can fail by allowing two processes to acquire the mutex at the
+ same time.
+
+ *Exercise 3.47:* A semaphore (of size n) is a generalization of a
+ mutex. Like a mutex, a semaphore supports acquire and release
+ operations, but it is more general in that up to n processes can
+ acquire it concurrently. Additional processes that attempt to
+ acquire the semaphore must wait for release operations. Give
+ implementations of semaphores
+
+ a. in terms of mutexes
+
+ b. in terms of atomic `test-and-set!' operations.
+
+
+Deadlock
+........
+
+Now that we have seen how to implement serializers, we can see that
+account exchanging still has a problem, even with the
+`serialized-exchange' procedure above. Imagine that Peter attempts to
+exchange a1 with a2 while Paul concurrently attempts to exchange a2
+with a1. Suppose that Peter's process reaches the point where it has
+entered a serialized procedure protecting a1 and, just after that,
+Paul's process enters a serialized procedure protecting a2. Now Peter
+cannot proceed (to enter a serialized procedure protecting a2) until
+Paul exits the serialized procedure protecting a2. Similarly, Paul
+cannot proceed until Peter exits the serialized procedure protecting
+a1. Each process is stalled forever, waiting for the other. This
+situation is called a "deadlock". Deadlock is always a danger in
+systems that provide concurrent access to multiple shared resources.
+
+ One way to avoid the deadlock in this situation is to give each
+account a unique identification number and rewrite
+`serialized-exchange' so that a process will always attempt to enter a
+procedure protecting the lowest-numbered account first. Although this
+method works well for the exchange problem, there are other situations
+that require more sophisticated deadlock-avoidance techniques, or where
+deadlock cannot be avoided at all. (See *Note Exercise 3-48:: and
+*Note Exercise 3-49::.)(9)
+
+ *Exercise 3.48:* Explain in detail why the deadlock-avoidance
+ method described above, (i.e., the accounts are numbered, and each
+ process attempts to acquire the smaller-numbered account first)
+ avoids deadlock in the exchange problem. Rewrite
+ `serialized-exchange' to incorporate this idea. (You will also
+ need to modify `make-account' so that each account is created with
+ a number, which can be accessed by sending an appropriate message.)
+
+ *Exercise 3.49:* Give a scenario where the deadlock-avoidance
+ mechanism described above does not work. (Hint: In the exchange
+ problem, each process knows in advance which accounts it will need
+ to get access to. Consider a situation where a process must get
+ access to some shared resources before it can know which
+ additional shared resources it will require.)
+
+Concurrency, time, and communication
+....................................
+
+We've seen how programming concurrent systems requires controlling the
+ordering of events when different processes access shared state, and
+we've seen how to achieve this control through judicious use of
+serializers. But the problems of concurrency lie deeper than this,
+because, from a fundamental point of view, it's not always clear what
+is meant by "shared state."
+
+ Mechanisms such as `test-and-set!' require processes to examine a
+global shared flag at arbitrary times. This is problematic and
+inefficient to implement in modern high-speed processors, where due to
+optimization techniques such as pipelining and cached memory, the
+contents of memory may not be in a consistent state at every instant.
+In contemporary multiprocessing systems, therefore, the serializer
+paradigm is being supplanted by new approaches to concurrency
+control.(10)
+
+ The problematic aspects of shared state also arise in large,
+distributed systems. For instance, imagine a distributed banking
+system where individual branch banks maintain local values for bank
+balances and periodically compare these with values maintained by other
+branches. In such a system the value of "the account balance" would be
+undetermined, except right after synchronization. If Peter deposits
+money in an account he holds jointly with Paul, when should we say that
+the account balance has changed--when the balance in the local branch
+changes, or not until after the synchronization? And if Paul accesses
+the account from a different branch, what are the reasonable
+constraints to place on the banking system such that the behavior is
+"correct"? The only thing that might matter for correctness is the
+behavior observed by Peter and Paul individually and the "state" of the
+account immediately after synchronization. Questions about the "real"
+account balance or the order of events between synchronizations may be
+irrelevant or meaningless.(11)
+
+ The basic phenomenon here is that synchronizing different processes,
+establishing shared state, or imposing an order on events requires
+communication among the processes. In essence, any notion of time in
+concurrency control must be intimately tied to communication.(12) It
+is intriguing that a similar connection between time and communication
+also arises in the Theory of Relativity, where the speed of light (the
+fastest signal that can be used to synchronize events) is a fundamental
+constant relating time and space. The complexities we encounter in
+dealing with time and state in our computational models may in fact
+mirror a fundamental complexity of the physical universe.
+
+ ---------- Footnotes ----------
+
+ (1) `Parallel-execute' is not part of standard Scheme, but it can be
+implemented in MIT Scheme. In our implementation, the new concurrent
+processes also run concurrently with the original Scheme process.
+Also, in our implementation, the value returned by `parallel-execute'
+is a special control object that can be used to halt the newly created
+processes.
+
+ (2) We have simplified `exchange' by exploiting the fact that our
+`deposit' message accepts negative amounts. (This is a serious bug in
+our banking system!)
+
+ (3) If the account balances start out as $10, $20, and $30, then
+after any number of concurrent exchanges, the balances should still be
+$10, $20, and $30 in some order. Serializing the deposits to
+individual accounts is not sufficient to guarantee this. See *Note
+Exercise 3-43::.
+
+ (4) *Note Exercise 3-45:: investigates why deposits and withdrawals
+are no longer automatically serialized by the account.
+
+ (5) The term "mutex" is an abbreviation for "mutual exclusion". The
+general problem of arranging a mechanism that permits concurrent
+processes to safely share resources is called the mutual exclusion
+problem. Our mutex is a simple variant of the "semaphore" mechanism
+(see *Note Exercise 3-47::), which was introduced in the "THE"
+Multiprogramming System developed at the Technological University of
+Eindhoven and named for the university's initials in Dutch (Dijkstra
+1968a). The acquire and release operations were originally called P
+and V, from the Dutch words _passeren_ (to pass) and _vrijgeven_ (to
+release), in reference to the semaphores used on railroad systems.
+Dijkstra's classic exposition (1968b) was one of the first to clearly
+present the issues of concurrency control, and showed how to use
+semaphores to handle a variety of concurrency problems.
+
+ (6) In most time-shared operating systems, processes that are
+blocked by a mutex do not waste time "busy-waiting" as above. Instead,
+the system schedules another process to run while the first is waiting,
+and the blocked process is awakened when the mutex becomes available.
+
+ (7) In MIT Scheme for a single processor, which uses a time-slicing
+model, `test-and-set!' can be implemented as follows:
+
+ (define (test-and-set! cell)
+ (without-interrupts
+ (lambda ()
+ (if (car cell)
+ true
+ (begin (set-car! cell true)
+ false)))))
+
+ `Without-interrupts' disables time-slicing interrupts while its
+procedure argument is being executed.
+
+ (8) There are many variants of such instructions--including
+test-and-set, test-and-clear, swap, compare-and-exchange, load-reserve,
+and store-conditional--whose design must be carefully matched to the
+machine's processor-memory interface. One issue that arises here is to
+determine what happens if two processes attempt to acquire the same
+resource at exactly the same time by using such an instruction. This
+requires some mechanism for making a decision about which process gets
+control. Such a mechanism is called an "arbiter". Arbiters usually
+boil down to some sort of hardware device. Unfortunately, it is
+possible to prove that one cannot physically construct a fair arbiter
+that works 100% of the time unless one allows the arbiter an
+arbitrarily long time to make its decision. The fundamental phenomenon
+here was originally observed by the fourteenth-century French
+philosopher Jean Buridan in his commentary on Aristotle's De caelo.
+Buridan argued that a perfectly rational dog placed between two equally
+attractive sources of food will starve to death, because it is
+incapable of deciding which to go to first.
+
+ (9) The general technique for avoiding deadlock by numbering the
+shared resources and acquiring them in order is due to Havender (1968).
+Situations where deadlock cannot be avoided require "deadlock-recovery"
+methods, which entail having processes "back out" of the deadlocked
+state and try again. Deadlock-recovery mechanisms are widely used in
+database management systems, a topic that is treated in detail in Gray
+and Reuter 1993.
+
+ (10) One such alternative to serialization is called "barrier
+synchronization". The programmer permits concurrent processes to
+execute as they please, but establishes certain synchronization points
+("barriers") through which no process can proceed until all the
+processes have reached the barrier. Modern processors provide machine
+instructions that permit programmers to establish synchronization
+points at places where consistency is required. The PowerPC^( TM), for
+example, includes for this purpose two instructions called SYNC and
+EIEIO (Enforced In-order Execution of Input/Output).
+
+ (11) This may seem like a strange point of view, but there are
+systems that work this way. International charges to credit-card
+accounts, for example, are normally cleared on a per-country basis, and
+the charges made in different countries are periodically reconciled.
+Thus the account balance may be different in different countries.
+
+ (12) For distributed systems, this perspective was pursued by
+Lamport (1978), who showed how to use communication to establish
+"global clocks" that can be used to establish orderings on events in
+distributed systems.
+
+\1f
+File: sicp.info, Node: 3-5, Prev: 3-4, Up: Chapter 3
+
+3.5 Streams
+===========
+
+We've gained a good understanding of assignment as a tool in modeling,
+as well as an appreciation of the complex problems that assignment
+raises. It is time to ask whether we could have gone about things in a
+different way, so as to avoid some of these problems. In this section,
+we explore an alternative approach to modeling state, based on data
+structures called "streams". As we shall see, streams can mitigate
+some of the complexity of modeling state.
+
+ Let's step back and review where this complexity comes from. In an
+attempt to model real-world phenomena, we made some apparently
+reasonable decisions: We modeled real-world objects with local state by
+computational objects with local variables. We identified time
+variation in the real world with time variation in the computer. We
+implemented the time variation of the states of the model objects in
+the computer with assignments to the local variables of the model
+objects.
+
+ Is there another approach? Can we avoid identifying time in the
+computer with time in the modeled world? Must we make the model change
+with time in order to model phenomena in a changing world? Think about
+the issue in terms of mathematical functions. We can describe the
+time-varying behavior of a quantity x as a function of time x(t). If
+we concentrate on x instant by instant, we think of it as a changing
+quantity. Yet if we concentrate on the entire time history of values,
+we do not emphasize change--the function itself does not change.(1)
+
+ If time is measured in discrete steps, then we can model a time
+function as a (possibly infinite) sequence. In this section, we will
+see how to model change in terms of sequences that represent the time
+histories of the systems being modeled. To accomplish this, we
+introduce new data structures called "streams". From an abstract point
+of view, a stream is simply a sequence. However, we will find that the
+straightforward implementation of streams as lists (as in section *Note
+2-2-1::) doesn't fully reveal the power of stream processing. As an
+alternative, we introduce the technique of "delayed evaluation", which
+enables us to represent very large (even infinite) sequences as streams.
+
+ Stream processing lets us model systems that have state without ever
+using assignment or mutable data. This has important implications,
+both theoretical and practical, because we can build models that avoid
+the drawbacks inherent in introducing assignment. On the other hand,
+the stream framework raises difficulties of its own, and the question
+of which modeling technique leads to more modular and more easily
+maintained systems remains open.
+
+* Menu:
+
+* 3-5-1:: Streams Are Delayed Lists
+* 3-5-2:: Infinite Streams
+* 3-5-3:: Exploiting the Stream Paradigm
+* 3-5-4:: Streams and Delayed Evaluation
+* 3-5-5:: Modularity of Functional Programs and Modularity of
+ Objects
+
+ ---------- Footnotes ----------
+
+ (1) Physicists sometimes adopt this view by introducing the "world
+lines" of particles as a device for reasoning about motion. We've also
+already mentioned (section *Note 2-2-3::) that this is the natural way
+to think about signal-processing systems. We will explore applications
+of streams to signal processing in section *Note 3-5-3::.
+
+\1f
+File: sicp.info, Node: 3-5-1, Next: 3-5-2, Prev: 3-5, Up: 3-5
+
+3.5.1 Streams Are Delayed Lists
+-------------------------------
+
+As we saw in section *Note 2-2-3::, sequences can serve as standard
+interfaces for combining program modules. We formulated powerful
+abstractions for manipulating sequences, such as `map', `filter', and
+`accumulate', that capture a wide variety of operations in a manner that
+is both succinct and elegant.
+
+ Unfortunately, if we represent sequences as lists, this elegance is
+bought at the price of severe inefficiency with respect to both the
+time and space required by our computations. When we represent
+manipulations on sequences as transformations of lists, our programs
+must construct and copy data structures (which may be huge) at every
+step of a process.
+
+ To see why this is true, let us compare two programs for computing
+the sum of all the prime numbers in an interval. The first program is
+written in standard iterative style:(1)
+
+ (define (sum-primes a b)
+ (define (iter count accum)
+ (cond ((> count b) accum)
+ ((prime? count) (iter (+ count 1) (+ count accum)))
+ (else (iter (+ count 1) accum))))
+ (iter a 0))
+
+ The second program performs the same computation using the sequence
+operations of section *Note 2-2-3:::
+
+ (define (sum-primes a b)
+ (accumulate +
+ 0
+ (filter prime? (enumerate-interval a b))))
+
+ In carrying out the computation, the first program needs to store
+only the sum being accumulated. In contrast, the filter in the second
+program cannot do any testing until `enumerate-interval' has
+constructed a complete list of the numbers in the interval. The filter
+generates another list, which in turn is passed to `accumulate' before
+being collapsed to form a sum. Such large intermediate storage is not
+needed by the first program, which we can think of as enumerating the
+interval incrementally, adding each prime to the sum as it is generated.
+
+ The inefficiency in using lists becomes painfully apparent if we use
+the sequence paradigm to compute the second prime in the interval from
+10,000 to 1,000,000 by evaluating the expression
+
+ (car (cdr (filter prime?
+ (enumerate-interval 10000 1000000))))
+
+ This expression does find the second prime, but the computational
+overhead is outrageous. We construct a list of almost a million
+integers, filter this list by testing each element for primality, and
+then ignore almost all of the result. In a more traditional
+programming style, we would interleave the enumeration and the
+filtering, and stop when we reached the second prime.
+
+ Streams are a clever idea that allows one to use sequence
+manipulations without incurring the costs of manipulating sequences as
+lists. With streams we can achieve the best of both worlds: We can
+formulate programs elegantly as sequence manipulations, while attaining
+the efficiency of incremental computation. The basic idea is to
+arrange to construct a stream only partially, and to pass the partial
+construction to the program that consumes the stream. If the consumer
+attempts to access a part of the stream that has not yet been
+constructed, the stream will automatically construct just enough more
+of itself to produce the required part, thus preserving the illusion
+that the entire stream exists. In other words, although we will write
+programs as if we were processing complete sequences, we design our
+stream implementation to automatically and transparently interleave the
+construction of the stream with its use.
+
+ On the surface, streams are just lists with different names for the
+procedures that manipulate them. There is a constructor,
+`cons-stream', and two selectors, `stream-car' and `stream-cdr', which
+satisfy the constraints
+
+ (stream-car (cons-stream x y)) = x
+ (stream-cdr (cons-stream x y)) = y
+
+ There is a distinguishable object, `the-empty-stream', which cannot
+be the result of any `cons-stream' operation, and which can be
+identified with the predicate `stream-null?'.(2) Thus we can make and
+use streams, in just the same way as we can make and use lists, to
+represent aggregate data arranged in a sequence. In particular, we can
+build stream analogs of the list operations from *Note Chapter 2::,
+such as `list-ref', `map', and `for-each':(3)
+
+ (define (stream-ref s n)
+ (if (= n 0)
+ (stream-car s)
+ (stream-ref (stream-cdr s) (- n 1))))
+
+ (define (stream-map proc s)
+ (if (stream-null? s)
+ the-empty-stream
+ (cons-stream (proc (stream-car s))
+ (stream-map proc (stream-cdr s)))))
+
+ (define (stream-for-each proc s)
+ (if (stream-null? s)
+ 'done
+ (begin (proc (stream-car s))
+ (stream-for-each proc (stream-cdr s)))))
+
+ `Stream-for-each' is useful for viewing streams:
+
+ (define (display-stream s)
+ (stream-for-each display-line s))
+
+ (define (display-line x)
+ (newline)
+ (display x))
+
+ To make the stream implementation automatically and transparently
+interleave the construction of a stream with its use, we will arrange
+for the `cdr' of a stream to be evaluated when it is accessed by the
+`stream-cdr' procedure rather than when the stream is constructed by
+`cons-stream'. This implementation choice is reminiscent of our
+discussion of rational numbers in section *Note 2-1-2::, where we saw
+that we can choose to implement rational numbers so that the reduction
+of numerator and denominator to lowest terms is performed either at
+construction time or at selection time. The two rational-number
+implementations produce the same data abstraction, but the choice has
+an effect on efficiency. There is a similar relationship between
+streams and ordinary lists. As a data abstraction, streams are the
+same as lists. The difference is the time at which the elements are
+evaluated. With ordinary lists, both the `car' and the `cdr' are
+evaluated at construction time. With streams, the `cdr' is evaluated
+at selection time.
+
+ Our implementation of streams will be based on a special form called
+`delay'. Evaluating `(delay <EXP>)' does not evaluate the expression
+<EXP>, but rather returns a so-called object "delayed object", which we
+can think of as a "promise" to evaluate <EXP> at some future time. As
+a companion to `delay', there is a procedure called `force' that takes
+a delayed object as argument and performs the evaluation--in effect,
+forcing the `delay' to fulfill its promise. We will see below how
+`delay' and `force' can be implemented, but first let us use these to
+construct streams.
+
+ `Cons-stream' is a special form defined so that
+
+ (cons-stream <A> <B>)
+
+is equivalent to
+
+ (cons <A> (delay <B>))
+
+ What this means is that we will construct streams using pairs.
+However, rather than placing the value of the rest of the stream into
+the `cdr' of the pair we will put there a promise to compute the rest
+if it is ever requested. `Stream-car' and `stream-cdr' can now be
+defined as procedures:
+
+ (define (stream-car stream) (car stream))
+
+ (define (stream-cdr stream) (force (cdr stream)))
+
+ `Stream-car' selects the `car' of the pair; `stream-cdr' selects the
+`cdr' of the pair and evaluates the delayed expression found there to
+obtain the rest of the stream.(4)
+
+The stream implementation in action
+...................................
+
+To see how this implementation behaves, let us analyze the "outrageous"
+prime computation we saw above, reformulated in terms of streams:
+
+ (stream-car
+ (stream-cdr
+ (stream-filter prime?
+ (stream-enumerate-interval 10000 1000000))))
+
+ We will see that it does indeed work efficiently.
+
+ We begin by calling `stream-enumerate-interval' with the arguments
+10,000 and 1,000,000. `Stream-enumerate-interval' is the stream analog
+of `enumerate-interval' (section *Note 2-2-3::):
+
+ (define (stream-enumerate-interval low high)
+ (if (> low high)
+ the-empty-stream
+ (cons-stream
+ low
+ (stream-enumerate-interval (+ low 1) high))))
+
+and thus the result returned by `stream-enumerate-interval', formed by
+the `cons-stream', is(5)
+
+ (cons 10000
+ (delay (stream-enumerate-interval 10001 1000000)))
+
+ That is, `stream-enumerate-interval' returns a stream represented as
+a pair whose `car' is 10,000 and whose `cdr' is a promise to enumerate
+more of the interval if so requested. This stream is now filtered for
+primes, using the stream analog of the `filter' procedure (section
+*Note 2-2-3::):
+
+ (define (stream-filter pred stream)
+ (cond ((stream-null? stream) the-empty-stream)
+ ((pred (stream-car stream))
+ (cons-stream (stream-car stream)
+ (stream-filter pred
+ (stream-cdr stream))))
+ (else (stream-filter pred (stream-cdr stream)))))
+
+ `Stream-filter' tests the `stream-car' of the stream (the `car' of
+the pair, which is 10,000). Since this is not prime, `stream-filter'
+examines the `stream-cdr' of its input stream. The call to
+`stream-cdr' forces evaluation of the delayed
+`stream-enumerate-interval', which now returns
+
+ (cons 10001
+ (delay (stream-enumerate-interval 10002 1000000)))
+
+ `Stream-filter' now looks at the `stream-car' of this stream, 10,001,
+sees that this is not prime either, forces another `stream-cdr', and so
+on, until `stream-enumerate-interval' yields the prime 10,007, whereupon
+`stream-filter', according to its definition, returns
+
+ (cons-stream (stream-car stream)
+ (stream-filter pred (stream-cdr stream)))
+
+which in this case is
+
+ (cons 10007
+ (delay
+ (stream-filter
+ prime?
+ (cons 10008
+ (delay
+ (stream-enumerate-interval 10009
+ 1000000))))))
+
+ This result is now passed to `stream-cdr' in our original expression.
+This forces the delayed `stream-filter', which in turn keeps forcing the
+delayed `stream-enumerate-interval' until it finds the next prime, which
+is 10,009. Finally, the result passed to `stream-car' in our original
+expression is
+
+ (cons 10009
+ (delay
+ (stream-filter
+ prime?
+ (cons 10010
+ (delay
+ (stream-enumerate-interval 10011
+ 1000000))))))
+
+ `Stream-car' returns 10,009, and the computation is complete. Only
+as many integers were tested for primality as were necessary to find
+the second prime, and the interval was enumerated only as far as was
+necessary to feed the prime filter.
+
+ In general, we can think of delayed evaluation as "demand-driven"
+programming, whereby each stage in the stream process is activated only
+enough to satisfy the next stage. What we have done is to decouple the
+actual order of events in the computation from the apparent structure
+of our procedures. We write procedures as if the streams existed "all
+at once" when, in reality, the computation is performed incrementally,
+as in traditional programming styles.
+
+Implementing `delay' and `force'
+................................
+
+Although `delay' and `force' may seem like mysterious operations, their
+implementation is really quite straightforward. `Delay' must package
+an expression so that it can be evaluated later on demand, and we can
+accomplish this simply by treating the expression as the body of a
+procedure. `Delay' can be a special form such that
+
+ (delay <EXP>)
+
+is syntactic sugar for
+
+ (lambda () <EXP>)
+
+ `Force' simply calls the procedure (of no arguments) produced by
+`delay', so we can implement `force' as a procedure:
+
+ (define (force delayed-object)
+ (delayed-object))
+
+ This implementation suffices for `delay' and `force' to work as
+advertised, but there is an important optimization that we can include.
+In many applications, we end up forcing the same delayed object many
+times. This can lead to serious inefficiency in recursive programs
+involving streams. (See *Note Exercise 3-57::.) The solution is to
+build delayed objects so that the first time they are forced, they
+store the value that is computed. Subsequent forcings will simply
+return the stored value without repeating the computation. In other
+words, we implement `delay' as a special-purpose memoized procedure
+similar to the one described in *Note Exercise 3-27::. One way to
+accomplish this is to use the following procedure, which takes as
+argument a procedure (of no arguments) and returns a memoized version
+of the procedure. The first time the memoized procedure is run, it
+saves the computed result. On subsequent evaluations, it simply
+returns the result.
+
+ (define (memo-proc proc)
+ (let ((already-run? false) (result false))
+ (lambda ()
+ (if (not already-run?)
+ (begin (set! result (proc))
+ (set! already-run? true)
+ result)
+ result))))
+
+ `Delay' is then defined so that `(delay <EXP>)' is equivalent to
+
+ (memo-proc (lambda () <EXP>))
+
+and `force' is as defined previously.(6)
+
+ *Exercise 3.50:* Complete the following definition, which
+ generalizes `stream-map' to allow procedures that take multiple
+ arguments, analogous to `map' in section *Note 2-2-3::, footnote
+ *Note Footnote 12::.
+
+ (define (stream-map proc . argstreams)
+ (if (<??> (car argstreams))
+ the-empty-stream
+ (<??>
+ (apply proc (map <??> argstreams))
+ (apply stream-map
+ (cons proc (map <??> argstreams))))))
+
+ *Exercise 3.51:* In order to take a closer look at delayed
+ evaluation, we will use the following procedure, which simply
+ returns its argument after printing it:
+
+ (define (show x)
+ (display-line x)
+ x)
+
+ What does the interpreter print in response to evaluating each
+ expression in the following sequence?(7)
+
+ (define x (stream-map show (stream-enumerate-interval 0 10)))
+
+ (stream-ref x 5)
+
+ (stream-ref x 7)
+
+ *Exercise 3.52:* Consider the sequence of expressions
+
+ (define sum 0)
+
+ (define (accum x)
+ (set! sum (+ x sum))
+ sum)
+
+ (define seq (stream-map accum (stream-enumerate-interval 1 20)))
+ (define y (stream-filter even? seq))
+ (define z (stream-filter (lambda (x) (= (remainder x 5) 0))
+ seq))
+
+ (stream-ref y 7)
+
+ (display-stream z)
+
+ What is the value of `sum' after each of the above expressions is
+ evaluated? What is the printed response to evaluating the
+ `stream-ref' and `display-stream' expressions? Would these
+ responses differ if we had implemented `(delay <EXP>)' simply as
+ `(lambda () <EXP>)' without using the optimization provided by
+ `memo-proc'? Explain
+
+ ---------- Footnotes ----------
+
+ (1) Assume that we have a predicate `prime?' (e.g., as in section
+*Note 1-2-6::) that tests for primality.
+
+ (2) In the MIT implementation, `the-empty-stream' is the same as the
+empty list `'()', and `stream-null?' is the same as `null?'.
+
+ (3) This should bother you. The fact that we are defining such
+similar procedures for streams and lists indicates that we are missing
+some underlying abstraction. Unfortunately, in order to exploit this
+abstraction, we will need to exert finer control over the process of
+evaluation than we can at present. We will discuss this point further
+at the end of section *Note 3-5-4::. In section *Note 4-2::, we'll
+develop a framework that unifies lists and streams.
+
+ (4) Although `stream-car' and `stream-cdr' can be defined as
+procedures, `cons-stream' must be a special form. If `cons-stream'
+were a procedure, then, according to our model of evaluation,
+evaluating `(cons-stream <A> <B>)' would automatically cause <B> to be
+evaluated, which is precisely what we do not want to happen. For the
+same reason, `delay' must be a special form, though `force' can be an
+ordinary procedure.
+
+ (5) The numbers shown here do not really appear in the delayed
+expression. What actually appears is the original expression, in an
+environment in which the variables are bound to the appropriate numbers.
+For example, `(+ low 1)' with `low' bound to 10,000 actually appears
+where `10001' is shown.
+
+ (6) There are many possible implementations of streams other than
+the one described in this section. Delayed evaluation, which is the
+key to making streams practical, was inherent in Algol 60's "call-by-name"
+parameter-passing method. The use of this mechanism to implement
+streams was first described by Landin (1965). Delayed evaluation for
+streams was introduced into Lisp by Friedman and Wise (1976). In their
+implementation, `cons' always delays evaluating its arguments, so that
+lists automatically behave as streams. The memoizing optimization is
+also known as "call-by-need". The Algol community would refer to our
+original delayed objects as "call-by-name thunks" and to the optimized
+versions as "call-by-need thunks".
+
+ (7) Exercises such as *Note Exercise 3-51:: and *Note Exercise
+3-52:: are valuable for testing our understanding of how `delay' works.
+On the other hand, intermixing delayed evaluation with printing--and,
+even worse, with assignment--is extremely confusing, and instructors of
+courses on computer languages have traditionally tormented their
+students with examination questions such as the ones in this section.
+Needless to say, writing programs that depend on such subtleties is
+odious programming style. Part of the power of stream processing is
+that it lets us ignore the order in which events actually happen in our
+programs. Unfortunately, this is precisely what we cannot afford to do
+in the presence of assignment, which forces us to be concerned with
+time and change.
+
+\1f
+File: sicp.info, Node: 3-5-2, Next: 3-5-3, Prev: 3-5-1, Up: 3-5
+
+3.5.2 Infinite Streams
+----------------------
+
+We have seen how to support the illusion of manipulating streams as
+complete entities even though, in actuality, we compute only as much of
+the stream as we need to access. We can exploit this technique to
+represent sequences efficiently as streams, even if the sequences are
+very long. What is more striking, we can use streams to represent
+sequences that are infinitely long. For instance, consider the
+following definition of the stream of positive integers:
+
+ (define (integers-starting-from n)
+ (cons-stream n (integers-starting-from (+ n 1))))
+
+ (define integers (integers-starting-from 1))
+
+ This makes sense because `integers' will be a pair whose `car' is 1
+and whose `cdr' is a promise to produce the integers beginning with 2.
+This is an infinitely long stream, but in any given time we can examine
+only a finite portion of it. Thus, our programs will never know that
+the entire infinite stream is not there.
+
+ Using `integers' we can define other infinite streams, such as the
+stream of integers that are not divisible by 7:
+
+ (define (divisible? x y) (= (remainder x y) 0))
+
+ (define no-sevens
+ (stream-filter (lambda (x) (not (divisible? x 7)))
+ integers))
+
+ Then we can find integers not divisible by 7 simply by accessing
+elements of this stream:
+
+ (stream-ref no-sevens 100)
+ 117
+
+ In analogy with `integers', we can define the infinite stream of
+Fibonacci numbers:
+
+ (define (fibgen a b)
+ (cons-stream a (fibgen b (+ a b))))
+
+ (define fibs (fibgen 0 1))
+
+ `Fibs' is a pair whose `car' is 0 and whose `cdr' is a promise to
+evaluate `(fibgen 1 1)'. When we evaluate this delayed `(fibgen 1 1)',
+it will produce a pair whose `car' is 1 and whose `cdr' is a promise to
+evaluate `(fibgen 1 2)', and so on.
+
+ For a look at a more exciting infinite stream, we can generalize the
+`no-sevens' example to construct the infinite stream of prime numbers,
+using a method known as the Eratosthenes "sieve of Eratosthenes".(1) We
+start with the integers beginning with 2, which is the first prime. To
+get the rest of the primes, we start by filtering the multiples of 2
+from the rest of the integers. This leaves a stream beginning with 3,
+which is the next prime. Now we filter the multiples of 3 from the
+rest of this stream. This leaves a stream beginning with 5, which is
+the next prime, and so on. In other words, we construct the primes by
+a sieving process, described as follows: To sieve a stream `S', form a
+stream whose first element is the first element of `S' and the rest of
+which is obtained by filtering all multiples of the first element of
+`S' out of the rest of `S' and sieving the result. This process is
+readily described in terms of stream operations:
+
+ (define (sieve stream)
+ (cons-stream
+ (stream-car stream)
+ (sieve (stream-filter
+ (lambda (x)
+ (not (divisible? x (stream-car stream))))
+ (stream-cdr stream)))))
+
+ (define primes (sieve (integers-starting-from 2)))
+
+ Now to find a particular prime we need only ask for it:
+
+ (stream-ref primes 50)
+ 233
+
+ It is interesting to contemplate the signal-processing system set up
+by `sieve', shown in the "Henderson diagram" in *Note Figure 3-31::.(2)
+The input stream feeds into an "un`cons'er" that separates the first
+element of the stream from the rest of the stream. The first element
+is used to construct a divisibility filter, through which the rest is
+passed, and the output of the filter is fed to another sieve box. Then
+the original first element is `cons'ed onto the output of the internal
+sieve to form the output stream. Thus, not only is the stream
+infinite, but the signal processor is also infinite, because the sieve
+contains a sieve within it.
+
+ *Figure 3.31:* The prime sieve viewed as a signal-processing
+ system.
+
+ +---------------------------------------------------------------+
+ | sieve |
+ | |
+ | __/| |\__ |
+ | __/car|........................................| \__ |
+ | _/ | : | \_ |
+ ----><_ | V | cons _>---->
+ | \__ | +------------+ +------------+ | __/ |
+ | \cdr|--->| filter: | | sieve |--->| __/ |
+ | \| | |--->| | |/ |
+ | | not | | | |
+ | | divisible? | | | |
+ | +------------+ +------------+ |
+ +---------------------------------------------------------------+
+
+Defining streams implicitly
+...........................
+
+The `integers' and `fibs' streams above were defined by specifying
+"generating" procedures that explicitly compute the stream elements one
+by one. An alternative way to specify streams is to take advantage of
+delayed evaluation to define streams implicitly. For example, the
+following expression defines the stream `ones' to be an infinite stream
+of ones:
+
+ (define ones (cons-stream 1 ones))
+
+ This works much like the definition of a recursive procedure: `ones'
+is a pair whose `car' is 1 and whose `cdr' is a promise to evaluate
+`ones'. Evaluating the `cdr' gives us again a 1 and a promise to
+evaluate `ones', and so on.
+
+ We can do more interesting things by manipulating streams with
+operations such as `add-streams', which produces the elementwise sum of
+two given streams:(3)
+
+ (define (add-streams s1 s2)
+ (stream-map + s1 s2))
+
+ Now we can define the integers as follows:
+
+ (define integers (cons-stream 1 (add-streams ones integers)))
+
+ This defines `integers' to be a stream whose first element is 1 and
+the rest of which is the sum of `ones' and `integers'. Thus, the second
+element of `integers' is 1 plus the first element of `integers', or 2;
+the third element of `integers' is 1 plus the second element of
+`integers', or 3; and so on. This definition works because, at any
+point, enough of the `integers' stream has been generated so that we
+can feed it back into the definition to produce the next integer.
+
+ We can define the Fibonacci numbers in the same style:
+
+ (define fibs
+ (cons-stream 0
+ (cons-stream 1
+ (add-streams (stream-cdr fibs)
+ fibs))))
+
+ This definition says that `fibs' is a stream beginning with 0 and 1,
+such that the rest of the stream can be generated by adding `fibs' to
+itself shifted by one place:
+
+ 1 1 2 3 5 8 13 21 ... = `(stream-cdr fibs)'
+ 0 1 1 2 3 5 8 13 ... = `fibs'
+ 0 1 1 2 3 5 8 13 21 34 ... = `fibs'
+
+ `Scale-stream' is another useful procedure in formulating such stream
+definitions. This multiplies each item in a stream by a given constant:
+
+ (define (scale-stream stream factor)
+ (stream-map (lambda (x) (* x factor)) stream))
+
+ For example,
+
+ (define double (cons-stream 1 (scale-stream double 2)))
+
+produces the stream of powers of 2: 1, 2, 4, 8, 16, 32, ....
+
+ An alternate definition of the stream of primes can be given by
+starting with the integers and filtering them by testing for primality.
+We will need the first prime, 2, to get started:
+
+ (define primes
+ (cons-stream
+ 2
+ (stream-filter prime? (integers-starting-from 3))))
+
+ This definition is not so straightforward as it appears, because we
+will test whether a number n is prime by checking whether n is
+divisible by a prime (not by just any integer) less than or equal to
+_[sqrt]_(n):
+
+ (define (prime? n)
+ (define (iter ps)
+ (cond ((> (square (stream-car ps)) n) true)
+ ((divisible? n (stream-car ps)) false)
+ (else (iter (stream-cdr ps)))))
+ (iter primes))
+
+ This is a recursive definition, since `primes' is defined in terms
+of the `prime?' predicate, which itself uses the `primes' stream. The
+reason this procedure works is that, at any point, enough of the
+`primes' stream has been generated to test the primality of the numbers
+we need to check next. That is, for every n we test for primality,
+either n is not prime (in which case there is a prime already generated
+that divides it) or n is prime (in which case there is a prime already
+generated--i.e., a prime less than n--that is greater than
+_[sqrt]_(n)).(4)
+
+ *Exercise 3.53:* Without running the program, describe the
+ elements of the stream defined by
+
+ (define s (cons-stream 1 (add-streams s s)))
+
+ *Exercise 3.54:* Define a procedure `mul-streams', analogous to
+ `add-streams', that produces the elementwise product of its two
+ input streams. Use this together with the stream of `integers' to
+ complete the following definition of the stream whose nth element
+ (counting from 0) is n + 1 factorial:
+
+ (define factorials (cons-stream 1 (mul-streams <??> <??>)))
+
+ *Exercise 3.55:* Define a procedure `partial-sums' that takes as
+ argument a stream S and returns the stream whose elements are S_0,
+ S_0 + S_1, S_0 + S_1 + S_2, .... For example, `(partial-sums
+ integers)' should be the stream 1, 3, 6, 10, 15, ....
+
+ *Exercise 3.56:* A famous problem, first raised by R. Hamming, is
+ to enumerate, in ascending order with no repetitions, all positive
+ integers with no prime factors other than 2, 3, or 5. One obvious
+ way to do this is to simply test each integer in turn to see
+ whether it has any factors other than 2, 3, and 5. But this is
+ very inefficient, since, as the integers get larger, fewer and
+ fewer of them fit the requirement. As an alternative, let us call
+ the required stream of numbers `S' and notice the following facts
+ about it.
+
+ * `S' begins with 1.
+
+ * The elements of `(scale-stream S 2)' are also elements of `S'.
+
+ * The same is true for `(scale-stream S 3)' and `(scale-stream
+ 5 S)'.
+
+ * These are all the elements of `S'.
+
+
+ Now all we have to do is combine elements from these sources. For
+ this we define a procedure `merge' that combines two ordered
+ streams into one ordered result stream, eliminating repetitions:
+
+ (define (merge s1 s2)
+ (cond ((stream-null? s1) s2)
+ ((stream-null? s2) s1)
+ (else
+ (let ((s1car (stream-car s1))
+ (s2car (stream-car s2)))
+ (cond ((< s1car s2car)
+ (cons-stream s1car (merge (stream-cdr s1) s2)))
+ ((> s1car s2car)
+ (cons-stream s2car (merge s1 (stream-cdr s2))))
+ (else
+ (cons-stream s1car
+ (merge (stream-cdr s1)
+ (stream-cdr s2)))))))))
+
+ Then the required stream may be constructed with `merge', as
+ follows:
+
+ (define S (cons-stream 1 (merge <??> <??>)))
+
+ Fill in the missing expressions in the places marked <??> above.
+
+ *Exercise 3.57:* How many additions are performed when we compute
+ the nth Fibonacci number using the definition of `fibs' based on
+ the `add-streams' procedure? Show that the number of additions
+ would be exponentially greater if we had implemented `(delay
+ <EXP>)' simply as `(lambda () <EXP>)', without using the
+ optimization provided by the `memo-proc' procedure described in
+ section *Note 3-5-1::.(5)
+
+ *Exercise 3.58:* Give an interpretation of the stream computed by
+ the following procedure:
+
+ (define (expand num den radix)
+ (cons-stream
+ (quotient (* num radix) den)
+ (expand (remainder (* num radix) den) den radix)))
+
+ (`Quotient' is a primitive that returns the integer quotient of two
+ integers.) What are the successive elements produced by `(expand
+ 1 7 10)'? What is produced by `(expand 3 8 10)'?
+
+ *Exercise 3.59:* In section *Note 2-5-3:: we saw how to implement
+ a polynomial arithmetic system representing polynomials as lists
+ of terms. In a similar way, we can work with "power series", such
+ as
+
+ x^2 x^3 x^4
+ e^x = 1 + x + ----- + ----- + --------- + ...
+ 2 3 * 2 4 * 3 * 2
+
+ x^2 x^4
+ cos x = 1 - ----- + --------- - ...
+ 2 4 * 3 * 2
+
+ x^3 x^5
+ sin x = x - ----- + ------------- - ...
+ 3 * 2 5 * 4 * 3 * 2
+
+ represented as infinite streams. We will represent the series a_0
+ + a_1 x + a_2 x^2 + a_3 x^3 + ... as the stream whose elements are
+ the coefficients a_0, a_1, a_2, a_3, ....
+
+ a. The integral of the series a_0 + a_1 x + a_2 x^2 + a_3 x^3 +
+ ... is the series
+
+ 1 1 1
+ c + a_0 x + --- x_1 r^2 + --- a_2 r^3 + --- a_3 r^4 + ...
+ 2 3 4
+
+ where c is any constant. Define a procedure
+ `integrate-series' that takes as input a stream a_0, a_1,
+ a_2, ... representing a power series and returns the stream
+ a_0, (1/2)a_1, (1/3)a_2, ... of coefficients of the
+ non-constant terms of the integral of the series. (Since the
+ result has no constant term, it doesn't represent a power
+ series; when we use `integrate-series', we will `cons' on the
+ appropriate constant.)
+
+ b. The function x |-> e^x is its own derivative. This implies
+ that e^x and the integral of e^x are the same series, except
+ for the constant term, which is e^0 = 1. Accordingly, we can
+ generate the series for e^x as
+
+ (define exp-series
+ (cons-stream 1 (integrate-series exp-series)))
+
+ Show how to generate the series for sine and cosine, starting
+ from the facts that the derivative of sine is cosine and the
+ derivative of cosine is the negative of sine:
+
+ (define cosine-series
+ (cons-stream 1 <??>))
+
+ (define sine-series
+ (cons-stream 0 <??>))
+
+ *Exercise 3.60:* With power series represented as streams of
+ coefficients as in *Note Exercise 3-59::, adding series is
+ implemented by `add-streams'. Complete the definition of the
+ following procedure for multiplying series:
+
+ (define (mul-series s1 s2)
+ (cons-stream <??> (add-streams <??> <??>)))
+
+ You can test your procedure by verifying that sin^2 x + cos^2 x =
+ 1, using the series from *Note Exercise 3-59::.
+
+ *Exercise 3.61:* Let S be a power series (*Note Exercise 3-59::)
+ whose constant term is 1. Suppose we want to find the power
+ series 1/S, that is, the series X such that S * X = 1. Write S =
+ 1 + S_R where S_R is the part of S after the constant term. Then
+ we can solve for X as follows:
+
+ S * X = 1
+ (1 + S_R) * X = 1
+ X + S_R * X = 1
+ X = 1 - S_R * X
+
+ In other words, X is the power series whose constant term is 1 and
+ whose higher-order terms are given by the negative of S_R times X.
+ Use this idea to write a procedure `invert-unit-series' that
+ computes 1/S for a power series S with constant term 1. You will
+ need to use `mul-series' from *Note Exercise 3-60::.
+
+ *Exercise 3.62:* Use the results of *Note Exercise 3-60:: and
+ *Note Exercise 3-61:: to define a procedure `div-series' that
+ divides two power series. `Div-series' should work for any two
+ series, provided that the denominator series begins with a nonzero
+ constant term. (If the denominator has a zero constant term, then
+ `div-series' should signal an error.) Show how to use
+ `div-series' together with the result of *Note Exercise 3-59:: to
+ generate the power series for tangent.
+
+ ---------- Footnotes ----------
+
+ (1) Eratosthenes, a third-century B.C. Alexandrian Greek
+philosopher, is famous for giving the first accurate estimate of the
+circumference of the Earth, which he computed by observing shadows cast
+at noon on the day of the summer solstice. Eratosthenes's sieve method,
+although ancient, has formed the basis for special-purpose hardware
+"sieves" that, until recently, were the most powerful tools in
+existence for locating large primes. Since the 70s, however, these
+methods have been superseded by outgrowths of the probabilistic
+techniques discussed in section *Note 1-2-6::.
+
+ (2) We have named these figures after Peter Henderson, who was the
+first person to show us diagrams of this sort as a way of thinking
+about stream processing. Each solid line represents a stream of values
+being transmitted. The dashed line from the `car' to the `cons' and
+the `filter' indicates that this is a single value rather than a stream.
+
+ (3) This uses the generalized version of `stream-map' from *Note
+Exercise 3-50::.
+
+ (4) This last point is very subtle and relies on the fact that
+p_(n+1) <= p_n^2. (Here, p_k denotes the kth prime.) Estimates such
+as these are very difficult to establish. The ancient proof by Euclid
+that there are an infinite number of primes shows that p_(n+1)<= p_1
+p_2...p_n + 1, and no substantially better result was proved until
+1851, when the Russian mathematician P. L. Chebyshev established that
+p_(n+1)<= 2p_n for all n. This result, originally conjectured in 1845,
+is known as hypothesis "Bertrand's hypothesis". A proof can be found
+in section 22.3 of Hardy and Wright 1960.
+
+ (5) This exercise shows how call-by-need is closely related to
+ordinary memoization as described in *Note Exercise 3-27::. In that
+exercise, we used assignment to explicitly construct a local table.
+Our call-by-need stream optimization effectively constructs such a
+table automatically, storing values in the previously forced parts of
+the stream.
+
+\1f
+File: sicp.info, Node: 3-5-3, Next: 3-5-4, Prev: 3-5-2, Up: 3-5
+
+3.5.3 Exploiting the Stream Paradigm
+------------------------------------
+
+Streams with delayed evaluation can be a powerful modeling tool,
+providing many of the benefits of local state and assignment.
+Moreover, they avoid some of the theoretical tangles that accompany the
+introduction of assignment into a programming language.
+
+ The stream approach can be illuminating because it allows us to
+build systems with different module boundaries than systems organized
+around assignment to state variables. For example, we can think of an
+entire time series (or signal) as a focus of interest, rather than the
+values of the state variables at individual moments. This makes it
+convenient to combine and compare components of state from different
+moments.
+
+Formulating iterations as stream processes
+..........................................
+
+In section *Note 1-2-1::, we introduced iterative processes, which
+proceed by updating state variables. We know now that we can represent
+state as a "timeless" stream of values rather than as a set of
+variables to be updated. Let's adopt this perspective in revisiting
+the square-root procedure from section *Note 1-1-7::. Recall that the
+idea is to generate a sequence of better and better guesses for the
+square root of x by applying over and over again the procedure that
+improves guesses:
+
+ (define (sqrt-improve guess x)
+ (average guess (/ x guess)))
+
+ In our original `sqrt' procedure, we made these guesses be the
+successive values of a state variable. Instead we can generate the
+infinite stream of guesses, starting with an initial guess of 1:(1)
+
+ (define (sqrt-stream x)
+ (define guesses
+ (cons-stream 1.0
+ (stream-map (lambda (guess)
+ (sqrt-improve guess x))
+ guesses)))
+ guesses)
+
+ (display-stream (sqrt-stream 2))
+ 1.
+ 1.5
+ 1.4166666666666665
+ 1.4142156862745097
+ 1.4142135623746899
+ ...
+
+ We can generate more and more terms of the stream to get better and
+better guesses. If we like, we can write a procedure that keeps
+generating terms until the answer is good enough. (See *Note Exercise
+3-64::.)
+
+ Another iteration that we can treat in the same way is to generate an
+approximation to [pi], based upon the alternating series that we saw in
+section *Note 1-3-1:::
+
+ [pi] 1 1 1
+ ---- = 1 - --- + --- - --- + ...
+ 4 3 5 7
+
+ We first generate the stream of summands of the series (the
+reciprocals of the odd integers, with alternating signs). Then we take
+the stream of sums of more and more terms (using the `partial-sums'
+procedure of *Note Exercise 3-55::) and scale the result by 4:
+
+ (define (pi-summands n)
+ (cons-stream (/ 1.0 n)
+ (stream-map - (pi-summands (+ n 2)))))
+
+ (define pi-stream
+ (scale-stream (partial-sums (pi-summands 1)) 4))
+
+ (display-stream pi-stream)
+ 4.
+ 2.666666666666667
+ 3.466666666666667
+ 2.8952380952380956
+ 3.3396825396825403
+ 2.9760461760461765
+ 3.2837384837384844
+ 3.017071817071818
+ ...
+
+ This gives us a stream of better and better approximations to [pi],
+although the approximations converge rather slowly. Eight terms of the
+sequence bound the value of [pi] between 3.284 and 3.017.
+
+ So far, our use of the stream of states approach is not much
+different from updating state variables. But streams give us an
+opportunity to do some interesting tricks. For example, we can
+transform a stream with a "sequence accelerator" that converts a
+sequence of approximations to a new sequence that converges to the same
+value as the original, only faster.
+
+ One such accelerator, due to the eighteenth-century Swiss
+mathematician Leonhard Euler, works well with sequences that are
+partial sums of alternating series (series of terms with alternating
+signs). In Euler's technique, if S_n is the nth term of the original
+sum sequence, then the accelerated sequence has terms
+
+ (S_(n+1) - S_n)^2
+ S_(n+1) - ------------------------
+ S_(n-1) - 2S_n + S_(n+1)
+
+ Thus, if the original sequence is represented as a stream of values,
+the transformed sequence is given by
+
+ (define (euler-transform s)
+ (let ((s0 (stream-ref s 0)) ; S_(n-1)
+ (s1 (stream-ref s 1)) ; S_n
+ (s2 (stream-ref s 2))) ; S_(n+1)
+ (cons-stream (- s2 (/ (square (- s2 s1))
+ (+ s0 (* -2 s1) s2)))
+ (euler-transform (stream-cdr s)))))
+
+ We can demonstrate Euler acceleration with our sequence of
+approximations to [pi]:
+
+ (display-stream (euler-transform pi-stream))
+ 3.166666666666667
+ 3.1333333333333337
+ 3.1452380952380956
+ 3.13968253968254
+ 3.1427128427128435
+ 3.1408813408813416
+ 3.142071817071818
+ 3.1412548236077655
+ ...
+
+ Even better, we can accelerate the accelerated sequence, and
+recursively accelerate that, and so on. Namely, we create a stream of
+streams (a structure we'll call a "tableau") in which each stream is
+the transform of the preceding one:
+
+ (define (make-tableau transform s)
+ (cons-stream s
+ (make-tableau transform
+ (transform s))))
+
+ The tableau has the form
+
+ s_00 s_01 s_02 s_03 s_04 ...
+ s_10 s_11 s_12 s_13 ...
+ s_20 s_21 s_22 ...
+ ...
+
+ Finally, we form a sequence by taking the first term in each row of
+the tableau:
+
+ (define (accelerated-sequence transform s)
+ (stream-map stream-car
+ (make-tableau transform s)))
+
+ We can demonstrate this kind of "super-acceleration" of the [pi]
+sequence:
+
+ (display-stream (accelerated-sequence euler-transform
+ pi-stream))
+ 4.
+ 3.166666666666667
+ 3.142105263157895
+ 3.141599357319005
+ 3.1415927140337785
+ 3.1415926539752927
+ 3.1415926535911765
+ 3.141592653589778
+ ...
+
+ The result is impressive. Taking eight terms of the sequence yields
+the correct value of [pi] to 14 decimal places. If we had used only the
+original [pi] sequence, we would need to compute on the order of 10^13
+terms (i.e., expanding the series far enough so that the individual
+terms are less then 10^(-13)) to get that much accuracy!
+
+ We could have implemented these acceleration techniques without
+using streams. But the stream formulation is particularly elegant and
+convenient because the entire sequence of states is available to us as
+a data structure that can be manipulated with a uniform set of
+operations.
+
+ *Exercise 3.63:* Louis Reasoner asks why the `sqrt-stream'
+ procedure was not written in the following more straightforward
+ way, without the local variable `guesses':
+
+ (define (sqrt-stream x)
+ (cons-stream 1.0
+ (stream-map (lambda (guess)
+ (sqrt-improve guess x))
+ (sqrt-stream x))))
+
+ Alyssa P. Hacker replies that this version of the procedure is
+ considerably less efficient because it performs redundant
+ computation. Explain Alyssa's answer. Would the two versions
+ still differ in efficiency if our implementation of `delay' used
+ only `(lambda () <EXP>)' without using the optimization provided
+ by `memo-proc' (section *Note 3-5-1::)?
+
+ *Exercise 3.64:* Write a procedure `stream-limit' that takes as
+ arguments a stream and a number (the tolerance). It should
+ examine the stream until it finds two successive elements that
+ differ in absolute value by less than the tolerance, and return
+ the second of the two elements. Using this, we could compute
+ square roots up to a given tolerance by
+
+ (define (sqrt x tolerance)
+ (stream-limit (sqrt-stream x) tolerance))
+
+ *Exercise 3.65:* Use the series
+
+ 1 1 1
+ ln 2 = 1 - --- + --- - --- + ...
+ 2 3 4
+
+ to compute three sequences of approximations to the natural
+ logarithm of 2, in the same way we did above for [pi]. How
+ rapidly do these sequences converge?
+
+Infinite streams of pairs
+.........................
+
+In section *Note 2-2-3::, we saw how the sequence paradigm handles
+traditional nested loops as processes defined on sequences of pairs.
+If we generalize this technique to infinite streams, then we can write
+programs that are not easily represented as loops, because the
+"looping" must range over an infinite set.
+
+ For example, suppose we want to generalize the `prime-sum-pairs'
+procedure of section *Note 2-2-3:: to produce the stream of pairs of
+_all_ integers (i,j) with i <= j such that i + j is prime. If
+`int-pairs' is the sequence of all pairs of integers (i,j) with i <= j,
+then our required stream is simply(2)
+
+ (stream-filter (lambda (pair)
+ (prime? (+ (car pair) (cadr pair))))
+ int-pairs)
+
+ Our problem, then, is to produce the stream `int-pairs'. More
+generally, suppose we have two streams S = (S_i) and T = (T_j), and
+imagine the infinite rectangular array
+
+ (S_0, T_0) (S_0, T_1) (S_0, T_2) ...
+ (S_1, T_0) (S_1, T_1) (S_1, T_2) ...
+ (S_2, T_0) (S_2, T_1) (S_2, T_2) ...
+ ...
+
+ We wish to generate a stream that contains all the pairs in the
+array that lie on or above the diagonal, i.e., the pairs
+
+ (S_0, T_0) (S_0, T_1) (S_0, T_2) ...
+ (S_1, T_1) (S_1, T_2) ...
+ (S_2, T_2) ...
+ ...
+
+(If we take both S and T to be the stream of integers, then this will
+be our desired stream `int-pairs'.)
+
+ Call the general stream of pairs `(pairs S T)', and consider it to be
+composed of three parts: the pair (S_0,T_0), the rest of the pairs in
+the first row, and the remaining pairs:(3)
+
+ (S_0, T_0) | (S_0, T_1) (S_0, T_2) ...
+ -----------+-----------------------------
+ | (S_1, T_1) (S_1, T_2) ...
+ | (S_2, T_2) ...
+ | ...
+
+ Observe that the third piece in this decomposition (pairs that are
+not in the first row) is (recursively) the pairs formed from
+`(stream-cdr S)' and `(stream-cdr T)'. Also note that the second piece
+(the rest of the first row) is
+
+ (stream-map (lambda (x) (list (stream-car s) x))
+ (stream-cdr t))
+
+ Thus we can form our stream of pairs as follows:
+
+ (define (pairs s t)
+ (cons-stream
+ (list (stream-car s) (stream-car t))
+ (<COMBINE-IN-SOME-WAY>
+ (stream-map (lambda (x) (list (stream-car s) x))
+ (stream-cdr t))
+ (pairs (stream-cdr s) (stream-cdr t)))))
+
+ In order to complete the procedure, we must choose some way to
+combine the two inner streams. One idea is to use the stream analog of
+the `append' procedure from section *Note 2-2-1:::
+
+ (define (stream-append s1 s2)
+ (if (stream-null? s1)
+ s2
+ (cons-stream (stream-car s1)
+ (stream-append (stream-cdr s1) s2))))
+
+ This is unsuitable for infinite streams, however, because it takes
+all the elements from the first stream before incorporating the second
+stream. In particular, if we try to generate all pairs of positive
+integers using
+
+ (pairs integers integers)
+
+our stream of results will first try to run through all pairs with the
+first integer equal to 1, and hence will never produce pairs with any
+other value of the first integer.
+
+ To handle infinite streams, we need to devise an order of
+combination that ensures that every element will eventually be reached
+if we let our program run long enough. An elegant way to accomplish
+this is with the following `interleave' procedure:(4)
+
+ (define (interleave s1 s2)
+ (if (stream-null? s1)
+ s2
+ (cons-stream (stream-car s1)
+ (interleave s2 (stream-cdr s1)))))
+
+ Since `interleave' takes elements alternately from the two streams,
+every element of the second stream will eventually find its way into
+the interleaved stream, even if the first stream is infinite.
+
+ We can thus generate the required stream of pairs as
+
+ (define (pairs s t)
+ (cons-stream
+ (list (stream-car s) (stream-car t))
+ (interleave
+ (stream-map (lambda (x) (list (stream-car s) x))
+ (stream-cdr t))
+ (pairs (stream-cdr s) (stream-cdr t)))))
+
+ *Exercise 3.66:* Examine the stream `(pairs integers integers)'.
+ Can you make any general comments about the order in which the
+ pairs are placed into the stream? For example, about how many
+ pairs precede the pair (1,100)? the pair (99,100)? the pair
+ (100,100)? (If you can make precise mathematical statements here,
+ all the better. But feel free to give more qualitative answers if
+ you find yourself getting bogged down.)
+
+ *Exercise 3.67:* Modify the `pairs' procedure so that `(pairs
+ integers integers)' will produce the stream of _all_ pairs of
+ integers (i,j) (without the condition i <= j). Hint: You will
+ need to mix in an additional stream.
+
+ *Exercise 3.68:* Louis Reasoner thinks that building a stream of
+ pairs from three parts is unnecessarily complicated. Instead of
+ separating the pair (S_0,T_0) from the rest of the pairs in the
+ first row, he proposes to work with the whole first row, as
+ follows:
+
+ (define (pairs s t)
+ (interleave
+ (stream-map (lambda (x) (list (stream-car s) x))
+ t)
+ (pairs (stream-cdr s) (stream-cdr t))))
+
+ Does this work? Consider what happens if we evaluate `(pairs
+ integers integers)' using Louis's definition of `pairs'.
+
+ *Exercise 3.69:* Write a procedure `triples' that takes three
+ infinite streams, S, T, and U, and produces the stream of triples
+ (S_i,T_j,U_k) such that i <= j <= k. Use `triples' to generate
+ the stream of all Pythagorean triples of positive integers, i.e.,
+ the triples (i,j,k) such that i <= j and i^2 + j^2 = k^2.
+
+ *Exercise 3.70:* It would be nice to be able to generate streams
+ in which the pairs appear in some useful order, rather than in the
+ order that results from an _ad hoc_ interleaving process. We can
+ use a technique similar to the `merge' procedure of *Note Exercise
+ 3-56::, if we define a way to say that one pair of integers is
+ "less than" another. One way to do this is to define a "weighting
+ function" W(i,j) and stipulate that (i_1,j_1) is less than
+ (i_2,j_2) if W(i_1,j_1) < W(i_2,j_2). Write a procedure
+ `merge-weighted' that is like `merge', except that
+ `merge-weighted' takes an additional argument `weight', which is a
+ procedure that computes the weight of a pair, and is used to
+ determine the order in which elements should appear in the
+ resulting merged stream.(5) Using this, generalize `pairs' to a
+ procedure `weighted-pairs' that takes two streams, together with a
+ procedure that computes a weighting function, and generates the
+ stream of pairs, ordered according to weight. Use your procedure
+ to generate
+
+ a. the stream of all pairs of positive integers (i,j) with i <= j
+ ordered according to the sum i + j
+
+ b. the stream of all pairs of positive integers (i,j) with i <=
+ j, where neither i nor j is divisible by 2, 3, or 5, and the
+ pairs are ordered according to the sum 2 i + 3 j + 5 i j.
+
+
+ *Exercise 3.71:* Numbers that can be expressed as the sum of two
+ cubes in more than one way are sometimes called "Ramanujan
+ numbers", in honor of the mathematician Srinivasa Ramanujan.(6)
+ Ordered streams of pairs provide an elegant solution to the
+ problem of computing these numbers. To find a number that can be
+ written as the sum of two cubes in two different ways, we need
+ only generate the stream of pairs of integers (i,j) weighted
+ according to the sum i^3 + j^3 (see *Note Exercise 3-70::), then
+ search the stream for two consecutive pairs with the same weight.
+ Write a procedure to generate the Ramanujan numbers. The first
+ such number is 1,729. What are the next five?
+
+ *Exercise 3.72:* In a similar way to *Note Exercise 3-71::
+ generate a stream of all numbers that can be written as the sum of
+ two squares in three different ways (showing how they can be so
+ written).
+
+Streams as signals
+..................
+
+We began our discussion of streams by describing them as computational
+analogs of the "signals" in signal-processing systems. In fact, we can
+use streams to model signal-processing systems in a very direct way,
+representing the values of a signal at successive time intervals as
+consecutive elements of a stream. For instance, we can implement an "integrator"
+or "summer" that, for an input stream x = (x_i), an initial value C,
+and a small increment dt, accumulates the sum
+
+ i
+ ---
+ S_i = C + > x_j dt
+ ---
+ j=1
+
+and returns the stream of values S = (S_i). The following `integral'
+procedure is reminiscent of the "implicit style" definition of the
+stream of integers (section *Note 3-5-2::):
+
+ (define (integral integrand initial-value dt)
+ (define int
+ (cons-stream initial-value
+ (add-streams (scale-stream integrand dt)
+ int)))
+ int)
+
+ *Note Figure 3-32:: is a picture of a signal-processing system that
+corresponds to the `integral' procedure. The input stream is scaled by
+dt and passed through an adder, whose output is passed back through the
+same adder. The self-reference in the definition of `int' is reflected
+in the figure by the feedback loop that connects the output of the
+adder to one of the inputs.
+
+ *Figure 3.32:* The `integral' procedure viewed as a
+ signal-processing system.
+
+ initial-value
+ |
+ +-----------+ | |\__
+ input | | |\__ +-->| \_ integral
+ ------>| scale: dt +----->| \_ |cons_>--*------->
+ | | | add_>---->| __/ |
+ +-----------+ +-->| __/ |/ |
+ | |/ |
+ | |
+ +------------------------+
+
+ *Exercise 3.73:* We can model electrical circuits using streams to
+ represent the values of currents or voltages at a sequence of
+ times. For instance, suppose we have an "RC circuit" consisting
+ of a resistor of resistance R and a capacitor of capacitance C in
+ series. The voltage response v of the circuit to an injected
+ current i is determined by the formula in *Note Figure 3-33::,
+ whose structure is shown by the accompanying signal-flow diagram.
+
+ Write a procedure `RC' that models this circuit. `RC' should take
+ as inputs the values of R, C, and dt and should return a procedure
+ that takes as inputs a stream representing the current i and an
+ initial value for the capacitor voltage v_0 and produces as output
+ the stream of voltages v. For example, you should be able to use
+ `RC' to model an RC circuit with R = 5 ohms, C = 1 farad, and a
+ 0.5-second time step by evaluating `(define RC1 (RC 5 1 0.5))'.
+ This defines `RC1' as a procedure that takes a stream representing
+ the time sequence of currents and an initial capacitor voltage and
+ produces the output stream of voltages.
+
+ *Figure 3.33:* An RC circuit and the associated signal-flow
+ diagram.
+
+ + -
+ ->----'\/\/\,---| |---
+ i C
+
+ / t
+ | i
+ v = v + | dt + R i
+ 0 |
+ / 0
+
+ +--------------+
+ +-->| scale: R |---------------------+ |\_
+ | +--------------+ | | \_
+ | +-->| \ v
+ i | +--------------+ +------------+ | add >--->
+ ----+-->| scale: 1/C |---->| integral |----->| _/
+ +--------------+ +------------+ | _/
+ |/
+
+ *Exercise 3.74:* Alyssa P. Hacker is designing a system to process
+ signals coming from physical sensors. One important feature she
+ wishes to produce is a signal that describes the "zero crossings"
+ of the input signal. That is, the resulting signal should be + 1
+ whenever the input signal changes from negative to positive, - 1
+ whenever the input signal changes from positive to negative, and 0
+ otherwise. (Assume that the sign of a 0 input is positive.) For
+ example, a typical input signal with its associated zero-crossing
+ signal would be
+
+ ... 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 ...
+ ... 0 0 0 0 0 -1 0 0 0 0 1 0 0 ...
+
+ In Alyssa's system, the signal from the sensor is represented as a
+ stream `sense-data' and the stream `zero-crossings' is the
+ corresponding stream of zero crossings. Alyssa first writes a
+ procedure `sign-change-detector' that takes two values as
+ arguments and compares the signs of the values to produce an
+ appropriate 0, 1, or - 1. She then constructs her zero-crossing
+ stream as follows:
+
+ (define (make-zero-crossings input-stream last-value)
+ (cons-stream
+ (sign-change-detector (stream-car input-stream) last-value)
+ (make-zero-crossings (stream-cdr input-stream)
+ (stream-car input-stream))))
+
+ (define zero-crossings (make-zero-crossings sense-data 0))
+
+ Alyssa's boss, Eva Lu Ator, walks by and suggests that this
+ program is approximately equivalent to the following one, which
+ uses the generalized version of `stream-map' from *Note Exercise
+ 3-50:::
+
+ (define zero-crossings
+ (stream-map sign-change-detector sense-data <EXPRESSION>))
+
+ Complete the program by supplying the indicated <EXPRESSION>.
+
+ *Exercise 3.75:* Unfortunately, Alyssa's zero-crossing detector in
+ *Note Exercise 3-74:: proves to be insufficient, because the noisy
+ signal from the sensor leads to spurious zero crossings. Lem E.
+ Tweakit, a hardware specialist, suggests that Alyssa smooth the
+ signal to filter out the noise before extracting the zero
+ crossings. Alyssa takes his advice and decides to extract the
+ zero crossings from the signal constructed by averaging each value
+ of the sense data with the previous value. She explains the
+ problem to her assistant, Louis Reasoner, who attempts to
+ implement the idea, altering Alyssa's program as follows:
+
+ (define (make-zero-crossings input-stream last-value)
+ (let ((avpt (/ (+ (stream-car input-stream) last-value) 2)))
+ (cons-stream (sign-change-detector avpt last-value)
+ (make-zero-crossings (stream-cdr input-stream)
+ avpt))))
+
+ This does not correctly implement Alyssa's plan. Find the bug
+ that Louis has installed and fix it without changing the structure
+ of the program. (Hint: You will need to increase the number of
+ arguments to `make-zero-crossings'.)
+
+ *Exercise 3.76:* Eva Lu Ator has a criticism of Louis's approach
+ in *Note Exercise 3-75::. The program he wrote is not modular,
+ because it intermixes the operation of smoothing with the
+ zero-crossing extraction. For example, the extractor should not
+ have to be changed if Alyssa finds a better way to condition her
+ input signal. Help Louis by writing a procedure `smooth' that
+ takes a stream as input and produces a stream in which each
+ element is the average of two successive input stream elements.
+ Then use `smooth' as a component to implement the zero-crossing
+ detector in a more modular style.
+
+ ---------- Footnotes ----------
+
+ (1) We can't use `let' to bind the local variable `guesses', because
+the value of `guesses' depends on `guesses' itself. *Note Exercise
+3-63:: addresses why we want a local variable here.
+
+ (2) As in section *Note 2-2-3::, we represent a pair of integers as
+a list rather than a Lisp pair.
+
+ (3) See *Note Exercise 3-68:: for some insight into why we chose
+this decomposition.
+
+ (4) The precise statement of the required property on the order of
+combination is as follows: There should be a function f of two
+arguments such that the pair corresponding to element i of the first
+stream and element j of the second stream will appear as element number
+f(i,j) of the output stream. The trick of using `interleave' to
+accomplish this was shown to us by David Turner, who employed it in the
+language KRC (Turner 1981).
+
+ (5) We will require that the weighting function be such that the
+weight of a pair increases as we move out along a row or down along a
+column of the array of pairs.
+
+ (6) To quote from G. H. Hardy's obituary of Ramanujan (Hardy 1921):
+"It was Mr. Littlewood (I believe) who remarked that `every positive
+integer was one of his friends.' I remember once going to see him when
+he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and
+remarked that the number seemed to me a rather dull one, and that I
+hoped it was not an unfavorable omen. `No,' he replied, `it is a very
+interesting number; it is the smallest number expressible as the sum of
+two cubes in two different ways.' " The trick of using weighted pairs
+to generate the Ramanujan numbers was shown to us by Charles Leiserson.
+
+\1f
+File: sicp.info, Node: 3-5-4, Next: 3-5-5, Prev: 3-5-3, Up: 3-5
+
+3.5.4 Streams and Delayed Evaluation
+------------------------------------
+
+The `integral' procedure at the end of the preceding section shows how
+we can use streams to model signal-processing systems that contain
+feedback loops. The feedback loop for the adder shown in *Note Figure
+3-32:: is modeled by the fact that `integral''s internal stream `int'
+is defined in terms of itself:
+
+ (define int
+ (cons-stream initial-value
+ (add-streams (scale-stream integrand dt)
+ int)))
+
+ The interpreter's ability to deal with such an implicit definition
+depends on the `delay' that is incorporated into `cons-stream'.
+Without this `delay', the interpreter could not construct `int' before
+evaluating both arguments to `cons-stream', which would require that
+`int' already be defined. In general, `delay' is crucial for using
+streams to model signal-processing systems that contain loops. Without
+`delay', our models would have to be formulated so that the inputs to
+any signal-processing component would be fully evaluated before the
+output could be produced. This would outlaw loops.
+
+ Unfortunately, stream models of systems with loops may require uses
+of `delay' beyond the "hidden" `delay' supplied by `cons-stream'. For
+instance, *Note Figure 3-34:: shows a signal-processing system for
+solving the differential equation dy/dt = f(y) where f is a given
+function. The figure shows a mapping component, which applies f to its
+input signal, linked in a feedback loop to an integrator in a manner
+very similar to that of the analog computer circuits that are actually
+used to solve such equations.
+
+ *Figure 3.34:* An "analog computer circuit" that solves the
+ equation dy/dt = f(y).
+
+ y_0
+ |
+ V
+ +----------+ dy +----------+ y
+ +-->| map: f +------>| integral +--*----->
+ | +----------+ +----------+ |
+ | |
+ +------------------------------------+
+
+ Assuming we are given an initial value y_0 for y, we could try to
+model this system using the procedure
+
+ (define (solve f y0 dt)
+ (define y (integral dy y0 dt))
+ (define dy (stream-map f y))
+ y)
+
+ This procedure does not work, because in the first line of `solve'
+the call to `integral' requires that the input `dy' be defined, which
+does not happen until the second line of `solve'.
+
+ On the other hand, the intent of our definition does make sense,
+because we can, in principle, begin to generate the `y' stream without
+knowing `dy'. Indeed, `integral' and many other stream operations have
+properties similar to those of `cons-stream', in that we can generate
+part of the answer given only partial information about the arguments.
+For `integral', the first element of the output stream is the specified
+`initial-value'. Thus, we can generate the first element of the output
+stream without evaluating the integrand `dy'. Once we know the first
+element of `y', the `stream-map' in the second line of `solve' can
+begin working to generate the first element of `dy', which will produce
+the next element of `y', and so on.
+
+ To take advantage of this idea, we will redefine `integral' to
+expect the integrand stream to be a "delayed argument". `Integral' will
+`force' the integrand to be evaluated only when it is required to
+generate more than the first element of the output stream:
+
+ (define (integral delayed-integrand initial-value dt)
+ (define int
+ (cons-stream initial-value
+ (let ((integrand (force delayed-integrand)))
+ (add-streams (scale-stream integrand dt)
+ int))))
+ int)
+
+ Now we can implement our `solve' procedure by delaying the
+evaluation of `dy' in the definition of `y':(1)
+
+ (define (solve f y0 dt)
+ (define y (integral (delay dy) y0 dt))
+ (define dy (stream-map f y))
+ y)
+
+ In general, every caller of `integral' must now `delay' the integrand
+argument. We can demonstrate that the `solve' procedure works by
+approximating eapprox 2.718 by computing the value at y = 1 of the
+solution to the differential equation dy/dt = y with initial condition
+y(0) = 1:
+
+ (stream-ref (solve (lambda (y) y) 1 0.001) 1000)
+ 2.716924
+
+ *Exercise 3.77:* The `integral' procedure used above was analogous
+ to the "implicit" definition of the infinite stream of integers in
+ section *Note 3-5-2::. Alternatively, we can give a definition of
+ `integral' that is more like `integers-starting-from' (also in
+ section *Note 3-5-2::):
+
+ (define (integral integrand initial-value dt)
+ (cons-stream initial-value
+ (if (stream-null? integrand)
+ the-empty-stream
+ (integral (stream-cdr integrand)
+ (+ (* dt (stream-car integrand))
+ initial-value)
+ dt))))
+
+ When used in systems with loops, this procedure has the same
+ problem as does our original version of `integral'. Modify the
+ procedure so that it expects the `integrand' as a delayed argument
+ and hence can be used in the `solve' procedure shown above.
+
+ *Figure 3.35:* Signal-flow diagram for the solution to a
+ second-order linear differential equation.
+
+ dy_0 y_0
+ | |
+ V V
+ ddy +----------+ dy +----------+ y
+ +--------->| integral +-----*--+ integral +--*--->
+ | +----------+ | +----------+ |
+ | | |
+ | +----------+ | |
+ | __/|<--+ scale: a |<--+ |
+ | _/ | +----------+ |
+ +--<_add | |
+ \__ | +----------+ |
+ \|<--+ scale: b |<-------------------+
+ +----------+
+
+ *Exercise 3.78:* Consider the problem of designing a
+ signal-processing system to study the homogeneous second-order
+ linear differential equation
+
+ d^2 y d y
+ ----- - a ----- - by = 0
+ d t^2 d t
+
+ The output stream, modeling y, is generated by a network that
+ contains a loop. This is because the value of d^2y/dt^2 depends
+ upon the values of y and dy/dt and both of these are determined by
+ integrating d^2y/dt^2. The diagram we would like to encode is
+ shown in *Note Figure 3-35::. Write a procedure `solve-2nd' that
+ takes as arguments the constants a, b, and dt and the initial
+ values y_0 and dy_0 for y and dy/dt and generates the stream of
+ successive values of y.
+
+ *Exercise 3.79:* Generalize the `solve-2nd' procedure of *Note
+ Exercise 3-78:: so that it can be used to solve general
+ second-order differential equations d^2 y/dt^2 = f(dy/dt, y).
+
+ *Exercise 3.80:* A "series RLC circuit" consists of a resistor, a
+ capacitor, and an inductor connected in series, as shown in *Note
+ Figure 3-36::. If R, L, and C are the resistance, inductance, and
+ capacitance, then the relations between voltage (v) and current
+ (i) for the three components are described by the equations
+
+ v_R = i_R R
+
+ d_(i L)
+ v_L = L ---------
+ d t
+
+ d v_C
+ i_C = C -------
+ d t
+
+ and the circuit connections dictate the relations
+
+ i_R = i_L = -i_C
+
+ v_C = v_L + v_R
+
+ Combining these equations shows that the state of the circuit
+ (summarized by v_C, the voltage across the capacitor, and i_L, the
+ current in the inductor) is described by the pair of differential
+ equations
+
+ d v_C i_L
+ ----- = - ---
+ d t C
+
+ d i_L 1 R
+ ----- = --- v_C - --- i_L
+ d t L L
+
+ The signal-flow diagram representing this system of differential
+ equations is shown in *Note Figure 3-37::.
+
+ *Figure 3.36:* A series RLC circuit.
+ + v_R -
+ i_R
+ +--->----'\/\/\,--------+
+ | | i_L
+ \|/ R \|/
+ + | i_C |_ +
+ -+- __)
+ v_C -+- C (_) v_L
+ | __)
+ - | | -
+ +-----------------------+
+
+ *Figure 3.37:* A signal-flow diagram for the solution to a series
+ RLC circuit.
+
+ +-------------+
+ +----------------+ scale: l/L |<--+
+ | +-------------+ |
+ | |
+ | +-------------+ | v_C
+ | dv_C +-->| integral +---*------>
+ | | +-------------+
+ | | ^
+ | | | v_(C_0)
+ | |
+ | | +-------------+
+ | +---+ scale: -l/C |<--+
+ | +-------------+ |
+ | |\__ |
+ +->| \_ di_L +-------------+ | i_L
+ | add_>------>| integral +---*------>
+ +->| __/ +-------------+ |
+ | |/ ^ |
+ | | i_(L_0) |
+ | |
+ | +-------------+ |
+ +----------------+ scale: -R/L |<--+
+ +-------------+
+
+ Write a procedure `RLC' that takes as arguments the parameters R, L,
+and C of the circuit and the time increment dt. In a manner similar to
+that of the `RC' procedure of *Note Exercise 3-73::, `RLC' should
+produce a procedure that takes the initial values of the state
+variables, v_(C_0) and i_(L_0), and produces a pair (using `cons') of
+the streams of states v_C and i_L. Using `RLC', generate the pair of
+streams that models the behavior of a series RLC circuit with R = 1
+ohm, C = 0.2 farad, L = 1 henry, dt = 0.1 second, and initial values
+i_(L_0) = 0 amps and v_(C_0) = 10 volts.
+
+Normal-order evaluation
+.......................
+
+The examples in this section illustrate how the explicit use of `delay'
+and `force' provides great programming flexibility, but the same
+examples also show how this can make our programs more complex. Our
+new `integral' procedure, for instance, gives us the power to model
+systems with loops, but we must now remember that `integral' should be
+called with a delayed integrand, and every procedure that uses
+`integral' must be aware of this. In effect, we have created two
+classes of procedures: ordinary procedures and procedures that take
+delayed arguments. In general, creating separate classes of procedures
+forces us to create separate classes of higher-order procedures as
+well.(2)
+
+ One way to avoid the need for two different classes of procedures is
+to make all procedures take delayed arguments. We could adopt a model
+of evaluation in which all arguments to procedures are automatically
+delayed and arguments are forced only when they are actually needed
+(for example, when they are required by a primitive operation). This
+would transform our language to use normal-order evaluation, which we
+first described when we introduced the substitution model for
+evaluation in section *Note 1-1-5::. Converting to normal-order
+evaluation provides a uniform and elegant way to simplify the use of
+delayed evaluation, and this would be a natural strategy to adopt if we
+were concerned only with stream processing. In section *Note 4-2::,
+after we have studied the evaluator, we will see how to transform our
+language in just this way. Unfortunately, including delays in
+procedure calls wreaks havoc with our ability to design programs that
+depend on the order of events, such as programs that use assignment,
+mutate data, or perform input or output. Even the single `delay' in
+`cons-stream' can cause great confusion, as illustrated by *Note
+Exercise 3-51:: and *Note Exercise 3-52::. As far as anyone knows,
+mutability and delayed evaluation do not mix well in programming
+languages, and devising ways to deal with both of these at once is an
+active area of research.
+
+ ---------- Footnotes ----------
+
+ (1) This procedure is not guaranteed to work in all Scheme
+implementations, although for any implementation there is a simple
+variation that will work. The problem has to do with subtle
+differences in the ways that Scheme implementations handle internal
+definitions. (See section *Note 4-1-6::.)
+
+ (2) This is a small reflection, in Lisp, of the difficulties that
+conventional strongly typed languages such as Pascal have in coping with
+higher-order procedures. In such languages, the programmer must
+specify the data types of the arguments and the result of each
+procedure: number, logical value, sequence, and so on. Consequently,
+we could not express an abstraction such as "map a given procedure
+`proc' over all the elements in a sequence" by a single higher-order
+procedure such as `stream-map'. Rather, we would need a different
+mapping procedure for each different combination of argument and result
+data types that might be specified for a `proc'. Maintaining a
+practical notion of "data type" in the presence of higher-order
+procedures raises many difficult issues. One way of dealing with this
+problem is illustrated by the language ML (Gordon, Milner, and
+Wadsworth 1979), whose "polymorphic data types" include templates for
+higher-order transformations between data types. Moreover, data types
+for most procedures in ML are never explicitly declared by the
+programmer. Instead, ML includes a "type-inferencing" mechanism that
+uses information in the environment to deduce the data types for newly
+defined procedures.
+
+\1f
+File: sicp.info, Node: 3-5-5, Prev: 3-5-4, Up: 3-5
+
+3.5.5 Modularity of Functional Programs and Modularity of Objects
+-----------------------------------------------------------------
+
+As we saw in section *Note 3-1-2::, one of the major benefits of
+introducing assignment is that we can increase the modularity of our
+systems by encapsulating, or "hiding," parts of the state of a large
+system within local variables. Stream models can provide an equivalent
+modularity without the use of assignment. As an illustration, we can
+reimplement the Monte Carlo estimation of [pi], which we examined in
+section *Note 3-1-2::, from a stream-processing point of view.
+
+ The key modularity issue was that we wished to hide the internal
+state of a random-number generator from programs that used random
+numbers. We began with a procedure `rand-update', whose successive
+values furnished our supply of random numbers, and used this to produce
+a random-number generator:
+
+ (define rand
+ (let ((x random-init))
+ (lambda ()
+ (set! x (rand-update x))
+ x)))
+
+ In the stream formulation there is no random-number generator _per
+se_, just a stream of random numbers produced by successive calls to
+`rand-update':
+
+ (define random-numbers
+ (cons-stream random-init
+ (stream-map rand-update random-numbers)))
+
+ We use this to construct the stream of outcomes of the Cesa`ro
+experiment performed on consecutive pairs in the `random-numbers'
+stream:
+
+ (define cesaro-stream
+ (map-successive-pairs (lambda (r1 r2) (= (gcd r1 r2) 1))
+ random-numbers))
+
+ (define (map-successive-pairs f s)
+ (cons-stream
+ (f (stream-car s) (stream-car (stream-cdr s)))
+ (map-successive-pairs f (stream-cdr (stream-cdr s)))))
+
+ The `cesaro-stream' is now fed to a `monte-carlo' procedure, which
+produces a stream of estimates of probabilities. The results are then
+converted into a stream of estimates of [pi]. This version of the
+program doesn't need a parameter telling how many trials to perform.
+Better estimates of [pi] (from performing more experiments) are
+obtained by looking farther into the `pi' stream:
+
+ (define (monte-carlo experiment-stream passed failed)
+ (define (next passed failed)
+ (cons-stream
+ (/ passed (+ passed failed))
+ (monte-carlo
+ (stream-cdr experiment-stream) passed failed)))
+ (if (stream-car experiment-stream)
+ (next (+ passed 1) failed)
+ (next passed (+ failed 1))))
+
+ (define pi
+ (stream-map (lambda (p) (sqrt (/ 6 p)))
+ (monte-carlo cesaro-stream 0 0)))
+
+ There is considerable modularity in this approach, because we still
+can formulate a general `monte-carlo' procedure that can deal with
+arbitrary experiments. Yet there is no assignment or local state.
+
+ *Exercise 3.81:* *Note Exercise 3-6:: discussed generalizing the
+ random-number generator to allow one to reset the random-number
+ sequence so as to produce repeatable sequences of "random"
+ numbers. Produce a stream formulation of this same generator that
+ operates on an input stream of requests to `generate' a new random
+ number or to `reset' the sequence to a specified value and that
+ produces the desired stream of random numbers. Don't use
+ assignment in your solution.
+
+ *Exercise 3.82:* Redo *Note Exercise 3-5:: on Monte Carlo
+ integration in terms of streams. The stream version of
+ `estimate-integral' will not have an argument telling how many
+ trials to perform. Instead, it will produce a stream of estimates
+ based on successively more trials.
+
+A functional-programming view of time
+.....................................
+
+Let us now return to the issues of objects and state that were raised
+at the beginning of this chapter and examine them in a new light. We
+introduced assignment and mutable objects to provide a mechanism for
+modular construction of programs that model systems with state. We
+constructed computational objects with local state variables and used
+assignment to modify these variables. We modeled the temporal behavior
+of the objects in the world by the temporal behavior of the
+corresponding computational objects.
+
+ Now we have seen that streams provide an alternative way to model
+objects with local state. We can model a changing quantity, such as
+the local state of some object, using a stream that represents the time
+history of successive states. In essence, we represent time
+explicitly, using streams, so that we decouple time in our simulated
+world from the sequence of events that take place during evaluation.
+Indeed, because of the presence of `delay' there may be little relation
+between simulated time in the model and the order of events during the
+evaluation.
+
+ In order to contrast these two approaches to modeling, let us
+reconsider the implementation of a "withdrawal processor" that monitors
+the balance in a bank account. In section *Note 3-1-3:: we implemented
+a simplified version of such a processor:
+
+ (define (make-simplified-withdraw balance)
+ (lambda (amount)
+ (set! balance (- balance amount))
+ balance))
+
+ Calls to `make-simplified-withdraw' produce computational objects,
+each with a local state variable `balance' that is decremented by
+successive calls to the object. The object takes an `amount' as an
+argument and returns the new balance. We can imagine the user of a
+bank account typing a sequence of inputs to such an object and
+observing the sequence of returned values shown on a display screen.
+
+ Alternatively, we can model a withdrawal processor as a procedure
+that takes as input a balance and a stream of amounts to withdraw and
+produces the stream of successive balances in the account:
+
+ (define (stream-withdraw balance amount-stream)
+ (cons-stream
+ balance
+ (stream-withdraw (- balance (stream-car amount-stream))
+ (stream-cdr amount-stream))))
+
+ `Stream-withdraw' implements a well-defined mathematical function
+whose output is fully determined by its input. Suppose, however, that
+the input `amount-stream' is the stream of successive values typed by
+the user and that the resulting stream of balances is displayed. Then,
+from the perspective of the user who is typing values and watching
+results, the stream process has the same behavior as the object created
+by `make-simplified-withdraw'. However, with the stream version, there
+is no assignment, no local state variable, and consequently none of the
+theoretical difficulties that we encountered in section *Note 3-1-3::.
+Yet the system has state!
+
+ This is really remarkable. Even though `stream-withdraw' implements
+a well-defined mathematical function whose behavior does not change,
+the user's perception here is one of interacting with a system that has
+a changing state. One way to resolve this paradox is to realize that
+it is the user's temporal existence that imposes state on the system.
+If the user could step back from the interaction and think in terms of
+streams of balances rather than individual transactions, the system
+would appear stateless.(1)
+
+ From the point of view of one part of a complex process, the other
+parts appear to change with time. They have hidden time-varying local
+state. If we wish to write programs that model this kind of natural
+decomposition in our world (as we see it from our viewpoint as a part
+of that world) with structures in our computer, we make computational
+objects that are not functional--they must change with time. We model
+state with local state variables, and we model the changes of state
+with assignments to those variables. By doing this we make the time of
+execution of a computation model time in the world that we are part of,
+and thus we get "objects" in our computer.
+
+ Modeling with objects is powerful and intuitive, largely because
+this matches the perception of interacting with a world of which we are
+part. However, as we've seen repeatedly throughout this chapter, these
+models raise thorny problems of constraining the order of events and of
+synchronizing multiple processes. The possibility of avoiding these
+problems has stimulated the development of "functional programming
+languages", which do not include any provision for assignment or
+mutable data. In such a language, all procedures implement
+well-defined mathematical functions of their arguments, whose behavior
+does not change. The functional approach is extremely attractive for
+dealing with concurrent systems.(2)
+
+ On the other hand, if we look closely, we can see time-related
+problems creeping into functional models as well. One particularly
+troublesome area arises when we wish to design interactive systems,
+especially ones that model interactions between independent entities.
+For instance, consider once more the implementation a banking system
+that permits joint bank accounts. In a conventional system using
+assignment and objects, we would model the fact that Peter and Paul
+share an account by having both Peter and Paul send their transaction
+requests to the same bank-account object, as we saw in section *Note
+3-1-3::. From the stream point of view, where there are no "objects"
+_per se_, we have already indicated that a bank account can be modeled
+as a process that operates on a stream of transaction requests to
+produce a stream of responses. Accordingly, we could model the fact
+that Peter and Paul have a joint bank account by merging Peter's stream
+of transaction requests with Paul's stream of requests and feeding the
+result to the bank-account stream process, as shown in *Note Figure
+3-38::.
+
+ *Figure 3.38:* A joint bank account, modeled by merging two
+ streams of transaction requests.
+
+ Peter's requests +---------+ +---------+
+ ------------------>| | | |
+ Paul's requests | merge |---->| bank |---->
+ ------------------>| | | account |
+ +---------+ +---------+
+
+ The trouble with this formulation is in the notion of "merge". It
+will not do to merge the two streams by simply taking alternately one
+request from Peter and one request from Paul. Suppose Paul accesses the
+account only very rarely. We could hardly force Peter to wait for Paul
+to access the account before he could issue a second transaction.
+However such a merge is implemented, it must interleave the two
+transaction streams in some way that is constrained by "real time" as
+perceived by Peter and Paul, in the sense that, if Peter and Paul meet,
+they can agree that certain transactions were processed before the
+meeting, and other transactions were processed after the meeting.(3)
+This is precisely the same constraint that we had to deal with in
+section *Note 3-4-1::, where we found the need to introduce explicit
+synchronization to ensure a "correct" order of events in concurrent
+processing of objects with state. Thus, in an attempt to support the
+functional style, the need to merge inputs from different agents
+reintroduces the same problems that the functional style was meant to
+eliminate.
+
+ We began this chapter with the goal of building computational models
+whose structure matches our perception of the real world we are trying
+to model. We can model the world as a collection of separate,
+time-bound, interacting objects with state, or we can model the world
+as a single, timeless, stateless unity. Each view has powerful
+advantages, but neither view alone is completely satisfactory. A grand
+unification has yet to emerge.(4)
+
+ ---------- Footnotes ----------
+
+ (1) Similarly in physics, when we observe a moving particle, we say
+that the position (state) of the particle is changing. However, from
+the perspective of the particle's world line in space-time there is no
+change involved.
+
+ (2) John Backus, the inventor of Fortran, gave high visibility to
+functional programming when he was awarded the ACM Turing award in
+1978. His acceptance speech (Backus 1978) strongly advocated the
+functional approach. A good overview of functional programming is
+given in Henderson 1980 and in Darlington, Henderson, and Turner 1982.
+
+ (3) Observe that, for any two streams, there is in general more than
+one acceptable order of interleaving. Thus, technically, "merge" is a
+relation rather than a function--the answer is not a deterministic
+function of the inputs. We already mentioned (*Note Footnote 39::)
+that nondeterminism is essential when dealing with concurrency. The
+merge relation illustrates the same essential nondeterminism, from the
+functional perspective. In section *Note 4-3::, we will look at
+nondeterminism from yet another point of view.
+
+ (4) The object model approximates the world by dividing it into
+separate pieces. The functional model does not modularize along object
+boundaries. The object model is useful when the unshared state of the
+"objects" is much larger than the state that they share. An example of
+a place where the object viewpoint fails is quantum mechanics, where
+thinking of things as individual particles leads to paradoxes and
+confusions. Unifying the object view with the functional view may have
+little to do with programming, but rather with fundamental
+epistemological issues.
+
+\1f
+File: sicp.info, Node: Chapter 4, Next: Chapter 5, Prev: Chapter 3, Up: Top
+
+4 Metalinguistic Abstraction
+****************************
+
+ ... It's in words that the magic is--Abracadabra, Open Sesame, and
+ the rest--but the magic words in one story aren't magical in the
+ next. The real magic is to understand which words work, and when,
+ and for what; the trick is to learn the trick.
+
+ ... And those words are made from the letters of our alphabet: a
+ couple-dozen squiggles we can draw with the pen. This is the key!
+ And the treasure, too, if we can only get our hands on it! It's
+ as if--as if the key to the treasure _is_ the treasure!
+
+ --John Barth, `Chimera'
+
+ In our study of program design, we have seen that expert programmers
+control the complexity of their designs with the same general
+techniques used by designers of all complex systems. They combine
+primitive elements to form compound objects, they abstract compound
+objects to form higher-level building blocks, and they preserve
+modularity by adopting appropriate large-scale views of system
+structure. In illustrating these techniques, we have used Lisp as a
+language for describing processes and for constructing computational
+data objects and processes to model complex phenomena in the real
+world. However, as we confront increasingly complex problems, we will
+find that Lisp, or indeed any fixed programming language, is not
+sufficient for our needs. We must constantly turn to new languages in
+order to express our ideas more effectively. Establishing new
+languages is a powerful strategy for controlling complexity in
+engineering design; we can often enhance our ability to deal with a
+complex problem by adopting a new language that enables us to describe
+(and hence to think about) the problem in a different way, using
+primitives, means of combination, and means of abstraction that are
+particularly well suited to the problem at hand.(1)
+
+ Programming is endowed with a multitude of languages. There are
+physical languages, such as the machine languages for particular
+computers. These languages are concerned with the representation of
+data and control in terms of individual bits of storage and primitive
+machine instructions. The machine-language programmer is concerned
+with using the given hardware to erect systems and utilities for the
+efficient implementation of resource-limited computations. High-level
+languages, erected on a machine-language substrate, hide concerns about
+the representation of data as collections of bits and the
+representation of programs as sequences of primitive instructions.
+These languages have means of combination and abstraction, such as
+procedure definition, that are appropriate to the larger-scale
+organization of systems.
+
+ "Metalinguistic abstraction"--establishing new languages--plays an
+important role in all branches of engineering design. It is
+particularly important to computer programming, because in programming
+not only can we formulate new languages but we can also implement these
+languages by constructing evaluators. An "evaluator" (or "interpreter")
+for a programming language is a procedure that, when applied to an
+expression of the language, performs the actions required to evaluate
+that expression.
+
+ It is no exaggeration to regard this as the most fundamental idea in
+programming:
+
+ The evaluator, which determines the meaning of expressions in a
+ programming language, is just another program.
+
+ To appreciate this point is to change our images of ourselves as
+programmers. We come to see ourselves as designers of languages,
+rather than only users of languages designed by others.
+
+ In fact, we can regard almost any program as the evaluator for some
+language. For instance, the polynomial manipulation system of section
+*Note 2-5-3:: embodies the rules of polynomial arithmetic and
+implements them in terms of operations on list-structured data. If we
+augment this system with procedures to read and print polynomial
+expressions, we have the core of a special-purpose language for dealing
+with problems in symbolic mathematics. The digital-logic simulator of
+section *Note 3-3-4:: and the constraint propagator of section *Note
+3-3-5:: are legitimate languages in their own right, each with its own
+primitives, means of combination, and means of abstraction. Seen from
+this perspective, the technology for coping with large-scale computer
+systems merges with the technology for building new computer languages,
+and computer science itself becomes no more (and no less) than the
+discipline of constructing appropriate descriptive languages.
+
+ We now embark on a tour of the technology by which languages are
+established in terms of other languages. In this chapter we shall use
+Lisp as a base, implementing evaluators as Lisp procedures. Lisp is
+particularly well suited to this task, because of its ability to
+represent and manipulate symbolic expressions. We will take the first
+step in understanding how languages are implemented by building an
+evaluator for Lisp itself. The language implemented by our evaluator
+will be a subset of the Scheme dialect of Lisp that we use in this
+book. Although the evaluator described in this chapter is written for a
+particular dialect of Lisp, it contains the essential structure of an
+evaluator for any expression-oriented language designed for writing
+programs for a sequential machine. (In fact, most language processors
+contain, deep within them, a little "Lisp" evaluator.) The evaluator
+has been simplified for the purposes of illustration and discussion,
+and some features have been left out that would be important to include
+in a production-quality Lisp system. Nevertheless, this simple
+evaluator is adequate to execute most of the programs in this book.(2)
+
+ An important advantage of making the evaluator accessible as a Lisp
+program is that we can implement alternative evaluation rules by
+describing these as modifications to the evaluator program. One place
+where we can use this power to good effect is to gain extra control
+over the ways in which computational models embody the notion of time,
+which was so central to the discussion in *Note Chapter 3::. There, we
+mitigated some of the complexities of state and assignment by using
+streams to decouple the representation of time in the world from time
+in the computer. Our stream programs, however, were sometimes
+cumbersome, because they were constrained by the applicative-order
+evaluation of Scheme. In section *Note 4-2::, we'll change the
+underlying language to provide for a more elegant approach, by
+modifying the evaluator to provide for "normal-order evaluation".
+
+ Section *Note 4-3:: implements a more ambitious linguistic change,
+whereby expressions have many values, rather than just a single value.
+In this language of "nondeterministic computing", it is natural to
+express processes that generate all possible values for expressions and
+then search for those values that satisfy certain constraints. In
+terms of models of computation and time, this is like having time
+branch into a set of "possible futures" and then searching for
+appropriate time lines. With our nondeterministic evaluator, keeping
+track of multiple values and performing searches are handled
+automatically by the underlying mechanism of the language.
+
+ In section *Note 4-4:: we implement a "logic-programming" language in
+which knowledge is expressed in terms of relations, rather than in
+terms of computations with inputs and outputs. Even though this makes
+the language drastically different from Lisp, or indeed from any
+conventional language, we will see that the logic-programming evaluator
+shares the essential structure of the Lisp evaluator.
+
+* Menu:
+
+* 4-1:: The Metacircular Evaluator
+* 4-2:: Variations on a Scheme -- Lazy Evaluation
+* 4-3:: Variations on a Scheme -- Nondeterministic Computing
+* 4-4:: Logic Programming
+
+ ---------- Footnotes ----------
+
+ (1) The same idea is pervasive throughout all of engineering. For
+example, electrical engineers use many different languages for
+describing circuits. Two of these are the language of electrical "networks"
+and the language of electrical "systems". The network language
+emphasizes the physical modeling of devices in terms of discrete
+electrical elements. The primitive objects of the network language are
+primitive electrical components such as resistors, capacitors,
+inductors, and transistors, which are characterized in terms of
+physical variables called voltage and current. When describing
+circuits in the network language, the engineer is concerned with the
+physical characteristics of a design. In contrast, the primitive
+objects of the system language are signal-processing modules such as
+filters and amplifiers. Only the functional behavior of the modules is
+relevant, and signals are manipulated without concern for their
+physical realization as voltages and currents. The system language is
+erected on the network language, in the sense that the elements of
+signal-processing systems are constructed from electrical networks.
+Here, however, the concerns are with the large-scale organization of
+electrical devices to solve a given application problem; the physical
+feasibility of the parts is assumed. This layered collection of
+languages is another example of the stratified design technique
+illustrated by the picture language of section *Note 2-2-4::.
+
+ (2) The most important features that our evaluator leaves out are
+mechanisms for handling errors and supporting debugging. For a more
+extensive discussion of evaluators, see Friedman, Wand, and Haynes
+1992, which gives an exposition of programming languages that proceeds
+via a sequence of evaluators written in Scheme.
+
+\1f
+File: sicp.info, Node: 4-1, Next: 4-2, Prev: Chapter 4, Up: Chapter 4
+
+4.1 The Metacircular Evaluator
+==============================
+
+Our evaluator for Lisp will be implemented as a Lisp program. It may
+seem circular to think about evaluating Lisp programs using an
+evaluator that is itself implemented in Lisp. However, evaluation is a
+process, so it is appropriate to describe the evaluation process using
+Lisp, which, after all, is our tool for describing processes.(1) An
+evaluator that is written in the same language that it evaluates is
+said to be "metacircular".
+
+ The metacircular evaluator is essentially a Scheme formulation of the
+environment model of evaluation described in section *Note 3-2::.
+Recall that the model has two basic parts:
+
+ 1. To evaluate a combination (a compound expression other than a
+ special form), evaluate the subexpressions and then apply the
+ value of the operator subexpression to the values of the operand
+ subexpressions.
+
+ 2. To apply a compound procedure to a set of arguments, evaluate the
+ body of the procedure in a new environment. To construct this
+ environment, extend the environment part of the procedure object
+ by a frame in which the formal parameters of the procedure are
+ bound to the arguments to which the procedure is applied.
+
+
+ These two rules describe the essence of the evaluation process, a
+basic cycle in which expressions to be evaluated in environments are
+reduced to procedures to be applied to arguments, which in turn are
+reduced to new expressions to be evaluated in new environments, and so
+on, until we get down to symbols, whose values are looked up in the
+environment, and to primitive procedures, which are applied directly
+(see *Note Figure 4-1::).(2) This evaluation cycle will be embodied by
+the interplay between the two critical procedures in the evaluator,
+`eval' and `apply', which are described in section *Note 4-1-1:: (see
+*Note Figure 4-1::).
+
+ The implementation of the evaluator will depend upon procedures that
+define the "syntax" of the expressions to be evaluated. We will use
+data abstraction to make the evaluator independent of the
+representation of the language. For example, rather than committing to
+a choice that an assignment is to be represented by a list beginning
+with the symbol `set!' we use an abstract predicate `assignment?' to
+test for an assignment, and we use abstract selectors
+`assignment-variable' and `assignment-value' to access the parts of an
+assignment. Implementation of expressions will be described in detail
+in section *Note 4-1-2::. There are also operations, described in
+section *Note 4-1-3::, that specify the representation of procedures
+and environments. For example, `make-procedure' constructs compound
+procedures, `lookup-variable-value' accesses the values of variables,
+and `apply-primitive-procedure' applies a primitive procedure to a
+given list of arguments.
+
+* Menu:
+
+* 4-1-1:: The Core of the Evaluator
+* 4-1-2:: Representing Expressions
+* 4-1-3:: Evaluator Data Structures
+* 4-1-4:: Running the Evaluator as a Program
+* 4-1-5:: Data as Programs
+* 4-1-6:: Internal Definitions
+* 4-1-7:: Separating Syntactic Analysis from Execution
+
+ ---------- Footnotes ----------
+
+ (1) Even so, there will remain important aspects of the evaluation
+process that are not elucidated by our evaluator. The most important
+of these are the detailed mechanisms by which procedures call other
+procedures and return values to their callers. We will address these
+issues in *Note Chapter 5::, where we take a closer look at the
+evaluation process by implementing the evaluator as a simple register
+machine.
+
+ (2) If we grant ourselves the ability to apply primitives, then what
+remains for us to implement in the evaluator? The job of the evaluator
+is not to specify the primitives of the language, but rather to provide
+the connective tissue--the means of combination and the means of
+abstraction--that binds a collection of primitives to form a language.
+Specifically:
+
+ * The evaluator enables us to deal with nested expressions. For
+ example, although simply applying primitives would suffice for
+ evaluating the expression `(+ 1 6)', it is not adequate for
+ handling `(+ 1 (* 2 3))'. As far as the primitive procedure `+'
+ is concerned, its arguments must be numbers, and it would choke if
+ we passed it the expression `(* 2 3)' as an argument. One
+ important role of the evaluator is to choreograph procedure
+ composition so that `(* 2 3)' is reduced to 6 before being passed
+ as an argument to `+'.
+
+ * The evaluator allows us to use variables. For example, the
+ primitive procedure for addition has no way to deal with
+ expressions such as `(+ x 1)'. We need an evaluator to keep track
+ of variables and obtain their values before invoking the primitive
+ procedures.
+
+ * The evaluator allows us to define compound procedures. This
+ involves keeping track of procedure definitions, knowing how to
+ use these definitions in evaluating expressions, and providing a
+ mechanism that enables procedures to accept arguments.
+
+ * The evaluator provides the special forms, which must be evaluated
+ differently from procedure calls.
+
+
+\1f
+File: sicp.info, Node: 4-1-1, Next: 4-1-2, Prev: 4-1, Up: 4-1
+
+4.1.1 The Core of the Evaluator
+-------------------------------
+
+ *Figure 4.1:* The `eval'-`apply' cycle exposes the essence of a
+ computer language.
+
+ .,ad88888888baa,
+ _ ,d8P""" ""9888ba. _
+ / .a8" ,ad88888888888a |\
+ / aP' ,88888888888888888a \
+ / ,8" ,88888888888888888888, \
+ | ,8' (888888888888888888888, |
+ / ,8' `8888888888888888888888 \
+ | 8) `888888888888888888888, |
+ Procedure, | 8 "88888 Apply 8888888) | Expression
+ Arguments | 8 Eval `888888888888888888) | Environment
+ | 8) "8888888888888888 |
+ \ (b "88888888888888' /
+ | `8, 8888888888888) |
+ \ "8a ,888888888888) /
+ \ V8, d88888888888" /
+ _\| `8b, ,d8888888888P' _/
+ `V8a, ,ad8888888888P'
+ ""88888888888888888P"
+ """"""""""""
+
+ [graphic by Normand Veillux, modified]
+
+The evaluation process can be described as the interplay between two
+procedures: `eval' and `apply'.
+
+Eval
+....
+
+`Eval' takes as arguments an expression and an environment. It
+classifies the expression and directs its evaluation. `Eval' is
+structured as a case analysis of the syntactic type of the expression
+to be evaluated. In order to keep the procedure general, we express
+the determination of the type of an expression abstractly, making no
+commitment to any particular representation for the various types of
+expressions. Each type of expression has a predicate that tests for it
+and an abstract means for selecting its parts. This "abstract syntax"
+makes it easy to see how we can change the syntax of the language by
+using the same evaluator, but with a different collection of syntax
+procedures.
+
+Primitive expressions
+
+ * For self-evaluating expressions, such as numbers, `eval' returns
+ the expression itself.
+
+ * `Eval' must look up variables in the environment to find their
+ values.
+
+
+Special forms
+
+ * For quoted expressions, `eval' returns the expression that was
+ quoted.
+
+ * An assignment to (or a definition of) a variable must recursively
+ call `eval' to compute the new value to be associated with the
+ variable. The environment must be modified to change (or create)
+ the binding of the variable.
+
+ * An `if' expression requires special processing of its parts, so as
+ to evaluate the consequent if the predicate is true, and otherwise
+ to evaluate the alternative.
+
+ * A `lambda' expression must be transformed into an applicable
+ procedure by packaging together the parameters and body specified
+ by the `lambda' expression with the environment of the evaluation.
+
+ * A `begin' expression requires evaluating its sequence of
+ expressions in the order in which they appear.
+
+ * A case analysis (`cond') is transformed into a nest of `if'
+ expressions and then evaluated.
+
+
+Combinations
+
+ * For a procedure application, `eval' must recursively evaluate the
+ operator part and the operands of the combination. The resulting
+ procedure and arguments are passed to `apply', which handles the
+ actual procedure application.
+
+
+ Here is the definition of `eval':
+
+ (define (eval exp env)
+ (cond ((self-evaluating? exp) exp)
+ ((variable? exp) (lookup-variable-value exp env))
+ ((quoted? exp) (text-of-quotation exp))
+ ((assignment? exp) (eval-assignment exp env))
+ ((definition? exp) (eval-definition exp env))
+ ((if? exp) (eval-if exp env))
+ ((lambda? exp)
+ (make-procedure (lambda-parameters exp)
+ (lambda-body exp)
+ env))
+ ((begin? exp)
+ (eval-sequence (begin-actions exp) env))
+ ((cond? exp) (eval (cond->if exp) env))
+ ((application? exp)
+ (apply (eval (operator exp) env)
+ (list-of-values (operands exp) env)))
+ (else
+ (error "Unknown expression type -- EVAL" exp))))
+
+ For clarity, `eval' has been implemented as a case analysis using
+`cond'. The disadvantage of this is that our procedure handles only a
+few distinguishable types of expressions, and no new ones can be
+defined without editing the definition of `eval'. In most Lisp
+implementations, dispatching on the type of an expression is done in a
+data-directed style. This allows a user to add new types of
+expressions that `eval' can distinguish, without modifying the
+definition of `eval' itself. (See *Note Exercise 4-3::.)
+
+Apply
+.....
+
+`Apply' takes two arguments, a procedure and a list of arguments to
+which the procedure should be applied. `Apply' classifies procedures
+into two kinds: It calls `apply-primitive-procedure' to apply
+primitives; it applies compound procedures by sequentially evaluating
+the expressions that make up the body of the procedure. The
+environment for the evaluation of the body of a compound procedure is
+constructed by extending the base environment carried by the procedure
+to include a frame that binds the parameters of the procedure to the
+arguments to which the procedure is to be applied. Here is the
+definition of `apply':
+
+ (define (apply procedure arguments)
+ (cond ((primitive-procedure? procedure)
+ (apply-primitive-procedure procedure arguments))
+ ((compound-procedure? procedure)
+ (eval-sequence
+ (procedure-body procedure)
+ (extend-environment
+ (procedure-parameters procedure)
+ arguments
+ (procedure-environment procedure))))
+ (else
+ (error
+ "Unknown procedure type -- APPLY" procedure))))
+
+Procedure arguments
+...................
+
+When `eval' processes a procedure application, it uses `list-of-values'
+to produce the list of arguments to which the procedure is to be
+applied. `List-of-values' takes as an argument the operands of the
+combination. It evaluates each operand and returns a list of the
+corresponding values:(1)
+
+ (define (list-of-values exps env)
+ (if (no-operands? exps)
+ '()
+ (cons (eval (first-operand exps) env)
+ (list-of-values (rest-operands exps) env))))
+
+Conditionals
+............
+
+`Eval-if' evaluates the predicate part of an `if' expression in the
+given environment. If the result is true, `eval-if' evaluates the
+consequent, otherwise it evaluates the alternative:
+
+ (define (eval-if exp env)
+ (if (true? (eval (if-predicate exp) env))
+ (eval (if-consequent exp) env)
+ (eval (if-alternative exp) env)))
+
+ The use of `true?' in `eval-if' highlights the issue of the
+connection between an implemented language and an implementation
+language. The `if-predicate' is evaluated in the language being
+implemented and thus yields a value in that language. The interpreter
+predicate `true?' translates that value into a value that can be tested
+by the `if' in the implementation language: The metacircular
+representation of truth might not be the same as that of the underlying
+Scheme.(2)
+
+Sequences
+.........
+
+`Eval-sequence' is used by `apply' to evaluate the sequence of
+expressions in a procedure body and by `eval' to evaluate the sequence
+of expressions in a `begin' expression. It takes as arguments a
+sequence of expressions and an environment, and evaluates the
+expressions in the order in which they occur. The value returned is
+the value of the final expression.
+
+ (define (eval-sequence exps env)
+ (cond ((last-exp? exps) (eval (first-exp exps) env))
+ (else (eval (first-exp exps) env)
+ (eval-sequence (rest-exps exps) env))))
+
+Assignments and definitions
+...........................
+
+The following procedure handles assignments to variables. It calls
+`eval' to find the value to be assigned and transmits the variable and
+the resulting value to `set-variable-value!' to be installed in the
+designated environment.
+
+ (define (eval-assignment exp env)
+ (set-variable-value! (assignment-variable exp)
+ (eval (assignment-value exp) env)
+ env)
+ 'ok)
+
+Definitions of variables are handled in a similar manner.(3)
+
+ (define (eval-definition exp env)
+ (define-variable! (definition-variable exp)
+ (eval (definition-value exp) env)
+ env)
+ 'ok)
+
+ We have chosen here to return the symbol `ok' as the value of an
+assignment or a definition.(4)
+
+ *Exercise 4.1:* Notice that we cannot tell whether the
+ metacircular evaluator evaluates operands from left to right or
+ from right to left. Its evaluation order is inherited from the
+ underlying Lisp: If the arguments to `cons' in `list-of-values'
+ are evaluated from left to right, then `list-of-values' will
+ evaluate operands from left to right; and if the arguments to
+ `cons' are evaluated from right to left, then `list-of-values'
+ will evaluate operands from right to left.
+
+ Write a version of `list-of-values' that evaluates operands from
+ left to right regardless of the order of evaluation in the
+ underlying Lisp. Also write a version of `list-of-values' that
+ evaluates operands from right to left.
+
+ ---------- Footnotes ----------
+
+ (1) We could have simplified the `application?' clause in `eval' by
+using `map' (and stipulating that `operands' returns a list) rather
+than writing an explicit `list-of-values' procedure. We chose not to
+use `map' here to emphasize the fact that the evaluator can be
+implemented without any use of higher-order procedures (and thus could
+be written in a language that doesn't have higher-order procedures),
+even though the language that it supports will include higher-order
+procedures.
+
+ (2) In this case, the language being implemented and the
+implementation language are the same. Contemplation of the meaning of
+`true?' here yields expansion of consciousness without the abuse of
+substance.
+
+ (3) This implementation of `define' ignores a subtle issue in the
+handling of internal definitions, although it works correctly in most
+cases. We will see what the problem is and how to solve it in section
+*Note 4-1-6::.
+
+ (4) As we said when we introduced `define' and `set!', these values
+are implementation-dependent in Scheme--that is, the implementor can
+choose what value to return.
+
+\1f
+File: sicp.info, Node: 4-1-2, Next: 4-1-3, Prev: 4-1-1, Up: 4-1
+
+4.1.2 Representing Expressions
+------------------------------
+
+The evaluator is reminiscent of the symbolic differentiation program
+discussed in section *Note 2-3-2::. Both programs operate on symbolic
+expressions. In both programs, the result of operating on a compound
+expression is determined by operating recursively on the pieces of the
+expression and combining the results in a way that depends on the type
+of the expression. In both programs we used data abstraction to
+decouple the general rules of operation from the details of how
+expressions are represented. In the differentiation program this meant
+that the same differentiation procedure could deal with algebraic
+expressions in prefix form, in infix form, or in some other form. For
+the evaluator, this means that the syntax of the language being
+evaluated is determined solely by the procedures that classify and
+extract pieces of expressions.
+
+ Here is the specification of the syntax of our language:
+
+ * The only self-evaluating items are numbers and strings:
+
+ (define (self-evaluating? exp)
+ (cond ((number? exp) true)
+ ((string? exp) true)
+ (else false)))
+
+ * Variables are represented by symbols:
+
+ (define (variable? exp) (symbol? exp))
+
+ * Quotations have the form `(quote <TEXT-OF-QUOTATION>)':(1)
+
+ (define (quoted? exp)
+ (tagged-list? exp 'quote))
+
+ (define (text-of-quotation exp) (cadr exp))
+
+ `Quoted?' is defined in terms of the procedure `tagged-list?',
+ which identifies lists beginning with a designated symbol:
+
+ (define (tagged-list? exp tag)
+ (if (pair? exp)
+ (eq? (car exp) tag)
+ false))
+
+ * Assignments have the form `(set! <VAR> <VALUE>)':
+
+ (define (assignment? exp)
+ (tagged-list? exp 'set!))
+
+ (define (assignment-variable exp) (cadr exp))
+
+ (define (assignment-value exp) (caddr exp))
+
+ * Definitions have the form
+
+ (define <VAR> <VALUE>)
+
+ or the form
+
+ (define (<VAR> <PARAMETER_1> ... <PARAMETER_N>)
+ <BODY>)
+
+ The latter form (standard procedure definition) is syntactic sugar
+ for
+
+ (define <VAR>
+ (lambda (<PARAMETER_1> ... <PARAMETER_N>)
+ <BODY>))
+
+ The corresponding syntax procedures are the following:
+
+ (define (definition? exp)
+ (tagged-list? exp 'define))
+
+ (define (definition-variable exp)
+ (if (symbol? (cadr exp))
+ (cadr exp)
+ (caadr exp)))
+
+ (define (definition-value exp)
+ (if (symbol? (cadr exp))
+ (caddr exp)
+ (make-lambda (cdadr exp) ; formal parameters
+ (cddr exp)))) ; body
+
+ * `Lambda' expressions are lists that begin with the symbol `lambda':
+
+ (define (lambda? exp) (tagged-list? exp 'lambda))
+
+ (define (lambda-parameters exp) (cadr exp))
+
+ (define (lambda-body exp) (cddr exp))
+
+ We also provide a constructor for `lambda' expressions, which is
+ used by `definition-value', above:
+
+ (define (make-lambda parameters body)
+ (cons 'lambda (cons parameters body)))
+
+ * Conditionals begin with `if' and have a predicate, a consequent,
+ and an (optional) alternative. If the expression has no
+ alternative part, we provide `false' as the alternative.(2)
+
+ (define (if? exp) (tagged-list? exp 'if))
+
+ (define (if-predicate exp) (cadr exp))
+
+ (define (if-consequent exp) (caddr exp))
+
+ (define (if-alternative exp)
+ (if (not (null? (cdddr exp)))
+ (cadddr exp)
+ 'false))
+
+ We also provide a constructor for `if' expressions, to be used by
+ `cond->if' to transform `cond' expressions into `if' expressions:
+
+ (define (make-if predicate consequent alternative)
+ (list 'if predicate consequent alternative))
+
+ * `Begin' packages a sequence of expressions into a single
+ expression. We include syntax operations on `begin' expressions
+ to extract the actual sequence from the `begin' expression, as
+ well as selectors that return the first expression and the rest of
+ the expressions in the sequence.(3)
+
+ (define (begin? exp) (tagged-list? exp 'begin))
+
+ (define (begin-actions exp) (cdr exp))
+
+ (define (last-exp? seq) (null? (cdr seq)))
+
+ (define (first-exp seq) (car seq))
+
+ (define (rest-exps seq) (cdr seq))
+
+ We also include a constructor `sequence->exp' (for use by
+ `cond->if') that transforms a sequence into a single expression,
+ using `begin' if necessary:
+
+ (define (sequence->exp seq)
+ (cond ((null? seq) seq)
+ ((last-exp? seq) (first-exp seq))
+ (else (make-begin seq))))
+
+ (define (make-begin seq) (cons 'begin seq))
+
+ * A procedure application is any compound expression that is not one
+ of the above expression types. The `car' of the expression is the
+ operator, and the `cdr' is the list of operands:
+
+ (define (application? exp) (pair? exp))
+
+ (define (operator exp) (car exp))
+
+ (define (operands exp) (cdr exp))
+
+ (define (no-operands? ops) (null? ops))
+
+ (define (first-operand ops) (car ops))
+
+ (define (rest-operands ops) (cdr ops))
+
+
+Derived expressions
+...................
+
+Some special forms in our language can be defined in terms of
+expressions involving other special forms, rather than being
+implemented directly. One example is `cond', which can be implemented
+as a nest of `if' expressions. For example, we can reduce the problem
+of evaluating the expression
+
+ (cond ((> x 0) x)
+ ((= x 0) (display 'zero) 0)
+ (else (- x)))
+
+to the problem of evaluating the following expression involving `if' and
+`begin' expressions:
+
+ (if (> x 0)
+ x
+ (if (= x 0)
+ (begin (display 'zero)
+ 0)
+ (- x)))
+
+ Implementing the evaluation of `cond' in this way simplifies the
+evaluator because it reduces the number of special forms for which the
+evaluation process must be explicitly specified.
+
+ We include syntax procedures that extract the parts of a `cond'
+expression, and a procedure `cond->if' that transforms `cond'
+expressions into `if' expressions. A case analysis begins with `cond'
+and has a list of predicate-action clauses. A clause is an `else'
+clause if its predicate is the symbol `else'.(4)
+
+ (define (cond? exp) (tagged-list? exp 'cond))
+
+ (define (cond-clauses exp) (cdr exp))
+
+ (define (cond-else-clause? clause)
+ (eq? (cond-predicate clause) 'else))
+
+ (define (cond-predicate clause) (car clause))
+
+ (define (cond-actions clause) (cdr clause))
+
+ (define (cond->if exp)
+ (expand-clauses (cond-clauses exp)))
+
+ (define (expand-clauses clauses)
+ (if (null? clauses)
+ 'false ; no `else' clause
+ (let ((first (car clauses))
+ (rest (cdr clauses)))
+ (if (cond-else-clause? first)
+ (if (null? rest)
+ (sequence->exp (cond-actions first))
+ (error "ELSE clause isn't last -- COND->IF"
+ clauses))
+ (make-if (cond-predicate first)
+ (sequence->exp (cond-actions first))
+ (expand-clauses rest))))))
+
+ Expressions (such as `cond') that we choose to implement as syntactic
+transformations are called "derived expressions". `Let' expressions
+are also derived expressions (see *Note Exercise 4-6::).(5)
+
+ *Exercise 4.2:* Louis Reasoner plans to reorder the `cond' clauses
+ in `eval' so that the clause for procedure applications appears
+ before the clause for assignments. He argues that this will make
+ the interpreter more efficient: Since programs usually contain more
+ applications than assignments, definitions, and so on, his
+ modified `eval' will usually check fewer clauses than the original
+ `eval' before identifying the type of an expression.
+
+ a. What is wrong with Louis's plan? (Hint: What will Louis's
+ evaluator do with the expression `(define x 3)'?)
+
+ b. Louis is upset that his plan didn't work. He is willing to
+ go to any lengths to make his evaluator recognize procedure
+ applications before it checks for most other kinds of
+ expressions. Help him by changing the syntax of the
+ evaluated language so that procedure applications start with
+ `call'. For example, instead of `(factorial 3)' we will now
+ have to write `(call factorial 3)' and instead of `(+ 1 2)'
+ we will have to write `(call + 1 2)'.
+
+
+ *Exercise 4.3:* Rewrite `eval' so that the dispatch is done in
+ data-directed style. Compare this with the data-directed
+ differentiation procedure of *Note Exercise 2-73::. (You may use
+ the `car' of a compound expression as the type of the expression,
+ as is appropriate for the syntax implemented in this section.)
+
+ *Exercise 4.4:* Recall the definitions of the special forms `and'
+ and `or' from *Note Chapter 1:::
+
+ * `and': The expressions are evaluated from left to right. If
+ any expression evaluates to false, false is returned; any
+ remaining expressions are not evaluated. If all the
+ expressions evaluate to true values, the value of the last
+ expression is returned. If there are no expressions then
+ true is returned.
+
+ * `or': The expressions are evaluated from left to right. If
+ any expression evaluates to a true value, that value is
+ returned; any remaining expressions are not evaluated. If
+ all expressions evaluate to false, or if there are no
+ expressions, then false is returned.
+
+
+ Install `and' and `or' as new special forms for the evaluator by
+ defining appropriate syntax procedures and evaluation procedures
+ `eval-and' and `eval-or'. Alternatively, show how to implement
+ `and' and `or' as derived expressions.
+
+ *Exercise 4.5:* Scheme allows an additional syntax for `cond'
+ clauses, `(<TEST> => <RECIPIENT>)'. If <TEST> evaluates to a true
+ value, then <RECIPIENT> is evaluated. Its value must be a
+ procedure of one argument; this procedure is then invoked on the
+ value of the <TEST>, and the result is returned as the value of
+ the `cond' expression. For example
+
+ (cond ((assoc 'b '((a 1) (b 2))) => cadr)
+ (else false))
+
+ returns 2. Modify the handling of `cond' so that it supports this
+ extended syntax.
+
+ *Exercise 4.6:* `Let' expressions are derived expressions, because
+
+ (let ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
+ <BODY>)
+
+ is equivalent to
+
+ ((lambda (<VAR_1> ... <VAR_N>)
+ <BODY>)
+ <EXP_1>
+ ...
+ <EXP_N>)
+
+ Implement a syntactic transformation `let->combination' that
+ reduces evaluating `let' expressions to evaluating combinations of
+ the type shown above, and add the appropriate clause to `eval' to
+ handle `let' expressions.
+
+ *Exercise 4.7:* `Let*' is similar to `let', except that the
+ bindings of the `let' variables are performed sequentially from
+ left to right, and each binding is made in an environment in which
+ all of the preceding bindings are visible. For example
+
+ (let* ((x 3)
+ (y (+ x 2))
+ (z (+ x y 5)))
+ (* x z))
+
+ returns 39. Explain how a `let*' expression can be rewritten as a
+ set of nested `let' expressions, and write a procedure
+ `let*->nested-lets' that performs this transformation. If we have
+ already implemented `let' (*Note Exercise 4-6::) and we want to
+ extend the evaluator to handle `let*', is it sufficient to add a
+ clause to `eval' whose action is
+
+ (eval (let*->nested-lets exp) env)
+
+ or must we explicitly expand `let*' in terms of non-derived
+ expressions?
+
+ *Exercise 4.8:* "Named `let'" is a variant of `let' that has the
+ form
+
+ (let <VAR> <BINDINGS> <BODY>)
+
+ The <BINDINGS> and <BODY> are just as in ordinary `let', except
+ that <VAR> is bound within <BODY> to a procedure whose body is
+ <BODY> and whose parameters are the variables in the <BINDINGS>.
+ Thus, one can repeatedly execute the <BODY> by invoking the
+ procedure named <VAR>. For example, the iterative Fibonacci
+ procedure (section *Note 1-2-2::) can be rewritten using named
+ `let' as follows:
+
+ (define (fib n)
+ (let fib-iter ((a 1)
+ (b 0)
+ (count n))
+ (if (= count 0)
+ b
+ (fib-iter (+ a b) a (- count 1)))))
+
+ Modify `let->combination' of *Note Exercise 4-6:: to also support
+ named `let'.
+
+ *Exercise 4.9:* Many languages support a variety of iteration
+ constructs, such as `do', `for', `while', and `until'. In Scheme,
+ iterative processes can be expressed in terms of ordinary
+ procedure calls, so special iteration constructs provide no
+ essential gain in computational power. On the other hand, such
+ constructs are often convenient. Design some iteration
+ constructs, give examples of their use, and show how to implement
+ them as derived expressions.
+
+ *Exercise 4.10:* By using data abstraction, we were able to write
+ an `eval' procedure that is independent of the particular syntax
+ of the language to be evaluated. To illustrate this, design and
+ implement a new syntax for Scheme by modifying the procedures in
+ this section, without changing `eval' or `apply'.
+
+ ---------- Footnotes ----------
+
+ (1) As mentioned in section *Note 2-3-1::, the evaluator sees a
+quoted expression as a list beginning with `quote', even if the
+expression is typed with the quotation mark. For example, the
+expression `'a' would be seen by the evaluator as `(quote a)'. See
+*Note Exercise 2-55::.
+
+ (2) The value of an `if' expression when the predicate is false and
+there is no alternative is unspecified in Scheme; we have chosen here
+to make it false. We will support the use of the variables `true' and
+`false' in expressions to be evaluated by binding them in the global
+environment. See section *Note 4-1-4::.
+
+ (3) These selectors for a list of expressions--and the corresponding
+ones for a list of operands--are not intended as a data abstraction.
+They are introduced as mnemonic names for the basic list operations in
+order to make it easier to understand the explicit-control evaluator in
+section *Note 5-4::.
+
+ (4) The value of a `cond' expression when all the predicates are
+false and there is no `else' clause is unspecified in Scheme; we have
+chosen here to make it false.
+
+ (5) Practical Lisp systems provide a mechanism that allows a user to
+add new derived expressions and specify their implementation as
+syntactic transformations without modifying the evaluator. Such a
+user-defined transformation is called a "macro". Although it is easy
+to add an elementary mechanism for defining macros, the resulting
+language has subtle name-conflict problems. There has been much
+research on mechanisms for macro definition that do not cause these
+difficulties. See, for example, Kohlbecker 1986, Clinger and Rees
+1991, and Hanson 1991.
+
+\1f
+File: sicp.info, Node: 4-1-3, Next: 4-1-4, Prev: 4-1-2, Up: 4-1
+
+4.1.3 Evaluator Data Structures
+-------------------------------
+
+In addition to defining the external syntax of expressions, the
+evaluator implementation must also define the data structures that the
+evaluator manipulates internally, as part of the execution of a
+program, such as the representation of procedures and environments and
+the representation of true and false.
+
+Testing of predicates
+.....................
+
+For conditionals, we accept anything to be true that is not the explicit
+`false' object.
+
+ (define (true? x)
+ (not (eq? x false)))
+
+ (define (false? x)
+ (eq? x false))
+
+Representing procedures
+.......................
+
+To handle primitives, we assume that we have available the following
+procedures:
+
+ * `(apply-primitive-procedure <PROC> <ARGS>)'
+
+ applies the given primitive procedure to the argument values in
+ the list <ARGS> and returns the result of the application.
+
+ * `(primitive-procedure? <PROC>)'
+
+ tests whether <PROC> is a primitive procedure.
+
+
+ These mechanisms for handling primitives are further described in
+section *Note 4-1-4::.
+
+ Compound procedures are constructed from parameters, procedure
+bodies, and environments using the constructor `make-procedure':
+
+ (define (make-procedure parameters body env)
+ (list 'procedure parameters body env))
+
+ (define (compound-procedure? p)
+ (tagged-list? p 'procedure))
+
+ (define (procedure-parameters p) (cadr p))
+
+ (define (procedure-body p) (caddr p))
+
+ (define (procedure-environment p) (cadddr p))
+
+Operations on Environments
+..........................
+
+The evaluator needs operations for manipulating environments. As
+explained in section *Note 3-2::, an environment is a sequence of
+frames, where each frame is a table of bindings that associate
+variables with their corresponding values. We use the following
+operations for manipulating environments:
+
+ * `(lookup-variable-value <VAR> <ENV>)'
+
+ returns the value that is bound to the symbol <VAR> in the
+ environment <ENV>, or signals an error if the variable is unbound.
+
+ * `(extend-environment <VARIABLES> <VALUES> <BASE-ENV>)'
+
+ returns a new environment, consisting of a new frame in which the
+ symbols in the list <VARIABLES> are bound to the corresponding
+ elements in the list <VALUES>, where the enclosing environment is
+ the environment <BASE-ENV>.
+
+ * `(define-variable! <VAR> <VALUE> <ENV>)'
+
+ adds to the first frame in the environment <ENV> a new binding that
+ associates the variable <VAR> with the value <VALUE>.
+
+ * `(set-variable-value! <VAR> <VALUE> <ENV>)'
+
+ changes the binding of the variable <VAR> in the environment <ENV>
+ so that the variable is now bound to the value <VALUE>, or signals
+ an error if the variable is unbound.
+
+
+ To implement these operations we represent an environment as a list
+of frames. The enclosing environment of an environment is the `cdr' of
+the list. The empty environment is simply the empty list.
+
+ (define (enclosing-environment env) (cdr env))
+
+ (define (first-frame env) (car env))
+
+ (define the-empty-environment '())
+
+ Each frame of an environment is represented as a pair of lists: a
+list of the variables bound in that frame and a list of the associated
+values.(1)
+
+ (define (make-frame variables values)
+ (cons variables values))
+
+ (define (frame-variables frame) (car frame))
+
+ (define (frame-values frame) (cdr frame))
+
+ (define (add-binding-to-frame! var val frame)
+ (set-car! frame (cons var (car frame)))
+ (set-cdr! frame (cons val (cdr frame))))
+
+ To extend an environment by a new frame that associates variables
+with values, we make a frame consisting of the list of variables and
+the list of values, and we adjoin this to the environment. We signal
+an error if the number of variables does not match the number of values.
+
+ (define (extend-environment vars vals base-env)
+ (if (= (length vars) (length vals))
+ (cons (make-frame vars vals) base-env)
+ (if (< (length vars) (length vals))
+ (error "Too many arguments supplied" vars vals)
+ (error "Too few arguments supplied" vars vals))))
+
+ To look up a variable in an environment, we scan the list of
+variables in the first frame. If we find the desired variable, we
+return the corresponding element in the list of values. If we do not
+find the variable in the current frame, we search the enclosing
+environment, and so on. If we reach the empty environment, we signal
+an "unbound variable" error.
+
+ (define (lookup-variable-value var env)
+ (define (env-loop env)
+ (define (scan vars vals)
+ (cond ((null? vars)
+ (env-loop (enclosing-environment env)))
+ ((eq? var (car vars))
+ (car vals))
+ (else (scan (cdr vars) (cdr vals)))))
+ (if (eq? env the-empty-environment)
+ (error "Unbound variable" var)
+ (let ((frame (first-frame env)))
+ (scan (frame-variables frame)
+ (frame-values frame)))))
+ (env-loop env))
+
+ To set a variable to a new value in a specified environment, we scan
+for the variable, just as in `lookup-variable-value', and change the
+corresponding value when we find it.
+
+ (define (set-variable-value! var val env)
+ (define (env-loop env)
+ (define (scan vars vals)
+ (cond ((null? vars)
+ (env-loop (enclosing-environment env)))
+ ((eq? var (car vars))
+ (set-car! vals val))
+ (else (scan (cdr vars) (cdr vals)))))
+ (if (eq? env the-empty-environment)
+ (error "Unbound variable -- SET!" var)
+ (let ((frame (first-frame env)))
+ (scan (frame-variables frame)
+ (frame-values frame)))))
+ (env-loop env))
+
+ To define a variable, we search the first frame for a binding for
+the variable, and change the binding if it exists (just as in
+`set-variable-value!'). If no such binding exists, we adjoin one to
+the first frame.
+
+ (define (define-variable! var val env)
+ (let ((frame (first-frame env)))
+ (define (scan vars vals)
+ (cond ((null? vars)
+ (add-binding-to-frame! var val frame))
+ ((eq? var (car vars))
+ (set-car! vals val))
+ (else (scan (cdr vars) (cdr vals)))))
+ (scan (frame-variables frame)
+ (frame-values frame))))
+
+ The method described here is only one of many plausible ways to
+represent environments. Since we used data abstraction to isolate the
+rest of the evaluator from the detailed choice of representation, we
+could change the environment representation if we wanted to. (See
+*Note Exercise 4-11::.) In a production-quality Lisp system, the speed
+of the evaluator's environment operations--especially that of variable
+lookup--has a major impact on the performance of the system. The
+representation described here, although conceptually simple, is not
+efficient and would not ordinarily be used in a production system.(2)
+
+ *Exercise 4.11:* Instead of representing a frame as a pair of
+ lists, we can represent a frame as a list of bindings, where each
+ binding is a name-value pair. Rewrite the environment operations
+ to use this alternative representation.
+
+ *Exercise 4.12:* The procedures `set-variable-value!',
+ `define-variable!', and `lookup-variable-value' can be expressed
+ in terms of more abstract procedures for traversing the
+ environment structure. Define abstractions that capture the
+ common patterns and redefine the three procedures in terms of these
+ abstractions.
+
+ *Exercise 4.13:* Scheme allows us to create new bindings for
+ variables by means of `define', but provides no way to get rid of
+ bindings. Implement for the evaluator a special form
+ `make-unbound!' that removes the binding of a given symbol from the
+ environment in which the `make-unbound!' expression is evaluated.
+ This problem is not completely specified. For example, should we
+ remove only the binding in the first frame of the environment?
+ Complete the specification and justify any choices you make.
+
+ ---------- Footnotes ----------
+
+ (1) Frames are not really a data abstraction in the following code:
+`Set-variable-value!' and `define-variable!' use `set-car!' to
+directly modify the values in a frame. The purpose of the frame
+procedures is to make the environment-manipulation procedures easy to
+read.
+
+ (2) The drawback of this representation (as well as the variant in
+*Note Exercise 4-11::) is that the evaluator may have to search through
+many frames in order to find the binding for a given variable. (Such
+an approach is referred to as "deep binding".) One way to avoid this
+inefficiency is to make use of a strategy called "lexical addressing",
+which will be discussed in section *Note 5-5-6::.
+
+\1f
+File: sicp.info, Node: 4-1-4, Next: 4-1-5, Prev: 4-1-3, Up: 4-1
+
+4.1.4 Running the Evaluator as a Program
+----------------------------------------
+
+Given the evaluator, we have in our hands a description (expressed in
+Lisp) of the process by which Lisp expressions are evaluated. One
+advantage of expressing the evaluator as a program is that we can run
+the program. This gives us, running within Lisp, a working model of
+how Lisp itself evaluates expressions. This can serve as a framework
+for experimenting with evaluation rules, as we shall do later in this
+chapter.
+
+ Our evaluator program reduces expressions ultimately to the
+application of primitive procedures. Therefore, all that we need to
+run the evaluator is to create a mechanism that calls on the underlying
+Lisp system to model the application of primitive procedures.
+
+ There must be a binding for each primitive procedure name, so that
+when `eval' evaluates the operator of an application of a primitive, it
+will find an object to pass to `apply'. We thus set up a global
+environment that associates unique objects with the names of the
+primitive procedures that can appear in the expressions we will be
+evaluating. The global environment also includes bindings for the
+symbols `true' and `false', so that they can be used as variables in
+expressions to be evaluated.
+
+ (define (setup-environment)
+ (let ((initial-env
+ (extend-environment (primitive-procedure-names)
+ (primitive-procedure-objects)
+ the-empty-environment)))
+ (define-variable! 'true true initial-env)
+ (define-variable! 'false false initial-env)
+ initial-env))
+
+ (define the-global-environment (setup-environment))
+
+ It does not matter how we represent the primitive procedure objects,
+so long as `apply' can identify and apply them by using the procedures
+`primitive-procedure?' and `apply-primitive-procedure'. We have chosen
+to represent a primitive procedure as a list beginning with the symbol
+`primitive' and containing a procedure in the underlying Lisp that
+implements that primitive.
+
+ (define (primitive-procedure? proc)
+ (tagged-list? proc 'primitive))
+
+ (define (primitive-implementation proc) (cadr proc))
+
+ `Setup-environment' will get the primitive names and implementation
+procedures from a list:(1)
+
+ (define primitive-procedures
+ (list (list 'car car)
+ (list 'cdr cdr)
+ (list 'cons cons)
+ (list 'null? null?)
+ <MORE PRIMITIVES>
+ ))
+
+ (define (primitive-procedure-names)
+ (map car
+ primitive-procedures))
+
+ (define (primitive-procedure-objects)
+ (map (lambda (proc) (list 'primitive (cadr proc)))
+ primitive-procedures))
+
+ To apply a primitive procedure, we simply apply the implementation
+procedure to the arguments, using the underlying Lisp system:(2)
+
+ (define (apply-primitive-procedure proc args)
+ (apply-in-underlying-scheme
+ (primitive-implementation proc) args))
+
+ For convenience in running the metacircular evaluator, we provide a "driver
+loop" that models the read-eval-print loop of the underlying Lisp
+system. It prints a "prompt", reads an input expression, evaluates
+this expression in the global environment, and prints the result. We
+precede each printed result by an "output prompt" so as to distinguish
+the value of the expression from other output that may be printed.(3)
+
+ (define input-prompt ";;; M-Eval input:")
+ (define output-prompt ";;; M-Eval value:")
+
+ (define (driver-loop)
+ (prompt-for-input input-prompt)
+ (let ((input (read)))
+ (let ((output (eval input the-global-environment)))
+ (announce-output output-prompt)
+ (user-print output)))
+ (driver-loop))
+
+ (define (prompt-for-input string)
+ (newline) (newline) (display string) (newline))
+
+ (define (announce-output string)
+ (newline) (display string) (newline))
+
+ We use a special printing procedure, `user-print', to avoid printing
+the environment part of a compound procedure, which may be a very long
+list (or may even contain cycles).
+
+ (define (user-print object)
+ (if (compound-procedure? object)
+ (display (list 'compound-procedure
+ (procedure-parameters object)
+ (procedure-body object)
+ '<procedure-env>))
+ (display object)))
+
+ Now all we need to do to run the evaluator is to initialize the
+global environment and start the driver loop. Here is a sample
+interaction:
+
+ (define the-global-environment (setup-environment))
+
+ (driver-loop)
+
+ ;;; M-Eval input:
+ (define (append x y)
+ (if (null? x)
+ y
+ (cons (car x)
+ (append (cdr x) y))))
+ ;;; M-Eval value:
+ ok
+
+ ;;; M-Eval input:
+ (append '(a b c) '(d e f))
+ ;;; M-Eval value:
+ (a b c d e f)
+
+ *Exercise 4.14:* Eva Lu Ator and Louis Reasoner are each
+ experimenting with the metacircular evaluator. Eva types in the
+ definition of `map', and runs some test programs that use it.
+ They work fine. Louis, in contrast, has installed the system
+ version of `map' as a primitive for the metacircular evaluator.
+ When he tries it, things go terribly wrong. Explain why Louis's
+ `map' fails even though Eva's works.
+
+ ---------- Footnotes ----------
+
+ (1) Any procedure defined in the underlying Lisp can be used as a
+primitive for the metacircular evaluator. The name of a primitive
+installed in the evaluator need not be the same as the name of its
+implementation in the underlying Lisp; the names are the same here
+because the metacircular evaluator implements Scheme itself. Thus, for
+example, we could put `(list 'first car)' or `(list 'square (lambda (x)
+(* x x)))' in the list of `primitive-procedures'.
+
+ (2) `Apply-in-underlying-scheme' is the `apply' procedure we have
+used in earlier chapters. The metacircular evaluator's `apply'
+procedure (section *Note 4-1-1::) models the working of this primitive.
+Having two different things called `apply' leads to a technical
+problem in running the metacircular evaluator, because defining the
+metacircular evaluator's `apply' will mask the definition of the
+primitive. One way around this is to rename the metacircular `apply' to
+avoid conflict with the name of the primitive procedure. We have
+assumed instead that we have saved a reference to the underlying
+`apply' by doing
+
+ (define apply-in-underlying-scheme apply)
+
+before defining the metacircular `apply'. This allows us to access the
+original version of `apply' under a different name.
+
+ (3) The primitive procedure `read' waits for input from the user,
+and returns the next complete expression that is typed. For example,
+if the user types `(+ 23 x)', `read' returns a three-element list
+containing the symbol `+', the number 23, and the symbol `x'. If the
+user types `'x', `read' returns a two-element list containing the
+symbol `quote' and the symbol `x'.
+
+\1f
+File: sicp.info, Node: 4-1-5, Next: 4-1-6, Prev: 4-1-4, Up: 4-1
+
+4.1.5 Data as Programs
+----------------------
+
+In thinking about a Lisp program that evaluates Lisp expressions, an
+analogy might be helpful. One operational view of the meaning of a
+program is that a program is a description of an abstract (perhaps
+infinitely large) machine. For example, consider the familiar program
+to compute factorials:
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+
+ We may regard this program as the description of a machine
+containing parts that decrement, multiply, and test for equality,
+together with a two-position switch and another factorial machine. (The
+factorial machine is infinite because it contains another factorial
+machine within it.) *Note Figure 4-2:: is a flow diagram for the
+factorial machine, showing how the parts are wired together.
+
+ *Figure 4.2:* The factorial program, viewed as an abstract machine.
+
+ +-----------------------------------+
+ | factorial |1 |
+ | |1 V |
+ | | +-----+ |
+ | V | # | |
+ | +-----+ | | |
+ 6 --------*-----| = |------->| #-+-----> 720
+ | | +-----+ | / | |
+ | | | # | |
+ | | +-----+ |
+ | | ^ |
+ | | | |
+ | | +--+--+ |
+ | *------------------->| * | |
+ | | +-----+ |
+ | V ^ |
+ | +-----+ +-----------+ | |
+ | | - +--->| factorial +---+ |
+ | +-----+ +-----------+ |
+ | ^ |
+ | |1 |
+ +-----------------------------------+
+
+ In a similar way, we can regard the evaluator as a very special
+machine that takes as input a description of a machine. Given this
+input, the evaluator configures itself to emulate the machine
+described. For example, if we feed our evaluator the definition of
+`factorial', as shown in *Note Figure 4-3::, the evaluator will be able
+to compute factorials.
+
+ *Figure 4.3:* The evaluator emulating a factorial machine.
+
+ +--------+
+ 6 ---->| eval |----> 720
+ +--------+
+ /
+ . . . / . . .
+ . . . ../. . .
+ . ..
+ . (define (factorial n) . . .
+ . (if (= n 1) . .
+ . 1 .
+ . (* (factorial (- n 1)) n))) .
+ . . . . .
+ . . . . . . . . . . . .
+ . ..
+
+ From this perspective, our evaluator is seen to be a machine
+"universal machine". It mimics other machines when these are described
+as Lisp programs.(1) This is striking. Try to imagine an analogous
+evaluator for electrical circuits. This would be a circuit that takes
+as input a signal encoding the plans for some other circuit, such as a
+filter. Given this input, the circuit evaluator would then behave like
+a filter with the same description. Such a universal electrical
+circuit is almost unimaginably complex. It is remarkable that the
+program evaluator is a rather simple program.(2)
+
+ Another striking aspect of the evaluator is that it acts as a bridge
+between the data objects that are manipulated by our programming
+language and the programming language itself. Imagine that the
+evaluator program (implemented in Lisp) is running, and that a user is
+typing expressions to the evaluator and observing the results. From
+the perspective of the user, an input expression such as `(* x x)' is
+an expression in the programming language, which the evaluator should
+execute. From the perspective of the evaluator, however, the
+expression is simply a list (in this case, a list of three symbols: `*',
+`x', and `x') that is to be manipulated according to a well-defined set
+of rules.
+
+ That the user's programs are the evaluator's data need not be a
+source of confusion. In fact, it is sometimes convenient to ignore
+this distinction, and to give the user the ability to explicitly
+evaluate a data object as a Lisp expression, by making `eval' available
+for use in programs. Many Lisp dialects provide a primitive `eval'
+procedure that takes as arguments an expression and an environment and
+evaluates the expression relative to the environment.(3) Thus,
+
+ (eval '(* 5 5) user-initial-environment)
+
+and
+
+ (eval (cons '* (list 5 5)) user-initial-environment)
+
+will both return 25.(4)
+
+ *Exercise 4.15:* Given a one-argument procedure `p' and an object
+ `a', `p' is said to "halt" on `a' if evaluating the expression `(p
+ a)' returns a value (as opposed to terminating with an error
+ message or running forever). Show that it is impossible to write
+ a procedure `halts?' that correctly determines whether `p' halts
+ on `a' for any procedure `p' and object `a'. Use the following
+ reasoning: If you had such a procedure `halts?', you could
+ implement the following program:
+
+ (define (run-forever) (run-forever))
+
+ (define (try p)
+ (if (halts? p p)
+ (run-forever)
+ 'halted))
+
+ Now consider evaluating the expression `(try try)' and show that
+ any possible outcome (either halting or running forever) violates
+ the intended behavior of `halts?'.(5)
+
+ ---------- Footnotes ----------
+
+ (1) The fact that the machines are described in Lisp is inessential.
+If we give our evaluator a Lisp program that behaves as an evaluator
+for some other language, say C, the Lisp evaluator will emulate the C
+evaluator, which in turn can emulate any machine described as a C
+program. Similarly, writing a Lisp evaluator in C produces a C program
+that can execute any Lisp program. The deep idea here is that any
+evaluator can emulate any other. Thus, the notion of "what can in
+principle be computed" (ignoring practicalities of time and memory
+required) is independent of the language or the computer, and instead
+reflects an underlying notion of "computability". This was first
+demonstrated in a clear way by Alan M. Turing (1912-1954), whose 1936
+paper laid the foundations for theoretical computer science. In the
+paper, Turing presented a simple computational model--now known as a "Turing
+machine"--and argued that any "effective process" can be formulated as
+a program for such a machine. (This argument is known as the "Church-Turing
+thesis".) Turing then implemented a universal machine, i.e., a Turing
+machine that behaves as an evaluator for Turing-machine programs. He
+used this framework to demonstrate that there are well-posed problems
+that cannot be computed by Turing machines (see *Note Exercise 4-15::),
+and so by implication cannot be formulated as "effective processes."
+Turing went on to make fundamental contributions to practical computer
+science as well. For example, he invented the idea of structuring
+programs using general-purpose subroutines. See Hodges 1983 for a
+biography of Turing.
+
+ (2) Some people find it counterintuitive that an evaluator, which is
+implemented by a relatively simple procedure, can emulate programs that
+are more complex than the evaluator itself. The existence of a
+universal evaluator machine is a deep and wonderful property of
+computation. theory "Recursion theory", a branch of mathematical
+logic, is concerned with logical limits of computation. Douglas
+Hofstadter's beautiful book `Go"del, Escher, Bach' (1979) explores some
+of these ideas.
+
+ (3) Warning: This `eval' primitive is not identical to the `eval'
+procedure we implemented in section *Note 4-1-1::, because it uses
+_actual_ Scheme environments rather than the sample environment
+structures we built in section *Note 4-1-3::. These actual
+environments cannot be manipulated by the user as ordinary lists; they
+must be accessed via `eval' or other special operations. Similarly,
+the `apply' primitive we saw earlier is not identical to the
+metacircular `apply', because it uses actual Scheme procedures rather
+than the procedure objects we constructed in sections *Note 4-1-3:: and
+*Note 4-1-4::.
+
+ (4) The MIT implementation of Scheme includes `eval', as well as a
+symbol `user-initial-environment' that is bound to the initial
+environment in which the user's input expressions are evaluated.
+
+ (5) Although we stipulated that `halts?' is given a procedure
+object, notice that this reasoning still applies even if `halts?' can
+gain access to the procedure's text and its environment. This is
+Turing's celebrated "Halting Theorem", which gave the first clear
+example of a "non-computable" problem, i.e., a well-posed task that
+cannot be carried out as a computational procedure.
+
+\1f
+File: sicp.info, Node: 4-1-6, Next: 4-1-7, Prev: 4-1-5, Up: 4-1
+
+4.1.6 Internal Definitions
+--------------------------
+
+Our environment model of evaluation and our metacircular evaluator
+execute definitions in sequence, extending the environment frame one
+definition at a time. This is particularly convenient for interactive
+program development, in which the programmer needs to freely mix the
+application of procedures with the definition of new procedures.
+However, if we think carefully about the internal definitions used to
+implement block structure (introduced in section *Note 1-1-8::), we
+will find that name-by-name extension of the environment may not be the
+best way to define local variables.
+
+ Consider a procedure with internal definitions, such as
+
+ (define (f x)
+ (define (even? n)
+ (if (= n 0)
+ true
+ (odd? (- n 1))))
+ (define (odd? n)
+ (if (= n 0)
+ false
+ (even? (- n 1))))
+ <REST OF BODY OF `F'>)
+
+ Our intention here is that the name `odd?' in the body of the
+procedure `even?' should refer to the procedure `odd?' that is defined
+after `even?'. The scope of the name `odd?' is the entire body of `f',
+not just the portion of the body of `f' starting at the point where the
+`define' for `odd?' occurs. Indeed, when we consider that `odd?' is
+itself defined in terms of `even?'--so that `even?' and `odd?' are
+mutually recursive procedures--we see that the only satisfactory
+interpretation of the two `define's is to regard them as if the names
+`even?' and `odd?' were being added to the environment simultaneously.
+More generally, in block structure, the scope of a local name is the
+entire procedure body in which the `define' is evaluated.
+
+ As it happens, our interpreter will evaluate calls to `f' correctly,
+but for an "accidental" reason: Since the definitions of the internal
+procedures come first, no calls to these procedures will be evaluated
+until all of them have been defined. Hence, `odd?' will have been
+defined by the time `even?' is executed. In fact, our sequential
+evaluation mechanism will give the same result as a mechanism that
+directly implements simultaneous definition for any procedure in which
+the internal definitions come first in a body and evaluation of the
+value expressions for the defined variables doesn't actually use any of
+the defined variables. (For an example of a procedure that doesn't
+obey these restrictions, so that sequential definition isn't equivalent
+to simultaneous definition, see *Note Exercise 4-19::.)(1)
+
+ There is, however, a simple way to treat definitions so that
+internally defined names have truly simultaneous scope--just create all
+local variables that will be in the current environment before
+evaluating any of the value expressions. One way to do this is by a
+syntax transformation on `lambda' expressions. Before evaluating the
+body of a `lambda' expression, we "scan out" and eliminate all the
+internal definitions in the body. The internally defined variables
+will be created with a `let' and then set to their values by
+assignment. For example, the procedure
+
+ (lambda <VARS>
+ (define u <E1>)
+ (define v <E2>)
+ <E3>)
+
+would be transformed into
+
+ (lambda <VARS>
+ (let ((u '*unassigned*)
+ (v '*unassigned*))
+ (set! u <E1>)
+ (set! v <E2>)
+ <E3>))
+
+where `*unassigned*' is a special symbol that causes looking up a
+variable to signal an error if an attempt is made to use the value of
+the not-yet-assigned variable.
+
+ An alternative strategy for scanning out internal definitions is
+shown in *Note Exercise 4-18::. Unlike the transformation shown above,
+this enforces the restriction that the defined variables' values can be
+evaluated without using any of the variables' values.(2)
+
+ *Exercise 4.16:* In this exercise we implement the method just
+ described for interpreting internal definitions. We assume that
+ the evaluator supports `let' (see *Note Exercise 4-6::).
+
+ a. Change `lookup-variable-value' (section *Note 4-1-3::) to
+ signal an error if the value it finds is the symbol
+ `*unassigned*'.
+
+ b. Write a procedure `scan-out-defines' that takes a procedure
+ body and returns an equivalent one that has no internal
+ definitions, by making the transformation described above.
+
+ c. Install `scan-out-defines' in the interpreter, either in
+ `make-procedure' or in `procedure-body' (see section *Note
+ 4-1-3::). Which place is better? Why?
+
+
+ *Exercise 4.17:* Draw diagrams of the environment in effect when
+ evaluating the expression <E3> in the procedure in the text,
+ comparing how this will be structured when definitions are
+ interpreted sequentially with how it will be structured if
+ definitions are scanned out as described. Why is there an extra
+ frame in the transformed program? Explain why this difference in
+ environment structure can never make a difference in the behavior
+ of a correct program. Design a way to make the interpreter
+ implement the "simultaneous" scope rule for internal definitions
+ without constructing the extra frame.
+
+ *Exercise 4.18:* Consider an alternative strategy for scanning out
+ definitions that translates the example in the text to
+
+ (lambda <VARS>
+ (let ((u '*unassigned*)
+ (v '*unassigned*))
+ (let ((a <E1>)
+ (b <E2>))
+ (set! u a)
+ (set! v b))
+ <E3>))
+
+ Here `a' and `b' are meant to represent new variable names, created
+ by the interpreter, that do not appear in the user's program.
+ Consider the `solve' procedure from section *Note 3-5-4:::
+
+ (define (solve f y0 dt)
+ (define y (integral (delay dy) y0 dt))
+ (define dy (stream-map f y))
+ y)
+
+ Will this procedure work if internal definitions are scanned out
+ as shown in this exercise? What if they are scanned out as shown
+ in the text? Explain.
+
+ *Exercise 4.19:* Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator
+ are arguing about the desired result of evaluating the expression
+
+ (let ((a 1))
+ (define (f x)
+ (define b (+ a x))
+ (define a 5)
+ (+ a b))
+ (f 10))
+
+ Ben asserts that the result should be obtained using the
+ sequential rule for `define': `b' is defined to be 11, then `a' is
+ defined to be 5, so the result is 16. Alyssa objects that mutual
+ recursion requires the simultaneous scope rule for internal
+ procedure definitions, and that it is unreasonable to treat
+ procedure names differently from other names. Thus, she argues
+ for the mechanism implemented in *Note Exercise 4-16::. This
+ would lead to `a' being unassigned at the time that the value for
+ `b' is to be computed. Hence, in Alyssa's view the procedure
+ should produce an error. Eva has a third opinion. She says that
+ if the definitions of `a' and `b' are truly meant to be
+ simultaneous, then the value 5 for `a' should be used in
+ evaluating `b'. Hence, in Eva's view `a' should be 5, `b' should
+ be 15, and the result should be 20. Which (if any) of these
+ viewpoints do you support? Can you devise a way to implement
+ internal definitions so that they behave as Eva prefers?(3)
+
+ *Exercise 4.20:* Because internal definitions look sequential but
+ are actually simultaneous, some people prefer to avoid them
+ entirely, and use the special form `letrec' instead. `Letrec'
+ looks like `let', so it is not surprising that the variables it
+ binds are bound simultaneously and have the same scope as each
+ other. The sample procedure `f' above can be written without
+ internal definitions, but with exactly the same meaning, as
+
+ (define (f x)
+ (letrec ((even?
+ (lambda (n)
+ (if (= n 0)
+ true
+ (odd? (- n 1)))))
+ (odd?
+ (lambda (n)
+ (if (= n 0)
+ false
+ (even? (- n 1))))))
+ <REST OF BODY OF `F'>))
+
+ `Letrec' expressions, which have the form
+
+ (letrec ((<VAR_1> <EXP_1>) ... (<VAR_N> <EXP_N>))
+ <BODY>)
+
+ are a variation on `let' in which the expressions <EXP_K> that
+ provide the initial values for the variables <VAR_K> are evaluated
+ in an environment that includes all the `letrec' bindings. This
+ permits recursion in the bindings, such as the mutual recursion of
+ `even?' and `odd?' in the example above, or the evaluation of 10
+ factorial with
+
+ (letrec ((fact
+ (lambda (n)
+ (if (= n 1)
+ 1
+ (* n (fact (- n 1)))))))
+ (fact 10))
+
+ a. Implement `letrec' as a derived expression, by transforming a
+ `letrec' expression into a `let' expression as shown in the
+ text above or in *Note Exercise 4-18::. That is, the
+ `letrec' variables should be created with a `let' and then be
+ assigned their values with `set!'.
+
+ b. Louis Reasoner is confused by all this fuss about internal
+ definitions. The way he sees it, if you don't like to use
+ `define' inside a procedure, you can just use `let'.
+ Illustrate what is loose about his reasoning by drawing an
+ environment diagram that shows the environment in which the
+ <REST OF BODY OF `F'> is evaluated during evaluation of the
+ expression `(f 5)', with `f' defined as in this exercise.
+ Draw an environment diagram for the same evaluation, but with
+ `let' in place of `letrec' in the definition of `f'.
+
+
+ *Exercise 4.21:* Amazingly, Louis's intuition in *Note Exercise
+ 4-20:: is correct. It is indeed possible to specify recursive
+ procedures without using `letrec' (or even `define'), although the
+ method for accomplishing this is much more subtle than Louis
+ imagined. The following expression computes 10 factorial by
+ applying a recursive factorial procedure:(4)
+
+ ((lambda (n)
+ ((lambda (fact)
+ (fact fact n))
+ (lambda (ft k)
+ (if (= k 1)
+ 1
+ (* k (ft ft (- k 1)))))))
+ 10)
+
+ a. Check (by evaluating the expression) that this really does
+ compute factorials. Devise an analogous expression for
+ computing Fibonacci numbers.
+
+ b. Consider the following procedure, which includes mutually
+ recursive internal definitions:
+
+ (define (f x)
+ (define (even? n)
+ (if (= n 0)
+ true
+ (odd? (- n 1))))
+ (define (odd? n)
+ (if (= n 0)
+ false
+ (even? (- n 1))))
+ (even? x))
+
+ Fill in the missing expressions to complete an alternative
+ definition of `f', which uses neither internal definitions
+ nor `letrec':
+
+ (define (f x)
+ ((lambda (even? odd?)
+ (even? even? odd? x))
+ (lambda (ev? od? n)
+ (if (= n 0) true (od? <??> <??> <??>)))
+ (lambda (ev? od? n)
+ (if (= n 0) false (ev? <??> <??> <??>)))))
+
+ ---------- Footnotes ----------
+
+ (1) Wanting programs to not depend on this evaluation mechanism is
+the reason for the "management is not responsible" remark in *Note
+Footnote 28:: of *Note Chapter 1::. By insisting that internal
+definitions come first and do not use each other while the definitions
+are being evaluated, the IEEE standard for Scheme leaves implementors
+some choice in the mechanism used to evaluate these definitions. The
+choice of one evaluation rule rather than another here may seem like a
+small issue, affecting only the interpretation of "badly formed"
+programs. However, we will see in section *Note 5-5-6:: that moving to
+a model of simultaneous scoping for internal definitions avoids some
+nasty difficulties that would otherwise arise in implementing a
+compiler.
+
+ (2) The IEEE standard for Scheme allows for different implementation
+strategies by specifying that it is up to the programmer to obey this
+restriction, not up to the implementation to enforce it. Some Scheme
+implementations, including MIT Scheme, use the transformation shown
+above. Thus, some programs that don't obey this restriction will in
+fact run in such implementations.
+
+ (3) The MIT implementors of Scheme support Alyssa on the following
+grounds: Eva is in principle correct - the definitions should be
+regarded as simultaneous. But it seems difficult to implement a
+general, efficient mechanism that does what Eva requires. In the
+absence of such a mechanism, it is better to generate an error in the
+difficult cases of simultaneous definitions (Alyssa's notion) than to
+produce an incorrect answer (as Ben would have it).
+
+ (4) This example illustrates a programming trick for formulating
+recursive procedures without using `define'. The most general trick of
+this sort is the Y "operator", which can be used to give a "pure
+[lambda]-calculus" implementation of recursion. (See Stoy 1977 for
+details on the [lambda] calculus, and Gabriel 1988 for an exposition of
+the Y operator in Scheme.)
+
+\1f
+File: sicp.info, Node: 4-1-7, Prev: 4-1-6, Up: 4-1
+
+4.1.7 Separating Syntactic Analysis from Execution
+--------------------------------------------------
+
+The evaluator implemented above is simple, but it is very inefficient,
+because the syntactic analysis of expressions is interleaved with their
+execution. Thus if a program is executed many times, its syntax is
+analyzed many times. Consider, for example, evaluating `(factorial 4)'
+using the following definition of `factorial':
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+
+ Each time `factorial' is called, the evaluator must determine that
+the body is an `if' expression and extract the predicate. Only then
+can it evaluate the predicate and dispatch on its value. Each time it
+evaluates the expression `(* (factorial (- n 1)) n)', or the
+subexpressions `(factorial (- n 1))' and `(- n 1)', the evaluator must
+perform the case analysis in `eval' to determine that the expression is
+an application, and must extract its operator and operands. This
+analysis is expensive. Performing it repeatedly is wasteful.
+
+ We can transform the evaluator to be significantly more efficient by
+arranging things so that syntactic analysis is performed only once.(1)
+We split `eval', which takes an expression and an environment, into two
+parts. The procedure `analyze' takes only the expression. It performs
+the syntactic analysis and returns a new procedure, the "execution
+procedure", that encapsulates the work to be done in executing the
+analyzed expression. The execution procedure takes an environment as
+its argument and completes the evaluation. This saves work because
+`analyze' will be called only once on an expression, while the
+execution procedure may be called many times.
+
+ With the separation into analysis and execution, `eval' now becomes
+
+ (define (eval exp env)
+ ((analyze exp) env))
+
+ The result of calling `analyze' is the execution procedure to be
+applied to the environment. The `analyze' procedure is the same case
+analysis as performed by the original `eval' of section *Note 4-1-1::,
+except that the procedures to which we dispatch perform only analysis,
+not full evaluation:
+
+ (define (analyze exp)
+ (cond ((self-evaluating? exp)
+ (analyze-self-evaluating exp))
+ ((quoted? exp) (analyze-quoted exp))
+ ((variable? exp) (analyze-variable exp))
+ ((assignment? exp) (analyze-assignment exp))
+ ((definition? exp) (analyze-definition exp))
+ ((if? exp) (analyze-if exp))
+ ((lambda? exp) (analyze-lambda exp))
+ ((begin? exp) (analyze-sequence (begin-actions exp)))
+ ((cond? exp) (analyze (cond->if exp)))
+ ((application? exp) (analyze-application exp))
+ (else
+ (error "Unknown expression type -- ANALYZE" exp))))
+
+ Here is the simplest syntactic analysis procedure, which handles
+self-evaluating expressions. It returns an execution procedure that
+ignores its environment argument and just returns the expression:
+
+ (define (analyze-self-evaluating exp)
+ (lambda (env) exp))
+
+ For a quoted expression, we can gain a little efficiency by
+extracting the text of the quotation only once, in the analysis phase,
+rather than in the execution phase.
+
+ (define (analyze-quoted exp)
+ (let ((qval (text-of-quotation exp)))
+ (lambda (env) qval)))
+
+ Looking up a variable value must still be done in the execution
+phase, since this depends upon knowing the environment.(2)
+
+ (define (analyze-variable exp)
+ (lambda (env) (lookup-variable-value exp env)))
+
+ `Analyze-assignment' also must defer actually setting the variable
+until the execution, when the environment has been supplied. However,
+the fact that the `assignment-value' expression can be analyzed
+(recursively) during analysis is a major gain in efficiency, because
+the `assignment-value' expression will now be analyzed only once. The
+same holds true for definitions.
+
+ (define (analyze-assignment exp)
+ (let ((var (assignment-variable exp))
+ (vproc (analyze (assignment-value exp))))
+ (lambda (env)
+ (set-variable-value! var (vproc env) env)
+ 'ok)))
+
+ (define (analyze-definition exp)
+ (let ((var (definition-variable exp))
+ (vproc (analyze (definition-value exp))))
+ (lambda (env)
+ (define-variable! var (vproc env) env)
+ 'ok)))
+
+ For `if' expressions, we extract and analyze the predicate,
+consequent, and alternative at analysis time.
+
+ (define (analyze-if exp)
+ (let ((pproc (analyze (if-predicate exp)))
+ (cproc (analyze (if-consequent exp)))
+ (aproc (analyze (if-alternative exp))))
+ (lambda (env)
+ (if (true? (pproc env))
+ (cproc env)
+ (aproc env)))))
+
+ Analyzing a `lambda' expression also achieves a major gain in
+efficiency: We analyze the `lambda' body only once, even though
+procedures resulting from evaluation of the `lambda' may be applied
+many times.
+
+ (define (analyze-lambda exp)
+ (let ((vars (lambda-parameters exp))
+ (bproc (analyze-sequence (lambda-body exp))))
+ (lambda (env) (make-procedure vars bproc env))))
+
+ Analysis of a sequence of expressions (as in a `begin' or the body
+of a `lambda' expression) is more involved.(3) Each expression in the
+sequence is analyzed, yielding an execution procedure. These execution
+procedures are combined to produce an execution procedure that takes an
+environment as argument and sequentially calls each individual execution
+procedure with the environment as argument.
+
+ (define (analyze-sequence exps)
+ (define (sequentially proc1 proc2)
+ (lambda (env) (proc1 env) (proc2 env)))
+ (define (loop first-proc rest-procs)
+ (if (null? rest-procs)
+ first-proc
+ (loop (sequentially first-proc (car rest-procs))
+ (cdr rest-procs))))
+ (let ((procs (map analyze exps)))
+ (if (null? procs)
+ (error "Empty sequence -- ANALYZE"))
+ (loop (car procs) (cdr procs))))
+
+ To analyze an application, we analyze the operator and operands and
+construct an execution procedure that calls the operator execution
+procedure (to obtain the actual procedure to be applied) and the
+operand execution procedures (to obtain the actual arguments). We then
+pass these to `execute-application', which is the analog of `apply' in
+section *Note 4-1-1::. `Execute-application' differs from `apply' in
+that the procedure body for a compound procedure has already been
+analyzed, so there is no need to do further analysis. Instead, we just
+call the execution procedure for the body on the extended environment.
+
+ (define (analyze-application exp)
+ (let ((fproc (analyze (operator exp)))
+ (aprocs (map analyze (operands exp))))
+ (lambda (env)
+ (execute-application (fproc env)
+ (map (lambda (aproc) (aproc env))
+ aprocs)))))
+
+ (define (execute-application proc args)
+ (cond ((primitive-procedure? proc)
+ (apply-primitive-procedure proc args))
+ ((compound-procedure? proc)
+ ((procedure-body proc)
+ (extend-environment (procedure-parameters proc)
+ args
+ (procedure-environment proc))))
+ (else
+ (error
+ "Unknown procedure type -- EXECUTE-APPLICATION"
+ proc))))
+
+ Our new evaluator uses the same data structures, syntax procedures,
+and run-time support procedures as in sections *Note 4-1-2::, *Note
+4-1-3::, and *Note 4-1-4::.
+
+ *Exercise 4.22:* Extend the evaluator in this section to support
+ the special form `let'. (See *Note Exercise 4-6::.)
+
+ *Exercise 4.23:* Alyssa P. Hacker doesn't understand why
+ `analyze-sequence' needs to be so complicated. All the other
+ analysis procedures are straightforward transformations of the
+ corresponding evaluation procedures (or `eval' clauses) in section
+ *Note 4-1-1::. She expected `analyze-sequence' to look like this:
+
+ (define (analyze-sequence exps)
+ (define (execute-sequence procs env)
+ (cond ((null? (cdr procs)) ((car procs) env))
+ (else ((car procs) env)
+ (execute-sequence (cdr procs) env))))
+ (let ((procs (map analyze exps)))
+ (if (null? procs)
+ (error "Empty sequence -- ANALYZE"))
+ (lambda (env) (execute-sequence procs env))))
+
+ Eva Lu Ator explains to Alyssa that the version in the text does
+ more of the work of evaluating a sequence at analysis time.
+ Alyssa's sequence-execution procedure, rather than having the
+ calls to the individual execution procedures built in, loops
+ through the procedures in order to call them: In effect, although
+ the individual expressions in the sequence have been analyzed, the
+ sequence itself has not been.
+
+ Compare the two versions of `analyze-sequence'. For example,
+ consider the common case (typical of procedure bodies) where the
+ sequence has just one expression. What work will the execution
+ procedure produced by Alyssa's program do? What about the
+ execution procedure produced by the program in the text above?
+ How do the two versions compare for a sequence with two
+ expressions?
+
+ *Exercise 4.24:* Design and carry out some experiments to compare
+ the speed of the original metacircular evaluator with the version
+ in this section. Use your results to estimate the fraction of time
+ that is spent in analysis versus execution for various procedures.
+
+ ---------- Footnotes ----------
+
+ (1) This technique is an integral part of the compilation process,
+which we shall discuss in *Note Chapter 5::. Jonathan Rees wrote a
+Scheme interpreter like this in about 1982 for the T project (Rees and
+Adams 1982). Marc Feeley (1986) (see also Feeley and Lapalme 1987)
+independently invented this technique in his master's thesis.
+
+ (2) There is, however, an important part of the variable search that
+_can_ be done as part of the syntactic analysis. As we will show in
+section *Note 5-5-6::, one can determine the position in the
+environment structure where the value of the variable will be found,
+thus obviating the need to scan the environment for the entry that
+matches the variable.
+
+ (3) See *Note Exercise 4-23:: for some insight into the processing
+of sequences.
+
+\1f
+File: sicp.info, Node: 4-2, Next: 4-3, Prev: 4-1, Up: Chapter 4
+
+4.2 Variations on a Scheme - Lazy Evaluation
+============================================
+
+Now that we have an evaluator expressed as a Lisp program, we can
+experiment with alternative choices in language design simply by
+modifying the evaluator. Indeed, new languages are often invented by
+first writing an evaluator that embeds the new language within an
+existing high-level language. For example, if we wish to discuss some
+aspect of a proposed modification to Lisp with another member of the
+Lisp community, we can supply an evaluator that embodies the change.
+The recipient can then experiment with the new evaluator and send back
+comments as further modifications. Not only does the high-level
+implementation base make it easier to test and debug the evaluator; in
+addition, the embedding enables the designer to snarf(1) features from
+the underlying language, just as our embedded Lisp evaluator uses
+primitives and control structure from the underlying Lisp. Only later
+(if ever) need the designer go to the trouble of building a complete
+implementation in a low-level language or in hardware. In this section
+and the next we explore some variations on Scheme that provide
+significant additional expressive power.
+
+* Menu:
+
+* 4-2-1:: Normal Order and Applicative Order
+* 4-2-2:: An Interpreter with Lazy Evaluation
+* 4-2-3:: Streams as Lazy Lists
+
+ ---------- Footnotes ----------
+
+ (1) Snarf: "To grab, especially a large document or file for the
+purpose of using it either with or without the owner's permission."
+Snarf down: "To snarf, sometimes with the connotation of absorbing,
+processing, or understanding." (These definitions were snarfed from
+Steele et al. 1983. See also Raymond 1993.)
+
+\1f
+File: sicp.info, Node: 4-2-1, Next: 4-2-2, Prev: 4-2, Up: 4-2
+
+4.2.1 Normal Order and Applicative Order
+----------------------------------------
+
+In section *Note 1-1::, where we began our discussion of models of
+evaluation, we noted that Scheme is an "applicative-order" language,
+namely, that all the arguments to Scheme procedures are evaluated when
+the procedure is applied. In contrast, "normal-order" languages delay
+evaluation of procedure arguments until the actual argument values are
+needed. Delaying evaluation of procedure arguments until the last
+possible moment (e.g., until they are required by a primitive
+operation) is called evaluation "lazy evaluation".(1) Consider the
+procedure
+
+ (define (try a b)
+ (if (= a 0) 1 b))
+
+ Evaluating `(try 0 (/ 1 0))' generates an error in Scheme. With lazy
+evaluation, there would be no error. Evaluating the expression would
+return 1, because the argument `(/ 1 0)' would never be evaluated.
+
+ An example that exploits lazy evaluation is the definition of a
+procedure `unless'
+
+ (define (unless condition usual-value exceptional-value)
+ (if condition exceptional-value usual-value))
+
+that can be used in expressions such as
+
+ (unless (= b 0)
+ (/ a b)
+ (begin (display "exception: returning 0")
+ 0))
+
+ This won't work in an applicative-order language because both the
+usual value and the exceptional value will be evaluated before `unless'
+is called (compare *Note Exercise 1-6::). An advantage of lazy
+evaluation is that some procedures, such as `unless', can do useful
+computation even if evaluation of some of their arguments would produce
+errors or would not terminate.
+
+ If the body of a procedure is entered before an argument has been
+evaluated we say that the procedure is "non-strict" in that argument.
+If the argument is evaluated before the body of the procedure is
+entered we say that the procedure is "strict" in that argument.(2) In
+a purely applicative-order language, all procedures are strict in each
+argument. In a purely normal-order language, all compound procedures
+are non-strict in each argument, and primitive procedures may be either
+strict or non-strict. There are also languages (see *Note Exercise
+4-31::) that give programmers detailed control over the strictness of
+the procedures they define.
+
+ A striking example of a procedure that can usefully be made
+non-strict is `cons' (or, in general, almost any constructor for data
+structures). One can do useful computation, combining elements to form
+data structures and operating on the resulting data structures, even if
+the values of the elements are not known. It makes perfect sense, for
+instance, to compute the length of a list without knowing the values of
+the individual elements in the list. We will exploit this idea in
+section *Note 4-2-3:: to implement the streams of *Note Chapter 3:: as
+lists formed of non-strict `cons' pairs.
+
+ *Exercise 4.25:* Suppose that (in ordinary applicative-order
+ Scheme) we define `unless' as shown above and then define
+ `factorial' in terms of `unless' as
+
+ (define (factorial n)
+ (unless (= n 1)
+ (* n (factorial (- n 1)))
+ 1))
+
+ What happens if we attempt to evaluate `(factorial 5)'? Will our
+ definitions work in a normal-order language?
+
+ *Exercise 4.26:* Ben Bitdiddle and Alyssa P. Hacker disagree over
+ the importance of lazy evaluation for implementing things such as
+ `unless'. Ben points out that it's possible to implement `unless'
+ in applicative order as a special form. Alyssa counters that, if
+ one did that, `unless' would be merely syntax, not a procedure
+ that could be used in conjunction with higher-order procedures.
+ Fill in the details on both sides of the argument. Show how to
+ implement `unless' as a derived expression (like `cond' or `let'),
+ and give an example of a situation where it might be useful to
+ have `unless' available as a procedure, rather than as a special
+ form.
+
+ ---------- Footnotes ----------
+
+ (1) The difference between the "lazy" terminology and the
+"normal-order" terminology is somewhat fuzzy. Generally, "lazy" refers
+to the mechanisms of particular evaluators, while "normal-order" refers
+to the semantics of languages, independent of any particular evaluation
+strategy. But this is not a hard-and-fast distinction, and the two
+terminologies are often used interchangeably.
+
+ (2) The "strict" versus "non-strict" terminology means essentially
+the same thing as "applicative-order" versus "normal-order," except
+that it refers to individual procedures and arguments rather than to
+the language as a whole. At a conference on programming languages you
+might hear someone say, "The normal-order language Hassle has certain
+strict primitives. Other procedures take their arguments by lazy
+evaluation."
+
+\1f
+File: sicp.info, Node: 4-2-2, Next: 4-2-3, Prev: 4-2-1, Up: 4-2
+
+4.2.2 An Interpreter with Lazy Evaluation
+-----------------------------------------
+
+In this section we will implement a normal-order language that is the
+same as Scheme except that compound procedures are non-strict in each
+argument. Primitive procedures will still be strict. It is not
+difficult to modify the evaluator of section *Note 4-1-1:: so that the
+language it interprets behaves this way. Almost all the required
+changes center around procedure application.
+
+ The basic idea is that, when applying a procedure, the interpreter
+must determine which arguments are to be evaluated and which are to be
+delayed. The delayed arguments are not evaluated; instead, they are
+transformed into objects called "thunks".(1) The thunk must contain the
+information required to produce the value of the argument when it is
+needed, as if it had been evaluated at the time of the application.
+Thus, the thunk must contain the argument expression and the
+environment in which the procedure application is being evaluated.
+
+ The process of evaluating the expression in a thunk is called "forcing".(2)
+In general, a thunk will be forced only when its value is needed: when
+it is passed to a primitive procedure that will use the value of the
+thunk; when it is the value of a predicate of a conditional; and when
+it is the value of an operator that is about to be applied as a
+procedure. One design choice we have available is whether or not to "memoize"
+thunks, as we did with delayed objects in section *Note 3-5-1::. With
+memoization, the first time a thunk is forced, it stores the value that
+is computed. Subsequent forcings simply return the stored value
+without repeating the computation. We'll make our interpreter memoize,
+because this is more efficient for many applications. There are tricky
+considerations here, however.(3)
+
+Modifying the evaluator
+.......................
+
+The main difference between the lazy evaluator and the one in section
+*Note 4-1:: is in the handling of procedure applications in `eval' and
+`apply'.
+
+ The `application?' clause of `eval' becomes
+
+ ((application? exp)
+ (apply (actual-value (operator exp) env)
+ (operands exp)
+ env))
+
+ This is almost the same as the `application?' clause of `eval' in
+section *Note 4-1-1::. For lazy evaluation, however, we call `apply'
+with the operand expressions, rather than the arguments produced by
+evaluating them. Since we will need the environment to construct
+thunks if the arguments are to be delayed, we must pass this as well.
+We still evaluate the operator, because `apply' needs the actual
+procedure to be applied in order to dispatch on its type (primitive
+versus compound) and apply it.
+
+ Whenever we need the actual value of an expression, we use
+
+ (define (actual-value exp env)
+ (force-it (eval exp env)))
+
+instead of just `eval', so that if the expression's value is a thunk, it
+will be forced.
+
+ Our new version of `apply' is also almost the same as the version in
+section *Note 4-1-1::. The difference is that `eval' has passed in
+unevaluated operand expressions: For primitive procedures (which are
+strict), we evaluate all the arguments before applying the primitive;
+for compound procedures (which are non-strict) we delay all the
+arguments before applying the procedure.
+
+ (define (apply procedure arguments env)
+ (cond ((primitive-procedure? procedure)
+ (apply-primitive-procedure
+ procedure
+ (list-of-arg-values arguments env))) ; changed
+ ((compound-procedure? procedure)
+ (eval-sequence
+ (procedure-body procedure)
+ (extend-environment
+ (procedure-parameters procedure)
+ (list-of-delayed-args arguments env) ; changed
+ (procedure-environment procedure))))
+ (else
+ (error
+ "Unknown procedure type -- APPLY" procedure))))
+
+ The procedures that process the arguments are just like
+`list-of-values' from section *Note 4-1-1::, except that
+`list-of-delayed-args' delays the arguments instead of evaluating them,
+and `list-of-arg-values' uses `actual-value' instead of `eval':
+
+ (define (list-of-arg-values exps env)
+ (if (no-operands? exps)
+ '()
+ (cons (actual-value (first-operand exps) env)
+ (list-of-arg-values (rest-operands exps)
+ env))))
+
+ (define (list-of-delayed-args exps env)
+ (if (no-operands? exps)
+ '()
+ (cons (delay-it (first-operand exps) env)
+ (list-of-delayed-args (rest-operands exps)
+ env))))
+
+ The other place we must change the evaluator is in the handling of
+`if', where we must use `actual-value' instead of `eval' to get the
+value of the predicate expression before testing whether it is true or
+false:
+
+ (define (eval-if exp env)
+ (if (true? (actual-value (if-predicate exp) env))
+ (eval (if-consequent exp) env)
+ (eval (if-alternative exp) env)))
+
+ Finally, we must change the `driver-loop' procedure (section *Note
+4-1-4::) to use `actual-value' instead of `eval', so that if a delayed
+value is propagated back to the read-eval-print loop, it will be forced
+before being printed. We also change the prompts to indicate that this
+is the lazy evaluator:
+
+ (define input-prompt ";;; L-Eval input:")
+ (define output-prompt ";;; L-Eval value:")
+
+ (define (driver-loop)
+ (prompt-for-input input-prompt)
+ (let ((input (read)))
+ (let ((output
+ (actual-value input the-global-environment)))
+ (announce-output output-prompt)
+ (user-print output)))
+ (driver-loop))
+
+ With these changes made, we can start the evaluator and test it. The
+successful evaluation of the `try' expression discussed in section
+*Note 4-2-1:: indicates that the interpreter is performing lazy
+evaluation:
+
+ (define the-global-environment (setup-environment))
+
+ (driver-loop)
+
+ ;;; L-Eval input:
+ (define (try a b)
+ (if (= a 0) 1 b))
+ ;;; L-Eval value:
+ ok
+
+ ;;; L-Eval input:
+ (try 0 (/ 1 0))
+ ;;; L-Eval value:
+ 1
+
+Representing thunks
+...................
+
+Our evaluator must arrange to create thunks when procedures are applied
+to arguments and to force these thunks later. A thunk must package an
+expression together with the environment, so that the argument can be
+produced later. To force the thunk, we simply extract the expression
+and environment from the thunk and evaluate the expression in the
+environment. We use `actual-value' rather than `eval' so that in case
+the value of the expression is itself a thunk, we will force that, and
+so on, until we reach something that is not a thunk:
+
+ (define (force-it obj)
+ (if (thunk? obj)
+ (actual-value (thunk-exp obj) (thunk-env obj))
+ obj))
+
+ One easy way to package an expression with an environment is to make
+a list containing the expression and the environment. Thus, we create
+a thunk as follows:
+
+ (define (delay-it exp env)
+ (list 'thunk exp env))
+
+ (define (thunk? obj)
+ (tagged-list? obj 'thunk))
+
+ (define (thunk-exp thunk) (cadr thunk))
+
+ (define (thunk-env thunk) (caddr thunk))
+
+ Actually, what we want for our interpreter is not quite this, but
+rather thunks that have been memoized. When a thunk is forced, we will
+turn it into an evaluated thunk by replacing the stored expression with
+its value and changing the `thunk' tag so that it can be recognized as
+already evaluated.(4)
+
+ (define (evaluated-thunk? obj)
+ (tagged-list? obj 'evaluated-thunk))
+
+ (define (thunk-value evaluated-thunk) (cadr evaluated-thunk))
+
+ (define (force-it obj)
+ (cond ((thunk? obj)
+ (let ((result (actual-value
+ (thunk-exp obj)
+ (thunk-env obj))))
+ (set-car! obj 'evaluated-thunk)
+ (set-car! (cdr obj) result) ; replace `exp' with its value
+ (set-cdr! (cdr obj) '()) ; forget unneeded `env'
+ result))
+ ((evaluated-thunk? obj)
+ (thunk-value obj))
+ (else obj)))
+
+ Notice that the same `delay-it' procedure works both with and without
+memoization.
+
+ *Exercise 4.27:* Suppose we type in the following definitions to
+ the lazy evaluator:
+
+ (define count 0)
+
+ (define (id x)
+ (set! count (+ count 1))
+ x)
+
+ Give the missing values in the following sequence of interactions,
+ and explain your answers.(5)
+
+ (define w (id (id 10)))
+
+ ;;; L-Eval input:
+ count
+ ;;; L-Eval value:
+ <RESPONSE>
+
+ ;;; L-Eval input:
+ w
+ ;;; L-Eval value:
+ <RESPONSE>
+
+ ;;; L-Eval input:
+ count
+ ;;; L-Eval value:
+ <RESPONSE>
+
+ *Exercise 4.28:* `Eval' uses `actual-value' rather than `eval' to
+ evaluate the operator before passing it to `apply', in order to
+ force the value of the operator. Give an example that
+ demonstrates the need for this forcing.
+
+ *Exercise 4.29:* Exhibit a program that you would expect to run
+ much more slowly without memoization than with memoization. Also,
+ consider the following interaction, where the `id' procedure is
+ defined as in *Note Exercise 4-27:: and `count' starts at 0:
+
+ (define (square x)
+ (* x x))
+
+ ;;; L-Eval input:
+ (square (id 10))
+ ;;; L-Eval value:
+ <RESPONSE>
+
+ ;;; L-Eval input:
+ count
+ ;;; L-Eval value:
+ <RESPONSE>
+
+ Give the responses both when the evaluator memoizes and when it
+ does not.
+
+ *Exercise 4.30:* Cy D. Fect, a reformed C programmer, is worried
+ that some side effects may never take place, because the lazy
+ evaluator doesn't force the expressions in a sequence. Since the
+ value of an expression in a sequence other than the last one is
+ not used (the expression is there only for its effect, such as
+ assigning to a variable or printing), there can be no subsequent
+ use of this value (e.g., as an argument to a primitive procedure)
+ that will cause it to be forced. Cy thus thinks that when
+ evaluating sequences, we must force all expressions in the
+ sequence except the final one. He proposes to modify
+ `eval-sequence' from section *Note 4-1-1:: to use `actual-value'
+ rather than `eval':
+
+ (define (eval-sequence exps env)
+ (cond ((last-exp? exps) (eval (first-exp exps) env))
+ (else (actual-value (first-exp exps) env)
+ (eval-sequence (rest-exps exps) env))))
+
+ a. Ben Bitdiddle thinks Cy is wrong. He shows Cy the `for-each'
+ procedure described in *Note Exercise 2-23::, which gives an
+ important example of a sequence with side effects:
+
+ (define (for-each proc items)
+ (if (null? items)
+ 'done
+ (begin (proc (car items))
+ (for-each proc (cdr items)))))
+
+ He claims that the evaluator in the text (with the original
+ `eval-sequence') handles this correctly:
+
+ ;;; L-Eval input:
+ (for-each (lambda (x) (newline) (display x))
+ (list 57 321 88))
+ 57
+ 321
+ 88
+ ;;; L-Eval value:
+ done
+
+ Explain why Ben is right about the behavior of `for-each'.
+
+ b. Cy agrees that Ben is right about the `for-each' example, but
+ says that that's not the kind of program he was thinking
+ about when he proposed his change to `eval-sequence'. He
+ defines the following two procedures in the lazy evaluator:
+
+ (define (p1 x)
+ (set! x (cons x '(2)))
+ x)
+
+ (define (p2 x)
+ (define (p e)
+ e
+ x)
+ (p (set! x (cons x '(2)))))
+
+ What are the values of `(p1 1)' and `(p2 1)' with the original
+ `eval-sequence'? What would the values be with Cy's proposed
+ change to `eval-sequence'?
+
+ c. Cy also points out that changing `eval-sequence' as he
+ proposes does not affect the behavior of the example in part
+ a. Explain why this is true.
+
+ d. How do you think sequences ought to be treated in the lazy
+ evaluator? Do you like Cy's approach, the approach in the
+ text, or some other approach?
+
+
+ *Exercise 4.31:* The approach taken in this section is somewhat
+ unpleasant, because it makes an incompatible change to Scheme. It
+ might be nicer to implement lazy evaluation as an "upward-compatible
+ extension", that is, so that ordinary Scheme programs will work as
+ before. We can do this by extending the syntax of procedure
+ declarations to let the user control whether or not arguments are
+ to be delayed. While we're at it, we may as well also give the
+ user the choice between delaying with and without memoization.
+ For example, the definition
+
+ (define (f a (b lazy) c (d lazy-memo))
+ ...)
+
+ would define `f' to be a procedure of four arguments, where the
+ first and third arguments are evaluated when the procedure is
+ called, the second argument is delayed, and the fourth argument is
+ both delayed and memoized. Thus, ordinary procedure definitions
+ will produce the same behavior as ordinary Scheme, while adding
+ the `lazy-memo' declaration to each parameter of every compound
+ procedure will produce the behavior of the lazy evaluator defined
+ in this section. Design and implement the changes required to
+ produce such an extension to Scheme. You will have to implement
+ new syntax procedures to handle the new syntax for `define'. You
+ must also arrange for `eval' or `apply' to determine when
+ arguments are to be delayed, and to force or delay arguments
+ accordingly, and you must arrange for forcing to memoize or not,
+ as appropriate.
+
+ ---------- Footnotes ----------
+
+ (1) The word "thunk" was invented by an informal working group that
+was discussing the implementation of call-by-name in Algol 60. They
+observed that most of the analysis of ("thinking about") the expression
+could be done at compile time; thus, at run time, the expression would
+already have been "thunk" about (Ingerman et al. 1960).
+
+ (2) This is analogous to the use of `force' on the delayed objects
+that were introduced in *Note Chapter 3:: to represent streams. The
+critical difference between what we are doing here and what we did in
+*Note Chapter 3:: is that we are building delaying and forcing into the
+evaluator, and thus making this uniform and automatic throughout the
+language.
+
+ (3) Lazy evaluation combined with memoization is sometimes referred
+to as "call-by-need" argument passing, in contrast to "call-by-name"
+argument passing. (Call-by-name, introduced in Algol 60, is similar to
+non-memoized lazy evaluation.) As language designers, we can build our
+evaluator to memoize, not to memoize, or leave this an option for
+programmers (*Note Exercise 4-31::). As you might expect from *Note
+Chapter 3::, these choices raise issues that become both subtle and
+confusing in the presence of assignments. (See *Note Exercise 4-27::
+and *Note Exercise 4-29::.) An excellent article by Clinger (1982)
+attempts to clarify the multiple dimensions of confusion that arise
+here.
+
+ (4) Notice that we also erase the `env' from the thunk once the
+expression's value has been computed. This makes no difference in the
+values returned by the interpreter. It does help save space, however,
+because removing the reference from the thunk to the `env' once it is
+no longer needed allows this structure to be "garbage-collected" and
+its space recycled, as we will discuss in section *Note 5-3::.
+
+ Similarly, we could have allowed unneeded environments in the
+memoized delayed objects of section *Note 3-5-1:: to be
+garbage-collected, by having `memo-proc' do something like `(set! proc
+'())' to discard the procedure `proc' (which includes the environment
+in which the `delay' was evaluated) after storing its value.
+
+ (5) This exercise demonstrates that the interaction between lazy
+evaluation and side effects can be very confusing. This is just what
+you might expect from the discussion in *Note Chapter 3::.
+
+\1f
+File: sicp.info, Node: 4-2-3, Prev: 4-2-2, Up: 4-2
+
+4.2.3 Streams as Lazy Lists
+---------------------------
+
+In section *Note 3-5-1::, we showed how to implement streams as delayed
+lists. We introduced special forms `delay' and `cons-stream', which
+allowed us to construct a "promise" to compute the `cdr' of a stream,
+without actually fulfilling that promise until later. We could use
+this general technique of introducing special forms whenever we need
+more control over the evaluation process, but this is awkward. For one
+thing, a special form is not a first-class object like a procedure, so
+we cannot use it together with higher-order procedures.(1)
+Additionally, we were forced to create streams as a new kind of data
+object similar but not identical to lists, and this required us to
+reimplement many ordinary list operations (`map', `append', and so on)
+for use with streams.
+
+ With lazy evaluation, streams and lists can be identical, so there
+is no need for special forms or for separate list and stream
+operations. All we need to do is to arrange matters so that `cons' is
+non-strict. One way to accomplish this is to extend the lazy evaluator
+to allow for non-strict primitives, and to implement `cons' as one of
+these. An easier way is to recall (section *Note 2-1-3::) that there
+is no fundamental need to implement `cons' as a primitive at all.
+Instead, we can represent pairs as procedures:(2)
+
+ (define (cons x y)
+ (lambda (m) (m x y)))
+
+ (define (car z)
+ (z (lambda (p q) p)))
+
+ (define (cdr z)
+ (z (lambda (p q) q)))
+
+ In terms of these basic operations, the standard definitions of the
+list operations will work with infinite lists (streams) as well as
+finite ones, and the stream operations can be implemented as list
+operations. Here are some examples:
+
+ (define (list-ref items n)
+ (if (= n 0)
+ (car items)
+ (list-ref (cdr items) (- n 1))))
+
+ (define (map proc items)
+ (if (null? items)
+ '()
+ (cons (proc (car items))
+ (map proc (cdr items)))))
+
+ (define (scale-list items factor)
+ (map (lambda (x) (* x factor))
+ items))
+
+ (define (add-lists list1 list2)
+ (cond ((null? list1) list2)
+ ((null? list2) list1)
+ (else (cons (+ (car list1) (car list2))
+ (add-lists (cdr list1) (cdr list2))))))
+
+ (define ones (cons 1 ones))
+
+ (define integers (cons 1 (add-lists ones integers)))
+
+ ;;; L-Eval input:
+ (list-ref integers 17)
+ ;;; L-Eval value:
+ 18
+
+ Note that these lazy lists are even lazier than the streams of *Note
+Chapter 3::: The `car' of the list, as well as the `cdr', is
+delayed.(3) In fact, even accessing the `car' or `cdr' of a lazy pair
+need not force the value of a list element. The value will be forced
+only when it is really needed - e.g., for use as the argument of a
+primitive, or to be printed as an answer.
+
+ Lazy pairs also help with the problem that arose with streams in
+section *Note 3-5-4::, where we found that formulating stream models of
+systems with loops may require us to sprinkle our programs with
+explicit `delay' operations, beyond the ones supplied by `cons-stream'.
+With lazy evaluation, all arguments to procedures are delayed
+uniformly. For instance, we can implement procedures to integrate
+lists and solve differential equations as we originally intended in
+section *Note 3-5-4:::
+
+ (define (integral integrand initial-value dt)
+ (define int
+ (cons initial-value
+ (add-lists (scale-list integrand dt)
+ int)))
+ int)
+
+ (define (solve f y0 dt)
+ (define y (integral dy y0 dt))
+ (define dy (map f y))
+ y)
+
+ ;;; L-Eval input:
+ (list-ref (solve (lambda (x) x) 1 0.001) 1000)
+ ;;; L-Eval value:
+ 2.716924
+
+ *Exercise 4.32:* Give some examples that illustrate the difference
+ between the streams of *Note Chapter 3:: and the "lazier" lazy
+ lists described in this section. How can you take advantage of
+ this extra laziness?
+
+ *Exercise 4.33:* Ben Bitdiddle tests the lazy list implementation
+ given above by evaluating the expression
+
+ (car '(a b c))
+
+ To his surprise, this produces an error. After some thought, he
+ realizes that the "lists" obtained by reading in quoted
+ expressions are different from the lists manipulated by the new
+ definitions of `cons', `car', and `cdr'. Modify the evaluator's
+ treatment of quoted expressions so that quoted lists typed at the
+ driver loop will produce true lazy lists.
+
+ *Exercise 4.34:* Modify the driver loop for the evaluator so that
+ lazy pairs and lists will print in some reasonable way. (What are
+ you going to do about infinite lists?) You may also need to modify
+ the representation of lazy pairs so that the evaluator can
+ identify them in order to print them.
+
+ ---------- Footnotes ----------
+
+ (1) This is precisely the issue with the `unless' procedure, as in
+*Note Exercise 4-26::.
+
+ (2) This is the procedural representation described in *Note
+Exercise 2-4::. Essentially any procedural representation (e.g., a
+message-passing implementation) would do as well. Notice that we can
+install these definitions in the lazy evaluator simply by typing them
+at the driver loop. If we had originally included `cons', `car', and
+`cdr' as primitives in the global environment, they will be redefined.
+(Also see *Note Exercise 4-33:: and *Note Exercise 4-34::.)
+
+ (3) This permits us to create delayed versions of more general kinds
+of list structures, not just sequences. Hughes 1990 discusses some
+applications of "lazy trees."
+
+\1f
+File: sicp.info, Node: 4-3, Next: 4-4, Prev: 4-2, Up: Chapter 4
+
+4.3 Variations on a Scheme - Nondeterministic Computing
+=======================================================
+
+In this section, we extend the Scheme evaluator to support a programming
+paradigm called "nondeterministic computing" by building into the
+evaluator a facility to support automatic search. This is a much more
+profound change to the language than the introduction of lazy
+evaluation in section *Note 4-2::.
+
+ Nondeterministic computing, like stream processing, is useful for
+"generate and test" applications. Consider the task of starting with
+two lists of positive integers and finding a pair of integers--one from
+the first list and one from the second list--whose sum is prime. We
+saw how to handle this with finite sequence operations in section *Note
+2-2-3:: and with infinite streams in section *Note 3-5-3::. Our
+approach was to generate the sequence of all possible pairs and filter
+these to select the pairs whose sum is prime. Whether we actually
+generate the entire sequence of pairs first as in *Note Chapter 2::, or
+interleave the generating and filtering as in *Note Chapter 3::, is
+immaterial to the essential image of how the computation is organized.
+
+ The nondeterministic approach evokes a different image. Imagine
+simply that we choose (in some way) a number from the first list and a
+number from the second list and require (using some mechanism) that
+their sum be prime. This is expressed by following procedure:
+
+ (define (prime-sum-pair list1 list2)
+ (let ((a (an-element-of list1))
+ (b (an-element-of list2)))
+ (require (prime? (+ a b)))
+ (list a b)))
+
+ It might seem as if this procedure merely restates the problem,
+rather than specifying a way to solve it. Nevertheless, this is a
+legitimate nondeterministic program.(1)
+
+ The key idea here is that expressions in a nondeterministic language
+can have more than one possible value. For instance, `an-element-of'
+might return any element of the given list. Our nondeterministic
+program evaluator will work by automatically choosing a possible value
+and keeping track of the choice. If a subsequent requirement is not
+met, the evaluator will try a different choice, and it will keep trying
+new choices until the evaluation succeeds, or until we run out of
+choices. Just as the lazy evaluator freed the programmer from the
+details of how values are delayed and forced, the nondeterministic
+program evaluator will free the programmer from the details of how
+choices are made.
+
+ It is instructive to contrast the different images of time evoked by
+nondeterministic evaluation and stream processing. Stream processing
+uses lazy evaluation to decouple the time when the stream of possible
+answers is assembled from the time when the actual stream elements are
+produced. The evaluator supports the illusion that all the possible
+answers are laid out before us in a timeless sequence. With
+nondeterministic evaluation, an expression represents the exploration
+of a set of possible worlds, each determined by a set of choices. Some
+of the possible worlds lead to dead ends, while others have useful
+values. The nondeterministic program evaluator supports the illusion
+that time branches, and that our programs have different possible
+execution histories. When we reach a dead end, we can revisit a
+previous choice point and proceed along a different branch.
+
+ The nondeterministic program evaluator implemented below is called
+the `amb' evaluator because it is based on a new special form called
+`amb'. We can type the above definition of `prime-sum-pair' at the
+`amb' evaluator driver loop (along with definitions of `prime?',
+`an-element-of', and `require') and run the procedure as follows:
+
+ ;;; Amb-Eval input:
+ (prime-sum-pair '(1 3 5 8) '(20 35 110))
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ (3 20)
+
+ The value returned was obtained after the evaluator repeatedly chose
+elements from each of the lists, until a successful choice was made.
+
+ Section *Note 4-3-1:: introduces `amb' and explains how it supports
+nondeterminism through the evaluator's automatic search mechanism.
+Section *Note 4-3-2:: presents examples of nondeterministic programs,
+and section *Note 4-3-3:: gives the details of how to implement the
+`amb' evaluator by modifying the ordinary Scheme evaluator.
+
+* Menu:
+
+* 4-3-1:: Amb and Search
+* 4-3-2:: Examples of Nondeterministic Programs
+* 4-3-3:: Implementing the `Amb' Evaluator
+
+ ---------- Footnotes ----------
+
+ (1) We assume that we have previously defined a procedure `prime?'
+that tests whether numbers are prime. Even with `prime?' defined, the
+`prime-sum-pair' procedure may look suspiciously like the unhelpful
+"pseudo-Lisp" attempt to define the square-root function, which we
+described at the beginning of section *Note 1-1-7::. In fact, a
+square-root procedure along those lines can actually be formulated as a
+nondeterministic program. By incorporating a search mechanism into the
+evaluator, we are eroding the distinction between purely declarative
+descriptions and imperative specifications of how to compute answers.
+We'll go even farther in this direction in section *Note 4-4::.
+
+\1f
+File: sicp.info, Node: 4-3-1, Next: 4-3-2, Prev: 4-3, Up: 4-3
+
+4.3.1 Amb and Search
+--------------------
+
+To extend Scheme to support nondeterminism, we introduce a new special
+form called `amb'.(1) The expression
+
+ (amb <E_1> <E_2> ... <E_N>)
+
+returns the value of one of the n expressions <E_I> "ambiguously." For
+example, the expression
+
+ (list (amb 1 2 3) (amb 'a 'b))
+
+can have six possible values:
+
+ `(1 a)' `(1 b)' `(2 a)' `(2 b)' `(3 a)' `(3 b)'
+
+ `Amb' with a single choice produces an ordinary (single) value.
+
+ `Amb' with no choices--the expression `(amb)'--is an expression with
+no acceptable values. Operationally, we can think of `(amb)' as an
+expression that when evaluated causes the computation to "fail": The
+computation aborts and no value is produced. Using this idea, we can
+express the requirement that a particular predicate expression `p' must
+be true as follows:
+
+ (define (require p)
+ (if (not p) (amb)))
+
+ With `amb' and `require', we can implement the `an-element-of'
+procedure used above:
+
+ (define (an-element-of items)
+ (require (not (null? items)))
+ (amb (car items) (an-element-of (cdr items))))
+
+ `An-element-of' fails if the list is empty. Otherwise it ambiguously
+returns either the first element of the list or an element chosen from
+the rest of the list.
+
+ We can also express infinite ranges of choices. The following
+procedure potentially returns any integer greater than or equal to some
+given n:
+
+ (define (an-integer-starting-from n)
+ (amb n (an-integer-starting-from (+ n 1))))
+
+ This is like the stream procedure `integers-starting-from' described
+in section *Note 3-5-2::, but with an important difference: The stream
+procedure returns an object that represents the sequence of all
+integers beginning with n, whereas the `amb' procedure returns a single
+integer.(2)
+
+ Abstractly, we can imagine that evaluating an `amb' expression
+causes time to split into branches, where the computation continues on
+each branch with one of the possible values of the expression. We say
+that `amb' represents a "nondeterministic choice point". If we had a
+machine with a sufficient number of processors that could be
+dynamically allocated, we could implement the search in a
+straightforward way. Execution would proceed as in a sequential
+machine, until an `amb' expression is encountered. At this point, more
+processors would be allocated and initialized to continue all of the
+parallel executions implied by the choice. Each processor would proceed
+sequentially as if it were the only choice, until it either terminates
+by encountering a failure, or it further subdivides, or it finishes.(3)
+
+ On the other hand, if we have a machine that can execute only one
+process (or a few concurrent processes), we must consider the
+alternatives sequentially. One could imagine modifying an evaluator to
+pick at random a branch to follow whenever it encounters a choice
+point. Random choice, however, can easily lead to failing values. We
+might try running the evaluator over and over, making random choices
+and hoping to find a non-failing value, but it is better to "systematically
+search" all possible execution paths. The `amb' evaluator that we will
+develop and work with in this section implements a systematic search as
+follows: When the evaluator encounters an application of `amb', it
+initially selects the first alternative. This selection may itself
+lead to a further choice. The evaluator will always initially choose
+the first alternative at each choice point. If a choice results in a
+failure, then the evaluator automagically(4) "backtracks" to the most
+recent choice point and tries the next alternative. If it runs out of
+alternatives at any choice point, the evaluator will back up to the
+previous choice point and resume from there. This process leads to a
+search strategy known as "depth-first search" or backtracking
+"chronological backtracking".(5)
+
+Driver loop
+...........
+
+The driver loop for the `amb' evaluator has some unusual properties. It
+reads an expression and prints the value of the first non-failing
+execution, as in the `prime-sum-pair' example shown above. If we want
+to see the value of the next successful execution, we can ask the
+interpreter to backtrack and attempt to generate a second non-failing
+execution. This is signaled by typing the symbol `try-again'. If any
+expression except `try-again' is given, the interpreter will start a
+new problem, discarding the unexplored alternatives in the previous
+problem. Here is a sample interaction:
+
+ ;;; Amb-Eval input:
+ (prime-sum-pair '(1 3 5 8) '(20 35 110))
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ (3 20)
+
+ ;;; Amb-Eval input:
+ try-again
+ ;;; Amb-Eval value:
+ (3 110)
+
+ ;;; Amb-Eval input:
+ try-again
+ ;;; Amb-Eval value:
+ (8 35)
+
+ ;;; Amb-Eval input:
+ try-again
+ ;;; There are no more values of
+ (prime-sum-pair (quote (1 3 5 8)) (quote (20 35 110)))
+
+ ;;; Amb-Eval input:
+ (prime-sum-pair '(19 27 30) '(11 36 58))
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ (30 11)
+
+ *Exercise 4.35:* Write a procedure `an-integer-between' that
+ returns an integer between two given bounds. This can be used to
+ implement a procedure that finds Pythagorean triples, i.e.,
+ triples of integers (i,j,k) between the given bounds such that i
+ <= j and i^2 + j^2 = k^2, as follows:
+
+ (define (a-pythagorean-triple-between low high)
+ (let ((i (an-integer-between low high)))
+ (let ((j (an-integer-between i high)))
+ (let ((k (an-integer-between j high)))
+ (require (= (+ (* i i) (* j j)) (* k k)))
+ (list i j k)))))
+
+ *Exercise 4.36:* *Note Exercise 3-69:: discussed how to generate
+ the stream of _all_ Pythagorean triples, with no upper bound on
+ the size of the integers to be searched. Explain why simply
+ replacing `an-integer-between' by `an-integer-starting-from' in
+ the procedure in *Note Exercise 4-35:: is not an adequate way to
+ generate arbitrary Pythagorean triples. Write a procedure that
+ actually will accomplish this. (That is, write a procedure for
+ which repeatedly typing `try-again' would in principle eventually
+ generate all Pythagorean triples.)
+
+ *Exercise 4.37:* Ben Bitdiddle claims that the following method
+ for generating Pythagorean triples is more efficient than the one
+ in *Note Exercise 4-35::. Is he correct? (Hint: Consider the
+ number of possibilities that must be explored.)
+
+ (define (a-pythagorean-triple-between low high)
+ (let ((i (an-integer-between low high))
+ (hsq (* high high)))
+ (let ((j (an-integer-between i high)))
+ (let ((ksq (+ (* i i) (* j j))))
+ (require (>= hsq ksq))
+ (let ((k (sqrt ksq)))
+ (require (integer? k))
+ (list i j k))))))
+
+ ---------- Footnotes ----------
+
+ (1) The idea of `amb' for nondeterministic programming was first
+described in 1961 by John McCarthy (see McCarthy 1967).
+
+ (2) In actuality, the distinction between nondeterministically
+returning a single choice and returning all choices depends somewhat on
+our point of view. From the perspective of the code that uses the
+value, the nondeterministic choice returns a single value. From the
+perspective of the programmer designing the code, the nondeterministic
+choice potentially returns all possible values, and the computation
+branches so that each value is investigated separately.
+
+ (3) One might object that this is a hopelessly inefficient
+mechanism. It might require millions of processors to solve some
+easily stated problem this way, and most of the time most of those
+processors would be idle. This objection should be taken in the
+context of history. Memory used to be considered just such an
+expensive commodity. In 1964 a megabyte of RAM cost about $400,000.
+Now every personal computer has many megabytes of RAM, and most of the
+time most of that RAM is unused. It is hard to underestimate the cost
+of mass-produced electronics.
+
+ (4) Automagically: "Automatically, but in a way which, for some
+reason (typically because it is too complicated, or too ugly, or
+perhaps even too trivial), the speaker doesn't feel like explaining."
+(Steele 1983, Raymond 1993)
+
+ (5) [Footnote 4.47] The integration of automatic search strategies
+into programming languages has had a long and checkered history. The
+first suggestions that nondeterministic algorithms might be elegantly
+encoded in a programming language with search and automatic
+backtracking came from Robert Floyd (1967). Carl Hewitt (1969)
+invented a programming language called Planner that explicitly
+supported automatic chronological backtracking, providing for a
+built-in depth-first search strategy. Sussman, Winograd, and Charniak
+(1971) implemented a subset of this language, called MicroPlanner,
+which was used to support work in problem solving and robot planning.
+Similar ideas, arising from logic and theorem proving, led to the
+genesis in Edinburgh and Marseille of the elegant language Prolog
+(which we will discuss in section *Note 4-4::). After sufficient
+frustration with automatic search, McDermott and Sussman (1972)
+developed a language called Conniver, which included mechanisms for
+placing the search strategy under programmer control. This proved
+unwieldy, however, and Sussman and Stallman (1975) found a more
+tractable approach while investigating methods of symbolic analysis for
+electrical circuits. They developed a non-chronological backtracking
+scheme that was based on tracing out the logical dependencies
+connecting facts, a technique that has come to be known as "dependency-directed
+backtracking". Although their method was complex, it produced
+reasonably efficient programs because it did little redundant search.
+Doyle (1979) and McAllester (1978, 1980) generalized and clarified the
+methods of Stallman and Sussman, developing a new paradigm for
+formulating search that is now called "truth maintenance". Modern
+problem-solving systems all use some form of truth-maintenance system
+as a substrate. See Forbus and deKleer 1993 for a discussion of
+elegant ways to build truth-maintenance systems and applications using
+truth maintenance. Zabih, McAllester, and Chapman 1987 describes a
+nondeterministic extension to Scheme that is based on `amb'; it is
+similar to the interpreter described in this section, but more
+sophisticated, because it uses dependency-directed backtracking rather
+than chronological backtracking. Winston 1992 gives an introduction to
+both kinds of backtracking.
+
+\1f
+File: sicp.info, Node: 4-3-2, Next: 4-3-3, Prev: 4-3-1, Up: 4-3
+
+4.3.2 Examples of Nondeterministic Programs
+-------------------------------------------
+
+Section *Note 4-3-3:: describes the implementation of the `amb'
+evaluator. First, however, we give some examples of how it can be
+used. The advantage of nondeterministic programming is that we can
+suppress the details of how search is carried out, thereby expressing
+our programs at a higher level of abstraction.
+
+Logic Puzzles
+.............
+
+The following puzzle (taken from Dinesman 1968) is typical of a large
+class of simple logic puzzles:
+
+ Baker, Cooper, Fletcher, Miller, and Smith live on different
+ floors of an apartment house that contains only five floors.
+ Baker does not live on the top floor. Cooper does not live on the
+ bottom floor. Fletcher does not live on either the top or the
+ bottom floor. Miller lives on a higher floor than does Cooper.
+ Smith does not live on a floor adjacent to Fletcher's. Fletcher
+ does not live on a floor adjacent to Cooper's. Where does
+ everyone live?
+
+ We can determine who lives on each floor in a straightforward way by
+enumerating all the possibilities and imposing the given
+restrictions:(1)
+
+ (define (multiple-dwelling)
+ (let ((baker (amb 1 2 3 4 5))
+ (cooper (amb 1 2 3 4 5))
+ (fletcher (amb 1 2 3 4 5))
+ (miller (amb 1 2 3 4 5))
+ (smith (amb 1 2 3 4 5)))
+ (require
+ (distinct? (list baker cooper fletcher miller smith)))
+ (require (not (= baker 5)))
+ (require (not (= cooper 1)))
+ (require (not (= fletcher 5)))
+ (require (not (= fletcher 1)))
+ (require (> miller cooper))
+ (require (not (= (abs (- smith fletcher)) 1)))
+ (require (not (= (abs (- fletcher cooper)) 1)))
+ (list (list 'baker baker)
+ (list 'cooper cooper)
+ (list 'fletcher fletcher)
+ (list 'miller miller)
+ (list 'smith smith))))
+
+ Evaluating the expression `(multiple-dwelling)' produces the result
+
+ ((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1))
+
+ Although this simple procedure works, it is very slow. *Note
+Exercise 4-39:: and *Note Exercise 4-40:: discuss some possible
+improvements.
+
+ *Exercise 4.38:* Modify the multiple-dwelling procedure to omit
+ the requirement that Smith and Fletcher do not live on adjacent
+ floors. How many solutions are there to this modified puzzle?
+
+ *Exercise 4.39:* Does the order of the restrictions in the
+ multiple-dwelling procedure affect the answer? Does it affect the
+ time to find an answer? If you think it matters, demonstrate a
+ faster program obtained from the given one by reordering the
+ restrictions. If you think it does not matter, argue your case.
+
+ *Exercise 4.40:* In the multiple dwelling problem, how many sets
+ of assignments are there of people to floors, both before and
+ after the requirement that floor assignments be distinct? It is
+ very inefficient to generate all possible assignments of people to
+ floors and then leave it to backtracking to eliminate them. For
+ example, most of the restrictions depend on only one or two of the
+ person-floor variables, and can thus be imposed before floors have
+ been selected for all the people. Write and demonstrate a much
+ more efficient nondeterministic procedure that solves this problem
+ based upon generating only those possibilities that are not already
+ ruled out by previous restrictions. (Hint: This will require a
+ nest of `let' expressions.)
+
+ *Exercise 4.41:* Write an ordinary Scheme program to solve the
+ multiple dwelling puzzle.
+
+ *Exercise 4.42:* Solve the following "Liars" puzzle (from Phillips
+ 1934):
+
+ Five schoolgirls sat for an examination. Their parents--so they
+ thought--showed an undue degree of interest in the result. They
+ therefore agreed that, in writing home about the examination, each
+ girl should make one true statement and one untrue one. The
+ following are the relevant passages from their letters:
+
+ * Betty: "Kitty was second in the examination. I was only
+ third."
+
+ * Ethel: "You'll be glad to hear that I was on top. Joan was
+ second."
+
+ * Joan: "I was third, and poor old Ethel was bottom."
+
+ * Kitty: "I came out second. Mary was only fourth."
+
+ * Mary: "I was fourth. Top place was taken by Betty."
+
+
+ What in fact was the order in which the five girls were placed?
+
+ *Exercise 4.43:* Use the `amb' evaluator to solve the following
+ puzzle:(2)
+
+ Mary Ann Moore's father has a yacht and so has each of his
+ four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood,
+ and Dr. Parker. Each of the five also has one daughter and
+ each has named his yacht after a daughter of one of the
+ others. Sir Barnacle's yacht is the Gabrielle, Mr. Moore
+ owns the Lorna; Mr. Hall the Rosalind. The Melissa, owned by
+ Colonel Downing, is named after Sir Barnacle's daughter.
+ Gabrielle's father owns the yacht that is named after Dr.
+ Parker's daughter. Who is Lorna's father?
+
+ Try to write the program so that it runs efficiently (see *Note
+ Exercise 4-40::). Also determine how many solutions there are if
+ we are not told that Mary Ann's last name is Moore.
+
+ *Exercise 4.44:* *Note Exercise 2-42:: described the "eight-queens
+ puzzle" of placing queens on a chessboard so that no two attack
+ each other. Write a nondeterministic program to solve this puzzle.
+
+Parsing natural language
+........................
+
+Programs designed to accept natural language as input usually start by
+attempting to "parse" the input, that is, to match the input against
+some grammatical structure. For example, we might try to recognize
+simple sentences consisting of an article followed by a noun followed
+by a verb, such as "The cat eats." To accomplish such an analysis, we
+must be able to identify the parts of speech of individual words. We
+could start with some lists that classify various words:(3)
+
+ (define nouns '(noun student professor cat class))
+
+ (define verbs '(verb studies lectures eats sleeps))
+
+ (define articles '(article the a))
+
+ We also need a "grammar", that is, a set of rules describing how
+grammatical elements are composed from simpler elements. A very simple
+grammar might stipulate that a sentence always consists of two
+pieces--a noun phrase followed by a verb--and that a noun phrase
+consists of an article followed by a noun. With this grammar, the
+sentence "The cat eats" is parsed as follows:
+
+ (sentence (noun-phrase (article the) (noun cat))
+ (verb eats))
+
+ We can generate such a parse with a simple program that has separate
+procedures for each of the grammatical rules. To parse a sentence, we
+identify its two constituent pieces and return a list of these two
+elements, tagged with the symbol `sentence':
+
+ (define (parse-sentence)
+ (list 'sentence
+ (parse-noun-phrase)
+ (parse-word verbs)))
+
+ A noun phrase, similarly, is parsed by finding an article followed
+by a noun:
+
+ (define (parse-noun-phrase)
+ (list 'noun-phrase
+ (parse-word articles)
+ (parse-word nouns)))
+
+ At the lowest level, parsing boils down to repeatedly checking that
+the next unparsed word is a member of the list of words for the
+required part of speech. To implement this, we maintain a global
+variable `*unparsed*', which is the input that has not yet been parsed.
+Each time we check a word, we require that `*unparsed*' must be
+non-empty and that it should begin with a word from the designated
+list. If so, we remove that word from `*unparsed*' and return the word
+together with its part of speech (which is found at the head of the
+list):(4)
+
+ (define (parse-word word-list)
+ (require (not (null? *unparsed*)))
+ (require (memq (car *unparsed*) (cdr word-list)))
+ (let ((found-word (car *unparsed*)))
+ (set! *unparsed* (cdr *unparsed*))
+ (list (car word-list) found-word)))
+
+ To start the parsing, all we need to do is set `*unparsed*' to be the
+entire input, try to parse a sentence, and check that nothing is left
+over:
+
+ (define *unparsed* '())
+
+ (define (parse input)
+ (set! *unparsed* input)
+ (let ((sent (parse-sentence)))
+ (require (null? *unparsed*))
+ sent))
+
+ We can now try the parser and verify that it works for our simple
+test sentence:
+
+ ;;; Amb-Eval input:
+ (parse '(the cat eats))
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ (sentence (noun-phrase (article the) (noun cat)) (verb eats))
+
+ The `amb' evaluator is useful here because it is convenient to
+express the parsing constraints with the aid of `require'. Automatic
+search and backtracking really pay off, however, when we consider more
+complex grammars where there are choices for how the units can be
+decomposed.
+
+ Let's add to our grammar a list of prepositions:
+
+ (define prepositions '(prep for to in by with))
+
+and define a prepositional phrase (e.g., "for the cat") to be a
+preposition followed by a noun phrase:
+
+ (define (parse-prepositional-phrase)
+ (list 'prep-phrase
+ (parse-word prepositions)
+ (parse-noun-phrase)))
+
+ Now we can define a sentence to be a noun phrase followed by a verb
+phrase, where a verb phrase can be either a verb or a verb phrase
+extended by a prepositional phrase:(5)
+
+ (define (parse-sentence)
+ (list 'sentence
+ (parse-noun-phrase)
+ (parse-verb-phrase)))
+
+ (define (parse-verb-phrase)
+ (define (maybe-extend verb-phrase)
+ (amb verb-phrase
+ (maybe-extend (list 'verb-phrase
+ verb-phrase
+ (parse-prepositional-phrase)))))
+ (maybe-extend (parse-word verbs)))
+
+ While we're at it, we can also elaborate the definition of noun
+phrases to permit such things as "a cat in the class." What we used to
+call a noun phrase, we'll now call a simple noun phrase, and a noun
+phrase will now be either a simple noun phrase or a noun phrase
+extended by a prepositional phrase:
+
+ (define (parse-simple-noun-phrase)
+ (list 'simple-noun-phrase
+ (parse-word articles)
+ (parse-word nouns)))
+
+ (define (parse-noun-phrase)
+ (define (maybe-extend noun-phrase)
+ (amb noun-phrase
+ (maybe-extend (list 'noun-phrase
+ noun-phrase
+ (parse-prepositional-phrase)))))
+ (maybe-extend (parse-simple-noun-phrase)))
+
+ Our new grammar lets us parse more complex sentences. For example
+
+ (parse '(the student with the cat sleeps in the class))
+
+produces
+
+ (sentence
+ (noun-phrase
+ (simple-noun-phrase (article the) (noun student))
+ (prep-phrase (prep with)
+ (simple-noun-phrase
+ (article the) (noun cat))))
+ (verb-phrase
+ (verb sleeps)
+ (prep-phrase (prep in)
+ (simple-noun-phrase
+ (article the) (noun class)))))
+
+ Observe that a given input may have more than one legal parse. In
+the sentence "The professor lectures to the student with the cat," it
+may be that the professor is lecturing with the cat, or that the
+student has the cat. Our nondeterministic program finds both
+possibilities:
+
+ (parse '(the professor lectures to the student with the cat))
+
+produces
+
+ (sentence
+ (simple-noun-phrase (article the) (noun professor))
+ (verb-phrase
+ (verb-phrase
+ (verb lectures)
+ (prep-phrase (prep to)
+ (simple-noun-phrase
+ (article the) (noun student))))
+ (prep-phrase (prep with)
+ (simple-noun-phrase
+ (article the) (noun cat)))))
+
+ Asking the evaluator to try again yields
+
+ (sentence
+ (simple-noun-phrase (article the) (noun professor))
+ (verb-phrase
+ (verb lectures)
+ (prep-phrase (prep to)
+ (noun-phrase
+ (simple-noun-phrase
+ (article the) (noun student))
+ (prep-phrase (prep with)
+ (simple-noun-phrase
+ (article the) (noun cat)))))))
+
+ *Exercise 4.45:* With the grammar given above, the following
+ sentence can be parsed in five different ways: "The professor
+ lectures to the student in the class with the cat." Give the five
+ parses and explain the differences in shades of meaning among them.
+
+ *Exercise 4.46:* The evaluators in sections *Note 4-1:: and *Note
+ 4-2:: do not determine what order operands are evaluated in. We
+ will see that the `amb' evaluator evaluates them from left to
+ right. Explain why our parsing program wouldn't work if the
+ operands were evaluated in some other order.
+
+ *Exercise 4.47:* Louis Reasoner suggests that, since a verb phrase
+ is either a verb or a verb phrase followed by a prepositional
+ phrase, it would be much more straightforward to define the
+ procedure `parse-verb-phrase' as follows (and similarly for noun
+ phrases):
+
+ (define (parse-verb-phrase)
+ (amb (parse-word verbs)
+ (list 'verb-phrase
+ (parse-verb-phrase)
+ (parse-prepositional-phrase))))
+
+ Does this work? Does the program's behavior change if we
+ interchange the order of expressions in the `amb'
+
+ *Exercise 4.48:* Extend the grammar given above to handle more
+ complex sentences. For example, you could extend noun phrases and
+ verb phrases to include adjectives and adverbs, or you could
+ handle compound sentences.(6)
+
+ *Exercise 4.49:* Alyssa P. Hacker is more interested in generating
+ interesting sentences than in parsing them. She reasons that by
+ simply changing the procedure `parse-word' so that it ignores the
+ "input sentence" and instead always succeeds and generates an
+ appropriate word, we can use the programs we had built for parsing
+ to do generation instead. Implement Alyssa's idea, and show the
+ first half-dozen or so sentences generated.(7)
+
+ ---------- Footnotes ----------
+
+ (1) Our program uses the following procedure to determine if the
+elements of a list are distinct:
+
+ (define (distinct? items)
+ (cond ((null? items) true)
+ ((null? (cdr items)) true)
+ ((member (car items) (cdr items)) false)
+ (else (distinct? (cdr items)))))
+
+ `Member' is like `memq' except that it uses `equal?' instead of
+`eq?' to test for equality.
+
+ (2) This is taken from a booklet called "Problematical Recreations,"
+published in the 1960s by Litton Industries, where it is attributed to
+the `Kansas State Engineer'.
+
+ (3) Here we use the convention that the first element of each list
+designates the part of speech for the rest of the words in the list.
+
+ (4) Notice that `parse-word' uses `set!' to modify the unparsed
+input list. For this to work, our `amb' evaluator must undo the
+effects of `set!' operations when it backtracks.
+
+ (5) Observe that this definition is recursive--a verb may be
+followed by any number of prepositional phrases.
+
+ (6) This kind of grammar can become arbitrarily complex, but it is
+only a toy as far as real language understanding is concerned. Real
+natural-language understanding by computer requires an elaborate
+mixture of syntactic analysis and interpretation of meaning. On the
+other hand, even toy parsers can be useful in supporting flexible
+command languages for programs such as information-retrieval systems.
+Winston 1992 discusses computational approaches to real language
+understanding and also the applications of simple grammars to command
+languages.
+
+ (7) Although Alyssa's idea works just fine (and is surprisingly
+simple), the sentences that it generates are a bit boring--they don't
+sample the possible sentences of this language in a very interesting
+way. In fact, the grammar is highly recursive in many places, and
+Alyssa's technique "falls into" one of these recursions and gets stuck.
+See *Note Exercise 4-50:: for a way to deal with this.
+
+\1f
+File: sicp.info, Node: 4-3-3, Prev: 4-3-2, Up: 4-3
+
+4.3.3 Implementing the `Amb' Evaluator
+--------------------------------------
+
+The evaluation of an ordinary Scheme expression may return a value, may
+never terminate, or may signal an error. In nondeterministic Scheme
+the evaluation of an expression may in addition result in the discovery
+of a dead end, in which case evaluation must backtrack to a previous
+choice point. The interpretation of nondeterministic Scheme is
+complicated by this extra case.
+
+ We will construct the `amb' evaluator for nondeterministic Scheme by
+modifying the analyzing evaluator of section *Note 4-1-7::.(1) As in
+the analyzing evaluator, evaluation of an expression is accomplished by
+calling an execution procedure produced by analysis of that expression.
+The difference between the interpretation of ordinary Scheme and the
+interpretation of nondeterministic Scheme will be entirely in the
+execution procedures.
+
+Execution procedures and continuations
+......................................
+
+Recall that the execution procedures for the ordinary evaluator take one
+argument: the environment of execution. In contrast, the execution
+procedures in the `amb' evaluator take three arguments: the
+environment, and two procedures called "continuation procedures". The
+evaluation of an expression will finish by calling one of these two
+continuations: If the evaluation results in a value, the "success
+continuation" is called with that value; if the evaluation results in
+the discovery of a dead end, the "failure continuation" is called.
+Constructing and calling appropriate continuations is the mechanism by
+which the nondeterministic evaluator implements backtracking.
+
+ It is the job of the success continuation to receive a value and
+proceed with the computation. Along with that value, the success
+continuation is passed another failure continuation, which is to be
+called subsequently if the use of that value leads to a dead end.
+
+ It is the job of the failure continuation to try another branch of
+the nondeterministic process. The essence of the nondeterministic
+language is in the fact that expressions may represent choices among
+alternatives. The evaluation of such an expression must proceed with
+one of the indicated alternative choices, even though it is not known
+in advance which choices will lead to acceptable results. To deal with
+this, the evaluator picks one of the alternatives and passes this value
+to the success continuation. Together with this value, the evaluator
+constructs and passes along a failure continuation that can be called
+later to choose a different alternative.
+
+ A failure is triggered during evaluation (that is, a failure
+continuation is called) when a user program explicitly rejects the
+current line of attack (for example, a call to `require' may result in
+execution of `(amb)', an expression that always fails--see section
+*Note 4-3-1::). The failure continuation in hand at that point will
+cause the most recent choice point to choose another alternative. If
+there are no more alternatives to be considered at that choice point, a
+failure at an earlier choice point is triggered, and so on. Failure
+continuations are also invoked by the driver loop in response to a
+`try-again' request, to find another value of the expression.
+
+ In addition, if a side-effect operation (such as assignment to a
+variable) occurs on a branch of the process resulting from a choice, it
+may be necessary, when the process finds a dead end, to undo the side
+effect before making a new choice. This is accomplished by having the
+side-effect operation produce a failure continuation that undoes the
+side effect and propagates the failure.
+
+ In summary, failure continuations are constructed by
+
+ * `amb' expressions--to provide a mechanism to make alternative
+ choices if the current choice made by the `amb' expression leads
+ to a dead end;
+
+ * the top-level driver--to provide a mechanism to report failure
+ when the choices are exhausted;
+
+ * assignments--to intercept failures and undo assignments during
+ backtracking.
+
+
+ Failures are initiated only when a dead end is encountered. This
+occurs
+
+ * if the user program executes `(amb)';
+
+ * if the user types `try-again' at the top-level driver.
+
+
+ Failure continuations are also called during processing of a failure:
+
+ * When the failure continuation created by an assignment finishes
+ undoing a side effect, it calls the failure continuation it
+ intercepted, in order to propagate the failure back to the choice
+ point that led to this assignment or to the top level.
+
+ * When the failure continuation for an `amb' runs out of choices, it
+ calls the failure continuation that was originally given to the
+ `amb', in order to propagate the failure back to the previous
+ choice point or to the top level.
+
+
+Structure of the evaluator
+..........................
+
+The syntax- and data-representation procedures for the `amb' evaluator,
+and also the basic `analyze' procedure, are identical to those in the
+evaluator of section *Note 4-1-7::, except for the fact that we need
+additional syntax procedures to recognize the `amb' special form:(2)
+
+ (define (amb? exp) (tagged-list? exp 'amb))
+
+ (define (amb-choices exp) (cdr exp))
+
+ We must also add to the dispatch in `analyze' a clause that will
+recognize this special form and generate an appropriate execution
+procedure:
+
+ ((amb? exp) (analyze-amb exp))
+
+ The top-level procedure `ambeval' (similar to the version of `eval'
+given in section *Note 4-1-7::) analyzes the given expression and
+applies the resulting execution procedure to the given environment,
+together with two given continuations:
+
+ (define (ambeval exp env succeed fail)
+ ((analyze exp) env succeed fail))
+
+ A success continuation is a procedure of two arguments: the value
+just obtained and another failure continuation to be used if that value
+leads to a subsequent failure. A failure continuation is a procedure of
+no arguments. So the general form of an execution procedure is
+
+ (lambda (env succeed fail)
+ ;; `succeed' is `(lambda (value fail) ...)'
+ ;; `fail' is `(lambda () ...)'
+ ...)
+
+ For example, executing
+
+ (ambeval <EXP>
+ the-global-environment
+ (lambda (value fail) value)
+ (lambda () 'failed))
+
+will attempt to evaluate the given expression and will return either the
+expression's value (if the evaluation succeeds) or the symbol `failed'
+(if the evaluation fails). The call to `ambeval' in the driver loop
+shown below uses much more complicated continuation procedures, which
+continue the loop and support the `try-again' request.
+
+ Most of the complexity of the `amb' evaluator results from the
+mechanics of passing the continuations around as the execution
+procedures call each other. In going through the following code, you
+should compare each of the execution procedures with the corresponding
+procedure for the ordinary evaluator given in section *Note 4-1-7::.
+
+Simple expressions
+..................
+
+The execution procedures for the simplest kinds of expressions are
+essentially the same as those for the ordinary evaluator, except for
+the need to manage the continuations. The execution procedures simply
+succeed with the value of the expression, passing along the failure
+continuation that was passed to them.
+
+ (define (analyze-self-evaluating exp)
+ (lambda (env succeed fail)
+ (succeed exp fail)))
+
+ (define (analyze-quoted exp)
+ (let ((qval (text-of-quotation exp)))
+ (lambda (env succeed fail)
+ (succeed qval fail))))
+
+ (define (analyze-variable exp)
+ (lambda (env succeed fail)
+ (succeed (lookup-variable-value exp env)
+ fail)))
+
+ (define (analyze-lambda exp)
+ (let ((vars (lambda-parameters exp))
+ (bproc (analyze-sequence (lambda-body exp))))
+ (lambda (env succeed fail)
+ (succeed (make-procedure vars bproc env)
+ fail))))
+
+ Notice that looking up a variable always "succeeds." If
+`lookup-variable-value' fails to find the variable, it signals an error,
+as usual. Such a "failure" indicates a program bug--a reference to an
+unbound variable; it is not an indication that we should try another
+nondeterministic choice instead of the one that is currently being
+tried.
+
+Conditionals and sequences
+..........................
+
+Conditionals are also handled in a similar way as in the ordinary
+evaluator. The execution procedure generated by `analyze-if' invokes
+the predicate execution procedure `pproc' with a success continuation
+that checks whether the predicate value is true and goes on to execute
+either the consequent or the alternative. If the execution of `pproc'
+fails, the original failure continuation for the `if' expression is
+called.
+
+ (define (analyze-if exp)
+ (let ((pproc (analyze (if-predicate exp)))
+ (cproc (analyze (if-consequent exp)))
+ (aproc (analyze (if-alternative exp))))
+ (lambda (env succeed fail)
+ (pproc env
+ ;; success continuation for evaluating the predicate
+ ;; to obtain `pred-value'
+ (lambda (pred-value fail2)
+ (if (true? pred-value)
+ (cproc env succeed fail2)
+ (aproc env succeed fail2)))
+ ;; failure continuation for evaluating the predicate
+ fail))))
+
+ Sequences are also handled in the same way as in the previous
+evaluator, except for the machinations in the subprocedure
+`sequentially' that are required for passing the continuations.
+Namely, to sequentially execute `a' and then `b', we call `a' with a
+success continuation that calls `b'.
+
+ (define (analyze-sequence exps)
+ (define (sequentially a b)
+ (lambda (env succeed fail)
+ (a env
+ ;; success continuation for calling `a'
+ (lambda (a-value fail2)
+ (b env succeed fail2))
+ ;; failure continuation for calling `a'
+ fail)))
+ (define (loop first-proc rest-procs)
+ (if (null? rest-procs)
+ first-proc
+ (loop (sequentially first-proc (car rest-procs))
+ (cdr rest-procs))))
+ (let ((procs (map analyze exps)))
+ (if (null? procs)
+ (error "Empty sequence -- ANALYZE"))
+ (loop (car procs) (cdr procs))))
+
+Definitions and assignments
+...........................
+
+Definitions are another case where we must go to some trouble to manage
+the continuations, because it is necessary to evaluate the
+definition-value expression before actually defining the new variable.
+To accomplish this, the definition-value execution procedure `vproc' is
+called with the environment, a success continuation, and the failure
+continuation. If the execution of `vproc' succeeds, obtaining a value
+`val' for the defined variable, the variable is defined and the success
+is propagated:
+
+ (define (analyze-definition exp)
+ (let ((var (definition-variable exp))
+ (vproc (analyze (definition-value exp))))
+ (lambda (env succeed fail)
+ (vproc env
+ (lambda (val fail2)
+ (define-variable! var val env)
+ (succeed 'ok fail2))
+ fail))))
+
+ Assignments are more interesting. This is the first place where we
+really use the continuations, rather than just passing them around.
+The execution procedure for assignments starts out like the one for
+definitions. It first attempts to obtain the new value to be assigned
+to the variable. If this evaluation of `vproc' fails, the assignment
+fails.
+
+ If `vproc' succeeds, however, and we go on to make the assignment,
+we must consider the possibility that this branch of the computation
+might later fail, which will require us to backtrack out of the
+assignment. Thus, we must arrange to undo the assignment as part of
+the backtracking process.(3)
+
+ This is accomplished by giving `vproc' a success continuation
+(marked with the comment "*1*" below) that saves the old value of the
+variable before assigning the new value to the variable and proceeding
+from the assignment. The failure continuation that is passed along
+with the value of the assignment (marked with the comment "*2*" below)
+restores the old value of the variable before continuing the failure.
+That is, a successful assignment provides a failure continuation that
+will intercept a subsequent failure; whatever failure would otherwise
+have called `fail2' calls this procedure instead, to undo the
+assignment before actually calling `fail2'.
+
+ (define (analyze-assignment exp)
+ (let ((var (assignment-variable exp))
+ (vproc (analyze (assignment-value exp))))
+ (lambda (env succeed fail)
+ (vproc env
+ (lambda (val fail2) ; *1*
+ (let ((old-value
+ (lookup-variable-value var env)))
+ (set-variable-value! var val env)
+ (succeed 'ok
+ (lambda () ; *2*
+ (set-variable-value! var
+ old-value
+ env)
+ (fail2)))))
+ fail))))
+
+Procedure applications
+......................
+
+The execution procedure for applications contains no new ideas except
+for the technical complexity of managing the continuations. This
+complexity arises in `analyze-application', due to the need to keep
+track of the success and failure continuations as we evaluate the
+operands. We use a procedure `get-args' to evaluate the list of
+operands, rather than a simple `map' as in the ordinary evaluator.
+
+ (define (analyze-application exp)
+ (let ((fproc (analyze (operator exp)))
+ (aprocs (map analyze (operands exp))))
+ (lambda (env succeed fail)
+ (fproc env
+ (lambda (proc fail2)
+ (get-args aprocs
+ env
+ (lambda (args fail3)
+ (execute-application
+ proc args succeed fail3))
+ fail2))
+ fail))))
+
+ In `get-args', notice how `cdr'ing down the list of `aproc'
+execution procedures and `cons'ing up the resulting list of `args' is
+accomplished by calling each `aproc' in the list with a success
+continuation that recursively calls `get-args'. Each of these recursive
+calls to `get-args' has a success continuation whose value is the
+`cons' of the newly obtained argument onto the list of accumulated
+arguments:
+
+ (define (get-args aprocs env succeed fail)
+ (if (null? aprocs)
+ (succeed '() fail)
+ ((car aprocs) env
+ ;; success continuation for this `aproc'
+ (lambda (arg fail2)
+ (get-args (cdr aprocs)
+ env
+ ;; success continuation for recursive
+ ;; call to `get-args'
+ (lambda (args fail3)
+ (succeed (cons arg args)
+ fail3))
+ fail2))
+ fail)))
+
+ The actual procedure application, which is performed by
+`execute-application', is accomplished in the same way as for the
+ordinary evaluator, except for the need to manage the continuations.
+
+ (define (execute-application proc args succeed fail)
+ (cond ((primitive-procedure? proc)
+ (succeed (apply-primitive-procedure proc args)
+ fail))
+ ((compound-procedure? proc)
+ ((procedure-body proc)
+ (extend-environment (procedure-parameters proc)
+ args
+ (procedure-environment proc))
+ succeed
+ fail))
+ (else
+ (error
+ "Unknown procedure type -- EXECUTE-APPLICATION"
+ proc))))
+
+Evaluating `amb' expressions
+............................
+
+The `amb' special form is the key element in the nondeterministic
+language. Here we see the essence of the interpretation process and
+the reason for keeping track of the continuations. The execution
+procedure for `amb' defines a loop `try-next' that cycles through the
+execution procedures for all the possible values of the `amb'
+expression. Each execution procedure is called with a failure
+continuation that will try the next one. When there are no more
+alternatives to try, the entire `amb' expression fails.
+
+ (define (analyze-amb exp)
+ (let ((cprocs (map analyze (amb-choices exp))))
+ (lambda (env succeed fail)
+ (define (try-next choices)
+ (if (null? choices)
+ (fail)
+ ((car choices) env
+ succeed
+ (lambda ()
+ (try-next (cdr choices))))))
+ (try-next cprocs))))
+
+Driver loop
+...........
+
+The driver loop for the `amb' evaluator is complex, due to the mechanism
+that permits the user to try again in evaluating an expression. The
+driver uses a procedure called `internal-loop', which takes as argument
+a procedure `try-again'. The intent is that calling `try-again' should
+go on to the next untried alternative in the nondeterministic
+evaluation. `Internal-loop' either calls `try-again' in response to
+the user typing `try-again' at the driver loop, or else starts a new
+evaluation by calling `ambeval'.
+
+ The failure continuation for this call to `ambeval' informs the user
+that there are no more values and re-invokes the driver loop.
+
+ The success continuation for the call to `ambeval' is more subtle.
+We print the obtained value and then invoke the internal loop again
+with a `try-again' procedure that will be able to try the next
+alternative. This `next-alternative' procedure is the second argument
+that was passed to the success continuation. Ordinarily, we think of
+this second argument as a failure continuation to be used if the
+current evaluation branch later fails. In this case, however, we have
+completed a successful evaluation, so we can invoke the "failure"
+alternative branch in order to search for additional successful
+evaluations.
+
+ (define input-prompt ";;; Amb-Eval input:")
+ (define output-prompt ";;; Amb-Eval value:")
+
+ (define (driver-loop)
+ (define (internal-loop try-again)
+ (prompt-for-input input-prompt)
+ (let ((input (read)))
+ (if (eq? input 'try-again)
+ (try-again)
+ (begin
+ (newline)
+ (display ";;; Starting a new problem ")
+ (ambeval input
+ the-global-environment
+ ;; `ambeval' success
+ (lambda (val next-alternative)
+ (announce-output output-prompt)
+ (user-print val)
+ (internal-loop next-alternative))
+ ;; `ambeval' failure
+ (lambda ()
+ (announce-output
+ ";;; There are no more values of")
+ (user-print input)
+ (driver-loop)))))))
+ (internal-loop
+ (lambda ()
+ (newline)
+ (display ";;; There is no current problem")
+ (driver-loop))))
+
+ The initial call to `internal-loop' uses a `try-again' procedure that
+complains that there is no current problem and restarts the driver
+loop. This is the behavior that will happen if the user types
+`try-again' when there is no evaluation in progress.
+
+ *Exercise 4.50:* Implement a new special form `ramb' that is like
+ `amb' except that it searches alternatives in a random order,
+ rather than from left to right. Show how this can help with
+ Alyssa's problem in *Note Exercise 4-49::.
+
+ *Exercise 4.51:* Implement a new kind of assignment called
+ `permanent-set!' that is not undone upon failure. For example, we
+ can choose two distinct elements from a list and count the number
+ of trials required to make a successful choice as follows:
+
+ (define count 0)
+
+ (let ((x (an-element-of '(a b c)))
+ (y (an-element-of '(a b c))))
+ (permanent-set! count (+ count 1))
+ (require (not (eq? x y)))
+ (list x y count))
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ (a b 2)
+
+ ;;; Amb-Eval input:
+ try-again
+ ;;; Amb-Eval value:
+ (a c 3)
+
+ What values would have been displayed if we had used `set!' here
+ rather than `permanent-set!' ?
+
+ *Exercise 4.52:* Implement a new construct called `if-fail' that
+ permits the user to catch the failure of an expression. `If-fail'
+ takes two expressions. It evaluates the first expression as usual
+ and returns as usual if the evaluation succeeds. If the evaluation
+ fails, however, the value of the second expression is returned, as
+ in the following example:
+
+ ;;; Amb-Eval input:
+ (if-fail (let ((x (an-element-of '(1 3 5))))
+ (require (even? x))
+ x)
+ 'all-odd)
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ all-odd
+
+ ;;; Amb-Eval input:
+ (if-fail (let ((x (an-element-of '(1 3 5 8))))
+ (require (even? x))
+ x)
+ 'all-odd)
+ ;;; Starting a new problem
+ ;;; Amb-Eval value:
+ 8
+
+ *Exercise 4.53:* With `permanent-set!' as described in *Note
+ Exercise 4-51:: and `if-fail' as in *Note Exercise 4-52::, what
+ will be the result of evaluating
+
+ (let ((pairs '()))
+ (if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110))))
+ (permanent-set! pairs (cons p pairs))
+ (amb))
+ pairs))
+
+ *Exercise 4.54:* If we had not realized that `require' could be
+ implemented as an ordinary procedure that uses `amb', to be
+ defined by the user as part of a nondeterministic program, we
+ would have had to implement it as a special form. This would
+ require syntax procedures
+
+ (define (require? exp) (tagged-list? exp 'require))
+
+ (define (require-predicate exp) (cadr exp))
+
+ and a new clause in the dispatch in `analyze'
+
+ ((require? exp) (analyze-require exp))
+
+ as well the procedure `analyze-require' that handles `require'
+ expressions. Complete the following definition of
+ `analyze-require'.
+
+ (define (analyze-require exp)
+ (let ((pproc (analyze (require-predicate exp))))
+ (lambda (env succeed fail)
+ (pproc env
+ (lambda (pred-value fail2)
+ (if <??>
+ <??>
+ (succeed 'ok fail2)))
+ fail))))
+
+ ---------- Footnotes ----------
+
+ (1) We chose to implement the lazy evaluator in section *Note 4-2::
+as a modification of the ordinary metacircular evaluator of section
+*Note 4-1-1::. In contrast, we will base the `amb' evaluator on the
+analyzing evaluator of section *Note 4-1-7::, because the execution
+procedures in that evaluator provide a convenient framework for
+implementing backtracking.
+
+ (2) We assume that the evaluator supports `let' (see *Note Exercise
+4-22::), which we have used in our nondeterministic programs.
+
+ (3) We didn't worry about undoing definitions, since we can assume
+that internal definitions are scanned out (section *Note 4-1-6::).
+
+\1f
+File: sicp.info, Node: 4-4, Prev: 4-3, Up: Chapter 4
+
+4.4 Logic Programming
+=====================
+
+In *Note Chapter 1:: we stressed that computer science deals with
+imperative (how to) knowledge, whereas mathematics deals with
+declarative (what is) knowledge. Indeed, programming languages require
+that the programmer express knowledge in a form that indicates the
+step-by-step methods for solving particular problems. On the other
+hand, high-level languages provide, as part of the language
+implementation, a substantial amount of methodological knowledge that
+frees the user from concern with numerous details of how a specified
+computation will progress.
+
+ Most programming languages, including Lisp, are organized around
+computing the values of mathematical functions. Expression-oriented
+languages (such as Lisp, Fortran, and Algol) capitalize on the "pun"
+that an expression that describes the value of a function may also be
+interpreted as a means of computing that value. Because of this, most
+programming languages are strongly biased toward unidirectional
+computations (computations with well-defined inputs and outputs).
+There are, however, radically different programming languages that
+relax this bias. We saw one such example in section *Note 3-3-5::,
+where the objects of computation were arithmetic constraints. In a
+constraint system the direction and the order of computation are not so
+well specified; in carrying out a computation the system must therefore
+provide more detailed "how to" knowledge than would be the case with an
+ordinary arithmetic computation. This does not mean, however, that the
+user is released altogether from the responsibility of providing
+imperative knowledge. There are many constraint networks that
+implement the same set of constraints, and the user must choose from
+the set of mathematically equivalent networks a suitable network to
+specify a particular computation.
+
+ The nondeterministic program evaluator of section *Note 4-3:: also
+moves away from the view that programming is about constructing
+algorithms for computing unidirectional functions. In a
+nondeterministic language, expressions can have more than one value,
+and, as a result, the computation is dealing with relations rather than
+with single-valued functions. Logic programming extends this idea by
+combining a relational vision of programming with a powerful kind of
+symbolic pattern matching called "unification".(1)
+
+ This approach, when it works, can be a very powerful way to write
+programs. Part of the power comes from the fact that a single "what
+is" fact can be used to solve a number of different problems that would
+have different "how to" components. As an example, consider the
+`append' operation, which takes two lists as arguments and combines
+their elements to form a single list. In a procedural language such as
+Lisp, we could define `append' in terms of the basic list constructor
+`cons', as we did in section *Note 2-2-1:::
+
+ (define (append x y)
+ (if (null? x)
+ y
+ (cons (car x) (append (cdr x) y))))
+
+ This procedure can be regarded as a translation into Lisp of the
+following two rules, the first of which covers the case where the first
+list is empty and the second of which handles the case of a nonempty
+list, which is a `cons' of two parts:
+
+ * For any list `y', the empty list and `y' `append' to form `y'.
+
+ * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
+ form `(cons u z)' if `v' and `y' `append' to form `z'.(2)
+
+
+ Using the `append' procedure, we can answer questions such as
+
+ Find the `append' of `(a b)' and `(c d)'.
+
+ But the same two rules are also sufficient for answering the
+following sorts of questions, which the procedure can't answer:
+
+ Find a list `y' that `append's with `(a b)' to produce `(a b c d)'.
+
+ Find all `x' and `y' that `append' to form `(a b c d)'.
+
+ In a logic programming language, the programmer writes an `append'
+"procedure" by stating the two rules about `append' given above. "How
+to" knowledge is provided automatically by the interpreter to allow this
+single pair of rules to be used to answer all three types of questions
+about `append'.(3)
+
+ Contemporary logic programming languages (including the one we
+implement here) have substantial deficiencies, in that their general
+"how to" methods can lead them into spurious infinite loops or other
+undesirable behavior. Logic programming is an active field of research
+in computer science.(4)
+
+ Earlier in this chapter we explored the technology of implementing
+interpreters and described the elements that are essential to an
+interpreter for a Lisp-like language (indeed, to an interpreter for any
+conventional language). Now we will apply these ideas to discuss an
+interpreter for a logic programming language. We call this language
+the "query language", because it is very useful for retrieving
+information from data bases by formulating "queries", or questions,
+expressed in the language. Even though the query language is very
+different from Lisp, we will find it convenient to describe the
+language in terms of the same general framework we have been using all
+along: as a collection of primitive elements, together with means of
+combination that enable us to combine simple elements to create more
+complex elements and means of abstraction that enable us to regard
+complex elements as single conceptual units. An interpreter for a
+logic programming language is considerably more complex than an
+interpreter for a language like Lisp. Nevertheless, we will see that
+our query-language interpreter contains many of the same elements found
+in the interpreter of section *Note 4-1::. In particular, there will
+be an "eval" part that classifies expressions according to type and an
+"apply" part that implements the language's abstraction mechanism
+(procedures in the case of Lisp, and "rules" in the case of logic
+programming). Also, a central role is played in the implementation by
+a frame data structure, which determines the correspondence between
+symbols and their associated values. One additional interesting aspect
+of our query-language implementation is that we make substantial use of
+streams, which were introduced in *Note Chapter 3::.
+
+* Menu:
+
+* 4-4-1:: Deductive Information Retrieval
+* 4-4-2:: How the Query System Works
+* 4-4-3:: Is Logic Programming Mathematical Logic?
+* 4-4-4:: Implementing the Query System
+
+ ---------- Footnotes ----------
+
+ (1) Logic programming has grown out of a long history of research in
+automatic theorem proving. Early theorem-proving programs could
+accomplish very little, because they exhaustively searched the space of
+possible proofs. The major breakthrough that made such a search
+plausible was the discovery in the early 1960s of the "unification
+algorithm" and the principle "resolution principle" (Robinson 1965).
+Resolution was used, for example, by Green and Raphael (1968) (see also
+Green 1969) as the basis for a deductive question-answering system.
+During most of this period, researchers concentrated on algorithms that
+are guaranteed to find a proof if one exists. Such algorithms were
+difficult to control and to direct toward a proof. Hewitt (1969)
+recognized the possibility of merging the control structure of a
+programming language with the operations of a logic-manipulation system,
+leading to the work in automatic search mentioned in section *Note
+4-3-1:: (footnote *Note Footnote 4-47::). At the same time that this
+was being done, Colmerauer, in Marseille, was developing rule-based
+systems for manipulating natural language (see Colmerauer et al. 1973).
+He invented a programming language called Prolog for representing
+those rules. Kowalski (1973; 1979), in Edinburgh, recognized that
+execution of a Prolog program could be interpreted as proving theorems
+(using a proof technique called linear Horn-clause resolution). The
+merging of the last two strands led to the logic-programming movement.
+Thus, in assigning credit for the development of logic programming, the
+French can point to Prolog's genesis at the University of Marseille,
+while the British can highlight the work at the University of
+Edinburgh. According to people at MIT, logic programming was developed
+by these groups in an attempt to figure out what Hewitt was talking
+about in his brilliant but impenetrable Ph.D. thesis. For a history of
+logic programming, see Robinson 1983.
+
+ (2) To see the correspondence between the rules and the procedure,
+let `x' in the procedure (where `x' is nonempty) correspond to `(cons u
+v)' in the rule. Then `z' in the rule corresponds to the `append' of
+`(cdr x)' and `y'.
+
+ (3) This certainly does not relieve the user of the entire problem
+of how to compute the answer. There are many different mathematically
+equivalent sets of rules for formulating the `append' relation, only
+some of which can be turned into effective devices for computing in any
+direction. In addition, sometimes "what is" information gives no clue
+"how to" compute an answer. For example, consider the problem of
+computing the y such that y^2 = x.
+
+ (4) Interest in logic programming peaked during the early 80s when
+the Japanese government began an ambitious project aimed at building
+superfast computers optimized to run logic programming languages. The
+speed of such computers was to be measured in LIPS (Logical Inferences
+Per Second) rather than the usual FLOPS (FLoating-point Operations Per
+Second). Although the project succeeded in developing hardware and
+software as originally planned, the international computer industry
+moved in a different direction. See Feigenbaum and Shrobe 1993 for an
+overview evaluation of the Japanese project. The logic programming
+community has also moved on to consider relational programming based on
+techniques other than simple pattern matching, such as the ability to
+deal with numerical constraints such as the ones illustrated in the
+constraint-propagation system of section *Note 3-3-5::.
+
+\1f
+File: sicp.info, Node: 4-4-1, Next: 4-4-2, Prev: 4-4, Up: 4-4
+
+4.4.1 Deductive Information Retrieval
+-------------------------------------
+
+Logic programming excels in providing interfaces to data bases for
+information retrieval. The query language we shall implement in this
+chapter is designed to be used in this way.
+
+ In order to illustrate what the query system does, we will show how
+it can be used to manage the data base of personnel records for
+Microshaft, a thriving high-technology company in the Boston area. The
+language provides pattern-directed access to personnel information and
+can also take advantage of general rules in order to make logical
+deductions.
+
+A sample data base
+..................
+
+The personnel data base for Microshaft contains "assertions" about
+company personnel. Here is the information about Ben Bitdiddle, the
+resident computer wizard:
+
+ (address (Bitdiddle Ben) (Slumerville (Ridge Road) 10))
+ (job (Bitdiddle Ben) (computer wizard))
+ (salary (Bitdiddle Ben) 60000)
+
+ Each assertion is a list (in this case a triple) whose elements can
+themselves be lists.
+
+ As resident wizard, Ben is in charge of the company's computer
+division, and he supervises two programmers and one technician. Here
+is the information about them:
+
+ (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))
+ (job (Hacker Alyssa P) (computer programmer))
+ (salary (Hacker Alyssa P) 40000)
+ (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
+
+ (address (Fect Cy D) (Cambridge (Ames Street) 3))
+ (job (Fect Cy D) (computer programmer))
+ (salary (Fect Cy D) 35000)
+ (supervisor (Fect Cy D) (Bitdiddle Ben))
+
+ (address (Tweakit Lem E) (Boston (Bay State Road) 22))
+ (job (Tweakit Lem E) (computer technician))
+ (salary (Tweakit Lem E) 25000)
+ (supervisor (Tweakit Lem E) (Bitdiddle Ben))
+
+ There is also a programmer trainee, who is supervised by Alyssa:
+
+ (address (Reasoner Louis) (Slumerville (Pine Tree Road) 80))
+ (job (Reasoner Louis) (computer programmer trainee))
+ (salary (Reasoner Louis) 30000)
+ (supervisor (Reasoner Louis) (Hacker Alyssa P))
+
+ All of these people are in the computer division, as indicated by
+the word `computer' as the first item in their job descriptions.
+
+ Ben is a high-level employee. His supervisor is the company's big
+wheel himself:
+
+ (supervisor (Bitdiddle Ben) (Warbucks Oliver))
+
+ (address (Warbucks Oliver) (Swellesley (Top Heap Road)))
+ (job (Warbucks Oliver) (administration big wheel))
+ (salary (Warbucks Oliver) 150000)
+
+ Besides the computer division supervised by Ben, the company has an
+accounting division, consisting of a chief accountant and his assistant:
+
+ (address (Scrooge Eben) (Weston (Shady Lane) 10))
+ (job (Scrooge Eben) (accounting chief accountant))
+ (salary (Scrooge Eben) 75000)
+ (supervisor (Scrooge Eben) (Warbucks Oliver))
+
+ (address (Cratchet Robert) (Allston (N Harvard Street) 16))
+ (job (Cratchet Robert) (accounting scrivener))
+ (salary (Cratchet Robert) 18000)
+ (supervisor (Cratchet Robert) (Scrooge Eben))
+
+ There is also a secretary for the big wheel:
+
+ (address (Aull DeWitt) (Slumerville (Onion Square) 5))
+ (job (Aull DeWitt) (administration secretary))
+ (salary (Aull DeWitt) 25000)
+ (supervisor (Aull DeWitt) (Warbucks Oliver))
+
+ The data base also contains assertions about which kinds of jobs can
+be done by people holding other kinds of jobs. For instance, a
+computer wizard can do the jobs of both a computer programmer and a
+computer technician:
+
+ (can-do-job (computer wizard) (computer programmer))
+ (can-do-job (computer wizard) (computer technician))
+
+ A computer programmer could fill in for a trainee:
+
+ (can-do-job (computer programmer)
+ (computer programmer trainee))
+
+ Also, as is well known,
+
+ (can-do-job (administration secretary)
+ (administration big wheel))
+
+Simple queries
+..............
+
+The query language allows users to retrieve information from the data
+base by posing queries in response to the system's prompt. For
+example, to find all computer programmers one can say
+
+ ;;; Query input:
+ (job ?x (computer programmer))
+
+ The system will respond with the following items:
+
+ ;;; Query results:
+ (job (Hacker Alyssa P) (computer programmer))
+ (job (Fect Cy D) (computer programmer))
+
+ The input query specifies that we are looking for entries in the
+data base that match a certain "pattern". In this example, the pattern
+specifies entries consisting of three items, of which the first is the
+literal symbol `job', the second can be anything, and the third is the
+literal list `(computer programmer)'. The "anything" that can be the
+second item in the matching list is specified by a "pattern variable",
+`?x'. The general form of a pattern variable is a symbol, taken to be
+the name of the variable, preceded by a question mark. We will see
+below why it is useful to specify names for pattern variables rather
+than just putting `?' into patterns to represent "anything." The
+system responds to a simple query by showing all entries in the data
+base that match the specified pattern.
+
+ A pattern can have more than one variable. For example, the query
+
+ (address ?x ?y)
+
+will list all the employees' addresses.
+
+ A pattern can have no variables, in which case the query simply
+determines whether that pattern is an entry in the data base. If so,
+there will be one match; if not, there will be no matches.
+
+ The same pattern variable can appear more than once in a query,
+specifying that the same "anything" must appear in each position. This
+is why variables have names. For example,
+
+ (supervisor ?x ?x)
+
+finds all people who supervise themselves (though there are no such
+assertions in our sample data base).
+
+ The query
+
+ (job ?x (computer ?type))
+
+matches all job entries whose third item is a two-element list whose
+first item is `computer':
+
+ (job (Bitdiddle Ben) (computer wizard))
+ (job (Hacker Alyssa P) (computer programmer))
+ (job (Fect Cy D) (computer programmer))
+ (job (Tweakit Lem E) (computer technician))
+
+ This same pattern does _not_ match
+
+ (job (Reasoner Louis) (computer programmer trainee))
+
+because the third item in the entry is a list of three elements, and the
+pattern's third item specifies that there should be two elements. If
+we wanted to change the pattern so that the third item could be any
+list beginning with `computer', we could specify(1)
+
+ (job ?x (computer . ?type))
+
+ For example,
+
+ (computer . ?type)
+
+matches the data
+
+ (computer programmer trainee)
+
+with `?type' as the list `(programmer trainee)'. It also matches the
+data
+
+ (computer programmer)
+
+with `?type' as the list `(programmer)', and matches the data
+
+ (computer)
+
+with `?type' as the empty list `()'.
+
+ We can describe the query language's processing of simple queries as
+follows:
+
+ * The system finds all assignments to variables in the query pattern
+ that "satisfy" the pattern--that is, all sets of values for the
+ variables such that if the pattern variables are "instantiated
+ with" (replaced by) the values, the result is in the data base.
+
+ * The system responds to the query by listing all instantiations of
+ the query pattern with the variable assignments that satisfy it.
+
+
+ Note that if the pattern has no variables, the query reduces to a
+determination of whether that pattern is in the data base. If so, the
+empty assignment, which assigns no values to variables, satisfies that
+pattern for that data base.
+
+ *Exercise 4.55:* Give simple queries that retrieve the following
+ information from the data base:
+
+ 1. all people supervised by Ben Bitdiddle;
+
+ 2. the names and jobs of all people in the accounting division;
+
+ 3. the names and addresses of all people who live in Slumerville.
+
+
+Compound queries
+................
+
+Simple queries form the primitive operations of the query language. In
+order to form compound operations, the query language provides means of
+combination. One thing that makes the query language a logic
+programming language is that the means of combination mirror the means
+of combination used in forming logical expressions: `and', `or', and
+`not'. (Here `and', `or', and `not' are not the Lisp primitives, but
+rather operations built into the query language.)
+
+ We can use `and' as follows to find the addresses of all the computer
+programmers:
+
+ (and (job ?person (computer programmer))
+ (address ?person ?where))
+
+ The resulting output is
+
+ (and (job (Hacker Alyssa P) (computer programmer))
+ (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)))
+
+ (and (job (Fect Cy D) (computer programmer))
+ (address (Fect Cy D) (Cambridge (Ames Street) 3)))
+
+ In general,
+
+ (and <QUERY_1> <QUERY_2> ... <QUERY_N>)
+
+is satisfied by all sets of values for the pattern variables that
+simultaneously satisfy <QUERY_1> ... <QUERY_N>.
+
+ As for simple queries, the system processes a compound query by
+finding all assignments to the pattern variables that satisfy the
+query, then displaying instantiations of the query with those values.
+
+ Another means of constructing compound queries is through `or'. For
+example,
+
+ (or (supervisor ?x (Bitdiddle Ben))
+ (supervisor ?x (Hacker Alyssa P)))
+
+will find all employees supervised by Ben Bitdiddle or Alyssa P.
+Hacker:
+
+ (or (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
+ (supervisor (Hacker Alyssa P) (Hacker Alyssa P)))
+
+ (or (supervisor (Fect Cy D) (Bitdiddle Ben))
+ (supervisor (Fect Cy D) (Hacker Alyssa P)))
+
+ (or (supervisor (Tweakit Lem E) (Bitdiddle Ben))
+ (supervisor (Tweakit Lem E) (Hacker Alyssa P)))
+
+ (or (supervisor (Reasoner Louis) (Bitdiddle Ben))
+ (supervisor (Reasoner Louis) (Hacker Alyssa P)))
+
+ In general,
+
+ (or <QUERY_1> <QUERY_2> ... <QUERY_N>)
+
+is satisfied by all sets of values for the pattern variables that
+satisfy at least one of <QUERY_1> ... <QUERY_N>.
+
+ Compound queries can also be formed with `not'. For example,
+
+ (and (supervisor ?x (Bitdiddle Ben))
+ (not (job ?x (computer programmer))))
+
+finds all people supervised by Ben Bitdiddle who are not computer
+programmers. In general,
+
+ (not <QUERY_1>)
+
+is satisfied by all assignments to the pattern variables that do not
+satisfy <QUERY_1>.(2)
+
+ The final combining form is called `lisp-value'. When `lisp-value'
+is the first element of a pattern, it specifies that the next element
+is a Lisp predicate to be applied to the rest of the (instantiated)
+elements as arguments. In general,
+
+ (lisp-value <PREDICATE> <ARG_1> ... <ARG_N>)
+
+will be satisfied by assignments to the pattern variables for which the
+<PREDICATE> applied to the instantiated <ARG_1> ... <ARG_N> is true.
+For example, to find all people whose salary is greater than $30,000 we
+could write(3)
+
+ (and (salary ?person ?amount)
+ (lisp-value > ?amount 30000))
+
+ *Exercise 4.56:* Formulate compound queries that retrieve the
+ following information:
+
+ a. the names of all people who are supervised by Ben Bitdiddle,
+ together with their addresses;
+
+ b. all people whose salary is less than Ben Bitdiddle's,
+ together with their salary and Ben Bitdiddle's salary;
+
+ c. all people who are supervised by someone who is not in the
+ computer division, together with the supervisor's name and
+ job.
+
+
+Rules
+.....
+
+In addition to primitive queries and compound queries, the query
+language provides means for abstracting queries. These are given by "rules".
+The rule
+
+ (rule (lives-near ?person-1 ?person-2)
+ (and (address ?person-1 (?town . ?rest-1))
+ (address ?person-2 (?town . ?rest-2))
+ (not (same ?person-1 ?person-2))))
+
+specifies that two people live near each other if they live in the same
+town. The final `not' clause prevents the rule from saying that all
+people live near themselves. The `same' relation is defined by a very
+simple rule:(4)
+
+ (rule (same ?x ?x))
+
+ The following rule declares that a person is a "wheel" in an
+organization if he supervises someone who is in turn a supervisor:
+
+ (rule (wheel ?person)
+ (and (supervisor ?middle-manager ?person)
+ (supervisor ?x ?middle-manager)))
+
+ The general form of a rule is
+
+ (rule <CONCLUSION> <BODY>)
+
+where <CONCLUSION> is a pattern and <BODY> is any query.(5) We can
+think of a rule as representing a large (even infinite) set of
+assertions, namely all instantiations of the rule conclusion with
+variable assignments that satisfy the rule body. When we described
+simple queries (patterns), we said that an assignment to variables
+satisfies a pattern if the instantiated pattern is in the data base.
+But the pattern needn't be explicitly in the data base as an assertion.
+It can be an implicit assertion implied by a rule. For example, the
+query
+
+ (lives-near ?x (Bitdiddle Ben))
+
+results in
+
+ (lives-near (Reasoner Louis) (Bitdiddle Ben))
+ (lives-near (Aull DeWitt) (Bitdiddle Ben))
+
+ To find all computer programmers who live near Ben Bitdiddle, we can
+ask
+
+ (and (job ?x (computer programmer))
+ (lives-near ?x (Bitdiddle Ben)))
+
+ As in the case of compound procedures, rules can be used as parts of
+other rules (as we saw with the `lives-near' rule above) or even be
+defined recursively. For instance, the rule
+
+ (rule (outranked-by ?staff-person ?boss)
+ (or (supervisor ?staff-person ?boss)
+ (and (supervisor ?staff-person ?middle-manager)
+ (outranked-by ?middle-manager ?boss))))
+
+says that a staff person is outranked by a boss in the organization if
+the boss is the person's supervisor or (recursively) if the person's
+supervisor is outranked by the boss.
+
+ *Exercise 4.57:* Define a rule that says that person 1 can replace
+ person 2 if either person 1 does the same job as person 2 or
+ someone who does person 1's job can also do person 2's job, and if
+ person 1 and person 2 are not the same person. Using your rule,
+ give queries that find the following:
+
+ a. all people who can replace Cy D. Fect;
+
+ b. all people who can replace someone who is being paid more
+ than they are, together with the two salaries.
+
+
+ *Exercise 4.58:* Define a rule that says that a person is a "big
+ shot" in a division if the person works in the division but does
+ not have a supervisor who works in the division.
+
+ *Exercise 4.59:* Ben Bitdiddle has missed one meeting too many.
+ Fearing that his habit of forgetting meetings could cost him his
+ job, Ben decides to do something about it. He adds all the weekly
+ meetings of the firm to the Microshaft data base by asserting the
+ following:
+
+ (meeting accounting (Monday 9am))
+ (meeting administration (Monday 10am))
+ (meeting computer (Wednesday 3pm))
+ (meeting administration (Friday 1pm))
+
+ Each of the above assertions is for a meeting of an entire
+ division. Ben also adds an entry for the company-wide meeting
+ that spans all the divisions. All of the company's employees
+ attend this meeting.
+
+ (meeting whole-company (Wednesday 4pm))
+
+ a. On Friday morning, Ben wants to query the data base for all
+ the meetings that occur that day. What query should he use?
+
+ b. Alyssa P. Hacker is unimpressed. She thinks it would be much
+ more useful to be able to ask for her meetings by specifying
+ her name. So she designs a rule that says that a person's
+ meetings include all `whole-company' meetings plus all
+ meetings of that person's division. Fill in the body of
+ Alyssa's rule.
+
+ (rule (meeting-time ?person ?day-and-time)
+ <RULE-BODY>)
+
+ c. Alyssa arrives at work on Wednesday morning and wonders what
+ meetings she has to attend that day. Having defined the
+ above rule, what query should she make to find this out?
+
+
+ *Exercise 4.60:* By giving the query
+
+ (lives-near ?person (Hacker Alyssa P))
+
+ Alyssa P. Hacker is able to find people who live near her, with
+ whom she can ride to work. On the other hand, when she tries to
+ find all pairs of people who live near each other by querying
+
+ (lives-near ?person-1 ?person-2)
+
+ she notices that each pair of people who live near each other is
+ listed twice; for example,
+
+ (lives-near (Hacker Alyssa P) (Fect Cy D))
+ (lives-near (Fect Cy D) (Hacker Alyssa P))
+
+ Why does this happen? Is there a way to find a list of people who
+ live near each other, in which each pair appears only once?
+ Explain.
+
+Logic as programs
+.................
+
+We can regard a rule as a kind of logical implication: _If_ an
+assignment of values to pattern variables satisfies the body, _then_ it
+satisfies the conclusion. Consequently, we can regard the query
+language as having the ability to perform "logical deductions" based
+upon the rules. As an example, consider the `append' operation
+described at the beginning of section *Note 4-4::. As we said,
+`append' can be characterized by the following two rules:
+
+ * For any list `y', the empty list and `y' `append' to form `y'.
+
+ * For any `u', `v', `y', and `z', `(cons u v)' and `y' `append' to
+ form `(cons u z)' if `v' and `y' `append' to form `z'.
+
+
+ To express this in our query language, we define two rules for a
+relation
+
+ (append-to-form x y z)
+
+which we can interpret to mean "`x' and `y' `append' to form `z'":
+
+ (rule (append-to-form () ?y ?y))
+
+ (rule (append-to-form (?u . ?v) ?y (?u . ?z))
+ (append-to-form ?v ?y ?z))
+
+ The first rule has no body, which means that the conclusion holds
+for any value of `?y'. Note how the second rule makes use of
+dotted-tail notation to name the `car' and `cdr' of a list.
+
+ Given these two rules, we can formulate queries that compute the
+`append' of two lists:
+
+ ;;; Query input:
+ (append-to-form (a b) (c d) ?z)
+ ;;; Query results:
+ (append-to-form (a b) (c d) (a b c d))
+
+ What is more striking, we can use the same rules to ask the question
+"Which list, when `append'ed to `(a b)', yields `(a b c d)'?" This is
+done as follows:
+
+ ;;; Query input:
+ (append-to-form (a b) ?y (a b c d))
+ ;;; Query results:
+ (append-to-form (a b) (c d) (a b c d))
+
+ We can also ask for all pairs of lists that `append' to form `(a b c
+d)':
+
+ ;;; Query input:
+ (append-to-form ?x ?y (a b c d))
+ ;;; Query results:
+ (append-to-form () (a b c d) (a b c d))
+ (append-to-form (a) (b c d) (a b c d))
+ (append-to-form (a b) (c d) (a b c d))
+ (append-to-form (a b c) (d) (a b c d))
+ (append-to-form (a b c d) () (a b c d))
+
+ The query system may seem to exhibit quite a bit of intelligence in
+using the rules to deduce the answers to the queries above. Actually,
+as we will see in the next section, the system is following a
+well-determined algorithm in unraveling the rules. Unfortunately,
+although the system works impressively in the `append' case, the
+general methods may break down in more complex cases, as we will see in
+section *Note 4-4-3::.
+
+ *Exercise 4.61:* The following rules implement a `next-to'
+ relation that finds adjacent elements of a list:
+
+ (rule (?x next-to ?y in (?x ?y . ?u)))
+
+ (rule (?x next-to ?y in (?v . ?z))
+ (?x next-to ?y in ?z))
+
+ What will the response be to the following queries?
+
+ (?x next-to ?y in (1 (2 3) 4))
+
+ (?x next-to 1 in (2 1 3 1))
+
+ *Exercise 4.62:* Define rules to implement the `last-pair'
+ operation of *Note Exercise 2-17::, which returns a list
+ containing the last element of a nonempty list. Check your rules
+ on queries such as `(last-pair (3) ?x)', `(last-pair (1 2 3) ?x)',
+ and `(last-pair (2 ?x) (3))'. Do your rules work correctly on
+ queries such as `(last-pair ?x (3))' ?
+
+ *Exercise 4.63:* The following data base (see Genesis 4) traces
+ the genealogy of the descendants of Ada back to Adam, by way of
+ Cain:
+
+ (son Adam Cain)
+ (son Cain Enoch)
+ (son Enoch Irad)
+ (son Irad Mehujael)
+ (son Mehujael Methushael)
+ (son Methushael Lamech)
+ (wife Lamech Ada)
+ (son Ada Jabal)
+ (son Ada Jubal)
+
+ Formulate rules such as "If S is the son of f, and f is the son of
+ G, then S is the grandson of G" and "If W is the wife of M, and S
+ is the son of W, then S is the son of M" (which was supposedly
+ more true in biblical times than today) that will enable the query
+ system to find the grandson of Cain; the sons of Lamech; the
+ grandsons of Methushael. (See *Note Exercise 4-69:: for some
+ rules to deduce more complicated relationships.)
+
+ ---------- Footnotes ----------
+
+ (1) This uses the dotted-tail notation introduced in *Note Exercise
+2-20::.
+
+ (2) Actually, this description of `not' is valid only for simple
+cases. The real behavior of `not' is more complex. We will examine
+`not''s peculiarities in sections *Note 4-4-2:: and *Note 4-4-3::.
+
+ (3) `Lisp-value' should be used only to perform an operation not
+provided in the query language. In particular, it should not be used
+to test equality (since that is what the matching in the query language
+is designed to do) or inequality (since that can be done with the
+`same' rule shown below).
+
+ (4) Notice that we do not need `same' in order to make two things be
+the same: We just use the same pattern variable for each--in effect, we
+have one thing instead of two things in the first place. For example,
+see `?town' in the `lives-near' rule and `?middle-manager' in the
+`wheel' rule below. `Same' is useful when we want to force two things
+to be different, such as `?person-1' and `?person-2' in the
+`lives-near' rule. Although using the same pattern variable in two
+parts of a query forces the same value to appear in both places, using
+different pattern variables does not force different values to appear.
+(The values assigned to different pattern variables may be the same or
+different.)
+
+ (5) We will also allow rules without bodies, as in `same', and we
+will interpret such a rule to mean that the rule conclusion is
+satisfied by any values of the variables.
+
+\1f
+File: sicp.info, Node: 4-4-2, Next: 4-4-3, Prev: 4-4-1, Up: 4-4
+
+4.4.2 How the Query System Works
+--------------------------------
+
+In section *Note 4-4-4:: we will present an implementation of the query
+interpreter as a collection of procedures. In this section we give an
+overview that explains the general structure of the system independent
+of low-level implementation details. After describing the
+implementation of the interpreter, we will be in a position to
+understand some of its limitations and some of the subtle ways in which
+the query language's logical operations differ from the operations of
+mathematical logic.
+
+ It should be apparent that the query evaluator must perform some
+kind of search in order to match queries against facts and rules in the
+data base. One way to do this would be to implement the query system
+as a nondeterministic program, using the `amb' evaluator of section
+*Note 4-3:: (see *Note Exercise 4-78::). Another possibility is to
+manage the search with the aid of streams. Our implementation follows
+this second approach.
+
+ The query system is organized around two central operations called "pattern
+matching" and "unification". We first describe pattern matching and
+explain how this operation, together with the organization of
+information in terms of streams of frames, enables us to implement both
+simple and compound queries. We next discuss unification, a
+generalization of pattern matching needed to implement rules. Finally,
+we show how the entire query interpreter fits together through a
+procedure that classifies expressions in a manner analogous to the way
+`eval' classifies expressions for the interpreter described in section
+*Note 4-1::.
+
+Pattern matching
+................
+
+A "pattern matcher" is a program that tests whether some datum fits a
+specified pattern. For example, the data list `((a b) c (a b))' matches
+the pattern `(?x c ?x)' with the pattern variable `?x' bound to `(a
+b)'. The same data list matches the pattern `(?x ?y ?z)' with `?x' and
+`?z' both bound to `(a b)' and `?y' bound to `c'. It also matches the
+pattern `((?x ?y) c (?x ?y))' with `?x' bound to `a' and `?y' bound to
+`b'. However, it does not match the pattern `(?x a ?y)', since that
+pattern specifies a list whose second element is the symbol `a'.
+
+ The pattern matcher used by the query system takes as inputs a
+pattern, a datum, and a "frame" that specifies bindings for various
+pattern variables. It checks whether the datum matches the pattern in
+a way that is consistent with the bindings already in the frame. If
+so, it returns the given frame augmented by any bindings that may have
+been determined by the match. Otherwise, it indicates that the match
+has failed.
+
+ For example, using the pattern `(?x ?y ?x)' to match `(a b a)' given
+an empty frame will return a frame specifying that `?x' is bound to `a'
+and `?y' is bound to `b'. Trying the match with the same pattern, the
+same datum, and a frame specifying that `?y' is bound to `a' will fail.
+Trying the match with the same pattern, the same datum, and a frame in
+which `?y' is bound to `b' and `?x' is unbound will return the given
+frame augmented by a binding of `?x' to `a'.
+
+ The pattern matcher is all the mechanism that is needed to process
+simple queries that don't involve rules. For instance, to process the
+query
+
+ (job ?x (computer programmer))
+
+we scan through all assertions in the data base and select those that
+match the pattern with respect to an initially empty frame. For each
+match we find, we use the frame returned by the match to instantiate
+the pattern with a value for `?x'.
+
+Streams of frames
+.................
+
+The testing of patterns against frames is organized through the use of
+streams. Given a single frame, the matching process runs through the
+data-base entries one by one. For each data-base entry, the matcher
+generates either a special symbol indicating that the match has failed
+or an extension to the frame. The results for all the data-base
+entries are collected into a stream, which is passed through a filter
+to weed out the failures. The result is a stream of all the frames
+that extend the given frame via a match to some assertion in the data
+base.(1)
+
+ In our system, a query takes an input stream of frames and performs
+the above matching operation for every frame in the stream, as
+indicated in *Note Figure 4-4::. That is, for each frame in the input
+stream, the query generates a new stream consisting of all extensions
+to that frame by matches to assertions in the data base. All these
+streams are then combined to form one huge stream, which contains all
+possible extensions of every frame in the input stream. This stream is
+the output of the query.
+
+ *Figure 4.4:* A query processes a stream of frames.
+
+ output stream
+ input stream +-------------+ of frames,
+ of frames | query | filtered and extended
+ ---------------->| +------------------------->
+ | (job ?x ?y) |
+ +-------------+
+ ^
+ |
+ stream of assertions
+ from data base
+
+ To answer a simple query, we use the query with an input stream
+consisting of a single empty frame. The resulting output stream
+contains all extensions to the empty frame (that is, all answers to our
+query). This stream of frames is then used to generate a stream of
+copies of the original query pattern with the variables instantiated by
+the values in each frame, and this is the stream that is finally
+printed.
+
+Compound queries
+................
+
+The real elegance of the stream-of-frames implementation is evident
+when we deal with compound queries. The processing of compound queries
+makes use of the ability of our matcher to demand that a match be
+consistent with a specified frame. For example, to handle the `and' of
+two queries, such as
+
+ (and (can-do-job ?x (computer programmer trainee))
+ (job ?person ?x))
+
+(informally, "Find all people who can do the job of a computer
+programmer trainee"), we first find all entries that match the pattern
+
+ (can-do-job ?x (computer programmer trainee))
+
+ This produces a stream of frames, each of which contains a binding
+for `?x'. Then for each frame in the stream we find all entries that
+match
+
+ (job ?person ?x)
+
+in a way that is consistent with the given binding for `?x'. Each such
+match will produce a frame containing bindings for `?x' and `?person'.
+The `and' of two queries can be viewed as a series combination of the
+two component queries, as shown in *Note Figure 4-5::. The frames that
+pass through the first query filter are filtered and further extended
+by the second query.
+
+ *Figure 4.5:* The `and' combination of two queries is produced by
+ operating on the stream of frames in series.
+
+ +----------------------+
+ | (and A B) |
+ input stream | | output stream
+ of frames | +---+ +---+ | of frames
+ ------------------->| A +------>| B +-------------------->
+ | +---+ +---+ |
+ | ^ ^ |
+ | | | |
+ | +-----*-----+ |
+ +-----------|----------+
+ |
+ data base
+
+ *Note Figure 4-6:: shows the analogous method for computing the `or'
+of two queries as a parallel combination of the two component queries.
+The input stream of frames is extended separately by each query. The
+two resulting streams are then merged to produce the final output
+stream.
+
+ *Figure 4.6:* The `or' combination of two queries is produced by
+ operating on the stream of frames in parallel and merging the
+ results.
+
+ +---------------------------+
+ | (or A B) |
+ | +---+ |
+ input | +->| A |------------+ | output
+ stream of | | +---+ V | stream of
+ frames | | ^ +-------+ | frames
+ -------------* | | merge +--------------->
+ | | | +-------+ |
+ | | | ^ |
+ | | | +---+ | |
+ | +------->| B +------+ |
+ | | +---+ |
+ | | ^ |
+ | | | |
+ | +--*--+ |
+ +---------|-----------------+
+ |
+ data base
+
+ Even from this high-level description, it is apparent that the
+processing of compound queries can be slow. For example, since a query
+may produce more than one output frame for each input frame, and each
+query in an `and' gets its input frames from the previous query, an
+`and' query could, in the worst case, have to perform a number of
+matches that is exponential in the number of queries (see *Note
+Exercise 4-76::).(2) Though systems for handling only simple queries
+are quite practical, dealing with complex queries is extremely
+difficult.(3)
+
+ From the stream-of-frames viewpoint, the `not' of some query acts as
+a filter that removes all frames for which the query can be satisfied.
+For instance, given the pattern
+
+ (not (job ?x (computer programmer)))
+
+we attempt, for each frame in the input stream, to produce extension
+frames that satisfy `(job ?x (computer programmer))'. We remove from
+the input stream all frames for which such extensions exist. The
+result is a stream consisting of only those frames in which the binding
+for `?x' does not satisfy `(job ?x (computer programmer))'. For
+example, in processing the query
+
+ (and (supervisor ?x ?y)
+ (not (job ?x (computer programmer))))
+
+the first clause will generate frames with bindings for `?x' and `?y'.
+The `not' clause will then filter these by removing all frames in which
+the binding for `?x' satisfies the restriction that `?x' is a computer
+programmer.(4)
+
+ The `lisp-value' special form is implemented as a similar filter on
+frame streams. We use each frame in the stream to instantiate any
+variables in the pattern, then apply the Lisp predicate. We remove
+from the input stream all frames for which the predicate fails.
+
+Unification
+...........
+
+In order to handle rules in the query language, we must be able to find
+the rules whose conclusions match a given query pattern. Rule
+conclusions are like assertions except that they can contain variables,
+so we will need a generalization of pattern matching--called "unification"--in
+which both the "pattern" and the "datum" may contain variables.
+
+ A unifier takes two patterns, each containing constants and
+variables, and determines whether it is possible to assign values to
+the variables that will make the two patterns equal. If so, it returns
+a frame containing these bindings. For example, unifying `(?x a ?y)'
+and `(?y ?z a)' will specify a frame in which `?x', `?y', and `?z' must
+all be bound to `a'. On the other hand, unifying `(?x ?y a)' and `(?x
+b ?y)' will fail, because there is no value for `?y' that can make the
+two patterns equal. (For the second elements of the patterns to be
+equal, `?y' would have to be `b'; however, for the third elements to be
+equal, `?y' would have to be `a'.) The unifier used in the query
+system, like the pattern matcher, takes a frame as input and performs
+unifications that are consistent with this frame.
+
+ The unification algorithm is the most technically difficult part of
+the query system. With complex patterns, performing unification may
+seem to require deduction. To unify `(?x ?x)' and `((a ?y c) (a b
+?z))', for example, the algorithm must infer that `?x' should be `(a b
+c)', `?y' should be `b', and `?z' should be `c'. We may think of this
+process as solving a set of equations among the pattern components. In
+general, these are simultaneous equations, which may require substantial
+manipulation to solve.(5) For example, unifying `(?x ?x)' and `((a ?y
+c) (a b ?z))' may be thought of as specifying the simultaneous equations
+
+ ?x = (a ?y c)
+ ?x = (a b ?z)
+
+ These equations imply that
+
+ (a ?y c) = (a b ?z)
+
+which in turn implies that
+
+ a = a, ?y = b, c = ?z,
+
+and hence that
+
+ ?x = (a b c)
+
+ In a successful pattern match, all pattern variables become bound,
+and the values to which they are bound contain only constants. This is
+also true of all the examples of unification we have seen so far. In
+general, however, a successful unification may not completely determine
+the variable values; some variables may remain unbound and others may
+be bound to values that contain variables.
+
+ Consider the unification of `(?x a)' and `((b ?y) ?z)'. We can
+deduce that `?x = (b ?y)' and `a = ?z', but we cannot further solve for
+`?x' or `?y'. The unification doesn't fail, since it is certainly
+possible to make the two patterns equal by assigning values to `?x' and
+`?y'. Since this match in no way restricts the values `?y' can take
+on, no binding for `?y' is put into the result frame. The match does,
+however, restrict the value of `?x'. Whatever value `?y' has, `?x'
+must be `(b ?y)'. A binding of `?x' to the pattern `(b ?y)' is thus
+put into the frame. If a value for `?y' is later determined and added
+to the frame (by a pattern match or unification that is required to be
+consistent with this frame), the previously bound `?x' will refer to
+this value.(6)
+
+Applying rules
+..............
+
+Unification is the key to the component of the query system that makes
+inferences from rules. To see how this is accomplished, consider
+processing a query that involves applying a rule, such as
+
+ (lives-near ?x (Hacker Alyssa P))
+
+ To process this query, we first use the ordinary pattern-match
+procedure described above to see if there are any assertions in the
+data base that match this pattern. (There will not be any in this
+case, since our data base includes no direct assertions about who lives
+near whom.) The next step is to attempt to unify the query pattern
+with the conclusion of each rule. We find that the pattern unifies
+with the conclusion of the rule
+
+ (rule (lives-near ?person-1 ?person-2)
+ (and (address ?person-1 (?town . ?rest-1))
+ (address ?person-2 (?town . ?rest-2))
+ (not (same ?person-1 ?person-2))))
+
+resulting in a frame specifying that `?person-2' is bound to `(Hacker
+Alyssa P)' and that `?x' should be bound to (have the same value as)
+`?person-1'. Now, relative to this frame, we evaluate the compound
+query given by the body of the rule. Successful matches will extend
+this frame by providing a binding for `?person-1', and consequently a
+value for `?x', which we can use to instantiate the original query
+pattern.
+
+ In general, the query evaluator uses the following method to apply a
+rule when trying to establish a query pattern in a frame that specifies
+bindings for some of the pattern variables:
+
+ * Unify the query with the conclusion of the rule to form, if
+ successful, an extension of the original frame.
+
+ * Relative to the extended frame, evaluate the query formed by the
+ body of the rule.
+
+
+ Notice how similar this is to the method for applying a procedure in
+the `eval'/`apply' evaluator for Lisp:
+
+ * Bind the procedure's parameters to its arguments to form a frame
+ that extends the original procedure environment.
+
+ * Relative to the extended environment, evaluate the expression
+ formed by the body of the procedure.
+
+
+ The similarity between the two evaluators should come as no
+surprise. Just as procedure definitions are the means of abstraction
+in Lisp, rule definitions are the means of abstraction in the query
+language. In each case, we unwind the abstraction by creating
+appropriate bindings and evaluating the rule or procedure body relative
+to these.
+
+Simple queries
+..............
+
+We saw earlier in this section how to evaluate simple queries in the
+absence of rules. Now that we have seen how to apply rules, we can
+describe how to evaluate simple queries by using both rules and
+assertions.
+
+ Given the query pattern and a stream of frames, we produce, for each
+frame in the input stream, two streams:
+
+ * a stream of extended frames obtained by matching the pattern
+ against all assertions in the data base (using the pattern
+ matcher), and
+
+ * a stream of extended frames obtained by applying all possible
+ rules (using the unifier).(7)
+
+
+ Appending these two streams produces a stream that consists of all
+the ways that the given pattern can be satisfied consistent with the
+original frame. These streams (one for each frame in the input stream)
+are now all combined to form one large stream, which therefore consists
+of all the ways that any of the frames in the original input stream can
+be extended to produce a match with the given pattern.
+
+The query evaluator and the driver loop
+.......................................
+
+Despite the complexity of the underlying matching operations, the
+system is organized much like an evaluator for any language. The
+procedure that coordinates the matching operations is called `qeval',
+and it plays a role analogous to that of the `eval' procedure for Lisp.
+`Qeval' takes as inputs a query and a stream of frames. Its output is
+a stream of frames, corresponding to successful matches to the query
+pattern, that extend some frame in the input stream, as indicated in
+*Note Figure 4-4::. Like `eval', `qeval' classifies the different
+types of expressions (queries) and dispatches to an appropriate
+procedure for each. There is a procedure for each special form (`and',
+`or', `not', and `lisp-value') and one for simple queries.
+
+ The driver loop, which is analogous to the `driver-loop' procedure
+for the other evaluators in this chapter, reads queries from the
+terminal. For each query, it calls `qeval' with the query and a stream
+that consists of a single empty frame. This will produce the stream of
+all possible matches (all possible extensions to the empty frame). For
+each frame in the resulting stream, it instantiates the original query
+using the values of the variables found in the frame. This stream of
+instantiated queries is then printed.(8)
+
+ The driver also checks for the special command `assert!', which
+signals that the input is not a query but rather an assertion or rule
+to be added to the data base. For instance,
+
+ (assert! (job (Bitdiddle Ben) (computer wizard)))
+
+ (assert! (rule (wheel ?person)
+ (and (supervisor ?middle-manager ?person)
+ (supervisor ?x ?middle-manager))))
+
+ ---------- Footnotes ----------
+
+ (1) Because matching is generally very expensive, we would like to
+avoid applying the full matcher to every element of the data base.
+This is usually arranged by breaking up the process into a fast, coarse
+match and the final match. The coarse match filters the data base to
+produce a small set of candidates for the final match. With care, we
+can arrange our data base so that some of the work of coarse matching
+can be done when the data base is constructed rather then when we want
+to select the candidates. This is called "indexing" the data base.
+There is a vast technology built around data-base-indexing schemes.
+Our implementation, described in section *Note 4-4-4::, contains a
+simple-minded form of such an optimization.
+
+ (2) But this kind of exponential explosion is not common in `and'
+queries because the added conditions tend to reduce rather than expand
+the number of frames produced.
+
+ (3) There is a large literature on data-base-management systems that
+is concerned with how to handle complex queries efficiently.
+
+ (4) There is a subtle difference between this filter implementation
+of `not' and the usual meaning of `not' in mathematical logic. See
+section *Note 4-4-3::.
+
+ (5) In one-sided pattern matching, all the equations that contain
+pattern variables are explicit and already solved for the unknown (the
+pattern variable).
+
+ (6) Another way to think of unification is that it generates the
+most general pattern that is a specialization of the two input
+patterns. That is, the unification of `(?x a)' and `((b ?y) ?z)' is
+`((b ?y) a)', and the unification of `(?x a ?y)' and `(?y ?z a)',
+discussed above, is `(a a a)'. For our implementation, it is more
+convenient to think of the result of unification as a frame rather than
+a pattern.
+
+ (7) Since unification is a generalization of matching, we could
+simplify the system by using the unifier to produce both streams.
+Treating the easy case with the simple matcher, however, illustrates
+how matching (as opposed to full-blown unification) can be useful in
+its own right.
+
+ (8) The reason we use streams (rather than lists) of frames is that
+the recursive application of rules can generate infinite numbers of
+values that satisfy a query. The delayed evaluation embodied in
+streams is crucial here: The system will print responses one by one as
+they are generated, regardless of whether there are a finite or
+infinite number of responses.
+
+\1f
+File: sicp.info, Node: 4-4-3, Next: 4-4-4, Prev: 4-4-2, Up: 4-4
+
+4.4.3 Is Logic Programming Mathematical Logic?
+----------------------------------------------
+
+The means of combination used in the query language may at first seem
+identical to the operations `and', `or', and `not' of mathematical
+logic, and the application of query-language rules is in fact
+accomplished through a legitimate method of inference.(1) This
+identification of the query language with mathematical logic is not
+really valid, though, because the query language provides a structure
+"control structure" that interprets the logical statements
+procedurally. We can often take advantage of this control structure.
+For example, to find all of the supervisors of programmers we could
+formulate a query in either of two logically equivalent forms:
+
+ (and (job ?x (computer programmer))
+ (supervisor ?x ?y))
+
+or
+
+ (and (supervisor ?x ?y)
+ (job ?x (computer programmer)))
+
+ If a company has many more supervisors than programmers (the usual
+case), it is better to use the first form rather than the second
+because the data base must be scanned for each intermediate result
+(frame) produced by the first clause of the `and'.
+
+ The aim of logic programming is to provide the programmer with
+techniques for decomposing a computational problem into two separate
+problems: "what" is to be computed, and "how" this should be computed.
+This is accomplished by selecting a subset of the statements of
+mathematical logic that is powerful enough to be able to describe
+anything one might want to compute, yet weak enough to have a
+controllable procedural interpretation. The intention here is that, on
+the one hand, a program specified in a logic programming language
+should be an effective program that can be carried out by a computer.
+Control ("how" to compute) is effected by using the order of evaluation
+of the language. We should be able to arrange the order of clauses and
+the order of subgoals within each clause so that the computation is
+done in an order deemed to be effective and efficient. At the same
+time, we should be able to view the result of the computation ("what"
+to compute) as a simple consequence of the laws of logic.
+
+ Our query language can be regarded as just such a procedurally
+interpretable subset of mathematical logic. An assertion represents a
+simple fact (an atomic proposition). A rule represents the implication
+that the rule conclusion holds for those cases where the rule body
+holds. A rule has a natural procedural interpretation: To establish
+the conclusion of the rule, establish the body of the rule. Rules,
+therefore, specify computations. However, because rules can also be
+regarded as statements of mathematical logic, we can justify any
+"inference" accomplished by a logic program by asserting that the same
+result could be obtained by working entirely within mathematical
+logic.(2)
+
+Infinite loops
+..............
+
+A consequence of the procedural interpretation of logic programs is
+that it is possible to construct hopelessly inefficient programs for
+solving certain problems. An extreme case of inefficiency occurs when
+the system falls into infinite loops in making deductions. As a simple
+example, suppose we are setting up a data base of famous marriages,
+including
+
+ (assert! (married Minnie Mickey))
+
+ If we now ask
+
+ (married Mickey ?who)
+
+we will get no response, because the system doesn't know that if A is
+married to B, then B is married to A. So we assert the rule
+
+ (assert! (rule (married ?x ?y)
+ (married ?y ?x)))
+
+and again query
+
+ (married Mickey ?who)
+
+ Unfortunately, this will drive the system into an infinite loop, as
+follows:
+
+ * The system finds that the `married' rule is applicable; that is,
+ the rule conclusion `(married ?x ?y)' successfully unifies with
+ the query pattern `(married Mickey ?who)' to produce a frame in
+ which `?x' is bound to `Mickey' and `?y' is bound to `?who'. So
+ the interpreter proceeds to evaluate the rule body `(married ?y
+ ?x)' in this frame--in effect, to process the query `(married ?who
+ Mickey)'.
+
+ * One answer appears directly as an assertion in the data base:
+ `(married Minnie Mickey)'.
+
+ * The `married' rule is also applicable, so the interpreter again
+ evaluates the rule body, which this time is equivalent to
+ `(married Mickey ?who)'.
+
+
+ The system is now in an infinite loop. Indeed, whether the system
+will find the simple answer `(married Minnie Mickey)' before it goes
+into the loop depends on implementation details concerning the order in
+which the system checks the items in the data base. This is a very
+simple example of the kinds of loops that can occur. Collections of
+interrelated rules can lead to loops that are much harder to
+anticipate, and the appearance of a loop can depend on the order of
+clauses in an `and' (see *Note Exercise 4-64::) or on low-level details
+concerning the order in which the system processes queries.(3)
+
+Problems with `not'
+...................
+
+Another quirk in the query system concerns `not'. Given the data base
+of section *Note 4-4-1::, consider the following two queries:
+
+ (and (supervisor ?x ?y)
+ (not (job ?x (computer programmer))))
+
+ (and (not (job ?x (computer programmer)))
+ (supervisor ?x ?y))
+
+ These two queries do not produce the same result. The first query
+begins by finding all entries in the data base that match `(supervisor
+?x ?y)', and then filters the resulting frames by removing the ones in
+which the value of `?x' satisfies `(job ?x (computer programmer))'.
+The second query begins by filtering the incoming frames to remove
+those that can satisfy `(job ?x (computer programmer))'. Since the
+only incoming frame is empty, it checks the data base to see if there
+are any patterns that satisfy `(job ?x (computer programmer))'. Since
+there generally are entries of this form, the `not' clause filters out
+the empty frame and returns an empty stream of frames. Consequently,
+the entire compound query returns an empty stream.
+
+ The trouble is that our implementation of `not' really is meant to
+serve as a filter on values for the variables. If a `not' clause is
+processed with a frame in which some of the variables remain unbound
+(as does `?x' in the example above), the system will produce unexpected
+results. Similar problems occur with the use of `lisp-value'--the Lisp
+predicate can't work if some of its arguments are unbound. See *Note
+Exercise 4-77::.
+
+ There is also a much more serious way in which the `not' of the query
+language differs from the `not' of mathematical logic. In logic, we
+interpret the statement "not P" to mean that P is not true. In the
+query system, however, "not P" means that P is not deducible from the
+knowledge in the data base. For example, given the personnel data base
+of section *Note 4-4-1::, the system would happily deduce all sorts of
+`not' statements, such as that Ben Bitdiddle is not a baseball fan,
+that it is not raining outside, and that 2 + 2 is not 4.(4) In other
+words, the `not' of logic programming languages reflects the so-called "closed
+world assumption" that all relevant information has been included in
+the data base.(5)
+
+ *Exercise 4.64:* Louis Reasoner mistakenly deletes the
+ `outranked-by' rule (section *Note 4-4-1::) from the data base.
+ When he realizes this, he quickly reinstalls it. Unfortunately,
+ he makes a slight change in the rule, and types it in as
+
+ (rule (outranked-by ?staff-person ?boss)
+ (or (supervisor ?staff-person ?boss)
+ (and (outranked-by ?middle-manager ?boss)
+ (supervisor ?staff-person ?middle-manager))))
+
+ Just after Louis types this information into the system, DeWitt
+ Aull comes by to find out who outranks Ben Bitdiddle. He issues
+ the query
+
+ (outranked-by (Bitdiddle Ben) ?who)
+
+ After answering, the system goes into an infinite loop. Explain
+ why.
+
+ *Exercise 4.65:* Cy D. Fect, looking forward to the day when he
+ will rise in the organization, gives a query to find all the
+ wheels (using the `wheel' rule of section *Note 4-4-1::):
+
+ (wheel ?who)
+
+ To his surprise, the system responds
+
+ ;;; Query results:
+ (wheel (Warbucks Oliver))
+ (wheel (Bitdiddle Ben))
+ (wheel (Warbucks Oliver))
+ (wheel (Warbucks Oliver))
+ (wheel (Warbucks Oliver))
+
+ Why is Oliver Warbucks listed four times?
+
+ *Exercise 4.66:* Ben has been generalizing the query system to
+ provide statistics about the company. For example, to find the
+ total salaries of all the computer programmers one will be able to
+ say
+
+ (sum ?amount
+ (and (job ?x (computer programmer))
+ (salary ?x ?amount)))
+
+ In general, Ben's new system allows expressions of the form
+
+ (accumulation-function <VARIABLE>
+ <QUERY PATTERN>)
+
+ where `accumulation-function' can be things like `sum', `average',
+ or `maximum'. Ben reasons that it should be a cinch to implement
+ this. He will simply feed the query pattern to `qeval'. This
+ will produce a stream of frames. He will then pass this stream
+ through a mapping function that extracts the value of the
+ designated variable from each frame in the stream and feed the
+ resulting stream of values to the accumulation function. Just as
+ Ben completes the implementation and is about to try it out, Cy
+ walks by, still puzzling over the `wheel' query result in exercise
+ *Note Exercise 4-65::. When Cy shows Ben the system's response,
+ Ben groans, "Oh, no, my simple accumulation scheme won't work!"
+
+ What has Ben just realized? Outline a method he can use to
+ salvage the situation.
+
+ *Exercise 4.67:* Devise a way to install a loop detector in the
+ query system so as to avoid the kinds of simple loops illustrated
+ in the text and in *Note Exercise 4-64::. The general idea is that
+ the system should maintain some sort of history of its current
+ chain of deductions and should not begin processing a query that
+ it is already working on. Describe what kind of information
+ (patterns and frames) is included in this history, and how the
+ check should be made. (After you study the details of the
+ query-system implementation in section *Note 4-4-4::, you may want
+ to modify the system to include your loop detector.)
+
+ *Exercise 4.68:* Define rules to implement the `reverse' operation
+ of *Note Exercise 2-18::, which returns a list containing the same
+ elements as a given list in reverse order. (Hint: Use
+ `append-to-form'.) Can your rules answer both `(reverse (1 2 3)
+ ?x)' and `(reverse ?x (1 2 3))' ?
+
+ *Exercise 4.69:* Beginning with the data base and the rules you
+ formulated in *Note Exercise 4-63::, devise a rule for adding
+ "greats" to a grandson relationship. This should enable the system
+ to deduce that Irad is the great-grandson of Adam, or that Jabal
+ and Jubal are the great-great-great-great-great-grandsons of Adam.
+ (Hint: Represent the fact about Irad, for example, as `((great
+ grandson) Adam Irad)'. Write rules that determine if a list ends
+ in the word `grandson'. Use this to express a rule that allows
+ one to derive the relationship `((great . ?rel) ?x ?y)', where
+ `?rel' is a list ending in `grandson'.) Check your rules on
+ queries such as `((great grandson) ?g ?ggs)' and `(?relationship
+ Adam Irad)'.
+
+ ---------- Footnotes ----------
+
+ (1) That a particular method of inference is legitimate is not a
+trivial assertion. One must prove that if one starts with true
+premises, only true conclusions can be derived. The method of
+inference represented by rule applications is "modus ponens", the
+familiar method of inference that says that if A is true and _A implies
+B_ is true, then we may conclude that B is true.
+
+ (2) We must qualify this statement by agreeing that, in speaking of
+the "inference" accomplished by a logic program, we assume that the
+computation terminates. Unfortunately, even this qualified statement
+is false for our implementation of the query language (and also false
+for programs in Prolog and most other current logic programming
+languages) because of our use of `not' and `lisp-value'. As we will
+describe below, the `not' implemented in the query language is not
+always consistent with the `not' of mathematical logic, and
+`lisp-value' introduces additional complications. We could implement a
+language consistent with mathematical logic by simply removing `not'
+and `lisp-value' from the language and agreeing to write programs using
+only simple queries, `and', and `or'. However, this would greatly
+restrict the expressive power of the language. One of the major
+concerns of research in logic programming is to find ways to achieve
+more consistency with mathematical logic without unduly sacrificing
+expressive power.
+
+ (3) This is not a problem of the logic but one of the procedural
+interpretation of the logic provided by our interpreter. We could
+write an interpreter that would not fall into a loop here. For
+example, we could enumerate all the proofs derivable from our
+assertions and our rules in a breadth-first rather than a depth-first
+order. However, such a system makes it more difficult to take
+advantage of the order of deductions in our programs. One attempt to
+build sophisticated control into such a program is described in deKleer
+et al. 1977. Another technique, which does not lead to such serious
+control problems, is to put in special knowledge, such as detectors for
+particular kinds of loops (*Note Exercise 4-67::). However, there can
+be no general scheme for reliably preventing a system from going down
+infinite paths in performing deductions. Imagine a diabolical rule of
+the form "To show P(x) is true, show that P(f(x)) is true," for some
+suitably chosen function f.
+
+ (4) Consider the query `(not (baseball-fan (Bitdiddle Ben)))'. The
+system finds that `(baseball-fan (Bitdiddle Ben))' is not in the data
+base, so the empty frame does not satisfy the pattern and is not
+filtered out of the initial stream of frames. The result of the query
+is thus the empty frame, which is used to instantiate the input query
+to produce `(not (baseball-fan (Bitdiddle Ben)))'.
+
+ (5) A discussion and justification of this treatment of `not' can be
+found in the article by Clark (1978).
+
+\1f
+File: sicp.info, Node: 4-4-4, Prev: 4-4-3, Up: 4-4
+
+4.4.4 Implementing the Query System
+-----------------------------------
+
+Section *Note 4-4-2:: described how the query system works. Now we fill
+in the details by presenting a complete implementation of the system.
+
+* Menu:
+
+* 4-4-4-1:: The Driver Loop and Instantiation
+* 4-4-4-2:: The Evaluator
+* 4-4-4-3:: Finding Assertions by Pattern Matching
+* 4-4-4-4:: Rules and Unification
+* 4-4-4-5:: Maintaining the Data Base
+* 4-4-4-6:: Stream Operations
+* 4-4-4-7:: Query Syntax Procedures
+* 4-4-4-8:: Frames and Bindings
+
+\1f
+File: sicp.info, Node: 4-4-4-1, Next: 4-4-4-2, Prev: 4-4-4, Up: 4-4-4
+
+4.4.4.1 The Driver Loop and Instantiation
+.........................................
+
+The driver loop for the query system repeatedly reads input
+expressions. If the expression is a rule or assertion to be added to
+the data base, then the information is added. Otherwise the expression
+is assumed to be a query. The driver passes this query to the
+evaluator `qeval' together with an initial frame stream consisting of a
+single empty frame. The result of the evaluation is a stream of frames
+generated by satisfying the query with variable values found in the
+data base. These frames are used to form a new stream consisting of
+copies of the original query in which the variables are instantiated
+with values supplied by the stream of frames, and this final stream is
+printed at the terminal:
+
+ (define input-prompt ";;; Query input:")
+ (define output-prompt ";;; Query results:")
+
+ (define (query-driver-loop)
+ (prompt-for-input input-prompt)
+ (let ((q (query-syntax-process (read))))
+ (cond ((assertion-to-be-added? q)
+ (add-rule-or-assertion! (add-assertion-body q))
+ (newline)
+ (display "Assertion added to data base.")
+ (query-driver-loop))
+ (else
+ (newline)
+ (display output-prompt)
+ (display-stream
+ (stream-map
+ (lambda (frame)
+ (instantiate q
+ frame
+ (lambda (v f)
+ (contract-question-mark v))))
+ (qeval q (singleton-stream '()))))
+ (query-driver-loop)))))
+
+ Here, as in the other evaluators in this chapter, we use an abstract
+syntax for the expressions of the query language. The implementation
+of the expression syntax, including the predicate
+`assertion-to-be-added?' and the selector `add-assertion-body', is
+given in section *Note 4-4-4-7::. `Add-rule-or-assertion!' is defined
+in section *Note 4-4-4-5::.
+
+ Before doing any processing on an input expression, the driver loop
+transforms it syntactically into a form that makes the processing more
+efficient. This involves changing the representation of pattern
+variables. When the query is instantiated, any variables that remain
+unbound are transformed back to the input representation before being
+printed. These transformations are performed by the two procedures
+`query-syntax-process' and `contract-question-mark' (section *Note
+4-4-4-7::).
+
+ To instantiate an expression, we copy it, replacing any variables in
+the expression by their values in a given frame. The values are
+themselves instantiated, since they could contain variables (for
+example, if `?x' in `exp' is bound to `?y' as the result of unification
+and `?y' is in turn bound to 5). The action to take if a variable
+cannot be instantiated is given by a procedural argument to
+`instantiate'.
+
+ (define (instantiate exp frame unbound-var-handler)
+ (define (copy exp)
+ (cond ((var? exp)
+ (let ((binding (binding-in-frame exp frame)))
+ (if binding
+ (copy (binding-value binding))
+ (unbound-var-handler exp frame))))
+ ((pair? exp)
+ (cons (copy (car exp)) (copy (cdr exp))))
+ (else exp)))
+ (copy exp))
+
+ The procedures that manipulate bindings are defined in section *Note
+4-4-4-8::.
+
+\1f
+File: sicp.info, Node: 4-4-4-2, Next: 4-4-4-3, Prev: 4-4-4-1, Up: 4-4-4
+
+4.4.4.2 The Evaluator
+.....................
+
+The `qeval' procedure, called by the `query-driver-loop', is the basic
+evaluator of the query system. It takes as inputs a query and a stream
+of frames, and it returns a stream of extended frames. It identifies
+special forms by a data-directed dispatch using `get' and `put', just
+as we did in implementing generic operations in *Note Chapter 2::. Any
+query that is not identified as a special form is assumed to be a
+simple query, to be processed by `simple-query'.
+
+ (define (qeval query frame-stream)
+ (let ((qproc (get (type query) 'qeval)))
+ (if qproc
+ (qproc (contents query) frame-stream)
+ (simple-query query frame-stream))))
+
+ `Type' and `contents', defined in section *Note 4-4-4-7::, implement
+the abstract syntax of the special forms.
+
+Simple queries
+..............
+
+The `simple-query' procedure handles simple queries. It takes as
+arguments a simple query (a pattern) together with a stream of frames,
+and it returns the stream formed by extending each frame by all
+data-base matches of the query.
+
+ (define (simple-query query-pattern frame-stream)
+ (stream-flatmap
+ (lambda (frame)
+ (stream-append-delayed
+ (find-assertions query-pattern frame)
+ (delay (apply-rules query-pattern frame))))
+ frame-stream))
+
+ For each frame in the input stream, we use `find-assertions' (section
+*Note 4-4-4-3::) to match the pattern against all assertions in the
+data base, producing a stream of extended frames, and we use
+`apply-rules' (section *Note 4-4-4-4::) to apply all possible rules,
+producing another stream of extended frames. These two streams are
+combined (using `stream-append-delayed', section *Note 4-4-4-6::) to
+make a stream of all the ways that the given pattern can be satisfied
+consistent with the original frame (see *Note Exercise 4-71::). The
+streams for the individual input frames are combined using
+`stream-flatmap' (section *Note 4-4-4-6::) to form one large stream of
+all the ways that any of the frames in the original input stream can be
+extended to produce a match with the given pattern.
+
+Compound queries
+................
+
+`And' queries are handled as illustrated in *Note Figure 4-5:: by the
+`conjoin' procedure. `Conjoin' takes as inputs the conjuncts and the
+frame stream and returns the stream of extended frames. First,
+`conjoin' processes the stream of frames to find the stream of all
+possible frame extensions that satisfy the first query in the
+conjunction. Then, using this as the new frame stream, it recursively
+applies `conjoin' to the rest of the queries.
+
+ (define (conjoin conjuncts frame-stream)
+ (if (empty-conjunction? conjuncts)
+ frame-stream
+ (conjoin (rest-conjuncts conjuncts)
+ (qeval (first-conjunct conjuncts)
+ frame-stream))))
+
+ The expression
+
+ (put 'and 'qeval conjoin)
+
+sets up `qeval' to dispatch to `conjoin' when an `and' form is
+encountered.
+
+ `Or' queries are handled similarly, as shown in *Note Figure 4-6::.
+The output streams for the various disjuncts of the `or' are computed
+separately and merged using the `interleave-delayed' procedure from
+section *Note 4-4-4-6::. (See *Note Exercise 4-71:: and *Note Exercise
+4-72::.)
+
+ (define (disjoin disjuncts frame-stream)
+ (if (empty-disjunction? disjuncts)
+ the-empty-stream
+ (interleave-delayed
+ (qeval (first-disjunct disjuncts) frame-stream)
+ (delay (disjoin (rest-disjuncts disjuncts)
+ frame-stream)))))
+
+ (put 'or 'qeval disjoin)
+
+ The predicates and selectors for the syntax of conjuncts and
+disjuncts are given in section *Note 4-4-4-7::.
+
+Filters
+.......
+
+`Not' is handled by the method outlined in section *Note 4-4-2::. We
+attempt to extend each frame in the input stream to satisfy the query
+being negated, and we include a given frame in the output stream only
+if it cannot be extended.
+
+ (define (negate operands frame-stream)
+ (stream-flatmap
+ (lambda (frame)
+ (if (stream-null? (qeval (negated-query operands)
+ (singleton-stream frame)))
+ (singleton-stream frame)
+ the-empty-stream))
+ frame-stream))
+
+ (put 'not 'qeval negate)
+
+ `Lisp-value' is a filter similar to `not'. Each frame in the stream
+is used to instantiate the variables in the pattern, the indicated
+predicate is applied, and the frames for which the predicate returns
+false are filtered out of the input stream. An error results if there
+are unbound pattern variables.
+
+ (define (lisp-value call frame-stream)
+ (stream-flatmap
+ (lambda (frame)
+ (if (execute
+ (instantiate
+ call
+ frame
+ (lambda (v f)
+ (error "Unknown pat var -- LISP-VALUE" v))))
+ (singleton-stream frame)
+ the-empty-stream))
+ frame-stream))
+
+ (put 'lisp-value 'qeval lisp-value)
+
+ `Execute', which applies the predicate to the arguments, must `eval'
+the predicate expression to get the procedure to apply. However, it
+must not evaluate the arguments, since they are already the actual
+arguments, not expressions whose evaluation (in Lisp) will produce the
+arguments. Note that `execute' is implemented using `eval' and `apply'
+from the underlying Lisp system.
+
+ (define (execute exp)
+ (apply (eval (predicate exp) user-initial-environment)
+ (args exp)))
+
+ The `always-true' special form provides for a query that is always
+satisfied. It ignores its contents (normally empty) and simply passes
+through all the frames in the input stream. `Always-true' is used by
+the `rule-body' selector (section *Note 4-4-4-7::) to provide bodies
+for rules that were defined without bodies (that is, rules whose
+conclusions are always satisfied).
+
+ (define (always-true ignore frame-stream) frame-stream)
+
+ (put 'always-true 'qeval always-true)
+
+ The selectors that define the syntax of `not' and `lisp-value' are
+given in section *Note 4-4-4-7::.
+
+\1f
+File: sicp.info, Node: 4-4-4-3, Next: 4-4-4-4, Prev: 4-4-4-2, Up: 4-4-4
+
+4.4.4.3 Finding Assertions by Pattern Matching
+..............................................
+
+`Find-assertions', called by `simple-query' (section *Note 4-4-4-2::),
+takes as input a pattern and a frame. It returns a stream of frames,
+each extending the given one by a data-base match of the given pattern.
+It uses `fetch-assertions' (section *Note 4-4-4-5::) to get a stream
+of all the assertions in the data base that should be checked for a
+match against the pattern and the frame. The reason for
+`fetch-assertions' here is that we can often apply simple tests that
+will eliminate many of the entries in the data base from the pool of
+candidates for a successful match. The system would still work if we
+eliminated `fetch-assertions' and simply checked a stream of all
+assertions in the data base, but the computation would be less efficient
+because we would need to make many more calls to the matcher.
+
+ (define (find-assertions pattern frame)
+ (stream-flatmap (lambda (datum)
+ (check-an-assertion datum pattern frame))
+ (fetch-assertions pattern frame)))
+
+ `Check-an-assertion' takes as arguments a pattern, a data object
+(assertion), and a frame and returns either a one-element stream
+containing the extended frame or `the-empty-stream' if the match fails.
+
+ (define (check-an-assertion assertion query-pat query-frame)
+ (let ((match-result
+ (pattern-match query-pat assertion query-frame)))
+ (if (eq? match-result 'failed)
+ the-empty-stream
+ (singleton-stream match-result))))
+
+ The basic pattern matcher returns either the symbol `failed' or an
+extension of the given frame. The basic idea of the matcher is to
+check the pattern against the data, element by element, accumulating
+bindings for the pattern variables. If the pattern and the data object
+are the same, the match succeeds and we return the frame of bindings
+accumulated so far. Otherwise, if the pattern is a variable we extend
+the current frame by binding the variable to the data, so long as this
+is consistent with the bindings already in the frame. If the pattern
+and the data are both pairs, we (recursively) match the `car' of the
+pattern against the `car' of the data to produce a frame; in this frame
+we then match the `cdr' of the pattern against the `cdr' of the data.
+If none of these cases are applicable, the match fails and we return
+the symbol `failed'.
+
+ (define (pattern-match pat dat frame)
+ (cond ((eq? frame 'failed) 'failed)
+ ((equal? pat dat) frame)
+ ((var? pat) (extend-if-consistent pat dat frame))
+ ((and (pair? pat) (pair? dat))
+ (pattern-match (cdr pat)
+ (cdr dat)
+ (pattern-match (car pat)
+ (car dat)
+ frame)))
+ (else 'failed)))
+
+ Here is the procedure that extends a frame by adding a new binding,
+if this is consistent with the bindings already in the frame:
+
+ (define (extend-if-consistent var dat frame)
+ (let ((binding (binding-in-frame var frame)))
+ (if binding
+ (pattern-match (binding-value binding) dat frame)
+ (extend var dat frame))))
+
+ If there is no binding for the variable in the frame, we simply add
+the binding of the variable to the data. Otherwise we match, in the
+frame, the data against the value of the variable in the frame. If the
+stored value contains only constants, as it must if it was stored
+during pattern matching by `extend-if-consistent', then the match
+simply tests whether the stored and new values are the same. If so, it
+returns the unmodified frame; if not, it returns a failure indication.
+The stored value may, however, contain pattern variables if it was
+stored during unification (see section *Note 4-4-4-4::). The recursive
+match of the stored pattern against the new data will add or check
+bindings for the variables in this pattern. For example, suppose we
+have a frame in which `?x' is bound to `(f ?y)' and `?y' is unbound,
+and we wish to augment this frame by a binding of `?x' to `(f b)'. We
+look up `?x' and find that it is bound to `(f ?y)'. This leads us to
+match `(f ?y)' against the proposed new value `(f b)' in the same
+frame. Eventually this match extends the frame by adding a binding of
+`?y' to `b'. `?X' remains bound to `(f ?y)'. We never modify a stored
+binding and we never store more than one binding for a given variable.
+
+ The procedures used by `extend-if-consistent' to manipulate bindings
+are defined in section *Note 4-4-4-8::.
+
+Patterns with dotted tails
+..........................
+
+If a pattern contains a dot followed by a pattern variable, the pattern
+variable matches the rest of the data list (rather than the next
+element of the data list), just as one would expect with the
+dotted-tail notation described in *Note Exercise 2-20::. Although the
+pattern matcher we have just implemented doesn't look for dots, it does
+behave as we want. This is because the Lisp `read' primitive, which is
+used by `query-driver-loop' to read the query and represent it as a
+list structure, treats dots in a special way.
+
+ When `read' sees a dot, instead of making the next item be the next
+element of a list (the `car' of a `cons' whose `cdr' will be the rest
+of the list) it makes the next item be the `cdr' of the list structure.
+For example, the list structure produced by `read' for the pattern
+`(computer ?type)' could be constructed by evaluating the expression
+`(cons 'computer (cons '?type '()))', and that for `(computer . ?type)'
+could be constructed by evaluating the expression `(cons 'computer
+'?type)'.
+
+ Thus, as `pattern-match' recursively compares `car's and `cdr's of a
+data list and a pattern that had a dot, it eventually matches the
+variable after the dot (which is a `cdr' of the pattern) against a
+sublist of the data list, binding the variable to that list. For
+example, matching the pattern `(computer . ?type)' against `(computer
+programmer trainee)' will match `?type' against the list `(programmer
+trainee)'.
+
+\1f
+File: sicp.info, Node: 4-4-4-4, Next: 4-4-4-5, Prev: 4-4-4-3, Up: 4-4-4
+
+4.4.4.4 Rules and Unification
+.............................
+
+`Apply-rules' is the rule analog of `find-assertions' (section *Note
+4-4-4-3::). It takes as input a pattern and a frame, and it forms a
+stream of extension frames by applying rules from the data base.
+`Stream-flatmap' maps `apply-a-rule' down the stream of possibly
+applicable rules (selected by `fetch-rules', section *Note 4-4-4-5::)
+and combines the resulting streams of frames.
+
+ (define (apply-rules pattern frame)
+ (stream-flatmap (lambda (rule)
+ (apply-a-rule rule pattern frame))
+ (fetch-rules pattern frame)))
+
+ `Apply-a-rule' applies rules using the method outlined in section
+*Note 4-4-2::. It first augments its argument frame by unifying the
+rule conclusion with the pattern in the given frame. If this succeeds,
+it evaluates the rule body in this new frame.
+
+ Before any of this happens, however, the program renames all the
+variables in the rule with unique new names. The reason for this is to
+prevent the variables for different rule applications from becoming
+confused with each other. For instance, if two rules both use a
+variable named `?x', then each one may add a binding for `?x' to the
+frame when it is applied. These two `?x''s have nothing to do with
+each other, and we should not be fooled into thinking that the two
+bindings must be consistent. Rather than rename variables, we could
+devise a more clever environment structure; however, the renaming
+approach we have chosen here is the most straightforward, even if not
+the most efficient. (See *Note Exercise 4-79::.) Here is the
+`apply-a-rule' procedure:
+
+ (define (apply-a-rule rule query-pattern query-frame)
+ (let ((clean-rule (rename-variables-in rule)))
+ (let ((unify-result
+ (unify-match query-pattern
+ (conclusion clean-rule)
+ query-frame)))
+ (if (eq? unify-result 'failed)
+ the-empty-stream
+ (qeval (rule-body clean-rule)
+ (singleton-stream unify-result))))))
+
+ The selectors `rule-body' and `conclusion' that extract parts of a
+rule are defined in section *Note 4-4-4-7::.
+
+ We generate unique variable names by associating a unique identifier
+(such as a number) with each rule application and combining this
+identifier with the original variable names. For example, if the
+rule-application identifier is 7, we might change each `?x' in the rule
+to `?x-7' and each `?y' in the rule to `?y-7'. (`Make-new-variable' and
+`new-rule-application-id' are included with the syntax procedures in
+section *Note 4-4-4-7::.)
+
+ (define (rename-variables-in rule)
+ (let ((rule-application-id (new-rule-application-id)))
+ (define (tree-walk exp)
+ (cond ((var? exp)
+ (make-new-variable exp rule-application-id))
+ ((pair? exp)
+ (cons (tree-walk (car exp))
+ (tree-walk (cdr exp))))
+ (else exp)))
+ (tree-walk rule)))
+
+ The unification algorithm is implemented as a procedure that takes
+as inputs two patterns and a frame and returns either the extended
+frame or the symbol `failed'. The unifier is like the pattern matcher
+except that it is symmetrical--variables are allowed on both sides of
+the match. `Unify-match' is basically the same as `pattern-match',
+except that there is extra code (marked "`***'" below) to handle the
+case where the object on the right side of the match is a variable.
+
+ (define (unify-match p1 p2 frame)
+ (cond ((eq? frame 'failed) 'failed)
+ ((equal? p1 p2) frame)
+ ((var? p1) (extend-if-possible p1 p2 frame))
+ ((var? p2) (extend-if-possible p2 p1 frame)) ; ***
+ ((and (pair? p1) (pair? p2))
+ (unify-match (cdr p1)
+ (cdr p2)
+ (unify-match (car p1)
+ (car p2)
+ frame)))
+ (else 'failed)))
+
+ In unification, as in one-sided pattern matching, we want to accept
+a proposed extension of the frame only if it is consistent with
+existing bindings. The procedure `extend-if-possible' used in
+unification is the same as the `extend-if-consistent' used in pattern
+matching except for two special checks, marked "`***'" in the program
+below. In the first case, if the variable we are trying to match is
+not bound, but the value we are trying to match it with is itself a
+(different) variable, it is necessary to check to see if the value is
+bound, and if so, to match its value. If both parties to the match are
+unbound, we may bind either to the other.
+
+ The second check deals with attempts to bind a variable to a pattern
+that includes that variable. Such a situation can occur whenever a
+variable is repeated in both patterns. Consider, for example, unifying
+the two patterns `(?x ?x)' and `(?y <EXPRESSION INVOLVING `?Y'>)' in a
+frame where both `?x' and `?y' are unbound. First `?x' is matched
+against `?y', making a binding of `?x' to `?y'. Next, the same `?x' is
+matched against the given expression involving `?y'. Since `?x' is
+already bound to `?y', this results in matching `?y' against the
+expression. If we think of the unifier as finding a set of values for
+the pattern variables that make the patterns the same, then these
+patterns imply instructions to find a `?y' such that `?y' is equal to
+the expression involving `?y'. There is no general method for solving
+such equations, so we reject such bindings; these cases are recognized
+by the predicate `depends-on?'.(1) On the other hand, we do not want
+to reject attempts to bind a variable to itself. For example, consider
+unifying `(?x ?x)' and `(?y ?y)'. The second attempt to bind `?x' to
+`?y' matches `?y' (the stored value of `?x') against `?y' (the new
+value of `?x'). This is taken care of by the `equal?' clause of
+`unify-match'.
+
+ (define (extend-if-possible var val frame)
+ (let ((binding (binding-in-frame var frame)))
+ (cond (binding
+ (unify-match
+ (binding-value binding) val frame))
+ ((var? val) ; ***
+ (let ((binding (binding-in-frame val frame)))
+ (if binding
+ (unify-match
+ var (binding-value binding) frame)
+ (extend var val frame))))
+ ((depends-on? val var frame) ; ***
+ 'failed)
+ (else (extend var val frame)))))
+
+ `Depends-on?' is a predicate that tests whether an expression
+proposed to be the value of a pattern variable depends on the variable.
+This must be done relative to the current frame because the expression
+may contain occurrences of a variable that already has a value that
+depends on our test variable. The structure of `depends-on?' is a
+simple recursive tree walk in which we substitute for the values of
+variables whenever necessary.
+
+ (define (depends-on? exp var frame)
+ (define (tree-walk e)
+ (cond ((var? e)
+ (if (equal? var e)
+ true
+ (let ((b (binding-in-frame e frame)))
+ (if b
+ (tree-walk (binding-value b))
+ false))))
+ ((pair? e)
+ (or (tree-walk (car e))
+ (tree-walk (cdr e))))
+ (else false)))
+ (tree-walk exp))
+
+ ---------- Footnotes ----------
+
+ (1) In general, unifying `?y' with an expression involving `?y'
+would require our being able to find a fixed point of the equation `?y'
+= <EXPRESSION INVOLVING `?Y'>. It is sometimes possible to
+syntactically form an expression that appears to be the solution. For
+example, `?y' = `(f ?y)' seems to have the fixed point `(f (f (f ...
+)))', which we can produce by beginning with the expression `(f ?y)'
+and repeatedly substituting `(f ?y)' for `?y'. Unfortunately, not
+every such equation has a meaningful fixed point. The issues that
+arise here are similar to the issues of manipulating infinite series in
+mathematics. For example, we know that 2 is the solution to the
+equation y = 1 + y/2. Beginning with the expression 1 + y/2 and
+repeatedly substituting 1 + y/2 for y gives
+
+ 2 = y = 1 + y/2 = 1 + (1 + y/2)/2 = 1 + 1/2 + y/4 = ...
+
+which leads to
+
+ 2 = 1 + 1/2 + 1/4 + 1/8 + ...
+
+However, if we try the same manipulation beginning with the observation
+that - 1 is the solution to the equation y = 1 + 2y, we obtain
+
+ -1 = y = 1 + 2y = 1 + 2(1 + 2y) = 1 + 2 + 4y = ...
+
+which leads to
+
+ -1 = 1 + 2 + 4 + 8 + ...
+
+Although the formal manipulations used in deriving these two equations
+are identical, the first result is a valid assertion about infinite
+series but the second is not. Similarly, for our unification results,
+reasoning with an arbitrary syntactically constructed expression may
+lead to errors.
+
+\1f
+File: sicp.info, Node: 4-4-4-5, Next: 4-4-4-6, Prev: 4-4-4-4, Up: 4-4-4
+
+4.4.4.5 Maintaining the Data Base
+.................................
+
+One important problem in designing logic programming languages is that
+of arranging things so that as few irrelevant data-base entries as
+possible will be examined in checking a given pattern. In our system,
+in addition to storing all assertions in one big stream, we store all
+assertions whose `car's are constant symbols in separate streams, in a
+table indexed by the symbol. To fetch an assertion that may match a
+pattern, we first check to see if the `car' of the pattern is a
+constant symbol. If so, we return (to be tested using the matcher) all
+the stored assertions that have the same `car'. If the pattern's `car'
+is not a constant symbol, we return all the stored assertions.
+Cleverer methods could also take advantage of information in the frame,
+or try also to optimize the case where the `car' of the pattern is not
+a constant symbol. We avoid building our criteria for indexing (using
+the `car', handling only the case of constant symbols) into the program;
+instead we call on predicates and selectors that embody our criteria.
+
+ (define THE-ASSERTIONS the-empty-stream)
+
+ (define (fetch-assertions pattern frame)
+ (if (use-index? pattern)
+ (get-indexed-assertions pattern)
+ (get-all-assertions)))
+
+ (define (get-all-assertions) THE-ASSERTIONS)
+
+ (define (get-indexed-assertions pattern)
+ (get-stream (index-key-of pattern) 'assertion-stream))
+
+ `Get-stream' looks up a stream in the table and returns an empty
+stream if nothing is stored there.
+
+ (define (get-stream key1 key2)
+ (let ((s (get key1 key2)))
+ (if s s the-empty-stream)))
+
+ Rules are stored similarly, using the `car' of the rule conclusion.
+Rule conclusions are arbitrary patterns, however, so they differ from
+assertions in that they can contain variables. A pattern whose `car'
+is a constant symbol can match rules whose conclusions start with a
+variable as well as rules whose conclusions have the same `car'. Thus,
+when fetching rules that might match a pattern whose `car' is a
+constant symbol we fetch all rules whose conclusions start with a
+variable as well as those whose conclusions have the same `car' as the
+pattern. For this purpose we store all rules whose conclusions start
+with a variable in a separate stream in our table, indexed by the
+symbol `?'.
+
+ (define THE-RULES the-empty-stream)
+
+ (define (fetch-rules pattern frame)
+ (if (use-index? pattern)
+ (get-indexed-rules pattern)
+ (get-all-rules)))
+
+ (define (get-all-rules) THE-RULES)
+
+ (define (get-indexed-rules pattern)
+ (stream-append
+ (get-stream (index-key-of pattern) 'rule-stream)
+ (get-stream '? 'rule-stream)))
+
+ `Add-rule-or-assertion!' is used by `query-driver-loop' to add
+assertions and rules to the data base. Each item is stored in the
+index, if appropriate, and in a stream of all assertions or rules in
+the data base.
+
+ (define (add-rule-or-assertion! assertion)
+ (if (rule? assertion)
+ (add-rule! assertion)
+ (add-assertion! assertion)))
+
+ (define (add-assertion! assertion)
+ (store-assertion-in-index assertion)
+ (let ((old-assertions THE-ASSERTIONS))
+ (set! THE-ASSERTIONS
+ (cons-stream assertion old-assertions))
+ 'ok))
+
+ (define (add-rule! rule)
+ (store-rule-in-index rule)
+ (let ((old-rules THE-RULES))
+ (set! THE-RULES (cons-stream rule old-rules))
+ 'ok))
+
+ To actually store an assertion or a rule, we check to see if it can
+be indexed. If so, we store it in the appropriate stream.
+
+ (define (store-assertion-in-index assertion)
+ (if (indexable? assertion)
+ (let ((key (index-key-of assertion)))
+ (let ((current-assertion-stream
+ (get-stream key 'assertion-stream)))
+ (put key
+ 'assertion-stream
+ (cons-stream assertion
+ current-assertion-stream))))))
+
+ (define (store-rule-in-index rule)
+ (let ((pattern (conclusion rule)))
+ (if (indexable? pattern)
+ (let ((key (index-key-of pattern)))
+ (let ((current-rule-stream
+ (get-stream key 'rule-stream)))
+ (put key
+ 'rule-stream
+ (cons-stream rule
+ current-rule-stream)))))))
+
+ The following procedures define how the data-base index is used. A
+pattern (an assertion or a rule conclusion) will be stored in the table
+if it starts with a variable or a constant symbol.
+
+ (define (indexable? pat)
+ (or (constant-symbol? (car pat))
+ (var? (car pat))))
+
+ The key under which a pattern is stored in the table is either `?'
+(if it starts with a variable) or the constant symbol with which it
+starts.
+
+ (define (index-key-of pat)
+ (let ((key (car pat)))
+ (if (var? key) '? key)))
+
+ The index will be used to retrieve items that might match a pattern
+if the pattern starts with a constant symbol.
+
+ (define (use-index? pat)
+ (constant-symbol? (car pat)))
+
+ *Exercise 4.70:* What is the purpose of the `let' bindings in the
+ procedures `add-assertion!' and `add-rule!' ? What would be wrong
+ with the following implementation of `add-assertion!' ? Hint:
+ Recall the definition of the infinite stream of ones in section
+ *Note 3-5-2::: `(define ones (cons-stream 1 ones))'.
+
+ (define (add-assertion! assertion)
+ (store-assertion-in-index assertion)
+ (set! THE-ASSERTIONS
+ (cons-stream assertion THE-ASSERTIONS))
+ 'ok)
+
+\1f
+File: sicp.info, Node: 4-4-4-6, Next: 4-4-4-7, Prev: 4-4-4-5, Up: 4-4-4
+
+4.4.4.6 Stream Operations
+.........................
+
+The query system uses a few stream operations that were not presented in
+*Note Chapter 3::.
+
+ `Stream-append-delayed' and `interleave-delayed' are just like
+`stream-append' and `interleave' (section *Note 3-5-3::), except that
+they take a delayed argument (like the `integral' procedure in section
+*Note 3-5-4::). This postpones looping in some cases (see *Note
+Exercise 4-71::).
+
+ (define (stream-append-delayed s1 delayed-s2)
+ (if (stream-null? s1)
+ (force delayed-s2)
+ (cons-stream
+ (stream-car s1)
+ (stream-append-delayed (stream-cdr s1) delayed-s2))))
+
+ (define (interleave-delayed s1 delayed-s2)
+ (if (stream-null? s1)
+ (force delayed-s2)
+ (cons-stream
+ (stream-car s1)
+ (interleave-delayed (force delayed-s2)
+ (delay (stream-cdr s1))))))
+
+ `Stream-flatmap', which is used throughout the query evaluator to
+map a procedure over a stream of frames and combine the resulting
+streams of frames, is the stream analog of the `flatmap' procedure
+introduced for ordinary lists in section *Note 2-2-3::. Unlike
+ordinary `flatmap', however, we accumulate the streams with an
+interleaving process, rather than simply appending them (see *Note
+Exercise 4-72:: and *Note Exercise 4-73::).
+
+ (define (stream-flatmap proc s)
+ (flatten-stream (stream-map proc s)))
+
+ (define (flatten-stream stream)
+ (if (stream-null? stream)
+ the-empty-stream
+ (interleave-delayed
+ (stream-car stream)
+ (delay (flatten-stream (stream-cdr stream))))))
+
+ The evaluator also uses the following simple procedure to generate a
+stream consisting of a single element:
+
+ (define (singleton-stream x)
+ (cons-stream x the-empty-stream))
+
+\1f
+File: sicp.info, Node: 4-4-4-7, Next: 4-4-4-8, Prev: 4-4-4-6, Up: 4-4-4
+
+4.4.4.7 Query Syntax Procedures
+...............................
+
+`Type' and `contents', used by `qeval' (section *Note 4-4-4-2::),
+specify that a special form is identified by the symbol in its `car'.
+They are the same as the `type-tag' and `contents' procedures in
+section *Note 2-4-2::, except for the error message.
+
+ (define (type exp)
+ (if (pair? exp)
+ (car exp)
+ (error "Unknown expression TYPE" exp)))
+
+ (define (contents exp)
+ (if (pair? exp)
+ (cdr exp)
+ (error "Unknown expression CONTENTS" exp)))
+
+ The following procedures, used by `query-driver-loop' (in section
+*Note 4-4-4-1::), specify that rules and assertions are added to the
+data base by expressions of the form `(assert! <RULE-OR-ASSERTION>)':
+
+ (define (assertion-to-be-added? exp)
+ (eq? (type exp) 'assert!))
+
+ (define (add-assertion-body exp)
+ (car (contents exp)))
+
+ Here are the syntax definitions for the `and', `or', `not', and
+`lisp-value' special forms (section *Note 4-4-4-2::):
+
+ (define (empty-conjunction? exps) (null? exps))
+ (define (first-conjunct exps) (car exps))
+ (define (rest-conjuncts exps) (cdr exps))
+
+ (define (empty-disjunction? exps) (null? exps))
+ (define (first-disjunct exps) (car exps))
+ (define (rest-disjuncts exps) (cdr exps))
+
+ (define (negated-query exps) (car exps))
+
+ (define (predicate exps) (car exps))
+ (define (args exps) (cdr exps))
+
+ The following three procedures define the syntax of rules:
+
+ (define (rule? statement)
+ (tagged-list? statement 'rule))
+
+ (define (conclusion rule) (cadr rule))
+
+ (define (rule-body rule)
+ (if (null? (cddr rule))
+ '(always-true)
+ (caddr rule)))
+
+ `Query-driver-loop' (section *Note 4-4-4-1::) calls
+`query-syntax-process' to transform pattern variables in the expression,
+which have the form `?symbol', into the internal format `(? symbol)'.
+That is to say, a pattern such as `(job ?x ?y)' is actually represented
+internally by the system as `(job (? x) (? y))'. This increases the
+efficiency of query processing, since it means that the system can
+check to see if an expression is a pattern variable by checking whether
+the `car' of the expression is the symbol `?', rather than having to
+extract characters from the symbol. The syntax transformation is
+accomplished by the following procedure:(1)
+
+ (define (query-syntax-process exp)
+ (map-over-symbols expand-question-mark exp))
+
+ (define (map-over-symbols proc exp)
+ (cond ((pair? exp)
+ (cons (map-over-symbols proc (car exp))
+ (map-over-symbols proc (cdr exp))))
+ ((symbol? exp) (proc exp))
+ (else exp)))
+
+ (define (expand-question-mark symbol)
+ (let ((chars (symbol->string symbol)))
+ (if (string=? (substring chars 0 1) "?")
+ (list '?
+ (string->symbol
+ (substring chars 1 (string-length chars))))
+ symbol)))
+
+ Once the variables are transformed in this way, the variables in a
+pattern are lists starting with `?', and the constant symbols (which
+need to be recognized for data-base indexing, section *Note 4-4-4-5::)
+are just the symbols.
+
+ (define (var? exp)
+ (tagged-list? exp '?))
+
+ (define (constant-symbol? exp) (symbol? exp))
+
+ Unique variables are constructed during rule application (in section
+*Note 4-4-4-4::) by means of the following procedures. The unique
+identifier for a rule application is a number, which is incremented
+each time a rule is applied.
+
+ (define rule-counter 0)
+
+ (define (new-rule-application-id)
+ (set! rule-counter (+ 1 rule-counter))
+ rule-counter)
+
+ (define (make-new-variable var rule-application-id)
+ (cons '? (cons rule-application-id (cdr var))))
+
+ When `query-driver-loop' instantiates the query to print the answer,
+it converts any unbound pattern variables back to the right form for
+printing, using
+
+ (define (contract-question-mark variable)
+ (string->symbol
+ (string-append "?"
+ (if (number? (cadr variable))
+ (string-append (symbol->string (caddr variable))
+ "-"
+ (number->string (cadr variable)))
+ (symbol->string (cadr variable))))))
+
+ ---------- Footnotes ----------
+
+ (1) Most Lisp systems give the user the ability to modify the
+ordinary `read' procedure to perform such transformations by defining "reader
+macro characters". Quoted expressions are already handled in this way:
+The reader automatically translates `'expression' into `(quote
+expression)' before the evaluator sees it. We could arrange for
+`?expression' to be transformed into `(? expression)' in the same way;
+however, for the sake of clarity we have included the transformation
+procedure here explicitly.
+
+ `Expand-question-mark' and `contract-question-mark' use several
+procedures with `string' in their names. These are Scheme primitives.
+
+\1f
+File: sicp.info, Node: 4-4-4-8, Prev: 4-4-4-7, Up: 4-4-4
+
+4.4.4.8 Frames and Bindings
+...........................
+
+Frames are represented as lists of bindings, which are variable-value
+pairs:
+
+ (define (make-binding variable value)
+ (cons variable value))
+
+ (define (binding-variable binding)
+ (car binding))
+
+ (define (binding-value binding)
+ (cdr binding))
+
+ (define (binding-in-frame variable frame)
+ (assoc variable frame))
+
+ (define (extend variable value frame)
+ (cons (make-binding variable value) frame))
+
+ *Exercise 4.71:* Louis Reasoner wonders why the `simple-query' and
+ `disjoin' procedures (section *Note 4-4-4-2::) are implemented
+ using explicit `delay' operations, rather than being defined as
+ follows:
+
+ (define (simple-query query-pattern frame-stream)
+ (stream-flatmap
+ (lambda (frame)
+ (stream-append (find-assertions query-pattern frame)
+ (apply-rules query-pattern frame)))
+ frame-stream))
+
+ (define (disjoin disjuncts frame-stream)
+ (if (empty-disjunction? disjuncts)
+ the-empty-stream
+ (interleave
+ (qeval (first-disjunct disjuncts) frame-stream)
+ (disjoin (rest-disjuncts disjuncts) frame-stream))))
+
+ Can you give examples of queries where these simpler definitions
+ would lead to undesirable behavior?
+
+ *Exercise 4.72:* Why do `disjoin' and `stream-flatmap' interleave
+ the streams rather than simply append them? Give examples that
+ illustrate why interleaving works better. (Hint: Why did we use
+ `interleave' in section *Note 3-5-3::?)
+
+ *Exercise 4.73:* Why does `flatten-stream' use `delay' explicitly?
+ What would be wrong with defining it as follows:
+
+ (define (flatten-stream stream)
+ (if (stream-null? stream)
+ the-empty-stream
+ (interleave
+ (stream-car stream)
+ (flatten-stream (stream-cdr stream)))))
+
+ *Exercise 4.74:* Alyssa P. Hacker proposes to use a simpler
+ version of `stream-flatmap' in `negate', `lisp-value', and
+ `find-assertions'. She observes that the procedure that is mapped
+ over the frame stream in these cases always produces either the
+ empty stream or a singleton stream, so no interleaving is needed
+ when combining these streams.
+
+ a. Fill in the missing expressions in Alyssa's program.
+
+ (define (simple-stream-flatmap proc s)
+ (simple-flatten (stream-map proc s)))
+
+ (define (simple-flatten stream)
+ (stream-map <??>
+ (stream-filter <??> stream)))
+
+ b. Does the query system's behavior change if we change it in
+ this way?
+
+
+ *Exercise 4.75:* Implement for the query language a new special
+ form called `unique'. `Unique' should succeed if there is
+ precisely one item in the data base satisfying a specified query.
+ For example,
+
+ (unique (job ?x (computer wizard)))
+
+ should print the one-item stream
+
+ (unique (job (Bitdiddle Ben) (computer wizard)))
+
+ since Ben is the only computer wizard, and
+
+ (unique (job ?x (computer programmer)))
+
+ should print the empty stream, since there is more than one
+ computer programmer. Moreover,
+
+ (and (job ?x ?j) (unique (job ?anyone ?j)))
+
+ should list all the jobs that are filled by only one person, and
+ the people who fill them.
+
+ There are two parts to implementing `unique'. The first is to
+ write a procedure that handles this special form, and the second
+ is to make `qeval' dispatch to that procedure. The second part is
+ trivial, since `qeval' does its dispatching in a data-directed
+ way. If your procedure is called `uniquely-asserted', all you
+ need to do is
+
+ (put 'unique 'qeval uniquely-asserted)
+
+ and `qeval' will dispatch to this procedure for every query whose
+ `type' (`car') is the symbol `unique'.
+
+ The real problem is to write the procedure `uniquely-asserted'.
+ This should take as input the `contents' (`cdr') of the `unique'
+ query, together with a stream of frames. For each frame in the
+ stream, it should use `qeval' to find the stream of all extensions
+ to the frame that satisfy the given query. Any stream that does
+ not have exactly one item in it should be eliminated. The
+ remaining streams should be passed back to be accumulated into one
+ big stream that is the result of the `unique' query. This is
+ similar to the implementation of the `not' special form.
+
+ Test your implementation by forming a query that lists all people
+ who supervise precisely one person.
+
+ *Exercise 4.76:* Our implementation of `and' as a series
+ combination of queries (*Note Figure 4-5::) is elegant, but it is
+ inefficient because in processing the second query of the `and' we
+ must scan the data base for each frame produced by the first
+ query. If the data base has n elements, and a typical query
+ produces a number of output frames proportional to n (say n/k),
+ then scanning the data base for each frame produced by the first
+ query will require n^2/k calls to the pattern matcher. Another
+ approach would be to process the two clauses of the `and'
+ separately, then look for all pairs of output frames that are
+ compatible. If each query produces n/k output frames, then this
+ means that we must perform n^2/k^2 compatibility checks--a factor
+ of k fewer than the number of matches required in our current
+ method.
+
+ Devise an implementation of `and' that uses this strategy. You
+ must implement a procedure that takes two frames as inputs, checks
+ whether the bindings in the frames are compatible, and, if so,
+ produces a frame that merges the two sets of bindings. This
+ operation is similar to unification.
+
+ *Exercise 4.77:* In section *Note 4-4-3:: we saw that `not' and
+ `lisp-value' can cause the query language to give "wrong" answers
+ if these filtering operations are applied to frames in which
+ variables are unbound. Devise a way to fix this shortcoming. One
+ idea is to perform the filtering in a "delayed" manner by
+ appending to the frame a "promise" to filter that is fulfilled
+ only when enough variables have been bound to make the operation
+ possible. We could wait to perform filtering until all other
+ operations have been performed. However, for efficiency's sake, we
+ would like to perform filtering as soon as possible so as to cut
+ down on the number of intermediate frames generated.
+
+ *Exercise 4.78:* Redesign the query language as a nondeterministic
+ program to be implemented using the evaluator of section *Note
+ 4-3::, rather than as a stream process. In this approach, each
+ query will produce a single answer (rather than the stream of all
+ answers) and the user can type `try-again' to see more answers.
+ You should find that much of the mechanism we built in this
+ section is subsumed by nondeterministic search and backtracking.
+ You will probably also find, however, that your new query language
+ has subtle differences in behavior from the one implemented here.
+ Can you find examples that illustrate this difference?
+
+ *Exercise 4.79:* When we implemented the Lisp evaluator in section
+ *Note 4-1::, we saw how to use local environments to avoid name
+ conflicts between the parameters of procedures. For example, in
+ evaluating
+
+ (define (square x)
+ (* x x))
+
+ (define (sum-of-squares x y)
+ (+ (square x) (square y)))
+
+ (sum-of-squares 3 4)
+
+ there is no confusion between the `x' in `square' and the `x' in
+ `sum-of-squares', because we evaluate the body of each procedure
+ in an environment that is specially constructed to contain
+ bindings for the local variables. In the query system, we used a
+ different strategy to avoid name conflicts in applying rules.
+ Each time we apply a rule we rename the variables with new names
+ that are guaranteed to be unique. The analogous strategy for the
+ Lisp evaluator would be to do away with local environments and
+ simply rename the variables in the body of a procedure each time
+ we apply the procedure.
+
+ Implement for the query language a rule-application method that
+ uses environments rather than renaming. See if you can build on
+ your environment structure to create constructs in the query
+ language for dealing with large systems, such as the rule analog
+ of block-structured procedures. Can you relate any of this to the
+ problem of making deductions in a context (e.g., "If I supposed
+ that P were true, then I would be able to deduce A and B.") as a
+ method of problem solving? (This problem is open-ended. A good
+ answer is probably worth a Ph.D.)
+
+\1f
+File: sicp.info, Node: Chapter 5, Next: References, Prev: Chapter 4, Up: Top
+
+5 Computing with Register Machines
+**********************************
+
+ My aim is to show that the heavenly machine is not a kind of
+ divine, live being, but a kind of clockwork (and he who believes
+ that a clock has soul attributes the maker's glory to the work),
+ insofar as nearly all the manifold motions are caused by a most
+ simple and material force, just as all motions of the clock are
+ caused by a single weight.
+
+ Johannes Kepler (letter to Herwart von Hohenburg, 1605)
+
+ We began this book by studying processes and by describing processes
+in terms of procedures written in Lisp. To explain the meanings of
+these procedures, we used a succession of models of evaluation: the
+substitution model of *Note Chapter 1::, the environment model of *Note
+Chapter 3::, and the metacircular evaluator of *Note Chapter 4::. Our
+examination of the metacircular evaluator, in particular, dispelled
+much of the mystery of how Lisp-like languages are interpreted. But
+even the metacircular evaluator leaves important questions unanswered,
+because it fails to elucidate the mechanisms of control in a Lisp
+system. For instance, the evaluator does not explain how the
+evaluation of a subexpression manages to return a value to the
+expression that uses this value, nor does the evaluator explain how
+some recursive procedures generate iterative processes (that is, are
+evaluated using constant space) whereas other recursive procedures
+generate recursive processes. These questions remain unanswered
+because the metacircular evaluator is itself a Lisp program and hence
+inherits the control structure of the underlying Lisp system. In order
+to provide a more complete description of the control structure of the
+Lisp evaluator, we must work at a more primitive level than Lisp itself.
+
+ In this chapter we will describe processes in terms of the
+step-by-step operation of a traditional computer. Such a computer, or machine
+"register machine", sequentially executes "instructions" that
+manipulate the contents of a fixed set of storage elements called "registers".
+A typical register-machine instruction applies a primitive operation
+to the contents of some registers and assigns the result to another
+register. Our descriptions of processes executed by register machines
+will look very much like "machine-language" programs for traditional
+computers. However, instead of focusing on the machine language of any
+particular computer, we will examine several Lisp procedures and design
+a specific register machine to execute each procedure. Thus, we will
+approach our task from the perspective of a hardware architect rather
+than that of a machine-language computer programmer. In designing
+register machines, we will develop mechanisms for implementing
+important programming constructs such as recursion. We will also
+present a language for describing designs for register machines. In
+section *Note 5-2:: we will implement a Lisp program that uses these
+descriptions to simulate the machines we design.
+
+ Most of the primitive operations of our register machines are very
+simple. For example, an operation might add the numbers fetched from
+two registers, producing a result to be stored into a third register.
+Such an operation can be performed by easily described hardware. In
+order to deal with list structure, however, we will also use the memory
+operations `car', `cdr', and `cons', which require an elaborate
+storage-allocation mechanism. In section *Note 5-3:: we study their
+implementation in terms of more elementary operations.
+
+ In section *Note 5-4::, after we have accumulated experience
+formulating simple procedures as register machines, we will design a
+machine that carries out the algorithm described by the metacircular
+evaluator of section *Note 4-1::. This will fill in the gap in our
+understanding of how Scheme expressions are interpreted, by providing
+an explicit model for the mechanisms of control in the evaluator. In
+section *Note 5-5:: we will study a simple compiler that translates
+Scheme programs into sequences of instructions that can be executed
+directly with the registers and operations of the evaluator register
+machine.
+
+* Menu:
+
+* 5-1:: Designing Register Machines
+* 5-2:: A Register-Machine Simulator
+* 5-3:: Storage Allocation and Garbage Collection
+* 5-4:: The Explicit-Control Evaluator
+* 5-5:: Compilation
+
+\1f
+File: sicp.info, Node: 5-1, Next: 5-2, Prev: Chapter 5, Up: Chapter 5
+
+5.1 Designing Register Machines
+===============================
+
+To design a register machine, we must design its "data paths"
+(registers and operations) and the "controller" that sequences these
+operations. To illustrate the design of a simple register machine, let
+us examine Euclid's Algorithm, which is used to compute the greatest
+common divisor (GCD) of two integers. As we saw in section *Note
+1-2-5::, Euclid's Algorithm can be carried out by an iterative process,
+as specified by the following procedure:
+
+ (define (gcd a b)
+ (if (= b 0)
+ a
+ (gcd b (remainder a b))))
+
+ A machine to carry out this algorithm must keep track of two
+numbers, a and b, so let us assume that these numbers are stored in two
+registers with those names. The basic operations required are testing
+whether the contents of register `b' is zero and computing the
+remainder of the contents of register `a' divided by the contents of
+register `b'. The remainder operation is a complex process, but assume
+for the moment that we have a primitive device that computes
+remainders. On each cycle of the GCD algorithm, the contents of
+register `a' must be replaced by the contents of register `b', and the
+contents of `b' must be replaced by the remainder of the old contents
+of `a' divided by the old contents of `b'. It would be convenient if
+these replacements could be done simultaneously, but in our model of
+register machines we will assume that only one register can be assigned
+a new value at each step. To accomplish the replacements, our machine
+will use a third "temporary" register, which we call `t'. (First the
+remainder will be placed in `t', then the contents of `b' will be
+placed in `a', and finally the remainder stored in `t' will be placed
+in `b'.)
+
+ We can illustrate the registers and operations required for this
+machine by using the data-path diagram shown in *Note Figure 5-1::. In
+this diagram, the registers (`a', `b', and `t') are represented by
+rectangles. Each way to assign a value to a register is indicated by
+an arrow with an `X' behind the head, pointing from the source of data
+to the register. We can think of the `X' as a button that, when
+pushed, allows the value at the source to "flow" into the designated
+register. The label next to each button is the name we will use to
+refer to the button. The names are arbitrary, and can be chosen to
+have mnemonic value (for example, `a<-b' denotes pushing the button
+that assigns the contents of register `b' to register `a'). The source
+of data for a register can be another register (as in the `a<-b'
+assignment), an operation result (as in the `t<-r' assignment), or a
+constant (a built-in value that cannot be changed, represented in a
+data-path diagram by a triangle containing the constant).
+
+ An operation that computes a value from constants and the contents
+of registers is represented in a data-path diagram by a trapezoid
+containing a name for the operation. For example, the box marked `rem'
+in *Note Figure 5-1:: represents an operation that computes the
+remainder of the contents of the registers `a' and `b' to which it is
+attached. Arrows (without buttons) point from the input registers and
+constants to the box, and arrows connect the operation's output value
+to registers. A test is represented by a circle containing a name for
+the test. For example, our GCD machine has an operation that tests
+whether the contents of register `b' is zero. A test also has arrows
+from its input registers and constants, but it has no output arrows;
+its value is used by the controller rather than by the data paths.
+Overall, the data-path diagram shows the registers and operations that
+are required for the machine and how they must be connected. If we
+view the arrows as wires and the `X' buttons as switches, the data-path
+diagram is very like the wiring diagram for a machine that could be
+constructed from electrical components.
+
+ *Figure 5.1:* Data paths for a GCD machine.
+
+ ___
+ +-----+ +-----+ / \
+ | a |<--(X)----| b +--->| = |
+ +--+--+ a<-b +-+---+ \___/
+ | | ^ ^
+ +------+ +----+ | |
+ | | (X) b<-t |
+ .--+---+--. | / \
+ \ rem / | / O \
+ \_____/ | +-----+
+ | |
+ (X) t<-r |
+ | |
+ V |
+ +-----+ |
+ | t +------+
+ +-----+
+
+ In order for the data paths to actually compute GCDs, the buttons
+must be pushed in the correct sequence. We will describe this sequence
+in terms of a controller diagram, as illustrated in *Note Figure 5-2::.
+The elements of the controller diagram indicate how the data-path
+components should be operated. The rectangular boxes in the controller
+diagram identify data-path buttons to be pushed, and the arrows
+describe the sequencing from one step to the next. The diamond in the
+diagram represents a decision. One of the two sequencing arrows will
+be followed, depending on the value of the data-path test identified in
+the diamond. We can interpret the controller in terms of a physical
+analogy: Think of the diagram as a maze in which a marble is rolling.
+When the marble rolls into a box, it pushes the data-path button that
+is named by the box. When the marble rolls into a decision node (such
+as the test for `b' = 0), it leaves the node on the path determined by
+the result of the indicated test. Taken together, the data paths and
+the controller completely describe a machine for computing GCDs. We
+start the controller (the rolling marble) at the place marked `start',
+after placing numbers in registers `a' and `b'. When the controller
+reaches `done', we will find the value of the GCD in register `a'.
+
+ *Figure 5.2:* Controller for a GCD machine.
+
+ start
+ |
+ V
+ / \ yes
+ +--->< = >-----> done
+ | \ /
+ | | no
+ | V
+ | +------+
+ | | t<-r |
+ | +---+--+
+ | |
+ | V
+ | +------+
+ | | a<-b |
+ | +---+--+
+ | |
+ | V
+ | +------+
+ +--+ b<-t |
+ +------+
+
+ *Exercise 5.1:* Design a register machine to compute factorials
+ using the iterative algorithm specified by the following
+ procedure. Draw data-path and controller diagrams for this
+ machine.
+
+ (define (factorial n)
+ (define (iter product counter)
+ (if (> counter n)
+ product
+ (iter (* counter product)
+ (+ counter 1))))
+ (iter 1 1))
+
+* Menu:
+
+* 5-1-1:: A Language for Describing Register Machines
+* 5-1-2:: Abstraction in Machine Design
+* 5-1-3:: Subroutines
+* 5-1-4:: Using a Stack to Implement Recursion
+* 5-1-5:: Instruction Summary
+
+\1f
+File: sicp.info, Node: 5-1-1, Next: 5-1-2, Prev: 5-1, Up: 5-1
+
+5.1.1 A Language for Describing Register Machines
+-------------------------------------------------
+
+Data-path and controller diagrams are adequate for representing simple
+machines such as GCD, but they are unwieldy for describing large
+machines such as a Lisp interpreter. To make it possible to deal with
+complex machines, we will create a language that presents, in textual
+form, all the information given by the data-path and controller
+diagrams. We will start with a notation that directly mirrors the
+diagrams.
+
+ We define the data paths of a machine by describing the registers
+and the operations. To describe a register, we give it a name and
+specify the buttons that control assignment to it. We give each of
+these buttons a name and specify the source of the data that enters the
+register under the button's control. (The source is a register, a
+constant, or an operation.) To describe an operation, we give it a
+name and specify its inputs (registers or constants).
+
+ We define the controller of a machine as a sequence of "instructions"
+together with "labels" that identify "entry points" in the sequence. An
+instruction is one of the following:
+
+ * The name of a data-path button to push to assign a value to a
+ register. (This corresponds to a box in the controller diagram.)
+
+ * A `test' instruction, that performs a specified test.
+
+ * A conditional branch (`branch' instruction) to a location
+ indicated by a controller label, based on the result of the
+ previous test. (The test and branch together correspond to a
+ diamond in the controller diagram.) If the test is false, the
+ controller should continue with the next instruction in the
+ sequence. Otherwise, the controller should continue with the
+ instruction after the label.
+
+ * An unconditional branch (`goto' instruction) naming a controller
+ label at which to continue execution.
+
+
+ The machine starts at the beginning of the controller instruction
+sequence and stops when execution reaches the end of the sequence.
+Except when a branch changes the flow of control, instructions are
+executed in the order in which they are listed.
+
+ *Figure 5.3:* A specification of the GCD machine.
+
+ (data-paths
+ (registers
+ ((name a)
+ (buttons ((name a<-b) (source (register b)))))
+ ((name b)
+ (buttons ((name b<-t) (source (register t)))))
+ ((name t)
+ (buttons ((name t<-r) (source (operation rem))))))
+
+ (operations
+ ((name rem)
+ (inputs (register a) (register b)))
+ ((name =)
+ (inputs (register b) (constant 0)))))
+
+ (controller
+ test-b ; label
+ (test =) ; test
+ (branch (label gcd-done)) ; conditional branch
+ (t<-r) ; button push
+ (a<-b) ; button push
+ (b<-t) ; button push
+ (goto (label test-b)) ; unconditional branch
+ gcd-done) ; label
+
+
+ *Note Figure 5-3:: shows the GCD machine described in this way. This
+example only hints at the generality of these descriptions, since the
+GCD machine is a very simple case: Each register has only one button,
+and each button and test is used only once in the controller.
+
+ Unfortunately, it is difficult to read such a description. In order
+to understand the controller instructions we must constantly refer back
+to the definitions of the button names and the operation names, and to
+understand what the buttons do we may have to refer to the definitions
+of the operation names. We will thus transform our notation to combine
+the information from the data-path and controller descriptions so that
+we see it all together.
+
+ To obtain this form of description, we will replace the arbitrary
+button and operation names by the definitions of their behavior. That
+is, instead of saying (in the controller) "Push button `t<-r'" and
+separately saying (in the data paths) "Button `t<-r' assigns the value
+of the `rem' operation to register `t'" and "The `rem' operation's
+inputs are the contents of registers `a' and `b'," we will say (in the
+controller) "Push the button that assigns to register `t' the value of
+the `rem' operation on the contents of registers `a' and `b'."
+Similarly, instead of saying (in the controller) "Perform the `=' test"
+and separately saying (in the data paths) "The `=' test operates on the
+contents of register `b' and the constant 0," we will say "Perform the
+`=' test on the contents of register `b' and the constant 0." We will
+omit the data-path description, leaving only the controller sequence.
+Thus, the GCD machine is described as follows:
+
+ (controller
+ test-b
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label test-b))
+ gcd-done)
+
+ This form of description is easier to read than the kind illustrated
+in *Note Figure 5-3::, but it also has disadvantages:
+
+ * It is more verbose for large machines, because complete
+ descriptions of the data-path elements are repeated whenever the
+ elements are mentioned in the controller instruction sequence.
+ (This is not a problem in the GCD example, because each operation
+ and button is used only once.) Moreover, repeating the data-path
+ descriptions obscures the actual data-path structure of the
+ machine; it is not obvious for a large machine how many registers,
+ operations, and buttons there are and how they are interconnected.
+
+ * Because the controller instructions in a machine definition look
+ like Lisp expressions, it is easy to forget that they are not
+ arbitrary Lisp expressions. They can notate only legal machine
+ operations. For example, operations can operate directly only on
+ constants and the contents of registers, not on the results of
+ other operations.
+
+
+ In spite of these disadvantages, we will use this register-machine
+language throughout this chapter, because we will be more concerned
+with understanding controllers than with understanding the elements and
+connections in data paths. We should keep in mind, however, that
+data-path design is crucial in designing real machines.
+
+ *Exercise 5.2:* Use the register-machine language to describe the
+ iterative factorial machine of *Note Exercise 5-1::.
+
+Actions
+.......
+
+Let us modify the GCD machine so that we can type in the numbers whose
+GCD we want and get the answer printed at our terminal. We will not
+discuss how to make a machine that can read and print, but will assume
+(as we do when we use `read' and `display' in Scheme) that they are
+available as primitive operations.(1)
+
+ `Read' is like the operations we have been using in that it produces
+a value that can be stored in a register. But `read' does not take
+inputs from any registers; its value depends on something that happens
+outside the parts of the machine we are designing. We will allow our
+machine's operations to have such behavior, and thus will draw and
+notate the use of `read' just as we do any other operation that
+computes a value.
+
+ `Print', on the other hand, differs from the operations we have been
+using in a fundamental way: It does not produce an output value to be
+stored in a register. Though it has an effect, this effect is not on a
+part of the machine we are designing. We will refer to this kind of
+operation as an "action". We will represent an action in a data-path
+diagram just as we represent an operation that computes a value--as a
+trapezoid that contains the name of the action. Arrows point to the
+action box from any inputs (registers or constants). We also associate
+a button with the action. Pushing the button makes the action happen.
+To make a controller push an action button we use a new kind of
+instruction called `perform'. Thus, the action of printing the
+contents of register `a' is represented in a controller sequence by the
+instruction
+
+ (perform (op print) (reg a))
+
+ *Note Figure 5-4:: shows the data paths and controller for the new
+GCD machine. Instead of having the machine stop after printing the
+answer, we have made it start over, so that it repeatedly reads a pair
+of numbers, computes their GCD, and prints the result. This structure
+is like the driver loops we used in the interpreters of *Note Chapter
+4::.
+
+ *Figure 5.4:* A GCD machine that reads inputs and prints results.
+
+ .--------.
+ \ read /
+ \____/
+ |
+ +-------*------+
+ | |
+ a<-rd (X) (X) b<-rd
+ | |
+ V V ___
+ +-----+ +-----+ / \
+ | a |<--(X)--+ b +----->| = |
+ +-+-+-+ a<-b +-+---+ \___/
+ | | | ^ ^
+ +--+ +----+ +--+ | |
+ | | | (X) b<-t / \
+ V V V | / O \
+ .---------. .---------. | /_____\
+ --(X)->\ print / \ rem / |
+ P \_____/ \_____/ |
+ | |
+ (X) t<-r |
+ | |
+ V |
+ +-----+ |
+ | t +----+
+ +-----+
+
+ (controller
+ gcd-loop
+ (assign a (op read))
+ (assign b (op read))
+ test-b
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label test-b))
+ gcd-done
+ (perform (op print) (reg a))
+ (goto (label gcd-loop)))
+
+
+ ---------- Footnotes ----------
+
+ (1) This assumption glosses over a great deal of complexity.
+Usually a large portion of the implementation of a Lisp system is
+dedicated to making reading and printing work.
+
+\1f
+File: sicp.info, Node: 5-1-2, Next: 5-1-3, Prev: 5-1-1, Up: 5-1
+
+5.1.2 Abstraction in Machine Design
+-----------------------------------
+
+We will often define a machine to include "primitive" operations that
+are actually very complex. For example, in sections *Note 5-4:: and
+*Note 5-5:: we will treat Scheme's environment manipulations as
+primitive. Such abstraction is valuable because it allows us to ignore
+the details of parts of a machine so that we can concentrate on other
+aspects of the design. The fact that we have swept a lot of complexity
+under the rug, however, does not mean that a machine design is
+unrealistic. We can always replace the complex "primitives" by simpler
+primitive operations.
+
+ Consider the GCD machine. The machine has an instruction that
+computes the remainder of the contents of registers `a' and `b' and
+assigns the result to register `t'. If we want to construct the GCD
+machine without using a primitive remainder operation, we must specify
+how to compute remainders in terms of simpler operations, such as
+subtraction. Indeed, we can write a Scheme procedure that finds
+remainders in this way:
+
+ (define (remainder n d)
+ (if (< n d)
+ n
+ (remainder (- n d) d)))
+
+ We can thus replace the remainder operation in the GCD machine's data
+paths with a subtraction operation and a comparison test. *Note Figure
+5-5:: shows the data paths and controller for the elaborated machine.
+The instruction
+
+ *Figure 5.5:* Data paths and controller for the elaborated GCD
+ machine.
+
+ ___
+ +-----+ +-----+ / \
+ | a |<--(X)---+ b +-------*-->| = |
+ +--+--+ a<-b +-+---+ | \___/
+ | | ^ |
+ (X) t<-a | | |
+ | | (X) b<-t |
+ V | | _V_
+ +-----+ | | / \
+ | t +-------*---|--*-----| < |
+ +-----+ | | \___/
+ ^ V V
+ | ---------
+ (X) t<-d \ - /
+ | --+--
+ | |
+ +------------+
+
+
+ start
+ |
+ V
+ / \ yes +-------+
+ +->< = >----> done | t<-d |<--+
+ | \ / +---+---+ |
+ | | no | |
+ | | V |
+ | | +------+ / \ no |
+ | +-->| t<-a +------->< < >-----+
+ | +------+ \ /
+ | | yes
+ | +-------------------+
+ | V
+ | +-------+
+ | | a<-b |
+ | +---+---+
+ | |
+ | V
+ | +-------+
+ +--+ b<-t |
+ +-------+
+
+ (assign t (op rem) (reg a) (reg b))
+
+in the GCD controller definition is replaced by a sequence of
+instructions that contains a loop, as shown in *Note Figure 5-6::.
+
+ *Figure 5.6:* Controller instruction sequence for the GCD machine
+ in *Note Figure 5-5::.
+
+ (controller
+ test-b
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (reg a))
+ rem-loop
+ (test (op <) (reg t) (reg b))
+ (branch (label rem-done))
+ (assign t (op -) (reg t) (reg b))
+ (goto (label rem-loop))
+ rem-done
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label test-b))
+ gcd-done)
+
+
+ *Exercise 5.3:* Design a machine to compute square roots using
+ Newton's method, as described in section *Note 1-1-7:::
+
+ (define (sqrt x)
+ (define (good-enough? guess)
+ (< (abs (- (square guess) x)) 0.001))
+ (define (improve guess)
+ (average guess (/ x guess)))
+ (define (sqrt-iter guess)
+ (if (good-enough? guess)
+ guess
+ (sqrt-iter (improve guess))))
+ (sqrt-iter 1.0))
+
+ Begin by assuming that `good-enough?' and `improve' operations are
+ available as primitives. Then show how to expand these in terms
+ of arithmetic operations. Describe each version of the `sqrt'
+ machine design by drawing a data-path diagram and writing a
+ controller definition in the register-machine language.
+
+\1f
+File: sicp.info, Node: 5-1-3, Next: 5-1-4, Prev: 5-1-2, Up: 5-1
+
+5.1.3 Subroutines
+-----------------
+
+When designing a machine to perform a computation, we would often
+prefer to arrange for components to be shared by different parts of the
+computation rather than duplicate the components. Consider a machine
+that includes two GCD computations--one that finds the GCD of the
+contents of registers `a' and `b' and one that finds the GCD of the
+contents of registers `c' and `d'. We might start by assuming we have
+a primitive `gcd' operation, then expand the two instances of `gcd' in
+terms of more primitive operations. *Note Figure 5-7:: shows just the
+GCD portions of the resulting machine's data paths, without showing how
+they connect to the rest of the machine. The figure also shows the
+corresponding portions of the machine's controller sequence.
+
+ *Figure 5.7:* Portions of the data paths and controller sequence
+ for a machine with two GCD computations.
+
+ ___ ___
+ +-----+ +-----+ / \ +-----+ +-----+ / \
+ | a |<-(X)---+ b |--->| = | | c |<-(X)---+ d |--->| = |
+ +--+--+ a<-b ++----+ \___/ +--+--+ c<-d ++----+ \___/
+ | | ^ ^ | | ^ ^
+ `----. .---' | | `----. .---' | |
+ V V (X) b<-t | V V (X) d<-t |
+ ------- | / \ ------- | / \
+ \ rem / | /_0_\ \ rem / | /_0_\
+ --+-- | --+-- |
+ | | | |
+ (X) t<-r | (X) s<-r |
+ | | | |
+ V | V |
+ +-----+ | +-----+ |
+ | t +-----' | s +-----'
+ +-----+ +-----+
+
+ gcd-1
+ (test (op =) (reg b) (const 0))
+ (branch (label after-gcd-1))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label gcd-1))
+ after-gcd-1
+ ...
+ gcd-2
+ (test (op =) (reg d) (const 0))
+ (branch (label after-gcd-2))
+ (assign s (op rem) (reg c) (reg d))
+ (assign c (reg d))
+ (assign d (reg s))
+ (goto (label gcd-2))
+ after-gcd-2
+
+
+ This machine has two remainder operation boxes and two boxes for
+testing equality. If the duplicated components are complicated, as is
+the remainder box, this will not be an economical way to build the
+machine. We can avoid duplicating the data-path components by using
+the same components for both GCD computations, provided that doing so
+will not affect the rest of the larger machine's computation. If the
+values in registers `a' and `b' are not needed by the time the
+controller gets to `gcd-2' (or if these values can be moved to other
+registers for safekeeping), we can change the machine so that it uses
+registers `a' and `b', rather than registers `c' and `d', in computing
+the second GCD as well as the first. If we do this, we obtain the
+controller sequence shown in *Note Figure 5-8::.
+
+ We have removed the duplicate data-path components (so that the data
+paths are again as in *Note Figure 5-1::), but the controller now has
+two GCD sequences that differ only in their entry-point labels. It
+would be better to replace these two sequences by branches to a single
+sequence--a `gcd' "subroutine"--at the end of which we branch back to
+the correct place in the main instruction sequence. We can accomplish
+this as follows: Before branching to `gcd', we place a distinguishing
+value (such as 0 or 1) into a special register, `continue'. At the end
+of the `gcd' subroutine we return either to `after-gcd-1' or to
+`after-gcd-2', depending on the value of the `continue' register.
+*Note Figure 5-9:: shows the relevant portion of the resulting
+controller sequence, which includes only a single copy of the `gcd'
+instructions.
+
+ *Figure 5.8:* Portions of the controller sequence for a machine
+ that uses the same data-path components for two different GCD
+ computations.
+
+ gcd-1
+ (test (op =) (reg b) (const 0))
+ (branch (label after-gcd-1))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label gcd-1))
+ after-gcd-1
+ ...
+ gcd-2
+ (test (op =) (reg b) (const 0))
+ (branch (label after-gcd-2))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label gcd-2))
+ after-gcd-2
+
+
+ *Figure 5.9:* Using a `continue' register to avoid the duplicate
+ controller sequence in *Note Figure 5-8::.
+
+ gcd
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label gcd))
+ gcd-done
+ (test (op =) (reg continue) (const 0))
+ (branch (label after-gcd-1))
+ (goto (label after-gcd-2))
+ ...
+ ;; Before branching to `gcd' from the first place where
+ ;; it is needed, we place 0 in the `continue' register
+ (assign continue (const 0))
+ (goto (label gcd))
+ after-gcd-1
+ ...
+ ;; Before the second use of `gcd', we place 1 in the `continue' register
+ (assign continue (const 1))
+ (goto (label gcd))
+ after-gcd-2
+
+
+ *Figure 5.10:* Assigning labels to the `continue' register
+ simplifies and generalizes the strategy shown in *Note Figure
+ 5-9::.
+
+ gcd
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label gcd))
+ gcd-done
+ (goto (reg continue))
+ ...
+ ;; Before calling `gcd', we assign to `continue'
+ ;; the label to which `gcd' should return.
+ (assign continue (label after-gcd-1))
+ (goto (label gcd))
+ after-gcd-1
+ ...
+ ;; Here is the second call to `gcd', with a different continuation.
+ (assign continue (label after-gcd-2))
+ (goto (label gcd))
+ after-gcd-2
+
+
+ This is a reasonable approach for handling small problems, but it
+would be awkward if there were many instances of GCD computations in the
+controller sequence. To decide where to continue executing after the
+`gcd' subroutine, we would need tests in the data paths and branch
+instructions in the controller for all the places that use `gcd'. A
+more powerful method for implementing subroutines is to have the
+`continue' register hold the label of the entry point in the controller
+sequence at which execution should continue when the subroutine is
+finished. Implementing this strategy requires a new kind of connection
+between the data paths and the controller of a register machine: There
+must be a way to assign to a register a label in the controller
+sequence in such a way that this value can be fetched from the register
+and used to continue execution at the designated entry point.
+
+ To reflect this ability, we will extend the `assign' instruction of
+the register-machine language to allow a register to be assigned as
+value a label from the controller sequence (as a special kind of
+constant). We will also extend the `goto' instruction to allow
+execution to continue at the entry point described by the contents of a
+register rather than only at an entry point described by a constant
+label. Using these new constructs we can terminate the `gcd'
+subroutine with a branch to the location stored in the `continue'
+register. This leads to the controller sequence shown in *Note Figure
+5-10::.
+
+ A machine with more than one subroutine could use multiple
+continuation registers (e.g., `gcd-continue', `factorial-continue') or
+we could have all subroutines share a single `continue' register.
+Sharing is more economical, but we must be careful if we have a
+subroutine (`sub1') that calls another subroutine (`sub2'). Unless
+`sub1' saves the contents of `continue' in some other register before
+setting up `continue' for the call to `sub2', `sub1' will not know
+where to go when it is finished. The mechanism developed in the next
+section to handle recursion also provides a better solution to this
+problem of nested subroutine calls.
+
+\1f
+File: sicp.info, Node: 5-1-4, Next: 5-1-5, Prev: 5-1-3, Up: 5-1
+
+5.1.4 Using a Stack to Implement Recursion
+------------------------------------------
+
+With the ideas illustrated so far, we can implement any iterative
+process by specifying a register machine that has a register
+corresponding to each state variable of the process. The machine
+repeatedly executes a controller loop, changing the contents of the
+registers, until some termination condition is satisfied. At each
+point in the controller sequence, the state of the machine
+(representing the state of the iterative process) is completely
+determined by the contents of the registers (the values of the state
+variables).
+
+ Implementing recursive processes, however, requires an additional
+mechanism. Consider the following recursive method for computing
+factorials, which we first examined in section *Note 1-2-1:::
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+
+ As we see from the procedure, computing n! requires computing (n -
+1)!. Our GCD machine, modeled on the procedure
+
+ (define (gcd a b)
+ (if (= b 0)
+ a
+ (gcd b (remainder a b))))
+
+similarly had to compute another GCD. But there is an important
+difference between the `gcd' procedure, which reduces the original
+computation to a new GCD computation, and `factorial', which requires
+computing another factorial as a subproblem. In GCD, the answer to the
+new GCD computation is the answer to the original problem. To compute
+the next GCD, we simply place the new arguments in the input registers
+of the GCD machine and reuse the machine's data paths by executing the
+same controller sequence. When the machine is finished solving the
+final GCD problem, it has completed the entire computation.
+
+ In the case of factorial (or any recursive process) the answer to
+the new factorial subproblem is not the answer to the original problem.
+The value obtained for (n - 1)! must be multiplied by n to get the
+final answer. If we try to imitate the GCD design, and solve the
+factorial subproblem by decrementing the `n' register and rerunning the
+factorial machine, we will no longer have available the old value of
+`n' by which to multiply the result. We thus need a second factorial
+machine to work on the subproblem. This second factorial computation
+itself has a factorial subproblem, which requires a third factorial
+machine, and so on. Since each factorial machine contains another
+factorial machine within it, the total machine contains an infinite
+nest of similar machines and hence cannot be constructed from a fixed,
+finite number of parts.
+
+ Nevertheless, we can implement the factorial process as a register
+machine if we can arrange to use the same components for each nested
+instance of the machine. Specifically, the machine that computes n!
+should use the same components to work on the subproblem of computing
+(n - 1)!, on the subproblem for (n - 2)!, and so on. This is plausible
+because, although the factorial process dictates that an unbounded
+number of copies of the same machine are needed to perform a
+computation, only one of these copies needs to be active at any given
+time. When the machine encounters a recursive subproblem, it can
+suspend work on the main problem, reuse the same physical parts to work
+on the subproblem, then continue the suspended computation.
+
+ In the subproblem, the contents of the registers will be different
+than they were in the main problem. (In this case the `n' register is
+decremented.) In order to be able to continue the suspended
+computation, the machine must save the contents of any registers that
+will be needed after the subproblem is solved so that these can be
+restored to continue the suspended computation. In the case of
+factorial, we will save the old value of `n', to be restored when we
+are finished computing the factorial of the decremented `n' register.(1)
+
+ Since there is no _a priori_ limit on the depth of nested recursive
+calls, we may need to save an arbitrary number of register values.
+These values must be restored in the reverse of the order in which they
+were saved, since in a nest of recursions the last subproblem to be
+entered is the first to be finished. This dictates the use of a "stack",
+or "last in, first out" data structure, to save register values. We
+can extend the register-machine language to include a stack by adding
+two kinds of instructions: Values are placed on the stack using a
+`save' instruction and restored from the stack using a `restore'
+instruction. After a sequence of values has been `save'd on the stack,
+a sequence of `restore's will retrieve these values in reverse order.(2)
+
+ With the aid of the stack, we can reuse a single copy of the
+factorial machine's data paths for each factorial subproblem. There is
+a similar design issue in reusing the controller sequence that operates
+the data paths. To reexecute the factorial computation, the controller
+cannot simply loop back to the beginning, as with an iterative process,
+because after solving the (n - 1)! subproblem the machine must still
+multiply the result by n. The controller must suspend its computation
+of n!, solve the (n - 1)! subproblem, then continue its computation of
+n!. This view of the factorial computation suggests the use of the
+subroutine mechanism described in section *Note 5-1-3::, which has the
+controller use a `continue' register to transfer to the part of the
+sequence that solves a subproblem and then continue where it left off
+on the main problem. We can thus make a factorial subroutine that
+returns to the entry point stored in the `continue' register. Around
+each subroutine call, we save and restore `continue' just as we do the
+`n' register, since each "level" of the factorial computation will use
+the same `continue' register. That is, the factorial subroutine must
+put a new value in `continue' when it calls itself for a subproblem,
+but it will need the old value in order to return to the place that
+called it to solve a subproblem.
+
+ *Note Figure 5-11:: shows the data paths and controller for a
+machine that implements the recursive `factorial' procedure. The
+machine has a stack and three registers, called `n', `val', and
+`continue'. To simplify the data-path diagram, we have not named the
+register-assignment buttons, only the stack-operation buttons (`sc' and
+`sn' to save registers, `rc' and `rn' to restore registers). To
+operate the machine, we put in register `n' the number whose factorial
+we wish to compute and start the machine. When the machine reaches
+`fact-done', the computation is finished and the answer will be found
+in the `val' register. In the controller sequence, `n' and `continue'
+are saved before each recursive call and restored upon return from the
+call. Returning from a call is accomplished by branching to the
+location stored in `continue'. `Continue' is initialized when the
+machine starts so that the last return will go to `fact-done'. The
+`val' register, which holds the result of the factorial computation, is
+not saved before the recursive call, because the old contents of `val'
+is not useful after the subroutine returns. Only the new value, which
+is the value produced by the subcomputation, is needed.
+
+ Although in principle the factorial computation requires an infinite
+machine, the machine in *Note Figure 5-11:: is actually finite except
+for the stack, which is potentially unbounded. Any particular physical
+implementation of a stack, however, will be of finite size, and this
+will limit the depth of recursive calls that can be handled by the
+machine. This implementation of factorial illustrates the general
+strategy for realizing recursive algorithms as ordinary register
+machines augmented by stacks. When a recursive subproblem is
+encountered, we save on the stack the registers whose current values
+will be required after the subproblem is solved, solve the recursive
+subproblem, then restore the saved registers and continue execution on
+the main problem. The `continue' register must always be saved.
+Whether there are other registers that need to be saved depends on the
+particular machine, since not all recursive computations need the
+original values of registers that are modified during solution of the
+subproblem (see *Note Exercise 5-4::).
+
+A double recursion
+..................
+
+Let us examine a more complex recursive process, the tree-recursive
+computation of the Fibonacci numbers, which we introduced in section
+*Note 1-2-2:::
+
+ (define (fib n)
+ (if (< n 2)
+ n
+ (+ (fib (- n 1)) (fib (- n 2)))))
+
+ Just as with factorial, we can implement the recursive Fibonacci
+computation as a register machine with registers `n', `val', and
+`continue'. The machine is more complex than the one for factorial,
+because there are two places in the controller sequence where we need
+to perform recursive calls--once to compute Fib(n - 1) and once to
+compute Fib(n - 2). To set up for each of these calls, we save the
+registers whose values will be needed later, set the `n' register to
+the number whose Fib we need to compute recursively (n - 1 or n - 2),
+and assign to `continue' the entry point in the main sequence to which
+to return (`afterfib-n-1' or `afterfib-n-2', respectively). We then go
+to `fib-loop'. When we return from the recursive call, the answer is
+in `val'. *Note Figure 5-12:: shows the controller sequence for this
+machine.
+
+ *Figure 5.11:* A recursive factorial machine.
+
+ ___
+ / \
+ +----------*-----------| = |
+ | | \___/
+ (X) | ^
+ | | |
+ V | +---+---+ sn +-------+
+ +-------+ | | +---(X)-->| |
+ | val |<-(X)-|----------+ n | | stack |
+ +-----+-+ | | |<--(X)---+ |
+ ^ | | +-------+ rn +-+-----+
+ | | | ^ | ^
+ (X) | | | | |
+ | | +----|--------* (X) | (X) sc
+ | | | | | | rc (X) |
+ | | | *----. | | | |
+ | V V | V V | V |
+ | ------- | ------- | +------+-+
+ | \ * / | \ - / | |continue+--> controller
+ | --+-- | --+-- | +--------+
+ | | | | | ^ ^
+ +-----+ | +-----+ | |
+ | (X) (X)
+ | | |
+ / \ after- / \ / \ fact-
+ /_1_\ fact /___\ /___\ done
+
+ (controller
+ (assign continue (label fact-done)) ; set up final return address
+ fact-loop
+ (test (op =) (reg n) (const 1))
+ (branch (label base-case))
+ ;; Set up for the recursive call by saving `n' and `continue'.
+ ;; Set up `continue' so that the computation will continue
+ ;; at `after-fact' when the subroutine returns.
+ (save continue)
+ (save n)
+ (assign n (op -) (reg n) (const 1))
+ (assign continue (label after-fact))
+ (goto (label fact-loop))
+ after-fact
+ (restore n)
+ (restore continue)
+ (assign val (op *) (reg n) (reg val)) ; `val' now contains n(n - 1)!
+ (goto (reg continue)) ; return to caller
+ base-case
+ (assign val (const 1)) ; base case: 1! = 1
+ (goto (reg continue)) ; return to caller
+ fact-done)
+
+
+ *Figure 5.12:* Controller for a machine to compute Fibonacci
+ numbers.
+
+ (controller
+ (assign continue (label fib-done))
+ fib-loop
+ (test (op <) (reg n) (const 2))
+ (branch (label immediate-answer))
+ ;; set up to compute _Fib_(n - 1)
+ (save continue)
+ (assign continue (label afterfib-n-1))
+ (save n) ; save old value of `n'
+ (assign n (op -) (reg n) (const 1)); clobber `n' to n - 1
+ (goto (label fib-loop)) ; perform recursive call
+ afterfib-n-1 ; upon return, `val' contains _Fib_(n - 1)
+ (restore n)
+ (restore continue)
+ ;; set up to compute _Fib_(n - 2)
+ (assign n (op -) (reg n) (const 2))
+ (save continue)
+ (assign continue (label afterfib-n-2))
+ (save val) ; save _Fib_(n - 1)
+ (goto (label fib-loop))
+ afterfib-n-2 ; upon return, `val' contains _Fib_(n - 2)
+ (assign n (reg val)) ; `n' now contains _Fib_(n - 2)
+ (restore val) ; `val' now contains _Fib_(n - 1)
+ (restore continue)
+ (assign val ; _Fib_(n - 1) + _Fib_(n - 2)
+ (op +) (reg val) (reg n))
+ (goto (reg continue)) ; return to caller, answer is in `val'
+ immediate-answer
+ (assign val (reg n)) ; base case: _Fib_(n) = n
+ (goto (reg continue))
+ fib-done)
+
+
+ *Exercise 5.4:* Specify register machines that implement each of
+ the following procedures. For each machine, write a controller
+ instruction sequence and draw a diagram showing the data paths.
+
+ a. Recursive exponentiation:
+
+ (define (expt b n)
+ (if (= n 0)
+ 1
+ (* b (expt b (- n 1)))))
+
+ b. Iterative exponentiation:
+
+ (define (expt b n)
+ (define (expt-iter counter product)
+ (if (= counter 0)
+ product
+ (expt-iter (- counter 1) (* b product))))
+ (expt-iter n 1))
+
+
+ *Exercise 5.5:* Hand-simulate the factorial and Fibonacci
+ machines, using some nontrivial input (requiring execution of at
+ least one recursive call). Show the contents of the stack at each
+ significant point in the execution.
+
+ *Exercise 5.6:* Ben Bitdiddle observes that the Fibonacci
+ machine's controller sequence has an extra `save' and an extra
+ `restore', which can be removed to make a faster machine. Where
+ are these instructions?
+
+ ---------- Footnotes ----------
+
+ (1) One might argue that we don't need to save the old `n'; after we
+decrement it and solve the subproblem, we could simply increment it to
+recover the old value. Although this strategy works for factorial, it
+cannot work in general, since the old value of a register cannot always
+be computed from the new one.
+
+ (2) In section *Note 5-3:: we will see how to implement a stack in
+terms of more primitive operations.
+
+\1f
+File: sicp.info, Node: 5-1-5, Prev: 5-1-4, Up: 5-1
+
+5.1.5 Instruction Summary
+-------------------------
+
+A controller instruction in our register-machine language has one of the
+following forms, where each <INPUT_I> is either `(reg <REGISTER-NAME>)'
+or `(const <CONSTANT-VALUE>)'. These instructions were introduced in
+section *Note 5-1-1:::
+
+ (assign <REGISTER-NAME> (reg <REGISTER-NAME>))
+
+ (assign <REGISTER-NAME> (const <CONSTANT-VALUE>))
+
+ (assign <REGISTER-NAME> (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
+
+ (perform (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
+
+ (test (op <OPERATION-NAME>) <INPUT_1> ... <INPUT_N>)
+
+ (branch (label <LABEL-NAME>))
+
+ (goto (label <LABEL-NAME>))
+
+ The use of registers to hold labels was introduced in section *Note
+5-1-3:::
+
+ (assign <REGISTER-NAME> (label <LABEL-NAME>))
+
+ (goto (reg <REGISTER-NAME>))
+
+ Instructions to use the stack were introduced in section *Note
+5-1-4:::
+
+ (save <REGISTER-NAME>)
+
+ (restore <REGISTER-NAME>)
+
+ The only kind of <CONSTANT-VALUE> we have seen so far is a number,
+but later we will use strings, symbols, and lists. For example,
+`(const "abc")' is the string `"abc"',
+`(const abc)' is the symbol `abc',
+`(const (a b c))' is the list `(a b c)',
+and `(const ())' is the empty list.
+
+\1f
+File: sicp.info, Node: 5-2, Next: 5-3, Prev: 5-1, Up: Chapter 5
+
+5.2 A Register-Machine Simulator
+================================
+
+In order to gain a good understanding of the design of register
+machines, we must test the machines we design to see if they perform as
+expected. One way to test a design is to hand-simulate the operation
+of the controller, as in *Note Exercise 5-5::. But this is extremely
+tedious for all but the simplest machines. In this section we
+construct a simulator for machines described in the register-machine
+language. The simulator is a Scheme program with four interface
+procedures. The first uses a description of a register machine to
+construct a model of the machine (a data structure whose parts
+correspond to the parts of the machine to be simulated), and the other
+three allow us to simulate the machine by manipulating the model:
+
+ (make-machine <REGISTER-NAMES> <OPERATIONS> <CONTROLLER>)
+
+ constructs and returns a model of the machine with the given
+ registers, operations, and controller.
+
+ (set-register-contents! <MACHINE-MODEL> <REGISTER-NAME> <VALUE>)
+
+ stores a value in a simulated register in the given machine.
+
+ (get-register-contents <MACHINE-MODEL> <REGISTER-NAME>)
+
+ returns the contents of a simulated register in the given machine.
+
+ (start <MACHINE-MODEL>)
+
+ simulates the execution of the given machine, starting from the
+ beginning of the controller sequence and stopping when it reaches
+ the end of the sequence.
+
+ As an example of how these procedures are used, we can define
+`gcd-machine' to be a model of the GCD machine of section *Note 5-1-1::
+as follows:
+
+ (define gcd-machine
+ (make-machine
+ '(a b t)
+ (list (list 'rem remainder) (list '= =))
+ '(test-b
+ (test (op =) (reg b) (const 0))
+ (branch (label gcd-done))
+ (assign t (op rem) (reg a) (reg b))
+ (assign a (reg b))
+ (assign b (reg t))
+ (goto (label test-b))
+ gcd-done)))
+
+ The first argument to `make-machine' is a list of register names.
+The next argument is a table (a list of two-element lists) that pairs
+each operation name with a Scheme procedure that implements the
+operation (that is, produces the same output value given the same input
+values). The last argument specifies the controller as a list of
+labels and machine instructions, as in section *Note 5-1::.
+
+ To compute GCDs with this machine, we set the input registers, start
+the machine, and examine the result when the simulation terminates:
+
+ (set-register-contents! gcd-machine 'a 206)
+ done
+
+ (set-register-contents! gcd-machine 'b 40)
+ done
+
+ (start gcd-machine)
+ done
+
+ (get-register-contents gcd-machine 'a)
+ 2
+
+ This computation will run much more slowly than a `gcd' procedure
+written in Scheme, because we will simulate low-level machine
+instructions, such as `assign', by much more complex operations.
+
+ *Exercise 5.7:* Use the simulator to test the machines you
+ designed in *Note Exercise 5-4::.
+
+* Menu:
+
+* 5-2-1:: The Machine Model
+* 5-2-2:: The Assembler
+* 5-2-3:: Generating Execution Procedures for Instructions
+* 5-2-4:: Monitoring Machine Performance
+
+\1f
+File: sicp.info, Node: 5-2-1, Next: 5-2-2, Prev: 5-2, Up: 5-2
+
+5.2.1 The Machine Model
+-----------------------
+
+The machine model generated by `make-machine' is represented as a
+procedure with local state using the message-passing techniques
+developed in *Note Chapter 3::. To build this model, `make-machine'
+begins by calling the procedure `make-new-machine' to construct the
+parts of the machine model that are common to all register machines.
+This basic machine model constructed by `make-new-machine' is
+essentially a container for some registers and a stack, together with
+an execution mechanism that processes the controller instructions one
+by one.
+
+ `Make-machine' then extends this basic model (by sending it
+messages) to include the registers, operations, and controller of the
+particular machine being defined. First it allocates a register in the
+new machine for each of the supplied register names and installs the
+designated operations in the machine. Then it uses an "assembler"
+(described below in section *Note 5-2-2::) to transform the controller
+list into instructions for the new machine and installs these as the
+machine's instruction sequence. `Make-machine' returns as its value
+the modified machine model.
+
+ (define (make-machine register-names ops controller-text)
+ (let ((machine (make-new-machine)))
+ (for-each (lambda (register-name)
+ ((machine 'allocate-register) register-name))
+ register-names)
+ ((machine 'install-operations) ops)
+ ((machine 'install-instruction-sequence)
+ (assemble controller-text machine))
+ machine))
+
+Registers
+.........
+
+We will represent a register as a procedure with local state, as in
+*Note Chapter 3::. The procedure `make-register' creates a register
+that holds a value that can be accessed or changed:
+
+ (define (make-register name)
+ (let ((contents '*unassigned*))
+ (define (dispatch message)
+ (cond ((eq? message 'get) contents)
+ ((eq? message 'set)
+ (lambda (value) (set! contents value)))
+ (else
+ (error "Unknown request -- REGISTER" message))))
+ dispatch))
+
+ The following procedures are used to access registers:
+
+ (define (get-contents register)
+ (register 'get))
+
+ (define (set-contents! register value)
+ ((register 'set) value))
+
+The stack
+.........
+
+We can also represent a stack as a procedure with local state. The
+procedure `make-stack' creates a stack whose local state consists of a
+list of the items on the stack. A stack accepts requests to `push' an
+item onto the stack, to `pop' the top item off the stack and return it,
+and to `initialize' the stack to empty.
+
+ (define (make-stack)
+ (let ((s '()))
+ (define (push x)
+ (set! s (cons x s)))
+ (define (pop)
+ (if (null? s)
+ (error "Empty stack -- POP")
+ (let ((top (car s)))
+ (set! s (cdr s))
+ top)))
+ (define (initialize)
+ (set! s '())
+ 'done)
+ (define (dispatch message)
+ (cond ((eq? message 'push) push)
+ ((eq? message 'pop) (pop))
+ ((eq? message 'initialize) (initialize))
+ (else (error "Unknown request -- STACK"
+ message))))
+ dispatch))
+
+ The following procedures are used to access stacks:
+
+ (define (pop stack)
+ (stack 'pop))
+
+ (define (push stack value)
+ ((stack 'push) value))
+
+The basic machine
+.................
+
+The `make-new-machine' procedure, shown in *Note Figure 5-13::,
+constructs an object whose local state consists of a stack, an
+initially empty instruction sequence, a list of operations that
+initially contains an operation to initialize the stack, and a "register
+table" that initially contains two registers, named `flag' and `pc'
+(for "program counter"). The internal procedure `allocate-register'
+adds new entries to the register table, and the internal procedure
+`lookup-register' looks up registers in the table.
+
+ The `flag' register is used to control branching in the simulated
+machine. `Test' instructions set the contents of `flag' to the result
+of the test (true or false). `Branch' instructions decide whether or
+not to branch by examining the contents of `flag'.
+
+ The `pc' register determines the sequencing of instructions as the
+machine runs. This sequencing is implemented by the internal procedure
+`execute'. In the simulation model, each machine instruction is a data
+structure that includes a procedure of no arguments, called the procedure
+"instruction execution procedure", such that calling this procedure
+simulates executing the instruction. As the simulation runs, `pc'
+points to the place in the instruction sequence beginning with the next
+instruction to be executed. `Execute' gets that instruction, executes
+it by calling the instruction execution procedure, and repeats this
+cycle until there are no more instructions to execute (i.e., until `pc'
+points to the end of the instruction sequence).
+
+ *Figure 5.13:* The `make-new-machine' procedure, which implements
+ the basic machine model.
+
+ (define (make-new-machine)
+ (let ((pc (make-register 'pc))
+ (flag (make-register 'flag))
+ (stack (make-stack))
+ (the-instruction-sequence '()))
+ (let ((the-ops
+ (list (list 'initialize-stack
+ (lambda () (stack 'initialize)))))
+ (register-table
+ (list (list 'pc pc) (list 'flag flag))))
+ (define (allocate-register name)
+ (if (assoc name register-table)
+ (error "Multiply defined register: " name)
+ (set! register-table
+ (cons (list name (make-register name))
+ register-table)))
+ 'register-allocated)
+ (define (lookup-register name)
+ (let ((val (assoc name register-table)))
+ (if val
+ (cadr val)
+ (error "Unknown register:" name))))
+ (define (execute)
+ (let ((insts (get-contents pc)))
+ (if (null? insts)
+ 'done
+ (begin
+ ((instruction-execution-proc (car insts)))
+ (execute)))))
+ (define (dispatch message)
+ (cond ((eq? message 'start)
+ (set-contents! pc the-instruction-sequence)
+ (execute))
+ ((eq? message 'install-instruction-sequence)
+ (lambda (seq) (set! the-instruction-sequence seq)))
+ ((eq? message 'allocate-register) allocate-register)
+ ((eq? message 'get-register) lookup-register)
+ ((eq? message 'install-operations)
+ (lambda (ops) (set! the-ops (append the-ops ops))))
+ ((eq? message 'stack) stack)
+ ((eq? message 'operations) the-ops)
+ (else (error "Unknown request -- MACHINE" message))))
+ dispatch)))
+
+ As part of its operation, each instruction execution procedure
+modifies `pc' to indicate the next instruction to be executed.
+`Branch' and `goto' instructions change `pc' to point to the new
+destination. All other instructions simply advance `pc', making it
+point to the next instruction in the sequence. Observe that each call
+to `execute' calls `execute' again, but this does not produce an
+infinite loop because running the instruction execution procedure
+changes the contents of `pc'.
+
+ `Make-new-machine' returns a `dispatch' procedure that implements
+message-passing access to the internal state. Notice that starting the
+machine is accomplished by setting `pc' to the beginning of the
+instruction sequence and calling `execute'.
+
+ For convenience, we provide an alternate procedural interface to a
+machine's `start' operation, as well as procedures to set and examine
+register contents, as specified at the beginning of section *Note 5-2:::
+
+ (define (start machine)
+ (machine 'start))
+
+ (define (get-register-contents machine register-name)
+ (get-contents (get-register machine register-name)))
+
+ (define (set-register-contents! machine register-name value)
+ (set-contents! (get-register machine register-name) value)
+ 'done)
+
+ These procedures (and many procedures in sections *Note 5-2-2:: and
+*Note 5-2-3::) use the following to look up the register with a given
+name in a given machine:
+
+ (define (get-register machine reg-name)
+ ((machine 'get-register) reg-name))
+
+\1f
+File: sicp.info, Node: 5-2-2, Next: 5-2-3, Prev: 5-2-1, Up: 5-2
+
+5.2.2 The Assembler
+-------------------
+
+The assembler transforms the sequence of controller expressions for a
+machine into a corresponding list of machine instructions, each with
+its execution procedure. Overall, the assembler is much like the
+evaluators we studied in *Note Chapter 4::--there is an input language
+(in this case, the register-machine language) and we must perform an
+appropriate action for each type of expression in the language.
+
+ The technique of producing an execution procedure for each
+instruction is just what we used in section *Note 4-1-7:: to speed up
+the evaluator by separating analysis from runtime execution. As we saw
+in *Note Chapter 4::, much useful analysis of Scheme expressions could
+be performed without knowing the actual values of variables. Here,
+analogously, much useful analysis of register-machine-language
+expressions can be performed without knowing the actual contents of
+machine registers. For example, we can replace references to registers
+by pointers to the register objects, and we can replace references to
+labels by pointers to the place in the instruction sequence that the
+label designates.
+
+ Before it can generate the instruction execution procedures, the
+assembler must know what all the labels refer to, so it begins by
+scanning the controller text to separate the labels from the
+instructions. As it scans the text, it constructs both a list of
+instructions and a table that associates each label with a pointer into
+that list. Then the assembler augments the instruction list by
+inserting the execution procedure for each instruction.
+
+ The `assemble' procedure is the main entry to the assembler. It
+takes the controller text and the machine model as arguments and
+returns the instruction sequence to be stored in the model. `Assemble'
+calls `extract-labels' to build the initial instruction list and label
+table from the supplied controller text. The second argument to
+`extract-labels' is a procedure to be called to process these results:
+This procedure uses `update-insts!' to generate the instruction
+execution procedures and insert them into the instruction list, and
+returns the modified list.
+
+ (define (assemble controller-text machine)
+ (extract-labels controller-text
+ (lambda (insts labels)
+ (update-insts! insts labels machine)
+ insts)))
+
+ `Extract-labels' takes as arguments a list `text' (the sequence of
+controller instruction expressions) and a `receive' procedure.
+`Receive' will be called with two values: (1) a list `insts' of
+instruction data structures, each containing an instruction from `text';
+and (2) a table called `labels', which associates each label from
+`text' with the position in the list `insts' that the label designates.
+
+ (define (extract-labels text receive)
+ (if (null? text)
+ (receive '() '())
+ (extract-labels (cdr text)
+ (lambda (insts labels)
+ (let ((next-inst (car text)))
+ (if (symbol? next-inst)
+ (receive insts
+ (cons (make-label-entry next-inst
+ insts)
+ labels))
+ (receive (cons (make-instruction next-inst)
+ insts)
+ labels)))))))
+
+ `Extract-labels' works by sequentially scanning the elements of the
+`text' and accumulating the `insts' and the `labels'. If an element is
+a symbol (and thus a label) an appropriate entry is added to the
+`labels' table. Otherwise the element is accumulated onto the `insts'
+list.(1)
+
+ `Update-insts!' modifies the instruction list, which initially
+contains only the text of the instructions, to include the
+corresponding execution procedures:
+
+ (define (update-insts! insts labels machine)
+ (let ((pc (get-register machine 'pc))
+ (flag (get-register machine 'flag))
+ (stack (machine 'stack))
+ (ops (machine 'operations)))
+ (for-each
+ (lambda (inst)
+ (set-instruction-execution-proc!
+ inst
+ (make-execution-procedure
+ (instruction-text inst) labels machine
+ pc flag stack ops)))
+ insts)))
+
+ The machine instruction data structure simply pairs the instruction
+text with the corresponding execution procedure. The execution
+procedure is not yet available when `extract-labels' constructs the
+instruction, and is inserted later by `update-insts!'.
+
+ (define (make-instruction text)
+ (cons text '()))
+
+ (define (instruction-text inst)
+ (car inst))
+
+ (define (instruction-execution-proc inst)
+ (cdr inst))
+
+ (define (set-instruction-execution-proc! inst proc)
+ (set-cdr! inst proc))
+
+ The instruction text is not used by our simulator, but it is handy
+to keep around for debugging (see *Note Exercise 5-16::).
+
+ Elements of the label table are pairs:
+
+ (define (make-label-entry label-name insts)
+ (cons label-name insts))
+
+ Entries will be looked up in the table with
+
+ (define (lookup-label labels label-name)
+ (let ((val (assoc label-name labels)))
+ (if val
+ (cdr val)
+ (error "Undefined label -- ASSEMBLE" label-name))))
+
+ *Exercise 5.8:* The following register-machine code is ambiguous,
+ because the label `here' is defined more than once:
+
+ start
+ (goto (label here))
+ here
+ (assign a (const 3))
+ (goto (label there))
+ here
+ (assign a (const 4))
+ (goto (label there))
+ there
+
+ With the simulator as written, what will the contents of register
+ `a' be when control reaches `there'? Modify the `extract-labels'
+ procedure so that the assembler will signal an error if the same
+ label name is used to indicate two different locations.
+
+ ---------- Footnotes ----------
+
+ (1) Using the `receive' procedure here is a way to get
+`extract-labels' to effectively return two values--`labels' and
+`insts'--without explicitly making a compound data structure to hold
+them. An alternative implementation, which returns an explicit pair of
+values, is
+
+ (define (extract-labels text)
+ (if (null? text)
+ (cons '() '())
+ (let ((result (extract-labels (cdr text))))
+ (let ((insts (car result)) (labels (cdr result)))
+ (let ((next-inst (car text)))
+ (if (symbol? next-inst)
+ (cons insts
+ (cons (make-label-entry next-inst insts) labels))
+ (cons (cons (make-instruction next-inst) insts)
+ labels)))))))
+
+which would be called by `assemble' as follows:
+
+ (define (assemble controller-text machine)
+ (let ((result (extract-labels controller-text)))
+ (let ((insts (car result)) (labels (cdr result)))
+ (update-insts! insts labels machine)
+ insts)))
+
+ You can consider our use of `receive' as demonstrating an elegant
+way to return multiple values, or simply an excuse to show off a
+programming trick. An argument like `receive' that is the next
+procedure to be invoked is called a "continuation." Recall that we
+also used continuations to implement the backtracking control structure
+in the `amb' evaluator in section *Note 4-3-3::.
+
+\1f
+File: sicp.info, Node: 5-2-3, Next: 5-2-4, Prev: 5-2-2, Up: 5-2
+
+5.2.3 Generating Execution Procedures for Instructions
+------------------------------------------------------
+
+The assembler calls `make-execution-procedure' to generate the execution
+procedure for an instruction. Like the `analyze' procedure in the
+evaluator of section *Note 4-1-7::, this dispatches on the type of
+instruction to generate the appropriate execution procedure.
+
+ (define (make-execution-procedure inst labels machine
+ pc flag stack ops)
+ (cond ((eq? (car inst) 'assign)
+ (make-assign inst machine labels ops pc))
+ ((eq? (car inst) 'test)
+ (make-test inst machine labels ops flag pc))
+ ((eq? (car inst) 'branch)
+ (make-branch inst machine labels flag pc))
+ ((eq? (car inst) 'goto)
+ (make-goto inst machine labels pc))
+ ((eq? (car inst) 'save)
+ (make-save inst machine stack pc))
+ ((eq? (car inst) 'restore)
+ (make-restore inst machine stack pc))
+ ((eq? (car inst) 'perform)
+ (make-perform inst machine labels ops pc))
+ (else (error "Unknown instruction type -- ASSEMBLE"
+ inst))))
+
+ For each type of instruction in the register-machine language, there
+is a generator that builds an appropriate execution procedure. The
+details of these procedures determine both the syntax and meaning of
+the individual instructions in the register-machine language. We use
+data abstraction to isolate the detailed syntax of register-machine
+expressions from the general execution mechanism, as we did for
+evaluators in section *Note 4-1-2::, by using syntax procedures to
+extract and classify the parts of an instruction.
+
+`Assign' instructions
+.....................
+
+The `make-assign' procedure handles `assign' instructions:
+
+ (define (make-assign inst machine labels operations pc)
+ (let ((target
+ (get-register machine (assign-reg-name inst)))
+ (value-exp (assign-value-exp inst)))
+ (let ((value-proc
+ (if (operation-exp? value-exp)
+ (make-operation-exp
+ value-exp machine labels operations)
+ (make-primitive-exp
+ (car value-exp) machine labels))))
+ (lambda () ; execution procedure for `assign'
+ (set-contents! target (value-proc))
+ (advance-pc pc)))))
+
+ `Make-assign' extracts the target register name (the second element
+of the instruction) and the value expression (the rest of the list that
+forms the instruction) from the `assign' instruction using the selectors
+
+ (define (assign-reg-name assign-instruction)
+ (cadr assign-instruction))
+
+ (define (assign-value-exp assign-instruction)
+ (cddr assign-instruction))
+
+ The register name is looked up with `get-register' to produce the
+target register object. The value expression is passed to
+`make-operation-exp' if the value is the result of an operation, and to
+`make-primitive-exp' otherwise. These procedures (shown below) parse
+the value expression and produce an execution procedure for the value.
+This is a procedure of no arguments, called `value-proc', which will be
+evaluated during the simulation to produce the actual value to be
+assigned to the register. Notice that the work of looking up the
+register name and parsing the value expression is performed just once,
+at assembly time, not every time the instruction is simulated. This
+saving of work is the reason we use execution procedures, and
+corresponds directly to the saving in work we obtained by separating
+program analysis from execution in the evaluator of section *Note
+4-1-7::.
+
+ The result returned by `make-assign' is the execution procedure for
+the `assign' instruction. When this procedure is called (by the machine
+model's `execute' procedure), it sets the contents of the target
+register to the result obtained by executing `value-proc'. Then it
+advances the `pc' to the next instruction by running the procedure
+
+ (define (advance-pc pc)
+ (set-contents! pc (cdr (get-contents pc))))
+
+ `Advance-pc' is the normal termination for all instructions except
+`branch' and `goto'.
+
+`Test', `branch', and `goto' instructions
+.........................................
+
+`Make-test' handles `test' instructions in a similar way. It extracts
+the expression that specifies the condition to be tested and generates
+an execution procedure for it. At simulation time, the procedure for
+the condition is called, the result is assigned to the `flag' register,
+and the `pc' is advanced:
+
+ (define (make-test inst machine labels operations flag pc)
+ (let ((condition (test-condition inst)))
+ (if (operation-exp? condition)
+ (let ((condition-proc
+ (make-operation-exp
+ condition machine labels operations)))
+ (lambda ()
+ (set-contents! flag (condition-proc))
+ (advance-pc pc)))
+ (error "Bad TEST instruction -- ASSEMBLE" inst))))
+
+ (define (test-condition test-instruction)
+ (cdr test-instruction))
+
+ The execution procedure for a `branch' instruction checks the
+contents of the `flag' register and either sets the contents of the
+`pc' to the branch destination (if the branch is taken) or else just
+advances the `pc' (if the branch is not taken). Notice that the
+indicated destination in a `branch' instruction must be a label, and
+the `make-branch' procedure enforces this. Notice also that the label
+is looked up at assembly time, not each time the `branch' instruction
+is simulated.
+
+ (define (make-branch inst machine labels flag pc)
+ (let ((dest (branch-dest inst)))
+ (if (label-exp? dest)
+ (let ((insts
+ (lookup-label labels (label-exp-label dest))))
+ (lambda ()
+ (if (get-contents flag)
+ (set-contents! pc insts)
+ (advance-pc pc))))
+ (error "Bad BRANCH instruction -- ASSEMBLE" inst))))
+
+ (define (branch-dest branch-instruction)
+ (cadr branch-instruction))
+
+ A `goto' instruction is similar to a branch, except that the
+destination may be specified either as a label or as a register, and
+there is no condition to check--the `pc' is always set to the new
+destination.
+
+ (define (make-goto inst machine labels pc)
+ (let ((dest (goto-dest inst)))
+ (cond ((label-exp? dest)
+ (let ((insts
+ (lookup-label labels
+ (label-exp-label dest))))
+ (lambda () (set-contents! pc insts))))
+ ((register-exp? dest)
+ (let ((reg
+ (get-register machine
+ (register-exp-reg dest))))
+ (lambda ()
+ (set-contents! pc (get-contents reg)))))
+ (else (error "Bad GOTO instruction -- ASSEMBLE"
+ inst)))))
+
+ (define (goto-dest goto-instruction)
+ (cadr goto-instruction))
+
+Other instructions
+..................
+
+The stack instructions `save' and `restore' simply use the stack with
+the designated register and advance the `pc':
+
+ (define (make-save inst machine stack pc)
+ (let ((reg (get-register machine
+ (stack-inst-reg-name inst))))
+ (lambda ()
+ (push stack (get-contents reg))
+ (advance-pc pc))))
+
+ (define (make-restore inst machine stack pc)
+ (let ((reg (get-register machine
+ (stack-inst-reg-name inst))))
+ (lambda ()
+ (set-contents! reg (pop stack))
+ (advance-pc pc))))
+
+ (define (stack-inst-reg-name stack-instruction)
+ (cadr stack-instruction))
+
+ The final instruction type, handled by `make-perform', generates an
+execution procedure for the action to be performed. At simulation
+time, the action procedure is executed and the `pc' advanced.
+
+ (define (make-perform inst machine labels operations pc)
+ (let ((action (perform-action inst)))
+ (if (operation-exp? action)
+ (let ((action-proc
+ (make-operation-exp
+ action machine labels operations)))
+ (lambda ()
+ (action-proc)
+ (advance-pc pc)))
+ (error "Bad PERFORM instruction -- ASSEMBLE" inst))))
+
+ (define (perform-action inst) (cdr inst))
+
+Execution procedures for subexpressions
+.......................................
+
+The value of a `reg', `label', or `const' expression may be needed for
+assignment to a register (`make-assign') or for input to an operation
+(`make-operation-exp', below). The following procedure generates
+execution procedures to produce values for these expressions during the
+simulation:
+
+ (define (make-primitive-exp exp machine labels)
+ (cond ((constant-exp? exp)
+ (let ((c (constant-exp-value exp)))
+ (lambda () c)))
+ ((label-exp? exp)
+ (let ((insts
+ (lookup-label labels
+ (label-exp-label exp))))
+ (lambda () insts)))
+ ((register-exp? exp)
+ (let ((r (get-register machine
+ (register-exp-reg exp))))
+ (lambda () (get-contents r))))
+ (else
+ (error "Unknown expression type -- ASSEMBLE" exp))))
+
+ The syntax of `reg', `label', and `const' expressions is determined
+by
+
+ (define (register-exp? exp) (tagged-list? exp 'reg))
+
+ (define (register-exp-reg exp) (cadr exp))
+
+ (define (constant-exp? exp) (tagged-list? exp 'const))
+
+ (define (constant-exp-value exp) (cadr exp))
+
+ (define (label-exp? exp) (tagged-list? exp 'label))
+
+ (define (label-exp-label exp) (cadr exp))
+
+ `Assign', `perform', and `test' instructions may include the
+application of a machine operation (specified by an `op' expression) to
+some operands (specified by `reg' and `const' expressions). The
+following procedure produces an execution procedure for an "operation
+expression"--a list containing the operation and operand expressions
+from the instruction:
+
+ (define (make-operation-exp exp machine labels operations)
+ (let ((op (lookup-prim (operation-exp-op exp) operations))
+ (aprocs
+ (map (lambda (e)
+ (make-primitive-exp e machine labels))
+ (operation-exp-operands exp))))
+ (lambda ()
+ (apply op (map (lambda (p) (p)) aprocs)))))
+
+ The syntax of operation expressions is determined by
+
+ (define (operation-exp? exp)
+ (and (pair? exp) (tagged-list? (car exp) 'op)))
+
+ (define (operation-exp-op operation-exp)
+ (cadr (car operation-exp)))
+
+ (define (operation-exp-operands operation-exp)
+ (cdr operation-exp))
+
+ Observe that the treatment of operation expressions is very much
+like the treatment of procedure applications by the
+`analyze-application' procedure in the evaluator of section *Note
+4-1-7:: in that we generate an execution procedure for each operand.
+At simulation time, we call the operand procedures and apply the Scheme
+procedure that simulates the operation to the resulting values. The
+simulation procedure is found by looking up the operation name in the
+operation table for the machine:
+
+ (define (lookup-prim symbol operations)
+ (let ((val (assoc symbol operations)))
+ (if val
+ (cadr val)
+ (error "Unknown operation -- ASSEMBLE" symbol))))
+
+ *Exercise 5.9:* The treatment of machine operations above permits
+ them to operate on labels as well as on constants and the contents
+ of registers. Modify the expression-processing procedures to
+ enforce the condition that operations can be used only with
+ registers and constants.
+
+ *Exercise 5.10:* Design a new syntax for register-machine
+ instructions and modify the simulator to use your new syntax. Can
+ you implement your new syntax without changing any part of the
+ simulator except the syntax procedures in this section?
+
+ *Exercise 5.11:* When we introduced `save' and `restore' in
+ section *Note 5-1-4::, we didn't specify what would happen if you
+ tried to restore a register that was not the last one saved, as in
+ the sequence
+
+ (save y)
+ (save x)
+ (restore y)
+
+ There are several reasonable possibilities for the meaning of
+ `restore':
+
+ a. `(restore y)' puts into `y' the last value saved on the stack,
+ regardless of what register that value came from. This is
+ the way our simulator behaves. Show how to take advantage of
+ this behavior to eliminate one instruction from the Fibonacci
+ machine of section *Note 5-1-4:: (*Note Figure 5-12::).
+
+ b. `(restore y)' puts into `y' the last value saved on the
+ stack, but only if that value was saved from `y'; otherwise,
+ it signals an error. Modify the simulator to behave this
+ way. You will have to change `save' to put the register name
+ on the stack along with the value.
+
+ c. `(restore y)' puts into `y' the last value saved from `y'
+ regardless of what other registers were saved after `y' and
+ not restored. Modify the simulator to behave this way. You
+ will have to associate a separate stack with each register.
+ You should make the `initialize-stack' operation initialize
+ all the register stacks.
+
+
+ *Exercise 5.12:* The simulator can be used to help determine the
+ data paths required for implementing a machine with a given
+ controller. Extend the assembler to store the following
+ information in the machine model:
+
+ * a list of all instructions, with duplicates removed, sorted
+ by instruction type (`assign', `goto', and so on);
+
+ * a list (without duplicates) of the registers used to hold
+ entry points (these are the registers referenced by `goto'
+ instructions);
+
+ * a list (without duplicates) of the registers that are `save'd
+ or `restore'd;
+
+ * for each register, a list (without duplicates) of the sources
+ from which it is assigned (for example, the sources for
+ register `val' in the factorial machine of *Note Figure
+ 5-11:: are `(const 1)' and `((op *) (reg n) (reg val))').
+
+
+ Extend the message-passing interface to the machine to provide
+ access to this new information. To test your analyzer, define the
+ Fibonacci machine from *Note Figure 5-12:: and examine the lists
+ you constructed.
+
+ *Exercise 5.13:* Modify the simulator so that it uses the
+ controller sequence to determine what registers the machine has
+ rather than requiring a list of registers as an argument to
+ `make-machine'. Instead of pre-allocating the registers in
+ `make-machine', you can allocate them one at a time when they are
+ first seen during assembly of the instructions.
+
+\1f
+File: sicp.info, Node: 5-2-4, Prev: 5-2-3, Up: 5-2
+
+5.2.4 Monitoring Machine Performance
+------------------------------------
+
+Simulation is useful not only for verifying the correctness of a
+proposed machine design but also for measuring the machine's
+performance. For example, we can install in our simulation program a
+"meter" that measures the number of stack operations used in a
+computation. To do this, we modify our simulated stack to keep track
+of the number of times registers are saved on the stack and the maximum
+depth reached by the stack, and add a message to the stack's interface
+that prints the statistics, as shown below. We also add an operation
+to the basic machine model to print the stack statistics, by
+initializing `the-ops' in `make-new-machine' to
+
+ (list (list 'initialize-stack
+ (lambda () (stack 'initialize)))
+ (list 'print-stack-statistics
+ (lambda () (stack 'print-statistics))))
+
+ Here is the new version of `make-stack':
+
+ (define (make-stack)
+ (let ((s '())
+ (number-pushes 0)
+ (max-depth 0)
+ (current-depth 0))
+ (define (push x)
+ (set! s (cons x s))
+ (set! number-pushes (+ 1 number-pushes))
+ (set! current-depth (+ 1 current-depth))
+ (set! max-depth (max current-depth max-depth)))
+ (define (pop)
+ (if (null? s)
+ (error "Empty stack -- POP")
+ (let ((top (car s)))
+ (set! s (cdr s))
+ (set! current-depth (- current-depth 1))
+ top)))
+ (define (initialize)
+ (set! s '())
+ (set! number-pushes 0)
+ (set! max-depth 0)
+ (set! current-depth 0)
+ 'done)
+ (define (print-statistics)
+ (newline)
+ (display (list 'total-pushes '= number-pushes
+ 'maximum-depth '= max-depth)))
+ (define (dispatch message)
+ (cond ((eq? message 'push) push)
+ ((eq? message 'pop) (pop))
+ ((eq? message 'initialize) (initialize))
+ ((eq? message 'print-statistics)
+ (print-statistics))
+ (else
+ (error "Unknown request -- STACK" message))))
+ dispatch))
+
+ *Note Exercise 5-15:: through *Note Exercise 5-19:: describe other
+useful monitoring and debugging features that can be added to the
+register-machine simulator.
+
+ *Exercise 5.14:* Measure the number of pushes and the maximum
+ stack depth required to compute n! for various small values of n
+ using the factorial machine shown in *Note Figure 5-11::. From
+ your data determine formulas in terms of n for the total number of
+ push operations and the maximum stack depth used in computing n!
+ for any n > 1. Note that each of these is a linear function of n
+ and is thus determined by two constants. In order to get the
+ statistics printed, you will have to augment the factorial machine
+ with instructions to initialize the stack and print the
+ statistics. You may want to also modify the machine so that it
+ repeatedly reads a value for n, computes the factorial, and prints
+ the result (as we did for the GCD machine in *Note Figure 5-4::),
+ so that you will not have to repeatedly invoke
+ `get-register-contents', `set-register-contents!', and `start'.
+
+ *Exercise 5.15:* Add counting "instruction counting" to the
+ register machine simulation. That is, have the machine model keep
+ track of the number of instructions executed. Extend the machine
+ model's interface to accept a new message that prints the value of
+ the instruction count and resets the count to zero.
+
+ *Exercise 5.16:* Augment the simulator to provide for "instruction
+ tracing". That is, before each instruction is executed, the
+ simulator should print the text of the instruction. Make the
+ machine model accept `trace-on' and `trace-off' messages to turn
+ tracing on and off.
+
+ *Exercise 5.17:* Extend the instruction tracing of *Note Exercise
+ 5-16:: so that before printing an instruction, the simulator
+ prints any labels that immediately precede that instruction in the
+ controller sequence. Be careful to do this in a way that does not
+ interfere with instruction counting (*Note Exercise 5-15::). You
+ will have to make the simulator retain the necessary label
+ information.
+
+ *Exercise 5.18:* Modify the `make-register' procedure of section
+ *Note 5-2-1:: so that registers can be traced. Registers should
+ accept messages that turn tracing on and off. When a register is
+ traced, assigning a value to the register should print the name of
+ the register, the old contents of the register, and the new
+ contents being assigned. Extend the interface to the machine
+ model to permit you to turn tracing on and off for designated
+ machine registers.
+
+ *Exercise 5.19:* Alyssa P. Hacker wants a "breakpoint" feature in
+ the simulator to help her debug her machine designs. You have
+ been hired to install this feature for her. She wants to be able
+ to specify a place in the controller sequence where the simulator
+ will stop and allow her to examine the state of the machine. You
+ are to implement a procedure
+
+ (set-breakpoint <MACHINE> <LABEL> <N>)
+
+ that sets a breakpoint just before the nth instruction after the
+ given label. For example,
+
+ (set-breakpoint gcd-machine 'test-b 4)
+
+ installs a breakpoint in `gcd-machine' just before the assignment
+ to register `a'. When the simulator reaches the breakpoint it
+ should print the label and the offset of the breakpoint and stop
+ executing instructions. Alyssa can then use
+ `get-register-contents' and `set-register-contents!' to manipulate
+ the state of the simulated machine. She should then be able to
+ continue execution by saying
+
+ (proceed-machine <MACHINE>)
+
+ She should also be able to remove a specific breakpoint by means of
+
+ (cancel-breakpoint <MACHINE> <LABEL> <N>)
+
+ or to remove all breakpoints by means of
+
+ (cancel-all-breakpoints <MACHINE>)
+
+\1f
+File: sicp.info, Node: 5-3, Next: 5-4, Prev: 5-2, Up: Chapter 5
+
+5.3 Storage Allocation and Garbage Collection
+=============================================
+
+In section *Note 5-4::, we will show how to implement a Scheme
+evaluator as a register machine. In order to simplify the discussion,
+we will assume that our register machines can be equipped with a "list-structured
+memory", in which the basic operations for manipulating list-structured
+data are primitive. Postulating the existence of such a memory is a
+useful abstraction when one is focusing on the mechanisms of control in
+a Scheme interpreter, but this does not reflect a realistic view of the
+actual primitive data operations of contemporary computers. To obtain
+a more complete picture of how a Lisp system operates, we must
+investigate how list structure can be represented in a way that is
+compatible with conventional computer memories.
+
+ There are two considerations in implementing list structure. The
+first is purely an issue of representation: how to represent the
+"box-and-pointer" structure of Lisp pairs, using only the storage and
+addressing capabilities of typical computer memories. The second issue
+concerns the management of memory as a computation proceeds. The
+operation of a Lisp system depends crucially on the ability to
+continually create new data objects. These include objects that are
+explicitly created by the Lisp procedures being interpreted as well as
+structures created by the interpreter itself, such as environments and
+argument lists. Although the constant creation of new data objects
+would pose no problem on a computer with an infinite amount of rapidly
+addressable memory, computer memories are available only in finite
+sizes (more's the pity). Lisp systems thus provide an "automatic
+storage allocation" facility to support the illusion of an infinite
+memory. When a data object is no longer needed, the memory allocated
+to it is automatically recycled and used to construct new data objects.
+There are various techniques for providing such automatic storage
+allocation. The method we shall discuss in this section is called "garbage
+collection".
+
+* Menu:
+
+* 5-3-1:: Memory as Vectors
+* 5-3-2:: Maintaining the Illusion of Infinite Memory
+
+\1f
+File: sicp.info, Node: 5-3-1, Next: 5-3-2, Prev: 5-3, Up: 5-3
+
+5.3.1 Memory as Vectors
+-----------------------
+
+A conventional computer memory can be thought of as an array of
+cubbyholes, each of which can contain a piece of information. Each
+cubbyhole has a unique name, called its "address" or "location".
+Typical memory systems provide two primitive operations: one that
+fetches the data stored in a specified location and one that assigns
+new data to a specified location. Memory addresses can be incremented
+to support sequential access to some set of the cubbyholes. More
+generally, many important data operations require that memory addresses
+be treated as data, which can be stored in memory locations and
+manipulated in machine registers. The representation of list structure
+is one application of such "address arithmetic".
+
+ To model computer memory, we use a new kind of data structure called
+a "vector". Abstractly, a vector is a compound data object whose
+individual elements can be accessed by means of an integer index in an
+amount of time that is independent of the index.(1) In order to
+describe memory operations, we use two primitive Scheme procedures for
+manipulating vectors:
+
+ * `(vector-ref <VECTOR> <N>)' returns the nth element of the vector.
+
+ * `(vector-set! <VECTOR> <N> <VALUE>)' sets the nth element of the
+ vector to the designated value.
+
+
+ For example, if `v' is a vector, then `(vector-ref v 5)' gets the
+fifth entry in the vector `v' and `(vector-set! v 5 7)' changes the
+value of the fifth entry of the vector `v' to 7.(2) For computer
+memory, this access can be implemented through the use of address
+arithmetic to combine a address "base address" that specifies the
+beginning location of a vector in memory with an "index" that specifies
+the offset of a particular element of the vector.
+
+Representing Lisp data
+......................
+
+We can use vectors to implement the basic pair structures required for a
+list-structured memory. Let us imagine that computer memory is divided
+into two vectors: `the-cars' and `the-cdrs'. We will represent list
+structure as follows: A pointer to a pair is an index into the two
+vectors. The `car' of the pair is the entry in `the-cars' with the
+designated index, and the `cdr' of the pair is the entry in `the-cdrs'
+with the designated index. We also need a representation for objects
+other than pairs (such as numbers and symbols) and a way to distinguish
+one kind of data from another. There are many methods of accomplishing
+this, but they all reduce to using "typed pointers", that is, to
+extending the notion of "pointer" to include information on data
+type.(3) The data type enables the system to distinguish a pointer to a
+pair (which consists of the "pair" data type and an index into the
+memory vectors) from pointers to other kinds of data (which consist of
+some other data type and whatever is being used to represent data of
+that type). Two data objects are considered to be the same (`eq?') if
+their pointers are identical.(4) *Note Figure 5-14:: illustrates the
+use of this method to represent the list `((1 2) 3 4)', whose
+box-and-pointer diagram is also shown. We use letter prefixes to
+denote the data-type information. Thus, a pointer to the pair with
+index 5 is denoted `p5', the empty list is denoted by the pointer `e0',
+and a pointer to the number 4 is denoted `n4'. In the box-and-pointer
+diagram, we have indicated at the lower left of each pair the vector
+index that specifies where the `car' and `cdr' of the pair are stored.
+The blank locations in `the-cars' and `the-cdrs' may contain parts of
+other list structures (not of interest here).
+
+ *Figure 5.14:* Box-and-pointer and memory-vector representations
+ of the list `((1 2) 3 4)'.
+
+ +---+---+ +---+---+ +---+---+
+ ((1 2) 3 4) -->| * | *-+-------------->| * | *-+--->| * | / |
+ +-|-+---+ +-|-+---+ +-|-+---+
+ 1 | 2 | 4 |
+ V V V
+ +---+---+ +---+---+ +---+ +---+
+ | * | *-+--->| * | / | | 3 | | 4 |
+ +-|-+---+ +-|-+---+ +---+ +---+
+ 5 | 7 |
+ V V
+ +---+ +---+
+ | 1 | | 2 |
+ +---+ +---+
+
+ Index 0 1 2 3 4 5 6 7 8 ...
+ +----+----+----+----+----+----+----+----+----+----
+ the-cars | | p5 | n3 | | n4 | n1 | | n2 | | ...
+ +----+----+----+----+----+----+----+----+----+----
+ the-cdrs | | p2 | p4 | | e0 | p7 | | e0 | | ...
+ +----+----+----+----+----+----+----+----+----+----
+
+ A pointer to a number, such as `n4', might consist of a type
+indicating numeric data together with the actual representation of the
+number 4.(5) To deal with numbers that are too large to be represented
+in the fixed amount of space allocated for a single pointer, we could
+use a distinct "bignum" data type, for which the pointer designates a
+list in which the parts of the number are stored.(6)
+
+ A symbol might be represented as a typed pointer that designates a
+sequence of the characters that form the symbol's printed
+representation. This sequence is constructed by the Lisp reader when
+the character string is initially encountered in input. Since we want
+two instances of a symbol to be recognized as the "same" symbol by
+`eq?' and we want `eq?' to be a simple test for equality of pointers,
+we must ensure that if the reader sees the same character string twice,
+it will use the same pointer (to the same sequence of characters) to
+represent both occurrences. To accomplish this, the reader maintains a
+table, traditionally called the "obarray", of all the symbols it has
+ever encountered. When the reader encounters a character string and is
+about to construct a symbol, it checks the obarray to see if it has ever
+before seen the same character string. If it has not, it uses the
+characters to construct a new symbol (a typed pointer to a new
+character sequence) and enters this pointer in the obarray. If the
+reader has seen the string before, it returns the symbol pointer stored
+in the obarray. This process of replacing character strings by unique
+pointers is called "interning" symbols.
+
+Implementing the primitive list operations
+..........................................
+
+Given the above representation scheme, we can replace each "primitive"
+list operation of a register machine with one or more primitive vector
+operations. We will use two registers, `the-cars' and `the-cdrs', to
+identify the memory vectors, and will assume that `vector-ref' and
+`vector-set!' are available as primitive operations. We also assume
+that numeric operations on pointers (such as incrementing a pointer,
+using a pair pointer to index a vector, or adding two numbers) use only
+the index portion of the typed pointer.
+
+ For example, we can make a register machine support the instructions
+
+ (assign <REG_1> (op car) (reg <REG_2>))
+
+ (assign <REG_1> (op cdr) (reg <REG_2>))
+
+if we implement these, respectively, as
+
+ (assign <REG_1> (op vector-ref) (reg the-cars) (reg <REG_2>))
+
+ (assign <REG_1> (op vector-ref) (reg the-cdrs) (reg <REG_2>))
+
+ The instructions
+
+ (perform (op set-car!) (reg <REG_1>) (reg <REG_2>))
+
+ (perform (op set-cdr!) (reg <REG_1>) (reg <REG_2>))
+
+are implemented as
+
+ (perform
+ (op vector-set!) (reg the-cars) (reg <REG_1>) (reg <REG_2>))
+
+ (perform
+ (op vector-set!) (reg the-cdrs) (reg <REG_1>) (reg <REG_2>))
+
+ `Cons' is performed by allocating an unused index and storing the
+arguments to `cons' in `the-cars' and `the-cdrs' at that indexed vector
+position. We presume that there is a special register, `free', that
+always holds a pair pointer containing the next available index, and
+that we can increment the index part of that pointer to find the next
+free location.(7) For example, the instruction
+
+ (assign <REG_1> (op cons) (reg <REG_2>) (reg <REG_3>))
+
+is implemented as the following sequence of vector operations:(8)
+
+ (perform
+ (op vector-set!) (reg the-cars) (reg free) (reg <REG_2>))
+ (perform
+ (op vector-set!) (reg the-cdrs) (reg free) (reg <REG_3>))
+ (assign <REG_1> (reg free))
+ (assign free (op +) (reg free) (const 1))
+
+ The `eq?' operation
+
+ (op eq?) (reg <REG_1>) (reg <REG_2>)
+
+simply tests the equality of all fields in the registers, and
+predicates such as `pair?', `null?', `symbol?', and `number?' need only
+check the type field.
+
+Implementing stacks
+...................
+
+Although our register machines use stacks, we need do nothing special
+here, since stacks can be modeled in terms of lists. The stack can be
+a list of the saved values, pointed to by a special register
+`the-stack'. Thus, ` (save <REG>)' can be implemented as
+
+ (assign the-stack (op cons) (reg <REG>) (reg the-stack))
+
+Similarly, `(restore <REG>)' can be implemented as
+
+ (assign <REG> (op car) (reg the-stack))
+ (assign the-stack (op cdr) (reg the-stack))
+
+and `(perform (op initialize-stack))' can be implemented as
+
+ (assign the-stack (const ()))
+
+ These operations can be further expanded in terms of the vector
+operations given above. In conventional computer architectures,
+however, it is usually advantageous to allocate the stack as a separate
+vector. Then pushing and popping the stack can be accomplished by
+incrementing or decrementing an index into that vector.
+
+ *Exercise 5.20:* Draw the box-and-pointer representation and the
+ memory-vector representation (as in *Note Figure 5-14::) of the
+ list structure produced by
+
+ (define x (cons 1 2))
+ (define y (list x x))
+
+ with the `free' pointer initially `p1'. What is the final value of
+ `free' ? What pointers represent the values of `x' and `y' ?
+
+ *Exercise 5.21:* Implement register machines for the following
+ procedures. Assume that the list-structure memory operations are
+ available as machine primitives.
+
+ a. Recursive `count-leaves':
+
+ (define (count-leaves tree)
+ (cond ((null? tree) 0)
+ ((not (pair? tree)) 1)
+ (else (+ (count-leaves (car tree))
+ (count-leaves (cdr tree))))))
+
+ b. Recursive `count-leaves' with explicit counter:
+
+ (define (count-leaves tree)
+ (define (count-iter tree n)
+ (cond ((null? tree) n)
+ ((not (pair? tree)) (+ n 1))
+ (else (count-iter (cdr tree)
+ (count-iter (car tree) n)))))
+ (count-iter tree 0))
+
+ *Exercise 5.22:* *Note Exercise 3-12:: of section *Note 3-3-1::
+ presented an `append' procedure that appends two lists to form a
+ new list and an `append!' procedure that splices two lists
+ together. Design a register machine to implement each of these
+ procedures. Assume that the list-structure memory operations are
+ available as primitive operations.
+
+ ---------- Footnotes ----------
+
+ (1) We could represent memory as lists of items. However, the
+access time would then not be independent of the index, since accessing
+the nth element of a list requires n - 1 `cdr' operations.
+
+ (2) For completeness, we should specify a `make-vector' operation
+that constructs vectors. However, in the present application we will
+use vectors only to model fixed divisions of the computer memory.
+
+ (3) This is precisely the same "tagged data" idea we introduced in
+*Note Chapter 2:: for dealing with generic operations. Here, however,
+the data types are included at the primitive machine level rather than
+constructed through the use of lists.
+
+ (4) Type information may be encoded in a variety of ways, depending
+on the details of the machine on which the Lisp system is to be
+implemented. The execution efficiency of Lisp programs will be
+strongly dependent on how cleverly this choice is made, but it is
+difficult to formulate general design rules for good choices. The most
+straightforward way to implement typed pointers is to allocate a fixed
+set of bits in each pointer to be a "type field" that encodes the data
+type. Important questions to be addressed in designing such a
+representation include the following: How many type bits are required?
+How large must the vector indices be? How efficiently can the
+primitive machine instructions be used to manipulate the type fields of
+pointers? Machines that include special hardware for the efficient
+handling of type fields are said to have architectures "tagged
+architectures".
+
+ (5) This decision on the representation of numbers determines whether
+`eq?', which tests equality of pointers, can be used to test for
+equality of numbers. If the pointer contains the number itself, then
+equal numbers will have the same pointer. But if the pointer contains
+the index of a location where the number is stored, equal numbers will
+be guaranteed to have equal pointers only if we are careful never to
+store the same number in more than one location.
+
+ (6) This is just like writing a number as a sequence of digits,
+except that each "digit" is a number between 0 and the largest number
+that can be stored in a single pointer.
+
+ (7) There are other ways of finding free storage. For example, we
+could link together all the unused pairs into a "free list". Our free
+locations are consecutive (and hence can be accessed by incrementing a
+pointer) because we are using a compacting garbage collector, as we
+will see in section *Note 5-3-2::.
+
+ (8) This is essentially the implementation of `cons' in terms of
+`set-car!' and `set-cdr!', as described in section *Note 3-3-1::. The
+operation `get-new-pair' used in that implementation is realized here
+by the `free' pointer.
+
+\1f
+File: sicp.info, Node: 5-3-2, Prev: 5-3-1, Up: 5-3
+
+5.3.2 Maintaining the Illusion of Infinite Memory
+-------------------------------------------------
+
+The representation method outlined in section *Note 5-3-1:: solves the
+problem of implementing list structure, provided that we have an
+infinite amount of memory. With a real computer we will eventually run
+out of free space in which to construct new pairs.(1) However, most of
+the pairs generated in a typical computation are used only to hold
+intermediate results. After these results are accessed, the pairs are
+no longer needed--they are "garbage". For instance, the computation
+
+ (accumulate + 0 (filter odd? (enumerate-interval 0 n)))
+
+constructs two lists: the enumeration and the result of filtering the
+enumeration. When the accumulation is complete, these lists are no
+longer needed, and the allocated memory can be reclaimed. If we can
+arrange to collect all the garbage periodically, and if this turns out
+to recycle memory at about the same rate at which we construct new
+pairs, we will have preserved the illusion that there is an infinite
+amount of memory.
+
+ In order to recycle pairs, we must have a way to determine which
+allocated pairs are not needed (in the sense that their contents can no
+longer influence the future of the computation). The method we shall
+examine for accomplishing this is known as "garbage collection".
+Garbage collection is based on the observation that, at any moment in a
+Lisp interpretation, the only objects that can affect the future of the
+computation are those that can be reached by some succession of `car'
+and `cdr' operations starting from the pointers that are currently in
+the machine registers.(2) Any memory cell that is not so accessible may
+be recycled.
+
+ There are many ways to perform garbage collection. The method we
+shall examine here is called "stop-and-copy". The basic idea is to
+divide memory into two halves: "working memory" and "free memory."
+When `cons' constructs pairs, it allocates these in working memory.
+When working memory is full, we perform garbage collection by locating
+all the useful pairs in working memory and copying these into
+consecutive locations in free memory. (The useful pairs are located by
+tracing all the `car' and `cdr' pointers, starting with the machine
+registers.) Since we do not copy the garbage, there will presumably be
+additional free memory that we can use to allocate new pairs. In
+addition, nothing in the working memory is needed, since all the useful
+pairs in it have been copied. Thus, if we interchange the roles of
+working memory and free memory, we can continue processing; new pairs
+will be allocated in the new working memory (which was the old free
+memory). When this is full, we can copy the useful pairs into the new
+free memory (which was the old working memory).(3)
+
+Implementation of a stop-and-copy garbage collector
+...................................................
+
+We now use our register-machine language to describe the stop-and-copy
+algorithm in more detail. We will assume that there is a register
+called `root' that contains a pointer to a structure that eventually
+points at all accessible data. This can be arranged by storing the
+contents of all the machine registers in a pre-allocated list pointed
+at by `root' just before starting garbage collection.(4) We also assume
+that, in addition to the current working memory, there is free memory
+available into which we can copy the useful data. The current working
+memory consists of vectors whose base addresses are in registers called
+`the-cars' and `the-cdrs', and the free memory is in registers called
+`new-cars' and `new-cdrs'.
+
+ Garbage collection is triggered when we exhaust the free cells in
+the current working memory, that is, when a `cons' operation attempts
+to increment the `free' pointer beyond the end of the memory vector.
+When the garbage-collection process is complete, the `root' pointer
+will point into the new memory, all objects accessible from the `root'
+will have been moved to the new memory, and the `free' pointer will
+indicate the next place in the new memory where a new pair can be
+allocated. In addition, the roles of working memory and new memory
+will have been interchanged--new pairs will be constructed in the new
+memory, beginning at the place indicated by `free', and the (previous)
+working memory will be available as the new memory for the next garbage
+collection. *Note Figure 5-15:: shows the arrangement of memory just
+before and just after garbage collection.
+
+ *Figure 5.15:* Reconfiguration of memory by the garbage-collection
+ process.
+
+ Just before garbage collection
+
+ +------------------------------------+
+ the-cars | | working
+ | mixture of useful data and garbage | memory
+ the-cdrs | |
+ +------------------------------------+
+ ^
+ | free
+
+ +------------------------------------+
+ new-cars | | free
+ | free memory | memory
+ new-cdrs | |
+ +------------------------------------+
+
+ Just after garbage collection
+
+ +------------------------------------+
+ new-cars | | new
+ | discarded memory | free
+ new-cdrs | | memory
+ +------------------------------------+
+
+ +------------------+-----------------+
+ the-cars | | | new
+ | useful data | free area | working
+ the-cdrs | | | memory
+ +------------------+-----------------+
+ ^
+ | free
+
+ The state of the garbage-collection process is controlled by
+maintaining two pointers: `free' and `scan'. These are initialized to
+point to the beginning of the new memory. The algorithm begins by
+relocating the pair pointed at by `root' to the beginning of the new
+memory. The pair is copied, the `root' pointer is adjusted to point to
+the new location, and the `free' pointer is incremented. In addition,
+the old location of the pair is marked to show that its contents have
+been moved. This marking is done as follows: In the `car' position, we
+place a special tag that signals that this is an already-moved object.
+(Such an object is traditionally called a "broken heart".)(5) In the
+`cdr' position we place a "forwarding address" that points at the
+location to which the object has been moved.
+
+ After relocating the root, the garbage collector enters its basic
+cycle. At each step in the algorithm, the `scan' pointer (initially
+pointing at the relocated root) points at a pair that has been moved to
+the new memory but whose `car' and `cdr' pointers still refer to
+objects in the old memory. These objects are each relocated, and the
+`scan' pointer is incremented. To relocate an object (for example, the
+object indicated by the `car' pointer of the pair we are scanning) we
+check to see if the object has already been moved (as indicated by the
+presence of a broken-heart tag in the `car' position of the object).
+If the object has not already been moved, we copy it to the place
+indicated by `free', update `free', set up a broken heart at the
+object's old location, and update the pointer to the object (in this
+example, the `car' pointer of the pair we are scanning) to point to the
+new location. If the object has already been moved, its forwarding
+address (found in the `cdr' position of the broken heart) is
+substituted for the pointer in the pair being scanned. Eventually, all
+accessible objects will have been moved and scanned, at which point the
+`scan' pointer will overtake the `free' pointer and the process will
+terminate.
+
+ We can specify the stop-and-copy algorithm as a sequence of
+instructions for a register machine. The basic step of relocating an
+object is accomplished by a subroutine called
+`relocate-old-result-in-new'. This subroutine gets its argument, a
+pointer to the object to be relocated, from a register named `old'. It
+relocates the designated object (incrementing `free' in the process),
+puts a pointer to the relocated object into a register called `new',
+and returns by branching to the entry point stored in the register
+`relocate-continue'. To begin garbage collection, we invoke this
+subroutine to relocate the `root' pointer, after initializing `free'
+and `scan'. When the relocation of `root' has been accomplished, we
+install the new pointer as the new `root' and enter the main loop of the
+garbage collector.
+
+ begin-garbage-collection
+ (assign free (const 0))
+ (assign scan (const 0))
+ (assign old (reg root))
+ (assign relocate-continue (label reassign-root))
+ (goto (label relocate-old-result-in-new))
+ reassign-root
+ (assign root (reg new))
+ (goto (label gc-loop))
+
+ In the main loop of the garbage collector we must determine whether
+there are any more objects to be scanned. We do this by testing
+whether the `scan' pointer is coincident with the `free' pointer. If
+the pointers are equal, then all accessible objects have been
+relocated, and we branch to `gc-flip', which cleans things up so that
+we can continue the interrupted computation. If there are still pairs
+to be scanned, we call the relocate subroutine to relocate the `car' of
+the next pair (by placing the `car' pointer in `old'). The
+`relocate-continue' register is set up so that the subroutine will
+return to update the `car' pointer.
+
+ gc-loop
+ (test (op =) (reg scan) (reg free))
+ (branch (label gc-flip))
+ (assign old (op vector-ref) (reg new-cars) (reg scan))
+ (assign relocate-continue (label update-car))
+ (goto (label relocate-old-result-in-new))
+
+ At `update-car', we modify the `car' pointer of the pair being
+scanned, then proceed to relocate the `cdr' of the pair. We return to
+`update-cdr' when that relocation has been accomplished. After
+relocating and updating the `cdr', we are finished scanning that pair,
+so we continue with the main loop.
+
+ update-car
+ (perform
+ (op vector-set!) (reg new-cars) (reg scan) (reg new))
+ (assign old (op vector-ref) (reg new-cdrs) (reg scan))
+ (assign relocate-continue (label update-cdr))
+ (goto (label relocate-old-result-in-new))
+
+ update-cdr
+ (perform
+ (op vector-set!) (reg new-cdrs) (reg scan) (reg new))
+ (assign scan (op +) (reg scan) (const 1))
+ (goto (label gc-loop))
+
+ The subroutine `relocate-old-result-in-new' relocates objects as
+follows: If the object to be relocated (pointed at by `old') is not a
+pair, then we return the same pointer to the object unchanged (in
+`new'). (For example, we may be scanning a pair whose `car' is the
+number 4. If we represent the `car' by `n4', as described in section
+*Note 5-3-1::, then we want the "relocated" `car' pointer to still be
+`n4'.) Otherwise, we must perform the relocation. If the `car'
+position of the pair to be relocated contains a broken-heart tag, then
+the pair has in fact already been moved, so we retrieve the forwarding
+address (from the `cdr' position of the broken heart) and return this
+in `new'. If the pointer in `old' points at a yet-unmoved pair, then
+we move the pair to the first free cell in new memory (pointed at by
+`free') and set up the broken heart by storing a broken-heart tag and
+forwarding address at the old location. `Relocate-old-result-in-new'
+uses a register `oldcr' to hold the `car' or the `cdr' of the object
+pointed at by `old'.(6)
+
+ relocate-old-result-in-new
+ (test (op pointer-to-pair?) (reg old))
+ (branch (label pair))
+ (assign new (reg old))
+ (goto (reg relocate-continue))
+ pair
+ (assign oldcr (op vector-ref) (reg the-cars) (reg old))
+ (test (op broken-heart?) (reg oldcr))
+ (branch (label already-moved))
+ (assign new (reg free)) ; new location for pair
+ ;; Update `free' pointer.
+ (assign free (op +) (reg free) (const 1))
+ ;; Copy the `car' and `cdr' to new memory.
+ (perform (op vector-set!)
+ (reg new-cars) (reg new) (reg oldcr))
+ (assign oldcr (op vector-ref) (reg the-cdrs) (reg old))
+ (perform (op vector-set!)
+ (reg new-cdrs) (reg new) (reg oldcr))
+ ;; Construct the broken heart.
+ (perform (op vector-set!)
+ (reg the-cars) (reg old) (const broken-heart))
+ (perform
+ (op vector-set!) (reg the-cdrs) (reg old) (reg new))
+ (goto (reg relocate-continue))
+ already-moved
+ (assign new (op vector-ref) (reg the-cdrs) (reg old))
+ (goto (reg relocate-continue))
+
+ At the very end of the garbage-collection process, we interchange
+the role of old and new memories by interchanging pointers:
+interchanging `the-cars' with `new-cars', and `the-cdrs' with
+`new-cdrs'. We will then be ready to perform another garbage
+collection the next time memory runs out.
+
+ gc-flip
+ (assign temp (reg the-cdrs))
+ (assign the-cdrs (reg new-cdrs))
+ (assign new-cdrs (reg temp))
+ (assign temp (reg the-cars))
+ (assign the-cars (reg new-cars))
+ (assign new-cars (reg temp))
+
+ ---------- Footnotes ----------
+
+ (1) This may not be true eventually, because memories may get large
+enough so that it would be impossible to run out of free memory in the
+lifetime of the computer. For example, there are about 3*(10^13),
+microseconds in a year, so if we were to `cons' once per microsecond we
+would need about 10^15 cells of memory to build a machine that could
+operate for 30 years without running out of memory. That much memory
+seems absurdly large by today's standards, but it is not physically
+impossible. On the other hand, processors are getting faster and a
+future computer may have large numbers of processors operating in
+parallel on a single memory, so it may be possible to use up memory
+much faster than we have postulated.
+
+ (2) We assume here that the stack is represented as a list as
+described in section *Note 5-3-1::, so that items on the stack are
+accessible via the pointer in the stack register.
+
+ (3) This idea was invented and first implemented by Minsky, as part
+of the implementation of Lisp for the PDP-1 at the MIT Research
+Laboratory of Electronics. It was further developed by Fenichel and
+Yochelson (1969) for use in the Lisp implementation for the Multics
+time-sharing system. Later, Baker (1978) developed a "real-time"
+version of the method, which does not require the computation to stop
+during garbage collection. Baker's idea was extended by Hewitt,
+Lieberman, and Moon (see Lieberman and Hewitt 1983) to take advantage
+of the fact that some structure is more volatile and other structure is
+more permanent.
+
+ An alternative commonly used garbage-collection technique is the "mark-sweep"
+method. This consists of tracing all the structure accessible from the
+machine registers and marking each pair we reach. We then scan all of
+memory, and any location that is unmarked is "swept up" as garbage and
+made available for reuse. A full discussion of the mark-sweep method
+can be found in Allen 1978.
+
+ The Minsky-Fenichel-Yochelson algorithm is the dominant algorithm in
+use for large-memory systems because it examines only the useful part
+of memory. This is in contrast to mark-sweep, in which the sweep phase
+must check all of memory. A second advantage of stop-and-copy is that
+it is a "compacting" garbage collector. That is, at the end of the
+garbage-collection phase the useful data will have been moved to
+consecutive memory locations, with all garbage pairs compressed out.
+This can be an extremely important performance consideration in
+machines with virtual memory, in which accesses to widely separated
+memory addresses may require extra paging operations.
+
+ (4) This list of registers does not include the registers used by
+the storage-allocation system--`root', `the-cars', `the-cdrs', and the
+other registers that will be introduced in this section.
+
+ (5) The term _broken heart_ was coined by David Cressey, who wrote a
+garbage collector for MDL, a dialect of Lisp developed at MIT during
+the early 1970s.
+
+ (6) The garbage collector uses the low-level predicate
+`pointer-to-pair?' instead of the list-structure `pair?' operation
+because in a real system there might be various things that are treated
+as pairs for garbage-collection purposes. For example, in a Scheme
+system that conforms to the IEEE standard a procedure object may be
+implemented as a special kind of "pair" that doesn't satisfy the
+`pair?' predicate. For simulation purposes, `pointer-to-pair?' can be
+implemented as `pair?'.
+
+\1f
+File: sicp.info, Node: 5-4, Next: 5-5, Prev: 5-3, Up: Chapter 5
+
+5.4 The Explicit-Control Evaluator
+==================================
+
+In section *Note 5-1:: we saw how to transform simple Scheme programs
+into descriptions of register machines. We will now perform this
+transformation on a more complex program, the metacircular evaluator of
+sections *Note 4-1-1::-*Note 4-1-4::, which shows how the behavior of a
+Scheme interpreter can be described in terms of the procedures `eval'
+and `apply'. The "explicit-control evaluator" that we develop in this
+section shows how the underlying procedure-calling and argument-passing
+mechanisms used in the evaluation process can be described in terms of
+operations on registers and stacks. In addition, the explicit-control
+evaluator can serve as an implementation of a Scheme interpreter,
+written in a language that is very similar to the native machine
+language of conventional computers. The evaluator can be executed by
+the register-machine simulator of section *Note 5-2::. Alternatively,
+it can be used as a starting point for building a machine-language
+implementation of a Scheme evaluator, or even a special-purpose machine
+for evaluating Scheme expressions. *Note Figure 5-16:: shows such a
+hardware implementation: a silicon chip that acts as an evaluator for
+Scheme. The chip designers started with the data-path and controller
+specifications for a register machine similar to the evaluator
+described in this section and used design automation programs to
+construct the integrated-circuit layout.(1)
+
+Registers and operations
+........................
+
+In designing the explicit-control evaluator, we must specify the
+operations to be used in our register machine. We described the
+metacircular evaluator in terms of abstract syntax, using procedures
+such as `quoted?' and `make-procedure'. In implementing the register
+machine, we could expand these procedures into sequences of elementary
+list-structure memory operations, and implement these operations on our
+register machine. However, this would make our evaluator very long,
+obscuring the basic structure with details. To clarify the
+presentation, we will include as primitive operations of the register
+machine the syntax procedures given in section *Note 4-1-2:: and the
+procedures for representing environments and other run-time data given
+in sections *Note 4-1-3:: and *Note 4-1-4::. In order to completely
+specify an evaluator that could be programmed in a low-level machine
+language or implemented in hardware, we would replace these operations
+by more elementary operations, using the list-structure implementation
+we described in section *Note 5-3::.
+
+ *Figure 5.16:* A silicon-chip implementation of an evaluator for
+ Scheme.
+
+ [This figure is missing.]
+
+
+ Our Scheme evaluator register machine includes a stack and seven
+registers: `exp', `env', `val', `continue', `proc', `argl', and `unev'.
+`Exp' is used to hold the expression to be evaluated, and `env'
+contains the environment in which the evaluation is to be performed.
+At the end of an evaluation, `val' contains the value obtained by
+evaluating the expression in the designated environment. The
+`continue' register is used to implement recursion, as explained in
+section *Note 5-1-4::. (The evaluator needs to call itself
+recursively, since evaluating an expression requires evaluating its
+subexpressions.) The registers `proc', `argl', and `unev' are used in
+evaluating combinations.
+
+ We will not provide a data-path diagram to show how the registers and
+operations of the evaluator are connected, nor will we give the
+complete list of machine operations. These are implicit in the
+evaluator's controller, which will be presented in detail.
+
+* Menu:
+
+* 5-4-1:: The Core of the Explicit-Control Evaluator
+* 5-4-2:: Sequence Evaluation and Tail Recursion
+* 5-4-3:: Conditionals, Assignments, and Definitions
+* 5-4-4:: Running the Evaluator
+
+ ---------- Footnotes ----------
+
+ (1) See Batali et al. 1982 for more information on the chip and the
+method by which it was designed.
+
+\1f
+File: sicp.info, Node: 5-4-1, Next: 5-4-2, Prev: 5-4, Up: 5-4
+
+5.4.1 The Core of the Explicit-Control Evaluator
+------------------------------------------------
+
+The central element in the evaluator is the sequence of instructions
+beginning at `eval-dispatch'. This corresponds to the `eval' procedure
+of the metacircular evaluator described in section *Note 4-1-1::. When
+the controller starts at `eval-dispatch', it evaluates the expression
+specified by `exp' in the environment specified by `env'. When
+evaluation is complete, the controller will go to the entry point
+stored in `continue', and the `val' register will hold the value of the
+expression. As with the metacircular `eval', the structure of
+`eval-dispatch' is a case analysis on the syntactic type of the
+expression to be evaluated.(1)
+
+ eval-dispatch
+ (test (op self-evaluating?) (reg exp))
+ (branch (label ev-self-eval))
+ (test (op variable?) (reg exp))
+ (branch (label ev-variable))
+ (test (op quoted?) (reg exp))
+ (branch (label ev-quoted))
+ (test (op assignment?) (reg exp))
+ (branch (label ev-assignment))
+ (test (op definition?) (reg exp))
+ (branch (label ev-definition))
+ (test (op if?) (reg exp))
+ (branch (label ev-if))
+ (test (op lambda?) (reg exp))
+ (branch (label ev-lambda))
+ (test (op begin?) (reg exp))
+ (branch (label ev-begin))
+ (test (op application?) (reg exp))
+ (branch (label ev-application))
+ (goto (label unknown-expression-type))
+
+Evaluating simple expressions
+.............................
+
+Numbers and strings (which are self-evaluating), variables, quotations,
+and `lambda' expressions have no subexpressions to be evaluated. For
+these, the evaluator simply places the correct value in the `val'
+register and continues execution at the entry point specified by
+`continue'. Evaluation of simple expressions is performed by the
+following controller code:
+
+ ev-self-eval
+ (assign val (reg exp))
+ (goto (reg continue))
+ ev-variable
+ (assign val (op lookup-variable-value) (reg exp) (reg env))
+ (goto (reg continue))
+ ev-quoted
+ (assign val (op text-of-quotation) (reg exp))
+ (goto (reg continue))
+ ev-lambda
+ (assign unev (op lambda-parameters) (reg exp))
+ (assign exp (op lambda-body) (reg exp))
+ (assign val (op make-procedure)
+ (reg unev) (reg exp) (reg env))
+ (goto (reg continue))
+
+ Observe how `ev-lambda' uses the `unev' and `exp' registers to hold
+the parameters and body of the lambda expression so that they can be
+passed to the `make-procedure' operation, along with the environment in
+`env'.
+
+Evaluating procedure applications
+.................................
+
+A procedure application is specified by a combination containing an
+operator and operands. The operator is a subexpression whose value is
+a procedure, and the operands are subexpressions whose values are the
+arguments to which the procedure should be applied. The metacircular
+`eval' handles applications by calling itself recursively to evaluate
+each element of the combination, and then passing the results to
+`apply', which performs the actual procedure application. The
+explicit-control evaluator does the same thing; these recursive calls
+are implemented by `goto' instructions, together with use of the stack
+to save registers that will be restored after the recursive call
+returns. Before each call we will be careful to identify which
+registers must be saved (because their values will be needed later).(2)
+
+ We begin the evaluation of an application by evaluating the operator
+to produce a procedure, which will later be applied to the evaluated
+operands. To evaluate the operator, we move it to the `exp' register
+and go to `eval-dispatch'. The environment in the `env' register is
+already the correct one in which to evaluate the operator. However, we
+save `env' because we will need it later to evaluate the operands. We
+also extract the operands into `unev' and save this on the stack. We
+set up `continue' so that `eval-dispatch' will resume at
+`ev-appl-did-operator' after the operator has been evaluated. First,
+however, we save the old value of `continue', which tells the controller
+where to continue after the application.
+
+ ev-application
+ (save continue)
+ (save env)
+ (assign unev (op operands) (reg exp))
+ (save unev)
+ (assign exp (op operator) (reg exp))
+ (assign continue (label ev-appl-did-operator))
+ (goto (label eval-dispatch))
+
+ Upon returning from evaluating the operator subexpression, we
+proceed to evaluate the operands of the combination and to accumulate
+the resulting arguments in a list, held in `argl'. First we restore
+the unevaluated operands and the environment. We initialize `argl' to
+an empty list. Then we assign to the `proc' register the procedure
+that was produced by evaluating the operator. If there are no
+operands, we go directly to `apply-dispatch'. Otherwise we save `proc'
+on the stack and start the argument-evaluation loop:(3)
+
+ ev-appl-did-operator
+ (restore unev) ; the operands
+ (restore env)
+ (assign argl (op empty-arglist))
+ (assign proc (reg val)) ; the operator
+ (test (op no-operands?) (reg unev))
+ (branch (label apply-dispatch))
+ (save proc)
+
+ Each cycle of the argument-evaluation loop evaluates an operand from
+the list in `unev' and accumulates the result into `argl'. To evaluate
+an operand, we place it in the `exp' register and go to `eval-dispatch',
+after setting `continue' so that execution will resume with the
+argument-accumulation phase. But first we save the arguments
+accumulated so far (held in `argl'), the environment (held in `env'),
+and the remaining operands to be evaluated (held in `unev'). A special
+case is made for the evaluation of the last operand, which is handled at
+`ev-appl-last-arg'.
+
+ ev-appl-operand-loop
+ (save argl)
+ (assign exp (op first-operand) (reg unev))
+ (test (op last-operand?) (reg unev))
+ (branch (label ev-appl-last-arg))
+ (save env)
+ (save unev)
+ (assign continue (label ev-appl-accumulate-arg))
+ (goto (label eval-dispatch))
+
+ When an operand has been evaluated, the value is accumulated into
+the list held in `argl'. The operand is then removed from the list of
+unevaluated operands in `unev', and the argument-evaluation continues.
+
+ ev-appl-accumulate-arg
+ (restore unev)
+ (restore env)
+ (restore argl)
+ (assign argl (op adjoin-arg) (reg val) (reg argl))
+ (assign unev (op rest-operands) (reg unev))
+ (goto (label ev-appl-operand-loop))
+
+ Evaluation of the last argument is handled differently. There is no
+need to save the environment or the list of unevaluated operands before
+going to `eval-dispatch', since they will not be required after the
+last operand is evaluated. Thus, we return from the evaluation to a
+special entry point `ev-appl-accum-last-arg', which restores the
+argument list, accumulates the new argument, restores the saved
+procedure, and goes off to perform the application.(4)
+
+ ev-appl-last-arg
+ (assign continue (label ev-appl-accum-last-arg))
+ (goto (label eval-dispatch))
+ ev-appl-accum-last-arg
+ (restore argl)
+ (assign argl (op adjoin-arg) (reg val) (reg argl))
+ (restore proc)
+ (goto (label apply-dispatch))
+
+ The details of the argument-evaluation loop determine the order in
+which the interpreter evaluates the operands of a combination (e.g.,
+left to right or right to left--see *Note Exercise 3-8::). This order
+is not determined by the metacircular evaluator, which inherits its
+control structure from the underlying Scheme in which it is
+implemented.(5) Because the `first-operand' selector (used in
+`ev-appl-operand-loop' to extract successive operands from `unev') is
+implemented as `car' and the `rest-operands' selector is implemented as
+`cdr', the explicit-control evaluator will evaluate the operands of a
+combination in left-to-right order.
+
+Procedure application
+.....................
+
+The entry point `apply-dispatch' corresponds to the `apply' procedure
+of the metacircular evaluator. By the time we get to `apply-dispatch',
+the `proc' register contains the procedure to apply and `argl' contains
+the list of evaluated arguments to which it must be applied. The saved
+value of `continue' (originally passed to `eval-dispatch' and saved at
+`ev-application'), which tells where to return with the result of the
+procedure application, is on the stack. When the application is
+complete, the controller transfers to the entry point specified by the
+saved `continue', with the result of the application in `val'. As with
+the metacircular `apply', there are two cases to consider. Either the
+procedure to be applied is a primitive or it is a compound procedure.
+
+ apply-dispatch
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-apply))
+ (test (op compound-procedure?) (reg proc))
+ (branch (label compound-apply))
+ (goto (label unknown-procedure-type))
+
+ We assume that each primitive is implemented so as to obtain its
+arguments from `argl' and place its result in `val'. To specify how
+the machine handles primitives, we would have to provide a sequence of
+controller instructions to implement each primitive and arrange for
+`primitive-apply' to dispatch to the instructions for the primitive
+identified by the contents of `proc'. Since we are interested in the
+structure of the evaluation process rather than the details of the
+primitives, we will instead just use an `apply-primitive-procedure'
+operation that applies the procedure in `proc' to the arguments in
+`argl'. For the purpose of simulating the evaluator with the simulator
+of section *Note 5-2:: we use the procedure
+`apply-primitive-procedure', which calls on the underlying Scheme system
+to perform the application, just as we did for the metacircular
+evaluator in section *Note 4-1-4::. After computing the value of the
+primitive application, we restore `continue' and go to the designated
+entry point.
+
+ primitive-apply
+ (assign val (op apply-primitive-procedure)
+ (reg proc)
+ (reg argl))
+ (restore continue)
+ (goto (reg continue))
+
+ To apply a compound procedure, we proceed just as with the
+metacircular evaluator. We construct a frame that binds the
+procedure's parameters to the arguments, use this frame to extend the
+environment carried by the procedure, and evaluate in this extended
+environment the sequence of expressions that forms the body of the
+procedure. `Ev-sequence', described below in section *Note 5-4-2::,
+handles the evaluation of the sequence.
+
+ compound-apply
+ (assign unev (op procedure-parameters) (reg proc))
+ (assign env (op procedure-environment) (reg proc))
+ (assign env (op extend-environment)
+ (reg unev) (reg argl) (reg env))
+ (assign unev (op procedure-body) (reg proc))
+ (goto (label ev-sequence))
+
+ `Compound-apply' is the only place in the interpreter where the `env'
+register is ever assigned a new value. Just as in the metacircular
+evaluator, the new environment is constructed from the environment
+carried by the procedure, together with the argument list and the
+corresponding list of variables to be bound.
+
+ ---------- Footnotes ----------
+
+ (1) In our controller, the dispatch is written as a sequence of
+`test' and `branch' instructions. Alternatively, it could have been
+written in a data-directed style (and in a real system it probably
+would have been) to avoid the need to perform sequential tests and to
+facilitate the definition of new expression types. A machine designed
+to run Lisp would probably include a `dispatch-on-type' instruction
+that would efficiently execute such data-directed dispatches.
+
+ (2) This is an important but subtle point in translating algorithms
+from a procedural language, such as Lisp, to a register-machine
+language. As an alternative to saving only what is needed, we could
+save all the registers (except `val') before each recursive call. This
+is called a "framed-stack" discipline. This would work but might save
+more registers than necessary; this could be an important consideration
+in a system where stack operations are expensive. Saving registers
+whose contents will not be needed later may also hold onto useless data
+that could otherwise be garbage-collected, freeing space to be reused.
+
+ (3) We add to the evaluator data-structure procedures in section
+*Note 4-1-3:: the following two procedures for manipulating argument
+lists:
+
+ (define (empty-arglist) '())
+
+ (define (adjoin-arg arg arglist)
+ (append arglist (list arg)))
+
+ We also use an additional syntax procedure to test for the last
+operand in a combination:
+
+ (define (last-operand? ops)
+ (null? (cdr ops)))
+
+ (4) The optimization of treating the last operand specially is known
+as "evlis tail recursion" (see Wand 1980). We could be somewhat more
+efficient in the argument evaluation loop if we made evaluation of the
+first operand a special case too. This would permit us to postpone
+initializing `argl' until after evaluating the first operand, so as to
+avoid saving `argl' in this case. The compiler in section *Note 5-5::
+performs this optimization. (Compare the `construct-arglist' procedure
+of section *Note 5-5-3::.)
+
+ (5) The order of operand evaluation in the metacircular evaluator is
+determined by the order of evaluation of the arguments to `cons' in the
+procedure `list-of-values' of section *Note 4-1-1:: (see *Note Exercise
+4-1::).
+
+\1f
+File: sicp.info, Node: 5-4-2, Next: 5-4-3, Prev: 5-4-1, Up: 5-4
+
+5.4.2 Sequence Evaluation and Tail Recursion
+--------------------------------------------
+
+The portion of the explicit-control evaluator at `ev-sequence' is
+analogous to the metacircular evaluator's `eval-sequence' procedure. It
+handles sequences of expressions in procedure bodies or in explicit
+`begin' expressions.
+
+ Explicit `begin' expressions are evaluated by placing the sequence of
+expressions to be evaluated in `unev', saving `continue' on the stack,
+and jumping to `ev-sequence'.
+
+ ev-begin
+ (assign unev (op begin-actions) (reg exp))
+ (save continue)
+ (goto (label ev-sequence))
+
+ The implicit sequences in procedure bodies are handled by jumping to
+`ev-sequence' from `compound-apply', at which point `continue' is
+already on the stack, having been saved at `ev-application'.
+
+ The entries at `ev-sequence' and `ev-sequence-continue' form a loop
+that successively evaluates each expression in a sequence. The list of
+unevaluated expressions is kept in `unev'. Before evaluating each
+expression, we check to see if there are additional expressions to be
+evaluated in the sequence. If so, we save the rest of the unevaluated
+expressions (held in `unev') and the environment in which these must be
+evaluated (held in `env') and call `eval-dispatch' to evaluate the
+expression. The two saved registers are restored upon the return from
+this evaluation, at `ev-sequence-continue'.
+
+ The final expression in the sequence is handled differently, at the
+entry point `ev-sequence-last-exp'. Since there are no more
+expressions to be evaluated after this one, we need not save `unev' or
+`env' before going to `eval-dispatch'. The value of the whole sequence
+is the value of the last expression, so after the evaluation of the
+last expression there is nothing left to do except continue at the
+entry point currently held on the stack (which was saved by
+`ev-application' or `ev-begin'.) Rather than setting up `continue' to
+arrange for `eval-dispatch' to return here and then restoring
+`continue' from the stack and continuing at that entry point, we
+restore `continue' from the stack before going to `eval-dispatch', so
+that `eval-dispatch' will continue at that entry point after evaluating
+the expression.
+
+ ev-sequence
+ (assign exp (op first-exp) (reg unev))
+ (test (op last-exp?) (reg unev))
+ (branch (label ev-sequence-last-exp))
+ (save unev)
+ (save env)
+ (assign continue (label ev-sequence-continue))
+ (goto (label eval-dispatch))
+ ev-sequence-continue
+ (restore env)
+ (restore unev)
+ (assign unev (op rest-exps) (reg unev))
+ (goto (label ev-sequence))
+ ev-sequence-last-exp
+ (restore continue)
+ (goto (label eval-dispatch))
+
+Tail recursion
+..............
+
+In *Note Chapter 1:: we said that the process described by a procedure
+such as
+
+ (define (sqrt-iter guess x)
+ (if (good-enough? guess x)
+ guess
+ (sqrt-iter (improve guess x)
+ x)))
+
+is an iterative process. Even though the procedure is syntactically
+recursive (defined in terms of itself), it is not logically necessary
+for an evaluator to save information in passing from one call to
+`sqrt-iter' to the next.(1) An evaluator that can execute a procedure
+such as `sqrt-iter' without requiring increasing storage as the
+procedure continues to call itself is called a "tail-recursive"
+evaluator. The metacircular implementation of the evaluator in *Note
+Chapter 4:: does not specify whether the evaluator is tail-recursive,
+because that evaluator inherits its mechanism for saving state from the
+underlying Scheme. With the explicit-control evaluator, however, we
+can trace through the evaluation process to see when procedure calls
+cause a net accumulation of information on the stack.
+
+ Our evaluator is tail-recursive, because in order to evaluate the
+final expression of a sequence we transfer directly to `eval-dispatch'
+without saving any information on the stack. Hence, evaluating the
+final expression in a sequence--even if it is a procedure call (as in
+`sqrt-iter', where the `if' expression, which is the last expression in
+the procedure body, reduces to a call to `sqrt-iter')--will not cause
+any information to be accumulated on the stack.(2)
+
+ If we did not think to take advantage of the fact that it was
+unnecessary to save information in this case, we might have implemented
+`eval-sequence' by treating all the expressions in a sequence in the
+same way--saving the registers, evaluating the expression, returning to
+restore the registers, and repeating this until all the expressions
+have been evaluated:(3)
+
+ ev-sequence
+ (test (op no-more-exps?) (reg unev))
+ (branch (label ev-sequence-end))
+ (assign exp (op first-exp) (reg unev))
+ (save unev)
+ (save env)
+ (assign continue (label ev-sequence-continue))
+ (goto (label eval-dispatch))
+ ev-sequence-continue
+ (restore env)
+ (restore unev)
+ (assign unev (op rest-exps) (reg unev))
+ (goto (label ev-sequence))
+ ev-sequence-end
+ (restore continue)
+ (goto (reg continue))
+
+ This may seem like a minor change to our previous code for
+evaluation of a sequence: The only difference is that we go through the
+save-restore cycle for the last expression in a sequence as well as for
+the others. The interpreter will still give the same value for any
+expression. But this change is fatal to the tail-recursive
+implementation, because we must now return after evaluating the final
+expression in a sequence in order to undo the (useless) register saves.
+These extra saves will accumulate during a nest of procedure calls.
+Consequently, processes such as `sqrt-iter' will require space
+proportional to the number of iterations rather than requiring constant
+space. This difference can be significant. For example, with tail
+recursion, an infinite loop can be expressed using only the
+procedure-call mechanism:
+
+ (define (count n)
+ (newline)
+ (display n)
+ (count (+ n 1)))
+
+ Without tail recursion, such a procedure would eventually run out of
+stack space, and expressing a true iteration would require some control
+mechanism other than procedure call.
+
+ ---------- Footnotes ----------
+
+ (1) We saw in section *Note 5-1:: how to implement such a process
+with a register machine that had no stack; the state of the process was
+stored in a fixed set of registers.
+
+ (2) This implementation of tail recursion in `ev-sequence' is one
+variety of a well-known optimization technique used by many compilers.
+In compiling a procedure that ends with a procedure call, one can
+replace the call by a jump to the called procedure's entry point.
+Building this strategy into the interpreter, as we have done in this
+section, provides the optimization uniformly throughout the language.
+
+ (3) We can define `no-more-exps?' as follows:
+
+ (define (no-more-exps? seq) (null? seq))
+
+\1f
+File: sicp.info, Node: 5-4-3, Next: 5-4-4, Prev: 5-4-2, Up: 5-4
+
+5.4.3 Conditionals, Assignments, and Definitions
+------------------------------------------------
+
+As with the metacircular evaluator, special forms are handled by
+selectively evaluating fragments of the expression. For an `if'
+expression, we must evaluate the predicate and decide, based on the
+value of predicate, whether to evaluate the consequent or the
+alternative.
+
+ Before evaluating the predicate, we save the `if' expression itself
+so that we can later extract the consequent or alternative. We also
+save the environment, which we will need later in order to evaluate the
+consequent or the alternative, and we save `continue', which we will
+need later in order to return to the evaluation of the expression that
+is waiting for the value of the `if'.
+
+ ev-if
+ (save exp) ; save expression for later
+ (save env)
+ (save continue)
+ (assign continue (label ev-if-decide))
+ (assign exp (op if-predicate) (reg exp))
+ (goto (label eval-dispatch)) ; evaluate the predicate
+
+ When we return from evaluating the predicate, we test whether it was
+true or false and, depending on the result, place either the consequent
+or the alternative in `exp' before going to `eval-dispatch'. Notice
+that restoring `env' and `continue' here sets up `eval-dispatch' to
+have the correct environment and to continue at the right place to
+receive the value of the `if' expression.
+
+ ev-if-decide
+ (restore continue)
+ (restore env)
+ (restore exp)
+ (test (op true?) (reg val))
+ (branch (label ev-if-consequent))
+
+ ev-if-alternative
+ (assign exp (op if-alternative) (reg exp))
+ (goto (label eval-dispatch))
+ ev-if-consequent
+ (assign exp (op if-consequent) (reg exp))
+ (goto (label eval-dispatch))
+
+Assignments and definitions
+...........................
+
+Assignments are handled by `ev-assignment', which is reached from
+`eval-dispatch' with the assignment expression in `exp'. The code at
+`ev-assignment' first evaluates the value part of the expression and
+then installs the new value in the environment. `Set-variable-value!'
+is assumed to be available as a machine operation.
+
+ ev-assignment
+ (assign unev (op assignment-variable) (reg exp))
+ (save unev) ; save variable for later
+ (assign exp (op assignment-value) (reg exp))
+ (save env)
+ (save continue)
+ (assign continue (label ev-assignment-1))
+ (goto (label eval-dispatch)) ; evaluate the assignment value
+ ev-assignment-1
+ (restore continue)
+ (restore env)
+ (restore unev)
+ (perform
+ (op set-variable-value!) (reg unev) (reg val) (reg env))
+ (assign val (const ok))
+ (goto (reg continue))
+
+ Definitions are handled in a similar way:
+
+ ev-definition
+ (assign unev (op definition-variable) (reg exp))
+ (save unev) ; save variable for later
+ (assign exp (op definition-value) (reg exp))
+ (save env)
+ (save continue)
+ (assign continue (label ev-definition-1))
+ (goto (label eval-dispatch)) ; evaluate the definition value
+ ev-definition-1
+ (restore continue)
+ (restore env)
+ (restore unev)
+ (perform
+ (op define-variable!) (reg unev) (reg val) (reg env))
+ (assign val (const ok))
+ (goto (reg continue))
+
+ *Exercise 5.23:* Extend the evaluator to handle derived
+ expressions such as `cond', `let', and so on (section *Note
+ 4-1-2::). You may "cheat" and assume that the syntax transformers
+ such as `cond->if' are available as machine operations.(1)
+
+ *Exercise 5.24:* Implement `cond' as a new basic special form
+ without reducing it to `if'. You will have to construct a loop
+ that tests the predicates of successive `cond' clauses until you
+ find one that is true, and then use `ev-sequence' to evaluate the
+ actions of the clause.
+
+ *Exercise 5.25:* Modify the evaluator so that it uses normal-order
+ evaluation, based on the lazy evaluator of section *Note 4-2::.
+
+ ---------- Footnotes ----------
+
+ (1) This isn't really cheating. In an actual implementation built
+from scratch, we would use our explicit-control evaluator to interpret
+a Scheme program that performs source-level transformations like
+`cond->if' in a syntax phase that runs before execution.
+
+\1f
+File: sicp.info, Node: 5-4-4, Prev: 5-4-3, Up: 5-4
+
+5.4.4 Running the Evaluator
+---------------------------
+
+With the implementation of the explicit-control evaluator we come to
+the end of a development, begun in *Note Chapter 1::, in which we have
+explored successively more precise models of the evaluation process.
+We started with the relatively informal substitution model, then
+extended this in *Note Chapter 3:: to the environment model, which
+enabled us to deal with state and change. In the metacircular
+evaluator of *Note Chapter 4::, we used Scheme itself as a language for
+making more explicit the environment structure constructed during
+evaluation of an expression. Now, with register machines, we have
+taken a close look at the evaluator's mechanisms for storage
+management, argument passing, and control. At each new level of
+description, we have had to raise issues and resolve ambiguities that
+were not apparent at the previous, less precise treatment of
+evaluation. To understand the behavior of the explicit-control
+evaluator, we can simulate it and monitor its performance.
+
+ We will install a driver loop in our evaluator machine. This plays
+the role of the `driver-loop' procedure of section *Note 4-1-4::. The
+evaluator will repeatedly print a prompt, read an expression, evaluate
+the expression by going to `eval-dispatch', and print the result. The
+following instructions form the beginning of the explicit-control
+evaluator's controller sequence:(1)
+
+ read-eval-print-loop
+ (perform (op initialize-stack))
+ (perform
+ (op prompt-for-input) (const ";;; EC-Eval input:"))
+ (assign exp (op read))
+ (assign env (op get-global-environment))
+ (assign continue (label print-result))
+ (goto (label eval-dispatch))
+ print-result
+ (perform
+ (op announce-output) (const ";;; EC-Eval value:"))
+ (perform (op user-print) (reg val))
+ (goto (label read-eval-print-loop))
+
+ When we encounter an error in a procedure (such as the "unknown
+procedure type error" indicated at `apply-dispatch'), we print an error
+message and return to the driver loop.(2)
+
+ unknown-expression-type
+ (assign val (const unknown-expression-type-error))
+ (goto (label signal-error))
+
+ unknown-procedure-type
+ (restore continue) ; clean up stack (from `apply-dispatch')
+ (assign val (const unknown-procedure-type-error))
+ (goto (label signal-error))
+
+ signal-error
+ (perform (op user-print) (reg val))
+ (goto (label read-eval-print-loop))
+
+ For the purposes of the simulation, we initialize the stack each
+time through the driver loop, since it might not be empty after an
+error (such as an undefined variable) interrupts an evaluation.(3)
+
+ If we combine all the code fragments presented in sections *Note
+5-4-1::-*Note 5-4-4::, we can create an evaluator machine model that we
+can run using the register-machine simulator of section *Note 5-2::.
+
+ (define eceval
+ (make-machine
+ '(exp env val proc argl continue unev)
+ eceval-operations
+ '(
+ read-eval-print-loop
+ <_entire machine controller as given above_>
+ )))
+
+ We must define Scheme procedures to simulate the operations used as
+primitives by the evaluator. These are the same procedures we used for
+the metacircular evaluator in section *Note 4-1::, together with the
+few additional ones defined in footnotes throughout section *Note 5-4::.
+
+ (define eceval-operations
+ (list (list 'self-evaluating? self-evaluating)
+ <_complete list of operations for eceval machine_>))
+
+ Finally, we can initialize the global environment and run the
+evaluator:
+
+ (define the-global-environment (setup-environment))
+
+ (start eceval)
+
+ ;;; EC-Eval input:
+ (define (append x y)
+ (if (null? x)
+ y
+ (cons (car x)
+ (append (cdr x) y))))
+ ;;; EC-Eval value:
+ ok
+
+ ;;; EC-Eval input:
+ (append '(a b c) '(d e f))
+ ;;; EC-Eval value:
+ (a b c d e f)
+
+ Of course, evaluating expressions in this way will take much longer
+than if we had directly typed them into Scheme, because of the multiple
+levels of simulation involved. Our expressions are evaluated by the
+explicit-control-evaluator machine, which is being simulated by a Scheme
+program, which is itself being evaluated by the Scheme interpreter.
+
+Monitoring the performance of the evaluator
+...........................................
+
+Simulation can be a powerful tool to guide the implementation of
+evaluators. Simulations make it easy not only to explore variations of
+the register-machine design but also to monitor the performance of the
+simulated evaluator. For example, one important factor in performance
+is how efficiently the evaluator uses the stack. We can observe the
+number of stack operations required to evaluate various expressions by
+defining the evaluator register machine with the version of the
+simulator that collects statistics on stack use (section *Note
+5-2-4::), and adding an instruction at the evaluator's `print-result'
+entry point to print the statistics:
+
+ print-result
+ (perform (op print-stack-statistics)); added instruction
+ (perform
+ (op announce-output) (const ";;; EC-Eval value:"))
+ ... ; same as before
+
+ Interactions with the evaluator now look like this:
+
+ ;;; EC-Eval input:
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+ (total-pushes = 3 maximum-depth = 3)
+ ;;; EC-Eval value:
+ ok
+
+ ;;; EC-Eval input:
+ (factorial 5)
+ (total-pushes = 144 maximum-depth = 28)
+ ;;; EC-Eval value:
+ 120
+
+ Note that the driver loop of the evaluator reinitializes the stack
+at the start of each interaction, so that the statistics printed will
+refer only to stack operations used to evaluate the previous expression.
+
+ *Exercise 5.26:* Use the monitored stack to explore the
+ tail-recursive property of the evaluator (section *Note 5-4-2::).
+ Start the evaluator and define the iterative `factorial' procedure
+ from section *Note 1-2-1:::
+
+ (define (factorial n)
+ (define (iter product counter)
+ (if (> counter n)
+ product
+ (iter (* counter product)
+ (+ counter 1))))
+ (iter 1 1))
+
+ Run the procedure with some small values of n. Record the maximum
+ stack depth and the number of pushes required to compute n! for
+ each of these values.
+
+ a. You will find that the maximum depth required to evaluate n!
+ is independent of n. What is that depth?
+
+ b. Determine from your data a formula in terms of n for the
+ total number of push operations used in evaluating n! for any
+ n >= 1. Note that the number of operations used is a linear
+ function of n and is thus determined by two constants.
+
+
+ *Exercise 5.27:* For comparison with *Note Exercise 5-26::,
+ explore the behavior of the following procedure for computing
+ factorials recursively:
+
+ (define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+
+ By running this procedure with the monitored stack, determine, as
+ a function of n, the maximum depth of the stack and the total
+ number of pushes used in evaluating n! for n >= 1. (Again, these
+ functions will be linear.) Summarize your experiments by filling
+ in the following table with the appropriate expressions in terms
+ of n:
+
+ Maximum depth Number of pushes
+
+ Recursive
+ factorial
+
+ Iterative
+ factorial
+
+ The maximum depth is a measure of the amount of space used by the
+ evaluator in carrying out the computation, and the number of
+ pushes correlates well with the time required.
+
+ *Exercise 5.28:* Modify the definition of the evaluator by
+ changing `eval-sequence' as described in section *Note 5-4-2:: so
+ that the evaluator is no longer tail-recursive. Rerun your
+ experiments from *Note Exercise 5-26:: and *Note Exercise 5-27::
+ to demonstrate that both versions of the `factorial' procedure now
+ require space that grows linearly with their input.
+
+ *Exercise 5.29:* Monitor the stack operations in the
+ tree-recursive Fibonacci computation:
+
+ (define (fib n)
+ (if (< n 2)
+ n
+ (+ (fib (- n 1)) (fib (- n 2)))))
+
+ a. Give a formula in terms of n for the maximum depth of the
+ stack required to compute _Fib_(n) for n >= 2. Hint: In
+ section *Note 1-2-2:: we argued that the space used by this
+ process grows linearly with n.
+
+ b. Give a formula for the total number of pushes used to compute
+ _Fib_(n) for n >= 2. You should find that the number of
+ pushes (which correlates well with the time used) grows
+ exponentially with n. Hint: Let S(n) be the number of pushes
+ used in computing _Fib_(n). You should be able to argue that
+ there is a formula that expresses S(n) in terms of S(n - 1),
+ S(n - 2), and some fixed "overhead" constant k that is
+ independent of n. Give the formula, and say what k is. Then
+ show that S(n) can be expressed as a _Fib_(n + 1) + b and
+ give the values of a and b.
+
+
+ *Exercise 5.30:* Our evaluator currently catches and signals only
+ two kinds of errors--unknown expression types and unknown
+ procedure types. Other errors will take us out of the evaluator
+ read-eval-print loop. When we run the evaluator using the
+ register-machine simulator, these errors are caught by the
+ underlying Scheme system. This is analogous to the computer
+ crashing when a user program makes an error.(4) It is a large
+ project to make a real error system work, but it is well worth the
+ effort to understand what is involved here.
+
+ a. Errors that occur in the evaluation process, such as an
+ attempt to access an unbound variable, could be caught by
+ changing the lookup operation to make it return a
+ distinguished condition code, which cannot be a possible
+ value of any user variable. The evaluator can test for this
+ condition code and then do what is necessary to go to
+ `signal-error'. Find all of the places in the evaluator
+ where such a change is necessary and fix them. This is lots
+ of work.
+
+ b. Much worse is the problem of handling errors that are
+ signaled by applying primitive procedures, such as an attempt
+ to divide by zero or an attempt to extract the `car' of a
+ symbol. In a professionally written high-quality system,
+ each primitive application is checked for safety as part of
+ the primitive. For example, every call to `car' could first
+ check that the argument is a pair. If the argument is not a
+ pair, the application would return a distinguished condition
+ code to the evaluator, which would then report the failure.
+ We could arrange for this in our register-machine simulator by
+ making each primitive procedure check for applicability and
+ returning an appropriate distinguished condition code on
+ failure. Then the `primitive-apply' code in the evaluator can
+ check for the condition code and go to `signal-error' if
+ necessary. Build this structure and make it work. This is a
+ major project.
+
+
+ ---------- Footnotes ----------
+
+ (1) We assume here that `read' and the various printing operations
+are available as primitive machine operations, which is useful for our
+simulation, but completely unrealistic in practice. These are actually
+extremely complex operations. In practice, they would be implemented
+using low-level input-output operations such as transferring single
+characters to and from a device.
+
+ To support the `get-global-environment' operation we define
+
+ (define the-global-environment (setup-environment))
+
+ (define (get-global-environment)
+ the-global-environment)
+
+ (2) There are other errors that we would like the interpreter to
+handle, but these are not so simple. See *Note Exercise 5-30::.
+
+ (3) We could perform the stack initialization only after errors, but
+doing it in the driver loop will be convenient for monitoring the
+evaluator's performance, as described below.
+
+ (4) Regrettably, this is the normal state of affairs in conventional
+compiler-based language systems such as C. In UNIX(tm) the system
+"dumps core," and in DOS/Windows(tm) it becomes catatonic. The
+Macintosh(tm) displays a picture of an exploding bomb and offers you
+the opportunity to reboot the computer--if you're lucky.
+
+\1f
+File: sicp.info, Node: 5-5, Prev: 5-4, Up: Chapter 5
+
+5.5 Compilation
+===============
+
+The explicit-control evaluator of section *Note 5-4:: is a register
+machine whose controller interprets Scheme programs. In this section
+we will see how to run Scheme programs on a register machine whose
+controller is not a Scheme interpreter.
+
+ The explicit-control evaluator machine is universal--it can carry
+out any computational process that can be described in Scheme. The
+evaluator's controller orchestrates the use of its data paths to
+perform the desired computation. Thus, the evaluator's data paths are
+universal: They are sufficient to perform any computation we desire,
+given an appropriate controller.(1)
+
+ Commercial general-purpose computers are register machines organized
+around a collection of registers and operations that constitute an
+efficient and convenient universal set of data paths. The controller
+for a general-purpose machine is an interpreter for a register-machine
+language like the one we have been using. This language is called the "native
+language" of the machine, or simply "machine language". Programs
+written in machine language are sequences of instructions that use the
+machine's data paths. For example, the explicit-control evaluator's
+instruction sequence can be thought of as a machine-language program
+for a general-purpose computer rather than as the controller for a
+specialized interpreter machine.
+
+ There are two common strategies for bridging the gap between
+higher-level languages and register-machine languages. The
+explicit-control evaluator illustrates the strategy of interpretation.
+An interpreter written in the native language of a machine configures
+the machine to execute programs written in a language (called the "source
+language") that may differ from the native language of the machine
+performing the evaluation. The primitive procedures of the source
+language are implemented as a library of subroutines written in the
+native language of the given machine. A program to be interpreted
+(called the "source program") is represented as a data structure. The
+interpreter traverses this data structure, analyzing the source
+program. As it does so, it simulates the intended behavior of the
+source program by calling appropriate primitive subroutines from the
+library.
+
+ In this section, we explore the alternative strategy of "compilation".
+A compiler for a given source language and machine translates a source
+program into an equivalent program (called the "object program")
+written in the machine's native language. The compiler that we
+implement in this section translates programs written in Scheme into
+sequences of instructions to be executed using the explicit-control
+evaluator machine's data paths.(2)
+
+ Compared with interpretation, compilation can provide a great
+increase in the efficiency of program execution, as we will explain
+below in the overview of the compiler. On the other hand, an
+interpreter provides a more powerful environment for interactive
+program development and debugging, because the source program being
+executed is available at run time to be examined and modified. In
+addition, because the entire library of primitives is present, new
+programs can be constructed and added to the system during debugging.
+
+ In view of the complementary advantages of compilation and
+interpretation, modern program-development environments pursue a mixed
+strategy. Lisp interpreters are generally organized so that
+interpreted procedures and compiled procedures can call each other.
+This enables a programmer to compile those parts of a program that are
+assumed to be debugged, thus gaining the efficiency advantage of
+compilation, while retaining the interpretive mode of execution for
+those parts of the program that are in the flux of interactive
+development and debugging. In section *Note 5-5-7::, after we have
+implemented the compiler, we will show how to interface it with our
+interpreter to produce an integrated interpreter-compiler development
+system.
+
+An overview of the compiler
+...........................
+
+Our compiler is much like our interpreter, both in its structure and in
+the function it performs. Accordingly, the mechanisms used by the
+compiler for analyzing expressions will be similar to those used by the
+interpreter. Moreover, to make it easy to interface compiled and
+interpreted code, we will design the compiler to generate code that
+obeys the same conventions of register usage as the interpreter: The
+environment will be kept in the `env' register, argument lists will be
+accumulated in `argl', a procedure to be applied will be in `proc',
+procedures will return their answers in `val', and the location to
+which a procedure should return will be kept in `continue'. In
+general, the compiler translates a source program into an object
+program that performs essentially the same register operations as would
+the interpreter in evaluating the same source program.
+
+ This description suggests a strategy for implementing a rudimentary
+compiler: We traverse the expression in the same way the interpreter
+does. When we encounter a register instruction that the interpreter
+would perform in evaluating the expression, we do not execute the
+instruction but instead accumulate it into a sequence. The resulting
+sequence of instructions will be the object code. Observe the
+efficiency advantage of compilation over interpretation. Each time the
+interpreter evaluates an expression--for example, `(f 84 96)'--it
+performs the work of classifying the expression (discovering that this
+is a procedure application) and testing for the end of the operand list
+(discovering that there are two operands). With a compiler, the
+expression is analyzed only once, when the instruction sequence is
+generated at compile time. The object code produced by the compiler
+contains only the instructions that evaluate the operator and the two
+operands, assemble the argument list, and apply the procedure (in
+`proc') to the arguments (in `argl').
+
+ This is the same kind of optimization we implemented in the
+analyzing evaluator of section *Note 4-1-7::. But there are further
+opportunities to gain efficiency in compiled code. As the interpreter
+runs, it follows a process that must be applicable to any expression in
+the language. In contrast, a given segment of compiled code is meant
+to execute some particular expression. This can make a big difference,
+for example in the use of the stack to save registers. When the
+interpreter evaluates an expression, it must be prepared for any
+contingency. Before evaluating a subexpression, the interpreter saves
+all registers that will be needed later, because the subexpression
+might require an arbitrary evaluation. A compiler, on the other hand,
+can exploit the structure of the particular expression it is processing
+to generate code that avoids unnecessary stack operations.
+
+ As a case in point, consider the combination `(f 84 96)'. Before the
+interpreter evaluates the operator of the combination, it prepares for
+this evaluation by saving the registers containing the operands and the
+environment, whose values will be needed later. The interpreter then
+evaluates the operator to obtain the result in `val', restores the
+saved registers, and finally moves the result from `val' to `proc'.
+However, in the particular expression we are dealing with, the operator
+is the symbol `f', whose evaluation is accomplished by the machine
+operation `lookup-variable-value', which does not alter any registers.
+The compiler that we implement in this section will take advantage of
+this fact and generate code that evaluates the operator using the
+instruction
+
+ (assign proc (op lookup-variable-value) (const f) (reg env))
+
+ This code not only avoids the unnecessary saves and restores but
+also assigns the value of the lookup directly to `proc', whereas the
+interpreter would obtain the result in `val' and then move this to
+`proc'.
+
+ A compiler can also optimize access to the environment. Having
+analyzed the code, the compiler can in many cases know in which frame a
+particular variable will be located and access that frame directly,
+rather than performing the `lookup-variable-value' search. We will
+discuss how to implement such variable access in section *Note 5-5-6::.
+Until then, however, we will focus on the kind of register and stack
+optimizations described above. There are many other optimizations that
+can be performed by a compiler, such as coding primitive operations "in
+line" instead of using a general `apply' mechanism (see *Note Exercise
+5-38::); but we will not emphasize these here. Our main goal in this
+section is to illustrate the compilation process in a simplified (but
+still interesting) context.
+
+* Menu:
+
+* 5-5-1:: Structure of the Compiler
+* 5-5-2:: Compiling Expressions
+* 5-5-3:: Compiling Combinations
+* 5-5-4:: Combining Instruction Sequences
+* 5-5-5:: An Example of Compiled Code
+* 5-5-6:: Lexical Addressing
+* 5-5-7:: Interfacing Compiled Code to the Evaluator
+
+ ---------- Footnotes ----------
+
+ (1) This is a theoretical statement. We are not claiming that the
+evaluator's data paths are a particularly convenient or efficient set of
+data paths for a general-purpose computer. For example, they are not
+very good for implementing high-performance floating-point calculations
+or calculations that intensively manipulate bit vectors.
+
+ (2) Actually, the machine that runs compiled code can be simpler
+than the interpreter machine, because we won't use the `exp' and `unev'
+registers. The interpreter used these to hold pieces of unevaluated
+expressions. With the compiler, however, these expressions get built
+into the compiled code that the register machine will run. For the same
+reason, we don't need the machine operations that deal with expression
+syntax. But compiled code will use a few additional machine operations
+(to represent compiled procedure objects) that didn't appear in the
+explicit-control evaluator machine.
+
+\1f
+File: sicp.info, Node: 5-5-1, Next: 5-5-2, Prev: 5-5, Up: 5-5
+
+5.5.1 Structure of the Compiler
+-------------------------------
+
+In section *Note 4-1-7:: we modified our original metacircular
+interpreter to separate analysis from execution. We analyzed each
+expression to produce an execution procedure that took an environment
+as argument and performed the required operations. In our compiler, we
+will do essentially the same analysis. Instead of producing execution
+procedures, however, we will generate sequences of instructions to be
+run by our register machine.
+
+ The procedure `compile' is the top-level dispatch in the compiler.
+It corresponds to the `eval' procedure of section *Note 4-1-1::, the
+`analyze' procedure of section *Note 4-1-7::, and the `eval-dispatch'
+entry point of the explicit-control-evaluator in section *Note 5-4-1::.
+The compiler, like the interpreters, uses the expression-syntax
+procedures defined in section *Note 4-1-2::.(1) `Compile' performs a
+case analysis on the syntactic type of the expression to be compiled.
+For each type of expression, it dispatches to a specialized "code
+generator":
+
+ (define (compile exp target linkage)
+ (cond ((self-evaluating? exp)
+ (compile-self-evaluating exp target linkage))
+ ((quoted? exp) (compile-quoted exp target linkage))
+ ((variable? exp)
+ (compile-variable exp target linkage))
+ ((assignment? exp)
+ (compile-assignment exp target linkage))
+ ((definition? exp)
+ (compile-definition exp target linkage))
+ ((if? exp) (compile-if exp target linkage))
+ ((lambda? exp) (compile-lambda exp target linkage))
+ ((begin? exp)
+ (compile-sequence (begin-actions exp)
+ target
+ linkage))
+ ((cond? exp) (compile (cond->if exp) target linkage))
+ ((application? exp)
+ (compile-application exp target linkage))
+ (else
+ (error "Unknown expression type -- COMPILE" exp))))
+
+Targets and linkages
+....................
+
+`Compile' and the code generators that it calls take two arguments in
+addition to the expression to compile. There is a "target", which
+specifies the register in which the compiled code is to return the
+value of the expression. There is also a "linkage descriptor", which
+describes how the code resulting from the compilation of the expression
+should proceed when it has finished its execution. The linkage
+descriptor can require that the code do one of the following three
+things:
+
+ * continue at the next instruction in sequence (this is specified by
+ the linkage descriptor `next'),
+
+ * return from the procedure being compiled (this is specified by the
+ linkage descriptor `return'), or
+
+ * jump to a named entry point (this is specified by using the
+ designated label as the linkage descriptor).
+
+
+ For example, compiling the expression `5' (which is self-evaluating)
+with a target of the `val' register and a linkage of `next' should
+produce the instruction
+
+ (assign val (const 5))
+
+Compiling the same expression with a linkage of `return' should produce
+the instructions
+
+ (assign val (const 5))
+ (goto (reg continue))
+
+ In the first case, execution will continue with the next instruction
+in the sequence. In the second case, we will return from a procedure
+call. In both cases, the value of the expression will be placed into
+the target `val' register.
+
+Instruction sequences and stack usage
+.....................................
+
+Each code generator returns an "instruction sequence" containing the
+object code it has generated for the expression. Code generation for a
+compound expression is accomplished by combining the output from
+simpler code generators for component expressions, just as evaluation
+of a compound expression is accomplished by evaluating the component
+expressions.
+
+ The simplest method for combining instruction sequences is a
+procedure called `append-instruction-sequences'. It takes as arguments
+any number of instruction sequences that are to be executed
+sequentially; it appends them and returns the combined sequence. That
+is, if <SEQ_1> and <SEQ_2> are sequences of instructions, then
+evaluating
+
+ (append-instruction-sequences <SEQ_1> <SEQ_2>)
+
+produces the sequence
+
+ <SEQ_1>
+ <SEQ_2>
+
+ Whenever registers might need to be saved, the compiler's code
+generators use `preserving', which is a more subtle method for
+combining instruction sequences. `Preserving' takes three arguments: a
+set of registers and two instruction sequences that are to be executed
+sequentially. It appends the sequences in such a way that the contents
+of each register in the set is preserved over the execution of the
+first sequence, if this is needed for the execution of the second
+sequence. That is, if the first sequence modifies the register and the
+second sequence actually needs the register's original contents, then
+`preserving' wraps a `save' and a `restore' of the register around the
+first sequence before appending the sequences. Otherwise, `preserving'
+simply returns the appended instruction sequences. Thus, for example,
+
+ (preserving (list <REG_1> <REG_2>) <SEQ_1> <SEQ_2>)
+
+produces one of the following four sequences of instructions, depending
+on how <SEQ_1> and <SEQ_2> use <REG_1> and <REG_2>:
+
+ <seq_1> | (save <reg_1>) | (save <reg_2>) | (save <reg_2>)
+ <seq_2> | <seq_1> | <seq_1> | (save <reg_1>)
+ | (restore <reg_1>) | (restore <reg_2>) | <seq_1>
+ | <seq_2> | <seq_2> | (restore <reg_1>)
+ | | | (restore <reg_2>)
+ | | | <seq_2>
+
+ By using `preserving' to combine instruction sequences the compiler
+avoids unnecessary stack operations. This also isolates the details of
+whether or not to generate `save' and `restore' instructions within the
+`preserving' procedure, separating them from the concerns that arise in
+writing each of the individual code generators. In fact no `save' or
+`restore' instructions are explicitly produced by the code generators.
+
+ In principle, we could represent an instruction sequence simply as a
+list of instructions. `Append-instruction-sequences' could then combine
+instruction sequences by performing an ordinary list `append'. However,
+`preserving' would then be a complex operation, because it would have to
+analyze each instruction sequence to determine how the sequence uses its
+registers. `Preserving' would be inefficient as well as complex,
+because it would have to analyze each of its instruction sequence
+arguments, even though these sequences might themselves have been
+constructed by calls to `preserving', in which case their parts would
+have already been analyzed. To avoid such repetitious analysis we will
+associate with each instruction sequence some information about its
+register use. When we construct a basic instruction sequence we will
+provide this information explicitly, and the procedures that combine
+instruction sequences will derive register-use information for the
+combined sequence from the information associated with the component
+sequences.
+
+ An instruction sequence will contain three pieces of information:
+
+ * the set of registers that must be initialized before the
+ instructions in the sequence are executed (these registers are
+ said to be "needed" by the sequence),
+
+ * the set of registers whose values are modified by the instructions
+ in the sequence, and
+
+ * the actual instructions (also called "statements") in the sequence.
+
+
+ We will represent an instruction sequence as a list of its three
+parts. The constructor for instruction sequences is thus
+
+ (define (make-instruction-sequence needs modifies statements)
+ (list needs modifies statements))
+
+ For example, the two-instruction sequence that looks up the value of
+the variable `x' in the current environment, assigns the result to
+`val', and then returns, requires registers `env' and `continue' to have
+been initialized, and modifies register `val'. This sequence would
+therefore be constructed as
+
+ (make-instruction-sequence '(env continue) '(val)
+ '((assign val
+ (op lookup-variable-value) (const x) (reg env))
+ (goto (reg continue))))
+
+ We sometimes need to construct an instruction sequence with no
+statements:
+
+ (define (empty-instruction-sequence)
+ (make-instruction-sequence '() '() '()))
+
+ The procedures for combining instruction sequences are shown in
+section *Note 5-5-4::.
+
+ *Exercise 5.31:* In evaluating a procedure application, the
+ explicit-control evaluator always saves and restores the `env'
+ register around the evaluation of the operator, saves and restores
+ `env' around the evaluation of each operand (except the final
+ one), saves and restores `argl' around the evaluation of each
+ operand, and saves and restores `proc' around the evaluation of
+ the operand sequence. For each of the following combinations, say
+ which of these `save' and `restore' operations are superfluous and
+ thus could be eliminated by the compiler's `preserving' mechanism:
+
+ (f 'x 'y)
+
+ ((f) 'x 'y)
+
+ (f (g 'x) y)
+
+ (f (g 'x) 'y)
+
+ *Exercise 5.32:* Using the `preserving' mechanism, the compiler
+ will avoid saving and restoring `env' around the evaluation of the
+ operator of a combination in the case where the operator is a
+ symbol. We could also build such optimizations into the
+ evaluator. Indeed, the explicit-control evaluator of section
+ *Note 5-4:: already performs a similar optimization, by treating
+ combinations with no operands as a special case.
+
+ a. Extend the explicit-control evaluator to recognize as a
+ separate class of expressions combinations whose operator is
+ a symbol, and to take advantage of this fact in evaluating
+ such expressions.
+
+ b. Alyssa P. Hacker suggests that by extending the evaluator to
+ recognize more and more special cases we could incorporate
+ all the compiler's optimizations, and that this would
+ eliminate the advantage of compilation altogether. What do
+ you think of this idea?
+
+
+ ---------- Footnotes ----------
+
+ (1) Notice, however, that our compiler is a Scheme program, and the
+syntax procedures that it uses to manipulate expressions are the actual
+Scheme procedures used with the metacircular evaluator. For the
+explicit-control evaluator, in contrast, we assumed that equivalent
+syntax operations were available as operations for the register
+machine. (Of course, when we simulated the register machine in Scheme,
+we used the actual Scheme procedures in our register machine
+simulation.)
+
+\1f
+File: sicp.info, Node: 5-5-2, Next: 5-5-3, Prev: 5-5-1, Up: 5-5
+
+5.5.2 Compiling Expressions
+---------------------------
+
+In this section and the next we implement the code generators to which
+the `compile' procedure dispatches.
+
+Compiling linkage code
+......................
+
+In general, the output of each code generator will end with
+instructions--generated by the procedure `compile-linkage'--that
+implement the required linkage. If the linkage is `return' then we must
+generate the instruction `(goto (reg continue))'. This needs the
+`continue' register and does not modify any registers. If the linkage
+is `next', then we needn't include any additional instructions.
+Otherwise, the linkage is a label, and we generate a `goto' to that
+label, an instruction that does not need or modify any registers.(1)
+
+ (define (compile-linkage linkage)
+ (cond ((eq? linkage 'return)
+ (make-instruction-sequence '(continue) '()
+ '((goto (reg continue)))))
+ ((eq? linkage 'next)
+ (empty-instruction-sequence))
+ (else
+ (make-instruction-sequence '() '()
+ `((goto (label ,linkage)))))))
+
+ The linkage code is appended to an instruction sequence by
+`preserving' the `continue' register, since a `return' linkage will
+require the `continue' register: If the given instruction sequence
+modifies `continue' and the linkage code needs it, `continue' will be
+saved and restored.
+
+ (define (end-with-linkage linkage instruction-sequence)
+ (preserving '(continue)
+ instruction-sequence
+ (compile-linkage linkage)))
+
+Compiling simple expressions
+............................
+
+The code generators for self-evaluating expressions, quotations, and
+variables construct instruction sequences that assign the required
+value to the target register and then proceed as specified by the
+linkage descriptor.
+
+ (define (compile-self-evaluating exp target linkage)
+ (end-with-linkage linkage
+ (make-instruction-sequence '() (list target)
+ `((assign ,target (const ,exp))))))
+
+ (define (compile-quoted exp target linkage)
+ (end-with-linkage linkage
+ (make-instruction-sequence '() (list target)
+ `((assign ,target (const ,(text-of-quotation exp)))))))
+
+ (define (compile-variable exp target linkage)
+ (end-with-linkage linkage
+ (make-instruction-sequence '(env) (list target)
+ `((assign ,target
+ (op lookup-variable-value)
+ (const ,exp)
+ (reg env))))))
+
+ All these assignment instructions modify the target register, and
+the one that looks up a variable needs the `env' register.
+
+ Assignments and definitions are handled much as they are in the
+interpreter. We recursively generate code that computes the value to
+be assigned to the variable, and append to it a two-instruction
+sequence that actually sets or defines the variable and assigns the
+value of the whole expression (the symbol `ok') to the target register.
+The recursive compilation has target `val' and linkage `next' so that
+the code will put its result into `val' and continue with the code that
+is appended after it. The appending is done preserving `env', since
+the environment is needed for setting or defining the variable and the
+code for the variable value could be the compilation of a complex
+expression that might modify the registers in arbitrary ways.
+
+ (define (compile-assignment exp target linkage)
+ (let ((var (assignment-variable exp))
+ (get-value-code
+ (compile (assignment-value exp) 'val 'next)))
+ (end-with-linkage linkage
+ (preserving '(env)
+ get-value-code
+ (make-instruction-sequence '(env val) (list target)
+ `((perform (op set-variable-value!)
+ (const ,var)
+ (reg val)
+ (reg env))
+ (assign ,target (const ok))))))))
+
+ (define (compile-definition exp target linkage)
+ (let ((var (definition-variable exp))
+ (get-value-code
+ (compile (definition-value exp) 'val 'next)))
+ (end-with-linkage linkage
+ (preserving '(env)
+ get-value-code
+ (make-instruction-sequence '(env val) (list target)
+ `((perform (op define-variable!)
+ (const ,var)
+ (reg val)
+ (reg env))
+ (assign ,target (const ok))))))))
+
+ The appended two-instruction sequence requires `env' and `val' and
+modifies the target. Note that although we preserve `env' for this
+sequence, we do not preserve `val', because the `get-value-code' is
+designed to explicitly place its result in `val' for use by this
+sequence. (In fact, if we did preserve `val', we would have a bug,
+because this would cause the previous contents of `val' to be restored
+right after the `get-value-code' is run.)
+
+Compiling conditional expressions
+.................................
+
+The code for an `if' expression compiled with a given target and linkage
+has the form
+
+ <_compilation of predicate, target `val', linkage `next'_>
+ (test (op false?) (reg val))
+ (branch (label false-branch))
+ true-branch
+ <_compilation of consequent with given target and given linkage or `after-if'_>
+ false-branch
+ <_compilation of alternative with given target and linkage_>
+ after-if
+
+ To generate this code, we compile the predicate, consequent, and
+alternative, and combine the resulting code with instructions to test
+the predicate result and with newly generated labels to mark the true
+and false branches and the end of the conditional.(2) In this
+arrangement of code, we must branch around the true branch if the test
+is false. The only slight complication is in how the linkage for the
+true branch should be handled. If the linkage for the conditional is
+`return' or a label, then the true and false branches will both use
+this same linkage. If the linkage is `next', the true branch ends with
+a jump around the code for the false branch to the label at the end of
+the conditional.
+
+ (define (compile-if exp target linkage)
+ (let ((t-branch (make-label 'true-branch))
+ (f-branch (make-label 'false-branch))
+ (after-if (make-label 'after-if)))
+ (let ((consequent-linkage
+ (if (eq? linkage 'next) after-if linkage)))
+ (let ((p-code (compile (if-predicate exp) 'val 'next))
+ (c-code
+ (compile
+ (if-consequent exp) target consequent-linkage))
+ (a-code
+ (compile (if-alternative exp) target linkage)))
+ (preserving '(env continue)
+ p-code
+ (append-instruction-sequences
+ (make-instruction-sequence '(val) '()
+ `((test (op false?) (reg val))
+ (branch (label ,f-branch))))
+ (parallel-instruction-sequences
+ (append-instruction-sequences t-branch c-code)
+ (append-instruction-sequences f-branch a-code))
+ after-if))))))
+
+ `Env' is preserved around the predicate code because it could be
+needed by the true and false branches, and `continue' is preserved
+because it could be needed by the linkage code in those branches. The
+code for the true and false branches (which are not executed
+sequentially) is appended using a special combiner
+`parallel-instruction-sequences' described in section *Note 5-5-4::.
+
+ Note that `cond' is a derived expression, so all that the compiler
+needs to do handle it is to apply the `cond->if' transformer (from
+section *Note 4-1-2::) and compile the resulting `if' expression.
+
+Compiling sequences
+...................
+
+The compilation of sequences (from procedure bodies or explicit `begin'
+expressions) parallels their evaluation. Each expression of the
+sequence is compiled--the last expression with the linkage specified
+for the sequence, and the other expressions with linkage `next' (to
+execute the rest of the sequence). The instruction sequences for the
+individual expressions are appended to form a single instruction
+sequence, such that `env' (needed for the rest of the sequence) and
+`continue' (possibly needed for the linkage at the end of the sequence)
+are preserved.
+
+ (define (compile-sequence seq target linkage)
+ (if (last-exp? seq)
+ (compile (first-exp seq) target linkage)
+ (preserving '(env continue)
+ (compile (first-exp seq) target 'next)
+ (compile-sequence (rest-exps seq) target linkage))))
+
+Compiling `lambda' expressions
+..............................
+
+`Lambda' expressions construct procedures. The object code for a
+`lambda' expression must have the form
+
+ <_construct procedure object and assign it to target register_>
+ <LINKAGE>
+
+ When we compile the `lambda' expression, we also generate the code
+for the procedure body. Although the body won't be executed at the
+time of procedure construction, it is convenient to insert it into the
+object code right after the code for the `lambda'. If the linkage for
+the `lambda' expression is a label or `return', this is fine. But if
+the linkage is `next', we will need to skip around the code for the
+procedure body by using a linkage that jumps to a label that is
+inserted after the body. The object code thus has the form
+
+ <_construct procedure object and assign it to target register_>
+ <_code for given linkage_> _or_ `(goto (label after-lambda))'
+ <_compilation of procedure body_>
+ after-lambda
+
+ `Compile-lambda' generates the code for constructing the procedure
+object followed by the code for the procedure body. The procedure
+object will be constructed at run time by combining the current
+environment (the environment at the point of definition) with the entry
+point to the compiled procedure body (a newly generated label).(3)
+
+ (define (compile-lambda exp target linkage)
+ (let ((proc-entry (make-label 'entry))
+ (after-lambda (make-label 'after-lambda)))
+ (let ((lambda-linkage
+ (if (eq? linkage 'next) after-lambda linkage)))
+ (append-instruction-sequences
+ (tack-on-instruction-sequence
+ (end-with-linkage lambda-linkage
+ (make-instruction-sequence '(env) (list target)
+ `((assign ,target
+ (op make-compiled-procedure)
+ (label ,proc-entry)
+ (reg env)))))
+ (compile-lambda-body exp proc-entry))
+ after-lambda))))
+
+ `Compile-lambda' uses the special combiner
+`tack-on-instruction-sequence' (section *Note 5-5-4::) rather than
+`append-instruction-sequences' to append the procedure body to the
+`lambda' expression code, because the body is not part of the sequence
+of instructions that will be executed when the combined sequence is
+entered; rather, it is in the sequence only because that was a
+convenient place to put it.
+
+ `Compile-lambda-body' constructs the code for the body of the
+procedure. This code begins with a label for the entry point. Next
+come instructions that will cause the run-time evaluation environment
+to switch to the correct environment for evaluating the procedure
+body--namely, the definition environment of the procedure, extended to
+include the bindings of the formal parameters to the arguments with
+which the procedure is called. After this comes the code for the
+sequence of expressions that makes up the procedure body. The sequence
+is compiled with linkage `return' and target `val' so that it will end
+by returning from the procedure with the procedure result in `val'.
+
+ (define (compile-lambda-body exp proc-entry)
+ (let ((formals (lambda-parameters exp)))
+ (append-instruction-sequences
+ (make-instruction-sequence '(env proc argl) '(env)
+ `(,proc-entry
+ (assign env (op compiled-procedure-env) (reg proc))
+ (assign env
+ (op extend-environment)
+ (const ,formals)
+ (reg argl)
+ (reg env))))
+ (compile-sequence (lambda-body exp) 'val 'return))))
+
+ ---------- Footnotes ----------
+
+ (1) This procedure uses a feature of Lisp called "backquote" (or "quasiquote")
+that is handy for constructing lists. Preceding a list with a
+backquote symbol is much like quoting it, except that anything in the
+list that is flagged with a comma is evaluated.
+
+ For example, if the value of `linkage' is the symbol
+`branch25', then the expression
+``((goto (label ,linkage)))'
+evaluates to the list
+`((goto (label branch25)))'.
+Similarly, if the value of `x' is the list `(a b c)', then
+``(1 2 ,(car x))'
+evaluates to the list
+`(1 2 a)'.
+
+ (2) We can't just use the labels `true-branch', `false-branch', and
+`after-if' as shown above, because there might be more than one `if' in
+the program. The compiler uses the procedure `make-label' to generate
+labels. `Make-label' takes a symbol as argument and returns a new
+symbol that begins with the given symbol. For example, successive
+calls to `(make-label 'a)' would return `a1', `a2', and so on.
+`Make-label' can be implemented similarly to the generation of unique
+variable names in the query language, as follows:
+
+ (define label-counter 0)
+
+ (define (new-label-number)
+ (set! label-counter (+ 1 label-counter))
+ label-counter)
+
+ (define (make-label name)
+ (string->symbol
+ (string-append (symbol->string name)
+ (number->string (new-label-number)))))
+
+ (3) [Footnote 38] We need machine operations to implement a data
+structure for representing compiled procedures, analogous to the
+structure for compound procedures described in section *Note 4-1-3:::
+
+ (define (make-compiled-procedure entry env)
+ (list 'compiled-procedure entry env))
+
+ (define (compiled-procedure? proc)
+ (tagged-list? proc 'compiled-procedure))
+
+ (define (compiled-procedure-entry c-proc) (cadr c-proc))
+
+ (define (compiled-procedure-env c-proc) (caddr c-proc))
+
+\1f
+File: sicp.info, Node: 5-5-3, Next: 5-5-4, Prev: 5-5-2, Up: 5-5
+
+5.5.3 Compiling Combinations
+----------------------------
+
+The essence of the compilation process is the compilation of procedure
+applications. The code for a combination compiled with a given target
+and linkage has the form
+
+ <_compilation of operator, target `proc', linkage `next'_>
+ <_evaluate operands and construct argument list in `argl'_>
+ <_compilation of procedure call with given target and linkage_>
+
+ The registers `env', `proc', and `argl' may have to be saved and
+restored during evaluation of the operator and operands. Note that
+this is the only place in the compiler where a target other than `val'
+is specified.
+
+ The required code is generated by `compile-application'. This
+recursively compiles the operator, to produce code that puts the
+procedure to be applied into `proc', and compiles the operands, to
+produce code that evaluates the individual operands of the application.
+The instruction sequences for the operands are combined (by
+`construct-arglist') with code that constructs the list of arguments in
+`argl', and the resulting argument-list code is combined with the
+procedure code and the code that performs the procedure call (produced
+by `compile-procedure-call'). In appending the code sequences, the
+`env' register must be preserved around the evaluation of the operator
+(since evaluating the operator might modify `env', which will be needed
+to evaluate the operands), and the `proc' register must be preserved
+around the construction of the argument list (since evaluating the
+operands might modify `proc', which will be needed for the actual
+procedure application). `Continue' must also be preserved throughout,
+since it is needed for the linkage in the procedure call.
+
+ (define (compile-application exp target linkage)
+ (let ((proc-code (compile (operator exp) 'proc 'next))
+ (operand-codes
+ (map (lambda (operand) (compile operand 'val 'next))
+ (operands exp))))
+ (preserving '(env continue)
+ proc-code
+ (preserving '(proc continue)
+ (construct-arglist operand-codes)
+ (compile-procedure-call target linkage)))))
+
+ The code to construct the argument list will evaluate each operand
+into `val' and then `cons' that value onto the argument list being
+accumulated in `argl'. Since we `cons' the arguments onto `argl' in
+sequence, we must start with the last argument and end with the first,
+so that the arguments will appear in order from first to last in the
+resulting list. Rather than waste an instruction by initializing `argl'
+to the empty list to set up for this sequence of evaluations, we make
+the first code sequence construct the initial `argl'. The general form
+of the argument-list construction is thus as follows:
+
+ <_compilation of last operand, targeted to `val'_>
+ (assign argl (op list) (reg val))
+ <_compilation of next operand, targeted to `val'_>
+ (assign argl (op cons) (reg val) (reg argl))
+ ...
+ <_compilation of first operand, targeted to `val'_>
+ (assign argl (op cons) (reg val) (reg argl))
+
+ `Argl' must be preserved around each operand evaluation except the
+first (so that arguments accumulated so far won't be lost), and `env'
+must be preserved around each operand evaluation except the last (for
+use by subsequent operand evaluations).
+
+ Compiling this argument code is a bit tricky, because of the special
+treatment of the first operand to be evaluated and the need to preserve
+`argl' and `env' in different places. The `construct-arglist'
+procedure takes as arguments the code that evaluates the individual
+operands. If there are no operands at all, it simply emits the
+instruction
+
+ (assign argl (const ()))
+
+ Otherwise, `construct-arglist' creates code that initializes `argl'
+with the last argument, and appends code that evaluates the rest of the
+arguments and adjoins them to `argl' in succession. In order to process
+the arguments from last to first, we must reverse the list of operand
+code sequences from the order supplied by `compile-application'.
+
+ (define (construct-arglist operand-codes)
+ (let ((operand-codes (reverse operand-codes)))
+ (if (null? operand-codes)
+ (make-instruction-sequence '() '(argl)
+ '((assign argl (const ()))))
+ (let ((code-to-get-last-arg
+ (append-instruction-sequences
+ (car operand-codes)
+ (make-instruction-sequence '(val) '(argl)
+ '((assign argl (op list) (reg val)))))))
+ (if (null? (cdr operand-codes))
+ code-to-get-last-arg
+ (preserving '(env)
+ code-to-get-last-arg
+ (code-to-get-rest-args
+ (cdr operand-codes))))))))
+
+ (define (code-to-get-rest-args operand-codes)
+ (let ((code-for-next-arg
+ (preserving '(argl)
+ (car operand-codes)
+ (make-instruction-sequence '(val argl) '(argl)
+ '((assign argl
+ (op cons) (reg val) (reg argl)))))))
+ (if (null? (cdr operand-codes))
+ code-for-next-arg
+ (preserving '(env)
+ code-for-next-arg
+ (code-to-get-rest-args (cdr operand-codes))))))
+
+Applying procedures
+...................
+
+After evaluating the elements of a combination, the compiled code must
+apply the procedure in `proc' to the arguments in `argl'. The code
+performs essentially the same dispatch as the `apply' procedure in the
+metacircular evaluator of section *Note 4-1-1:: or the `apply-dispatch'
+entry point in the explicit-control evaluator of section *Note 5-4-1::.
+It checks whether the procedure to be applied is a primitive procedure
+or a compiled procedure. For a primitive procedure, it uses
+`apply-primitive-procedure'; we will see shortly how it handles compiled
+procedures. The procedure-application code has the following form:
+
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch))
+ compiled-branch
+ <_code to apply compiled procedure with given target and appropriate linkage_>
+ primitive-branch
+ (assign <TARGET>
+ (op apply-primitive-procedure)
+ (reg proc)
+ (reg argl))
+ <LINKAGE>
+ after-call
+
+ Observe that the compiled branch must skip around the primitive
+branch. Therefore, if the linkage for the original procedure call was
+`next', the compound branch must use a linkage that jumps to a label
+that is inserted after the primitive branch. (This is similar to the
+linkage used for the true branch in `compile-if'.)
+
+ (define (compile-procedure-call target linkage)
+ (let ((primitive-branch (make-label 'primitive-branch))
+ (compiled-branch (make-label 'compiled-branch))
+ (after-call (make-label 'after-call)))
+ (let ((compiled-linkage
+ (if (eq? linkage 'next) after-call linkage)))
+ (append-instruction-sequences
+ (make-instruction-sequence '(proc) '()
+ `((test (op primitive-procedure?) (reg proc))
+ (branch (label ,primitive-branch))))
+ (parallel-instruction-sequences
+ (append-instruction-sequences
+ compiled-branch
+ (compile-proc-appl target compiled-linkage))
+ (append-instruction-sequences
+ primitive-branch
+ (end-with-linkage linkage
+ (make-instruction-sequence '(proc argl)
+ (list target)
+ `((assign ,target
+ (op apply-primitive-procedure)
+ (reg proc)
+ (reg argl)))))))
+ after-call))))
+
+ The primitive and compound branches, like the true and false
+branches in `compile-if', are appended using
+`parallel-instruction-sequences' rather than the ordinary
+`append-instruction-sequences', because they will not be executed
+sequentially.
+
+Applying compiled procedures
+............................
+
+The code that handles procedure application is the most subtle part of
+the compiler, even though the instruction sequences it generates are
+very short. A compiled procedure (as constructed by `compile-lambda')
+has an entry point, which is a label that designates where the code for
+the procedure starts. The code at this entry point computes a result
+in `val' and returns by executing the instruction `(goto (reg
+continue))'. Thus, we might expect the code for a compiled-procedure
+application (to be generated by `compile-proc-appl') with a given
+target and linkage to look like this if the linkage is a label
+
+ (assign continue (label proc-return))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ proc-return
+ (assign <TARGET> (reg val)) ; included if target is not `val'
+ (goto (label <LINKAGE>)) ; linkage code
+
+or like this if the linkage is `return'.
+
+ (save continue)
+ (assign continue (label proc-return))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ proc-return
+ (assign <TARGET> (reg val)) ; included if target is not `val'
+ (restore continue)
+ (goto (reg continue)) ; linkage code
+
+ This code sets up `continue' so that the procedure will return to a
+label `proc-return' and jumps to the procedure's entry point. The code
+at `proc-return' transfers the procedure's result from `val' to the
+target register (if necessary) and then jumps to the location specified
+by the linkage. (The linkage is always `return' or a label, because
+`compile-procedure-call' replaces a `next' linkage for the
+compound-procedure branch by an `after-call' label.)
+
+ In fact, if the target is not `val', that is exactly the code our
+compiler will generate.(1) Usually, however, the target is `val' (the
+only time the compiler specifies a different register is when targeting
+the evaluation of an operator to `proc'), so the procedure result is
+put directly into the target register and there is no need to return to
+a special location that copies it. Instead, we simplify the code by
+setting up `continue' so that the procedure will "return" directly to
+the place specified by the caller's linkage:
+
+ <_set up `continue' for linkage_>
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+
+ If the linkage is a label, we set up `continue' so that the
+procedure will return to that label. (That is, the `(goto (reg
+continue))' the procedure ends with becomes equivalent to the `(goto
+(label <LINKAGE>))' at `proc-return' above.)
+
+ (assign continue (label <LINKAGE>))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+
+ If the linkage is `return', we don't need to set up `continue' at
+all: It already holds the desired location. (That is, the `(goto (reg
+continue))' the procedure ends with goes directly to the place where the
+`(goto (reg continue))' at `proc-return' would have gone.)
+
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+
+ With this implementation of the `return' linkage, the compiler
+generates tail-recursive code. Calling a procedure as the final step
+in a procedure body does a direct transfer, without saving any
+information on the stack.
+
+ Suppose instead that we had handled the case of a procedure call
+with a linkage of `return' and a target of `val' as shown above for a
+non-`val' target. This would destroy tail recursion. Our system would
+still give the same value for any expression. But each time we called
+a procedure, we would save `continue' and return after the call to undo
+the (useless) save. These extra saves would accumulate during a nest
+of procedure calls.(2)
+
+ `Compile-proc-appl' generates the above procedure-application code by
+considering four cases, depending on whether the target for the call is
+`val' and whether the linkage is `return'. Observe that the
+instruction sequences are declared to modify all the registers, since
+executing the procedure body can change the registers in arbitrary
+ways.(3) Also note that the code sequence for the case with target
+`val' and linkage `return' is declared to need `continue': Even though
+`continue' is not explicitly used in the two-instruction sequence, we
+must be sure that `continue' will have the correct value when we enter
+the compiled procedure.
+
+ (define (compile-proc-appl target linkage)
+ (cond ((and (eq? target 'val) (not (eq? linkage 'return)))
+ (make-instruction-sequence '(proc) all-regs
+ `((assign continue (label ,linkage))
+ (assign val (op compiled-procedure-entry)
+ (reg proc))
+ (goto (reg val)))))
+ ((and (not (eq? target 'val))
+ (not (eq? linkage 'return)))
+ (let ((proc-return (make-label 'proc-return)))
+ (make-instruction-sequence '(proc) all-regs
+ `((assign continue (label ,proc-return))
+ (assign val (op compiled-procedure-entry)
+ (reg proc))
+ (goto (reg val))
+ ,proc-return
+ (assign ,target (reg val))
+ (goto (label ,linkage))))))
+ ((and (eq? target 'val) (eq? linkage 'return))
+ (make-instruction-sequence '(proc continue) all-regs
+ '((assign val (op compiled-procedure-entry)
+ (reg proc))
+ (goto (reg val)))))
+ ((and (not (eq? target 'val)) (eq? linkage 'return))
+ (error "return linkage, target not val -- COMPILE"
+ target))))
+
+ ---------- Footnotes ----------
+
+ (1) Actually, we signal an error when the target is not `val' and
+the linkage is `return', since the only place we request `return'
+linkages is in compiling procedures, and our convention is that
+procedures return their values in `val'.
+
+ (2) Making a compiler generate tail-recursive code might seem like a
+straightforward idea. But most compilers for common languages,
+including C and Pascal, do not do this, and therefore these languages
+cannot represent iterative processes in terms of procedure call alone.
+The difficulty with tail recursion in these languages is that their
+implementations use the stack to store procedure arguments and local
+variables as well as return addresses. The Scheme implementations
+described in this book store arguments and variables in memory to be
+garbage-collected. The reason for using the stack for variables and
+arguments is that it avoids the need for garbage collection in languages
+that would not otherwise require it, and is generally believed to be
+more efficient. Sophisticated Lisp compilers can, in fact, use the
+stack for arguments without destroying tail recursion. (See Hanson
+1990 for a description.) There is also some debate about whether stack
+allocation is actually more efficient than garbage collection in the
+first place, but the details seem to hinge on fine points of computer
+architecture. (See Appel 1987 and Miller and Rozas 1994 for opposing
+views on this issue.)
+
+ (3) The variable `all-regs' is bound to the list of names of all the
+registers:
+
+ (define all-regs '(env proc val argl continue))
+
+\1f
+File: sicp.info, Node: 5-5-4, Next: 5-5-5, Prev: 5-5-3, Up: 5-5
+
+5.5.4 Combining Instruction Sequences
+-------------------------------------
+
+This section describes the details on how instruction sequences are
+represented and combined. Recall from section *Note 5-5-1:: that an
+instruction sequence is represented as a list of the registers needed,
+the registers modified, and the actual instructions. We will also
+consider a label (symbol) to be a degenerate case of an instruction
+sequence, which doesn't need or modify any registers. So to determine
+the registers needed and modified by instruction sequences we use the
+selectors
+
+ (define (registers-needed s)
+ (if (symbol? s) '() (car s)))
+
+ (define (registers-modified s)
+ (if (symbol? s) '() (cadr s)))
+
+ (define (statements s)
+ (if (symbol? s) (list s) (caddr s)))
+
+and to determine whether a given sequence needs or modifies a given
+register we use the predicates
+
+ (define (needs-register? seq reg)
+ (memq reg (registers-needed seq)))
+
+ (define (modifies-register? seq reg)
+ (memq reg (registers-modified seq)))
+
+ In terms of these predicates and selectors, we can implement the
+various instruction sequence combiners used throughout the compiler.
+
+ The basic combiner is `append-instruction-sequences'. This takes as
+arguments an arbitrary number of instruction sequences that are to be
+executed sequentially and returns an instruction sequence whose
+statements are the statements of all the sequences appended together.
+The subtle point is to determine the registers that are needed and
+modified by the resulting sequence. It modifies those registers that
+are modified by any of the sequences; it needs those registers that
+must be initialized before the first sequence can be run (the registers
+needed by the first sequence), together with those registers needed by
+any of the other sequences that are not initialized (modified) by
+sequences preceding it.
+
+ The sequences are appended two at a time by `append-2-sequences'.
+This takes two instruction sequences `seq1' and `seq2' and returns the
+instruction sequence whose statements are the statements of `seq1'
+followed by the statements of `seq2', whose modified registers are those
+registers that are modified by either `seq1' or `seq2', and whose
+needed registers are the registers needed by `seq1' together with those
+registers needed by `seq2' that are not modified by `seq1'. (In terms
+of set operations, the new set of needed registers is the union of the
+set of registers needed by `seq1' with the set difference of the
+registers needed by `seq2' and the registers modified by `seq1'.) Thus,
+`append-instruction-sequences' is implemented as follows:
+
+ (define (append-instruction-sequences . seqs)
+ (define (append-2-sequences seq1 seq2)
+ (make-instruction-sequence
+ (list-union (registers-needed seq1)
+ (list-difference (registers-needed seq2)
+ (registers-modified seq1)))
+ (list-union (registers-modified seq1)
+ (registers-modified seq2))
+ (append (statements seq1) (statements seq2))))
+ (define (append-seq-list seqs)
+ (if (null? seqs)
+ (empty-instruction-sequence)
+ (append-2-sequences (car seqs)
+ (append-seq-list (cdr seqs)))))
+ (append-seq-list seqs))
+
+ This procedure uses some simple operations for manipulating sets
+represented as lists, similar to the (unordered) set representation
+described in section *Note 2-3-3:::
+
+ (define (list-union s1 s2)
+ (cond ((null? s1) s2)
+ ((memq (car s1) s2) (list-union (cdr s1) s2))
+ (else (cons (car s1) (list-union (cdr s1) s2)))))
+
+ (define (list-difference s1 s2)
+ (cond ((null? s1) '())
+ ((memq (car s1) s2) (list-difference (cdr s1) s2))
+ (else (cons (car s1)
+ (list-difference (cdr s1) s2)))))
+
+ `Preserving', the second major instruction sequence combiner, takes
+a list of registers `regs' and two instruction sequences `seq1' and
+`seq2' that are to be executed sequentially. It returns an instruction
+sequence whose statements are the statements of `seq1' followed by the
+statements of `seq2', with appropriate `save' and `restore'
+instructions around `seq1' to protect the registers in `regs' that are
+modified by `seq1' but needed by `seq2'. To accomplish this,
+`preserving' first creates a sequence that has the required `save's
+followed by the statements of `seq1' followed by the required
+`restore's. This sequence needs the registers being saved and restored
+in addition to the registers needed by `seq1', and modifies the
+registers modified by `seq1' except for the ones being saved and
+restored. This augmented sequence and `seq2' are then appended in the
+usual way. The following procedure implements this strategy
+recursively, walking down the list of registers to be preserved:(1)
+
+ (define (preserving regs seq1 seq2)
+ (if (null? regs)
+ (append-instruction-sequences seq1 seq2)
+ (let ((first-reg (car regs)))
+ (if (and (needs-register? seq2 first-reg)
+ (modifies-register? seq1 first-reg))
+ (preserving (cdr regs)
+ (make-instruction-sequence
+ (list-union (list first-reg)
+ (registers-needed seq1))
+ (list-difference (registers-modified seq1)
+ (list first-reg))
+ (append `((save ,first-reg))
+ (statements seq1)
+ `((restore ,first-reg))))
+ seq2)
+ (preserving (cdr regs) seq1 seq2)))))
+
+ Another sequence combiner, `tack-on-instruction-sequence', is used by
+`compile-lambda' to append a procedure body to another sequence.
+Because the procedure body is not "in line" to be executed as part of
+the combined sequence, its register use has no impact on the register
+use of the sequence in which it is embedded. We thus ignore the
+procedure body's sets of needed and modified registers when we tack it
+onto the other sequence.
+
+ (define (tack-on-instruction-sequence seq body-seq)
+ (make-instruction-sequence
+ (registers-needed seq)
+ (registers-modified seq)
+ (append (statements seq) (statements body-seq))))
+
+ `Compile-if' and `compile-procedure-call' use a special combiner
+called `parallel-instruction-sequences' to append the two alternative
+branches that follow a test. The two branches will never be executed
+sequentially; for any particular evaluation of the test, one branch or
+the other will be entered. Because of this, the registers needed by
+the second branch are still needed by the combined sequence, even if
+these are modified by the first branch.
+
+ (define (parallel-instruction-sequences seq1 seq2)
+ (make-instruction-sequence
+ (list-union (registers-needed seq1)
+ (registers-needed seq2))
+ (list-union (registers-modified seq1)
+ (registers-modified seq2))
+ (append (statements seq1) (statements seq2))))
+
+ ---------- Footnotes ----------
+
+ (1) Note that `preserving' calls `append' with three arguments.
+Though the definition of `append' shown in this book accepts only two
+arguments, Scheme standardly provides an `append' procedure that takes
+an arbitrary number of arguments.
+
+\1f
+File: sicp.info, Node: 5-5-5, Next: 5-5-6, Prev: 5-5-4, Up: 5-5
+
+5.5.5 An Example of Compiled Code
+---------------------------------
+
+Now that we have seen all the elements of the compiler, let us examine
+an example of compiled code to see how things fit together. We will
+compile the definition of a recursive `factorial' procedure by calling
+`compile':
+
+ (compile
+ '(define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n)))
+ 'val
+ 'next)
+
+ We have specified that the value of the `define' expression should be
+placed in the `val' register. We don't care what the compiled code does
+after executing the `define', so our choice of `next' as the linkage
+descriptor is arbitrary.
+
+ `Compile' determines that the expression is a definition, so it calls
+`compile-definition' to compile code to compute the value to be assigned
+(targeted to `val'), followed by code to install the definition,
+followed by code to put the value of the `define' (which is the symbol
+`ok') into the target register, followed finally by the linkage code.
+`Env' is preserved around the computation of the value, because it is
+needed in order to install the definition. Because the linkage is
+`next', there is no linkage code in this case. The skeleton of the
+compiled code is thus
+
+ <_save `env' if modified by code to compute value_>
+ <_compilation of definition value, target `val', linkage `next'_>
+ <_restore `env' if saved above_>
+ (perform (op define-variable!)
+ (const factorial)
+ (reg val)
+ (reg env))
+ (assign val (const ok))
+
+ The expression that is to be compiled to produce the value for the
+variable `factorial' is a `lambda' expression whose value is the
+procedure that computes factorials. `Compile' handles this by calling
+`compile-lambda', which compiles the procedure body, labels it as a new
+entry point, and generates the instruction that will combine the
+procedure body at the new entry point with the run-time environment and
+assign the result to `val'. The sequence then skips around the
+compiled procedure code, which is inserted at this point. The
+procedure code itself begins by extending the procedure's definition
+environment by a frame that binds the formal parameter `n' to the
+procedure argument. Then comes the actual procedure body. Since this
+code for the value of the variable doesn't modify the `env' register,
+the optional `save' and `restore' shown above aren't generated. (The
+procedure code at `entry2' isn't executed at this point, so its use of
+`env' is irrelevant.) Therefore, the skeleton for the compiled code
+becomes
+
+ (assign val (op make-compiled-procedure)
+ (label entry2)
+ (reg env))
+ (goto (label after-lambda1))
+ entry2
+ (assign env (op compiled-procedure-env) (reg proc))
+ (assign env (op extend-environment)
+ (const (n))
+ (reg argl)
+ (reg env))
+ <_compilation of procedure body_>
+ after-lambda1
+ (perform (op define-variable!)
+ (const factorial)
+ (reg val)
+ (reg env))
+ (assign val (const ok))
+
+ A procedure body is always compiled (by `compile-lambda-body') as a
+sequence with target `val' and linkage `return'. The sequence in this
+case consists of a single `if' expression:
+
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n))
+
+ `Compile-if' generates code that first computes the predicate
+(targeted to `val'), then checks the result and branches around the
+true branch if the predicate is false. `Env' and `continue' are
+preserved around the predicate code, since they may be needed for the
+rest of the `if' expression. Since the `if' expression is the final
+expression (and only expression) in the sequence making up the
+procedure body, its target is `val' and its linkage is `return', so the
+true and false branches are both compiled with target `val' and linkage
+`return'. (That is, the value of the conditional, which is the value
+computed by either of its branches, is the value of the procedure.)
+
+ <_save `continue', `env' if modified by predicate and needed by branches_>
+ <_compilation of predicate, target `val', linkage `next'_>
+ <_restore `continue', `env' if saved above_>
+ (test (op false?) (reg val))
+ (branch (label false-branch4))
+ true-branch5
+ <_compilation of true branch, target `val', linkage `return'_>
+ false-branch4
+ <_compilation of false branch, target `val', linkage `return'_>
+ after-if3
+
+ The predicate `(= n 1)' is a procedure call. This looks up the
+operator (the symbol `=') and places this value in `proc'. It then
+assembles the arguments `1' and the value of `n' into `argl'. Then it
+tests whether `proc' contains a primitive or a compound procedure, and
+dispatches to a primitive branch or a compound branch accordingly. Both
+branches resume at the `after-call' label. The requirements to preserve
+registers around the evaluation of the operator and operands don't
+result in any saving of registers, because in this case those
+evaluations don't modify the registers in question.
+
+ (assign proc
+ (op lookup-variable-value) (const =) (reg env))
+ (assign val (const 1))
+ (assign argl (op list) (reg val))
+ (assign val (op lookup-variable-value) (const n) (reg env))
+ (assign argl (op cons) (reg val) (reg argl))
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch17))
+ compiled-branch16
+ (assign continue (label after-call15))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch17
+ (assign val (op apply-primitive-procedure)
+ (reg proc)
+ (reg argl))
+ after-call15
+
+ The true branch, which is the constant 1, compiles (with target
+`val' and linkage `return') to
+
+ (assign val (const 1))
+ (goto (reg continue))
+
+ The code for the false branch is another a procedure call, where the
+procedure is the value of the symbol `*', and the arguments are `n' and
+the result of another procedure call (a call to `factorial'). Each of
+these calls sets up `proc' and `argl' and its own primitive and compound
+branches. *Note Figure 5-17:: shows the complete compilation of the
+definition of the `factorial' procedure. Notice that the possible
+`save' and `restore' of `continue' and `env' around the predicate, shown
+above, are in fact generated, because these registers are modified by
+the procedure call in the predicate and needed for the procedure call
+and the `return' linkage in the branches.
+
+ *Exercise 5.33:* Consider the following definition of a factorial
+ procedure, which is slightly different from the one given above:
+
+ (define (factorial-alt n)
+ (if (= n 1)
+ 1
+ (* n (factorial-alt (- n 1)))))
+
+ Compile this procedure and compare the resulting code with that
+ produced for `factorial'. Explain any differences you find. Does
+ either program execute more efficiently than the other?
+
+ *Exercise 5.34:* Compile the iterative factorial procedure
+
+ (define (factorial n)
+ (define (iter product counter)
+ (if (> counter n)
+ product
+ (iter (* counter product)
+ (+ counter 1))))
+ (iter 1 1))
+
+ Annotate the resulting code, showing the essential difference
+ between the code for iterative and recursive versions of
+ `factorial' that makes one process build up stack space and the
+ other run in constant stack space.
+
+ *Figure 5.17:* Compilation of the definition of the `factorial'
+ procedure
+
+ ;; construct the procedure and skip over code for the procedure body
+ (assign val
+ (op make-compiled-procedure) (label entry2) (reg env))
+ (goto (label after-lambda1))
+
+ entry2 ; calls to `factorial' will enter here
+ (assign env (op compiled-procedure-env) (reg proc))
+ (assign env
+ (op extend-environment) (const (n)) (reg argl) (reg env))
+ ;; begin actual procedure body
+ (save continue)
+ (save env)
+
+ ;; compute `(= n 1)'
+ (assign proc (op lookup-variable-value) (const =) (reg env))
+ (assign val (const 1))
+ (assign argl (op list) (reg val))
+ (assign val (op lookup-variable-value) (const n) (reg env))
+ (assign argl (op cons) (reg val) (reg argl))
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch17))
+ compiled-branch16
+ (assign continue (label after-call15))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch17
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+
+ after-call15 ; `val' now contains result of `(= n 1)'
+ (restore env)
+ (restore continue)
+ (test (op false?) (reg val))
+ (branch (label false-branch4))
+ true-branch5 ; return 1
+ (assign val (const 1))
+ (goto (reg continue))
+
+ false-branch4
+ ;; compute and return `(* (factorial (- n 1)) n)'
+ (assign proc (op lookup-variable-value) (const *) (reg env))
+ (save continue)
+ (save proc) ; save `*' procedure
+ (assign val (op lookup-variable-value) (const n) (reg env))
+ (assign argl (op list) (reg val))
+ (save argl) ; save partial argument list for `*'
+
+ ;; compute `(factorial (- n 1))', which is the other argument for `*'
+ (assign proc
+ (op lookup-variable-value) (const factorial) (reg env))
+ (save proc) ; save `factorial' procedure
+
+ ;; compute `(- n 1)', which is the argument for `factorial'
+ (assign proc (op lookup-variable-value) (const -) (reg env))
+ (assign val (const 1))
+ (assign argl (op list) (reg val))
+ (assign val (op lookup-variable-value) (const n) (reg env))
+ (assign argl (op cons) (reg val) (reg argl))
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch8))
+ compiled-branch7
+ (assign continue (label after-call6))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch8
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+
+ after-call6 ; `val' now contains result of `(- n 1)'
+ (assign argl (op list) (reg val))
+ (restore proc) ; restore `factorial'
+ ;; apply `factorial'
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch11))
+ compiled-branch10
+ (assign continue (label after-call9))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch11
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+
+ after-call9 ; `val' now contains result of `(factorial (- n 1))'
+ (restore argl) ; restore partial argument list for `*'
+ (assign argl (op cons) (reg val) (reg argl))
+ (restore proc) ; restore `*'
+ (restore continue)
+ ;; apply `*' and return its value
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch14))
+ compiled-branch13
+ ;; note that a compound procedure here is called tail-recursively
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch14
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+ (goto (reg continue))
+ after-call12
+ after-if3
+ after-lambda1
+ ;; assign the procedure to the variable `factorial'
+ (perform
+ (op define-variable!) (const factorial) (reg val) (reg env))
+ (assign val (const ok))
+
+
+ *Exercise 5.35:* What expression was compiled to produce the code
+ shown in *Note Figure 5-18::?
+
+ *Figure 5.18:* An example of compiler output. See *Note Exercise
+ 5-35::.
+
+ (assign val (op make-compiled-procedure) (label entry16)
+ (reg env))
+ (goto (label after-lambda15))
+ entry16
+ (assign env (op compiled-procedure-env) (reg proc))
+ (assign env
+ (op extend-environment) (const (x)) (reg argl) (reg env))
+ (assign proc (op lookup-variable-value) (const +) (reg env))
+ (save continue)
+ (save proc)
+ (save env)
+ (assign proc (op lookup-variable-value) (const g) (reg env))
+ (save proc)
+ (assign proc (op lookup-variable-value) (const +) (reg env))
+ (assign val (const 2))
+ (assign argl (op list) (reg val))
+ (assign val (op lookup-variable-value) (const x) (reg env))
+ (assign argl (op cons) (reg val) (reg argl))
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch19))
+ compiled-branch18
+ (assign continue (label after-call17))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch19
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+ after-call17
+ (assign argl (op list) (reg val))
+ (restore proc)
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch22))
+ compiled-branch21
+ (assign continue (label after-call20))
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch22
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+
+ after-call20
+ (assign argl (op list) (reg val))
+ (restore env)
+ (assign val (op lookup-variable-value) (const x) (reg env))
+ (assign argl (op cons) (reg val) (reg argl))
+ (restore proc)
+ (restore continue)
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-branch25))
+ compiled-branch24
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+ primitive-branch25
+ (assign val (op apply-primitive-procedure) (reg proc) (reg argl))
+ (goto (reg continue))
+ after-call23
+ after-lambda15
+ (perform (op define-variable!) (const f) (reg val) (reg env))
+ (assign val (const ok))
+
+
+ *Exercise 5.36:* What order of evaluation does our compiler
+ produce for operands of a combination? Is it left-to-right,
+ right-to-left, or some other order? Where in the compiler is this
+ order determined? Modify the compiler so that it produces some
+ other order of evaluation. (See the discussion of order of
+ evaluation for the explicit-control evaluator in section *Note
+ 5-4-1::.) How does changing the order of operand evaluation
+ affect the efficiency of the code that constructs the argument
+ list?
+
+ *Exercise 5.37:* One way to understand the compiler's `preserving'
+ mechanism for optimizing stack usage is to see what extra
+ operations would be generated if we did not use this idea. Modify
+ `preserving' so that it always generates the `save' and `restore'
+ operations. Compile some simple expressions and identify the
+ unnecessary stack operations that are generated. Compare the code
+ to that generated with the `preserving' mechanism intact.
+
+ *Exercise 5.38:* Our compiler is clever about avoiding unnecessary
+ stack operations, but it is not clever at all when it comes to
+ compiling calls to the primitive procedures of the language in
+ terms of the primitive operations supplied by the machine. For
+ example, consider how much code is compiled to compute `(+ a 1)':
+ The code sets up an argument list in `argl', puts the primitive
+ addition procedure (which it finds by looking up the symbol `+' in
+ the environment) into `proc', and tests whether the procedure is
+ primitive or compound. The compiler always generates code to
+ perform the test, as well as code for primitive and compound
+ branches (only one of which will be executed). We have not shown
+ the part of the controller that implements primitives, but we
+ presume that these instructions make use of primitive arithmetic
+ operations in the machine's data paths. Consider how much less
+ code would be generated if the compiler could "open-code"
+ primitives--that is, if it could generate code to directly use
+ these primitive machine operations. The expression `(+ a 1)'
+ might be compiled into something as simple as (1)
+
+ (assign val (op lookup-variable-value) (const a) (reg env))
+ (assign val (op +) (reg val) (const 1))
+
+ In this exercise we will extend our compiler to support open
+ coding of selected primitives. Special-purpose code will be
+ generated for calls to these primitive procedures instead of the
+ general procedure-application code. In order to support this, we
+ will augment our machine with special argument registers `arg1'
+ and `arg2'. The primitive arithmetic operations of the machine
+ will take their inputs from `arg1' and `arg2'. The results may be
+ put into `val', `arg1', or `arg2'.
+
+ The compiler must be able to recognize the application of an
+ open-coded primitive in the source program. We will augment the
+ dispatch in the `compile' procedure to recognize the names of
+ these primitives in addition to the reserved words (the special
+ forms) it currently recognizes.(2) For each special form our
+ compiler has a code generator. In this exercise we will construct
+ a family of code generators for the open-coded primitives.
+
+ a. The open-coded primitives, unlike the special forms, all need
+ their operands evaluated. Write a code generator
+ `spread-arguments' for use by all the open-coding code
+ generators. `Spread-arguments' should take an operand list
+ and compile the given operands targeted to successive
+ argument registers. Note that an operand may contain a call
+ to an open-coded primitive, so argument registers will have
+ to be preserved during operand evaluation.
+
+ b. For each of the primitive procedures `=', `*', `-', and `+',
+ write a code generator that takes a combination with that
+ operator, together with a target and a linkage descriptor,
+ and produces code to spread the arguments into the registers
+ and then perform the operation targeted to the given target
+ with the given linkage. You need only handle expressions
+ with two operands. Make `compile' dispatch to these code
+ generators.
+
+ c. Try your new compiler on the `factorial' example. Compare
+ the resulting code with the result produced without open
+ coding.
+
+ d. Extend your code generators for `+' and `*' so that they can
+ handle expressions with arbitrary numbers of operands. An
+ expression with more than two operands will have to be
+ compiled into a sequence of operations, each with only two
+ inputs.
+
+
+ ---------- Footnotes ----------
+
+ (1) We have used the same symbol `+' here to denote both the
+source-language procedure and the machine operation. In general there
+will not be a one-to-one correspondence between primitives of the
+source language and primitives of the machine.
+
+ (2) Making the primitives into reserved words is in general a bad
+idea, since a user cannot then rebind these names to different
+procedures. Moreover, if we add reserved words to a compiler that is
+in use, existing programs that define procedures with these names will
+stop working. See *Note Exercise 5-44:: for ideas on how to avoid this
+problem.
+
+\1f
+File: sicp.info, Node: 5-5-6, Next: 5-5-7, Prev: 5-5-5, Up: 5-5
+
+5.5.6 Lexical Addressing
+------------------------
+
+One of the most common optimizations performed by compilers is the
+optimization of variable lookup. Our compiler, as we have implemented
+it so far, generates code that uses the `lookup-variable-value'
+operation of the evaluator machine. This searches for a variable by
+comparing it with each variable that is currently bound, working frame
+by frame outward through the run-time environment. This search can be
+expensive if the frames are deeply nested or if there are many
+variables. For example, consider the problem of looking up the value
+of `x' while evaluating the expression `(* x y z)' in an application of
+the procedure that is returned by
+
+ (let ((x 3) (y 4))
+ (lambda (a b c d e)
+ (let ((y (* a b x))
+ (z (+ c d x)))
+ (* x y z))))
+
+ Since a `let' expression is just syntactic sugar for a `lambda'
+combination, this expression is equivalent to
+
+ ((lambda (x y)
+ (lambda (a b c d e)
+ ((lambda (y z) (* x y z))
+ (* a b x)
+ (+ c d x))))
+ 3
+ 4)
+
+ Each time `lookup-variable-value' searches for `x', it must determine
+that the symbol `x' is not `eq?' to `y' or `z' (in the first frame),
+nor to `a', `b', `c', `d', or `e' (in the second frame). We will
+assume, for the moment, that our programs do not use `define'--that
+variables are bound only with `lambda'. Because our language is
+lexically scoped, the run-time environment for any expression will have
+a structure that parallels the lexical structure of the program in
+which the expression appears.(1) Thus, the compiler can know, when it
+analyzes the above expression, that each time the procedure is applied
+the variable `x' in `(* x y z)' will be found two frames out from the
+current frame and will be the first variable in that frame.
+
+ We can exploit this fact by inventing a new kind of variable-lookup
+operation, `lexical-address-lookup', that takes as arguments an
+environment and a "lexical address" that consists of two numbers: a number
+"frame number", which specifies how many frames to pass over, and a "displacement
+number", which specifies how many variables to pass over in that frame.
+`Lexical-address-lookup' will produce the value of the variable stored
+at that lexical address relative to the current environment. If we add
+the `lexical-address-lookup' operation to our machine, we can make the
+compiler generate code that references variables using this operation,
+rather than `lookup-variable-value'. Similarly, our compiled code can
+use a new `lexical-address-set!' operation instead of
+`set-variable-value!'.
+
+ In order to generate such code, the compiler must be able to
+determine the lexical address of a variable it is about to compile a
+reference to. The lexical address of a variable in a program depends
+on where one is in the code. For example, in the following program,
+the address of `x' in expression <E1> is (2,0)--two frames back and the
+first variable in the frame. At that point `y' is at address (0,0) and
+`c' is at address (1,2). In expression <E2>, `x' is at (1,0), `y' is
+at (1,1), and `c' is at (0,2).
+
+ ((lambda (x y)
+ (lambda (a b c d e)
+ ((lambda (y z) <E1>)
+ <E2>
+ (+ c d x))))
+ 3
+ 4)
+
+ One way for the compiler to produce code that uses lexical
+addressing is to maintain a data structure called a "compile-time
+environment". This keeps track of which variables will be at which
+positions in which frames in the run-time environment when a particular
+variable-access operation is executed. The compile-time environment is
+a list of frames, each containing a list of variables. (There will of
+course be no values bound to the variables, since values are not
+computed at compile time.) The compile-time environment becomes an
+additional argument to `compile' and is passed along to each code
+generator. The top-level call to `compile' uses an empty compile-time
+environment. When a `lambda' body is compiled, `compile-lambda-body'
+extends the compile-time environment by a frame containing the
+procedure's parameters, so that the sequence making up the body is
+compiled with that extended environment. At each point in the
+compilation, `compile-variable' and `compile-assignment' use the
+compile-time environment in order to generate the appropriate lexical
+addresses.
+
+ *Note Exercise 5-39:: through *Note Exercise 5-43:: describe how to
+complete this sketch of the lexical-addressing strategy in order to
+incorporate lexical lookup into the compiler. *Note Exercise 5-44::
+describes another use for the compile-time environment.
+
+ *Exercise 5.39:* Write a procedure `lexical-address-lookup' that
+ implements the new lookup operation. It should take two
+ arguments--a lexical address and a run-time environment--and
+ return the value of the variable stored at the specified lexical
+ address. `Lexical-address-lookup' should signal an error if the
+ value of the variable is the symbol `*unassigned*'.(2) Also write
+ a procedure `lexical-address-set!' that implements the operation
+ that changes the value of the variable at a specified lexical
+ address.
+
+ *Exercise 5.40:* Modify the compiler to maintain the compile-time
+ environment as described above. That is, add a
+ compile-time-environment argument to `compile' and the various code
+ generators, and extend it in `compile-lambda-body'.
+
+ *Exercise 5.41:* Write a procedure `find-variable' that takes as
+ arguments a variable and a compile-time environment and returns
+ the lexical address of the variable with respect to that
+ environment. For example, in the program fragment that is shown
+ above, the compile-time environment during the compilation of
+ expression <E1> is `((y z) (a b c d e) (x y))'. `Find-variable'
+ should produce
+
+ (find-variable 'c '((y z) (a b c d e) (x y)))
+ (1 2)
+
+ (find-variable 'x '((y z) (a b c d e) (x y)))
+ (2 0)
+
+ (find-variable 'w '((y z) (a b c d e) (x y)))
+ not-found
+
+ *Exercise 5.42:* Using `find-variable' from *Note Exercise 5-41::,
+ rewrite `compile-variable' and `compile-assignment' to output
+ lexical-address instructions. In cases where `find-variable'
+ returns `not-found' (that is, where the variable is not in the
+ compile-time environment), you should have the code generators use
+ the evaluator operations, as before, to search for the binding.
+ (The only place a variable that is not found at compile time can
+ be is in the global environment, which is part of the run-time
+ environment but is not part of the compile-time environment.(3)
+ Thus, if you wish, you may have the evaluator operations look
+ directly in the global environment, which can be obtained with the
+ operation `(op get-global-environment)', instead of having them
+ search the whole run-time environment found in `env'.) Test the
+ modified compiler on a few simple cases, such as the nested
+ `lambda' combination at the beginning of this section.
+
+ *Exercise 5.43:* We argued in section *Note 4-1-6:: that internal
+ definitions for block structure should not be considered "real"
+ `define's. Rather, a procedure body should be interpreted as if
+ the internal variables being defined were installed as ordinary
+ `lambda' variables initialized to their correct values using
+ `set!'. Section *Note 4-1-6:: and *Note Exercise 4-16:: showed
+ how to modify the metacircular interpreter to accomplish this by
+ scanning out internal definitions. Modify the compiler to perform
+ the same transformation before it compiles a procedure body.
+
+ *Exercise 5.44:* In this section we have focused on the use of the
+ compile-time environment to produce lexical addresses. But there
+ are other uses for compile-time environments. For instance, in
+ *Note Exercise 5-38:: we increased the efficiency of compiled code
+ by open-coding primitive procedures. Our implementation treated
+ the names of open-coded procedures as reserved words. If a
+ program were to rebind such a name, the mechanism described in
+ *Note Exercise 5-38:: would still open-code it as a primitive,
+ ignoring the new binding. For example, consider the procedure
+
+ (lambda (+ * a b x y)
+ (+ (* a x) (* b y)))
+
+ which computes a linear combination of `x' and `y'. We might call
+ it with arguments `+matrix', `*matrix', and four matrices, but the
+ open-coding compiler would still open-code the `+' and the `*' in
+ `(+ (* a x) (* b y))' as primitive `+' and `*'. Modify the
+ open-coding compiler to consult the compile-time environment in
+ order to compile the correct code for expressions involving the
+ names of primitive procedures. (The code will work correctly as
+ long as the program does not `define' or `set!' these names.)
+
+ ---------- Footnotes ----------
+
+ (1) This is not true if we allow internal definitions, unless we
+scan them out. See *Note Exercise 5-43::.
+
+ (2) This is the modification to variable lookup required if we
+implement the scanning method to eliminate internal definitions (*Note
+Exercise 5-43::). We will need to eliminate these definitions in order
+for lexical addressing to work.
+
+ (3) Lexical addresses cannot be used to access variables in the
+global environment, because these names can be defined and redefined
+interactively at any time. With internal definitions scanned out, as
+in *Note Exercise 5-43::, the only definitions the compiler sees are
+those at top level, which act on the global environment. Compilation
+of a definition does not cause the defined name to be entered in the
+compile-time environment.
+
+\1f
+File: sicp.info, Node: 5-5-7, Prev: 5-5-6, Up: 5-5
+
+5.5.7 Interfacing Compiled Code to the Evaluator
+------------------------------------------------
+
+We have not yet explained how to load compiled code into the evaluator
+machine or how to run it. We will assume that the
+explicit-control-evaluator machine has been defined as in section *Note
+5-4-4::, with the additional operations specified in footnote *Note
+Footnote 38::. We will implement a procedure `compile-and-go' that
+compiles a Scheme expression, loads the resulting object code into the
+evaluator machine, and causes the machine to run the code in the
+evaluator global environment, print the result, and enter the
+evaluator's driver loop. We will also modify the evaluator so that
+interpreted expressions can call compiled procedures as well as
+interpreted ones. We can then put a compiled procedure into the
+machine and use the evaluator to call it:
+
+ (compile-and-go
+ '(define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n))))
+ ;;; EC-Eval value:
+ ok
+
+ ;;; EC-Eval input:
+ (factorial 5)
+ ;;; EC-Eval value:
+ 120
+
+ To allow the evaluator to handle compiled procedures (for example,
+to evaluate the call to `factorial' above), we need to change the code
+at `apply-dispatch' (section *Note 5-4-1::) so that it recognizes
+compiled procedures (as distinct from compound or primitive procedures)
+and transfers control directly to the entry point of the compiled
+code:(1)
+
+ apply-dispatch
+ (test (op primitive-procedure?) (reg proc))
+ (branch (label primitive-apply))
+ (test (op compound-procedure?) (reg proc))
+ (branch (label compound-apply))
+ (test (op compiled-procedure?) (reg proc))
+ (branch (label compiled-apply))
+ (goto (label unknown-procedure-type))
+
+ compiled-apply
+ (restore continue)
+ (assign val (op compiled-procedure-entry) (reg proc))
+ (goto (reg val))
+
+ Note the restore of `continue' at `compiled-apply'. Recall that the
+evaluator was arranged so that at `apply-dispatch', the continuation
+would be at the top of the stack. The compiled code entry point, on
+the other hand, expects the continuation to be in `continue', so
+`continue' must be restored before the compiled code is executed.
+
+ To enable us to run some compiled code when we start the evaluator
+machine, we add a `branch' instruction at the beginning of the
+evaluator machine, which causes the machine to go to a new entry point
+if the `flag' register is set.(2)
+
+ (branch (label external-entry)) ; branches if `flag' is set
+ read-eval-print-loop
+ (perform (op initialize-stack))
+ ...
+
+ `External-entry' assumes that the machine is started with `val'
+containing the location of an instruction sequence that puts a result
+into `val' and ends with `(goto (reg continue))'. Starting at this
+entry point jumps to the location designated by `val', but first assigns
+`continue' so that execution will return to `print-result', which
+prints the value in `val' and then goes to the beginning of the
+evaluator's read-eval-print loop.(3)
+
+ external-entry
+ (perform (op initialize-stack))
+ (assign env (op get-global-environment))
+ (assign continue (label print-result))
+ (goto (reg val))
+
+ Now we can use the following procedure to compile a procedure
+definition, execute the compiled code, and run the read-eval-print loop
+so we can try the procedure. Because we want the compiled code to
+return to the location in `continue' with its result in `val', we
+compile the expression with a target of `val' and a linkage of
+`return'. In order to transform the object code produced by the
+compiler into executable instructions for the evaluator register
+machine, we use the procedure `assemble' from the register-machine
+simulator (section *Note 5-2-2::). We then initialize the `val'
+register to point to the list of instructions, set the `flag' so that
+the evaluator will go to `external-entry', and start the evaluator.
+
+ (define (compile-and-go expression)
+ (let ((instructions
+ (assemble (statements
+ (compile expression 'val 'return))
+ eceval)))
+ (set! the-global-environment (setup-environment))
+ (set-register-contents! eceval 'val instructions)
+ (set-register-contents! eceval 'flag true)
+ (start eceval)))
+
+ If we have set up stack monitoring, as at the end of section *Note
+5-4-4::, we can examine the stack usage of compiled code:
+
+ (compile-and-go
+ '(define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n))))
+
+ (total-pushes = 0 maximum-depth = 0)
+ ;;; EC-Eval value:
+ ok
+
+ ;;; EC-Eval input:
+ (factorial 5)
+ (total-pushes = 31 maximum-depth = 14)
+ ;;; EC-Eval value:
+ 120
+
+ Compare this example with the evaluation of `(factorial 5)' using the
+interpreted version of the same procedure, shown at the end of section
+*Note 5-4-4::. The interpreted version required 144 pushes and a
+maximum stack depth of 28. This illustrates the optimization that
+results from our compilation strategy.
+
+Interpretation and compilation
+..............................
+
+With the programs in this section, we can now experiment with the
+alternative execution strategies of interpretation and compilation.(4)
+An interpreter raises the machine to the level of the user program; a
+compiler lowers the user program to the level of the machine language.
+We can regard the Scheme language (or any programming language) as a
+coherent family of abstractions erected on the machine language.
+Interpreters are good for interactive program development and debugging
+because the steps of program execution are organized in terms of these
+abstractions, and are therefore more intelligible to the programmer.
+Compiled code can execute faster, because the steps of program
+execution are organized in terms of the machine language, and the
+compiler is free to make optimizations that cut across the higher-level
+abstractions.(5)
+
+ The alternatives of interpretation and compilation also lead to
+different strategies for porting languages to new computers. Suppose
+that we wish to implement Lisp for a new machine. One strategy is to
+begin with the explicit-control evaluator of section *Note 5-4:: and
+translate its instructions to instructions for the new machine. A
+different strategy is to begin with the compiler and change the code
+generators so that they generate code for the new machine. The second
+strategy allows us to run any Lisp program on the new machine by first
+compiling it with the compiler running on our original Lisp system, and
+linking it with a compiled version of the run-time library.(6) Better
+yet, we can compile the compiler itself, and run this on the new
+machine to compile other Lisp programs.(7) Or we can compile one of
+the interpreters of section *Note 4-1:: to produce an interpreter that
+runs on the new machine.
+
+ *Exercise 5.45:* By comparing the stack operations used by
+ compiled code to the stack operations used by the evaluator for the
+ same computation, we can determine the extent to which the
+ compiler optimizes use of the stack, both in speed (reducing the
+ total number of stack operations) and in space (reducing the
+ maximum stack depth). Comparing this optimized stack use to the
+ performance of a special-purpose machine for the same computation
+ gives some indication of the quality of the compiler.
+
+ a. *Note Exercise 5-27:: asked you to determine, as a function
+ of n, the number of pushes and the maximum stack depth needed
+ by the evaluator to compute n! using the recursive factorial
+ procedure given above. *Note Exercise 5-14:: asked you to do
+ the same measurements for the special-purpose factorial
+ machine shown in *Note Figure 5-11::. Now perform the same
+ analysis using the compiled `factorial' procedure.
+
+ Take the ratio of the number of pushes in the compiled
+ version to the number of pushes in the interpreted version,
+ and do the same for the maximum stack depth. Since the
+ number of operations and the stack depth used to compute n!
+ are linear in n, these ratios should approach constants as n
+ becomes large. What are these constants? Similarly, find
+ the ratios of the stack usage in the special-purpose machine
+ to the usage in the interpreted version.
+
+ Compare the ratios for special-purpose versus interpreted
+ code to the ratios for compiled versus interpreted code. You
+ should find that the special-purpose machine does much better
+ than the compiled code, since the hand-tailored controller
+ code should be much better than what is produced by our
+ rudimentary general-purpose compiler.
+
+ b. Can you suggest improvements to the compiler that would help
+ it generate code that would come closer in performance to the
+ hand-tailored version?
+
+
+ *Exercise 5.46:* Carry out an analysis like the one in *Note
+ Exercise 5-45:: to determine the effectiveness of compiling the
+ tree-recursive Fibonacci procedure
+
+ (define (fib n)
+ (if (< n 2)
+ n
+ (+ (fib (- n 1)) (fib (- n 2)))))
+
+ compared to the effectiveness of using the special-purpose
+ Fibonacci machine of *Note Figure 5-12::. (For measurement of the
+ interpreted performance, see *Note Exercise 5-29::.) For
+ Fibonacci, the time resource used is not linear in n; hence the
+ ratios of stack operations will not approach a limiting value that
+ is independent of n.
+
+ *Exercise 5.47:* This section described how to modify the
+ explicit-control evaluator so that interpreted code can call
+ compiled procedures. Show how to modify the compiler so that
+ compiled procedures can call not only primitive procedures and
+ compiled procedures, but interpreted procedures as well. This
+ requires modifying `compile-procedure-call' to handle the case of
+ compound (interpreted) procedures. Be sure to handle all the same
+ `target' and `linkage' combinations as in `compile-proc-appl'. To
+ do the actual procedure application, the code needs to jump to the
+ evaluator's `compound-apply' entry point. This label cannot be
+ directly referenced in object code (since the assembler requires
+ that all labels referenced by the code it is assembling be defined
+ there), so we will add a register called `compapp' to the
+ evaluator machine to hold this entry point, and add an instruction
+ to initialize it:
+
+ (assign compapp (label compound-apply))
+ (branch (label external-entry)) ; branches if `flag' is set
+ read-eval-print-loop
+ ...
+
+ To test your code, start by defining a procedure `f' that calls a
+ procedure `g'. Use `compile-and-go' to compile the definition of
+ `f' and start the evaluator. Now, typing at the evaluator, define
+ `g' and try to call `f'.
+
+ *Exercise 5.48:* The `compile-and-go' interface implemented in
+ this section is awkward, since the compiler can be called only
+ once (when the evaluator machine is started). Augment the
+ compiler-interpreter interface by providing a `compile-and-run'
+ primitive that can be called from within the explicit-control
+ evaluator as follows:
+
+ ;;; EC-Eval input:
+ (compile-and-run
+ '(define (factorial n)
+ (if (= n 1)
+ 1
+ (* (factorial (- n 1)) n))))
+ ;;; EC-Eval value:
+ ok
+
+ ;;; EC-Eval input:
+ (factorial 5)
+ ;;; EC-Eval value:
+ 120
+
+ *Exercise 5.49:* As an alternative to using the explicit-control
+ evaluator's read-eval-print loop, design a register machine that
+ performs a read-compile-execute-print loop. That is, the machine
+ should run a loop that reads an expression, compiles it, assembles
+ and executes the resulting code, and prints the result. This is
+ easy to run in our simulated setup, since we can arrange to call
+ the procedures `compile' and `assemble' as "register-machine
+ operations."
+
+ *Exercise 5.50:* Use the compiler to compile the metacircular
+ evaluator of section *Note 4-1:: and run this program using the
+ register-machine simulator. (To compile more than one definition
+ at a time, you can package the definitions in a `begin'.) The
+ resulting interpreter will run very slowly because of the multiple
+ levels of interpretation, but getting all the details to work is
+ an instructive exercise.
+
+ *Exercise 5.51:* Develop a rudimentary implementation of Scheme in
+ C (or some other low-level language of your choice) by translating
+ the explicit-control evaluator of section *Note 5-4:: into C. In
+ order to run this code you will need to also provide appropriate
+ storage-allocation routines and other run-time support.
+
+ *Exercise 5.52:* As a counterpoint to exercise *Note Exercise
+ 5-51::, modify the compiler so that it compiles Scheme procedures
+ into sequences of C instructions. Compile the metacircular
+ evaluator of section *Note 4-1:: to produce a Scheme interpreter
+ written in C.
+
+ ---------- Footnotes ----------
+
+ (1) Of course, compiled procedures as well as interpreted procedures
+are compound (nonprimitive). For compatibility with the terminology
+used in the explicit-control evaluator, in this section we will use
+"compound" to mean interpreted (as opposed to compiled).
+
+ (2) Now that the evaluator machine starts with a `branch', we must
+always initialize the `flag' register before starting the evaluator
+machine. To start the machine at its ordinary read-eval-print loop, we
+could use
+
+ (define (start-eceval)
+ (set! the-global-environment (setup-environment))
+ (set-register-contents! eceval 'flag false)
+ (start eceval))
+
+ (3) Since a compiled procedure is an object that the system may try
+to print, we also modify the system print operation `user-print' (from
+section *Note 4-1-4::) so that it will not attempt to print the
+components of a compiled procedure:
+
+ (define (user-print object)
+ (cond ((compound-procedure? object)
+ (display (list 'compound-procedure
+ (procedure-parameters object)
+ (procedure-body object)
+ '<procedure-env>)))
+ ((compiled-procedure? object)
+ (display '<compiled-procedure>))
+ (else (display object))))
+
+ (4) We can do even better by extending the compiler to allow
+compiled code to call interpreted procedures. See *Note Exercise
+5-47::.
+
+ (5) Independent of the strategy of execution, we incur significant
+overhead if we insist that errors encountered in execution of a user
+program be detected and signaled, rather than being allowed to kill the
+system or produce wrong answers. For example, an out-of-bounds array
+reference can be detected by checking the validity of the reference
+before performing it. The overhead of checking, however, can be many
+times the cost of the array reference itself, and a programmer should
+weigh speed against safety in determining whether such a check is
+desirable. A good compiler should be able to produce code with such
+checks, should avoid redundant checks, and should allow programmers to
+control the extent and type of error checking in the compiled code.
+
+ Compilers for popular languages, such as C and C++, put hardly any
+error-checking operations into running code, so as to make things run
+as fast as possible. As a result, it falls to programmers to
+explicitly provide error checking. Unfortunately, people often neglect
+to do this, even in critical applications where speed is not a
+constraint. Their programs lead fast and dangerous lives. For
+example, the notorious "Worm" that paralyzed the Internet in 1988
+exploited the UNIX(tm) operating system's failure to check whether the
+input buffer has overflowed in the finger daemon. (See Spafford 1989.)
+
+ (6) Of course, with either the interpretation or the compilation
+strategy we must also implement for the new machine storage allocation,
+input and output, and all the various operations that we took as
+"primitive" in our discussion of the evaluator and compiler. One
+strategy for minimizing work here is to write as many of these
+operations as possible in Lisp and then compile them for the new
+machine. Ultimately, everything reduces to a small kernel (such as
+garbage collection and the mechanism for applying actual machine
+primitives) that is hand-coded for the new machine.
+
+ (7) This strategy leads to amusing tests of correctness of the
+compiler, such as checking whether the compilation of a program on the
+new machine, using the compiled compiler, is identical with the
+compilation of the program on the original Lisp system. Tracking down
+the source of differences is fun but often frustrating, because the
+results are extremely sensitive to minuscule details.
+
+\1f
+File: sicp.info, Node: References, Next: Index, Prev: Chapter 5, Up: Top
+
+References
+**********
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+ McDermott, Drew, and Gerald Jay Sussman. 1972. Conniver reference
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+ Miller, Gary L. 1976. Riemann's Hypothesis and tests for primality.
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+
+\1f
+File: sicp.info, Node: Index, Prev: References, Up: Top
+
+Index
+*****
+
+\0\b[index\0\b]
+* Menu:
+
+* abstract: 2-1-3. (line 139)
+* abstract syntax: 4-1-1. (line 44)
+* abstraction: 2-1-2. (line 16)
+* abstraction barriers: Chapter 2. (line 118)
+* accumulator <1>: 3-1-1. (line 216)
+* accumulator: 2-2-3. (line 76)
+* acquired: 3-4-2. (line 388)
+* action: 5-1-1. (line 164)
+* additive: 2-4-3. (line 26)
+* additively <1>: 2-4. (line 51)
+* additively: Chapter 2. (line 158)
+* address: 5-3-1. (line 8)
+* address arithmetic: 5-3-1. (line 16)
+* agenda: 3-3-4. (line 345)
+* algebraic specification: 2-1-3. (line 149)
+* aliasing: 3-1-3. (line 313)
+* and-gate: 3-3-4. (line 30)
+* applicative-order: 4-2-1. (line 7)
+* applicative-order evaluation: 1-1-5. (line 127)
+* arbiter: 3-4-2. (line 633)
+* arguments: 1-1-1. (line 47)
+* assembler: 5-2-1. (line 20)
+* assertions: 4-4-1. (line 20)
+* assignment: 3-1. (line 37)
+* atomically: 3-4-2. (line 447)
+* automatic storage allocation: 5-3. (line 30)
+* average damping: 1-3-3. (line 180)
+* B-trees: 2-3-3. (line 459)
+* backbone: 3-3-3. (line 19)
+* backquote: 5-5-2. (line 279)
+* backtracks: 4-3-1. (line 82)
+* balanced: 2-2-2. (line 195)
+* barrier synchronization: 3-4-2. (line 652)
+* base: 5-3-1. (line 35)
+* Bertrand's: 3-5-2. (line 411)
+* bignum: 5-3-1. (line 97)
+* bindings: 3-2. (line 24)
+* binds: 1-1-8. (line 109)
+* binomial coefficients: 1-2-2. (line 251)
+* block structure: 1-1-8. (line 182)
+* bound variable: 1-1-8. (line 108)
+* box-and-pointer notation: 2-2. (line 9)
+* breakpoint: 5-2-4. (line 109)
+* broken heart: 5-3-2. (line 125)
+* bugs: Chapter 1. (line 38)
+* cache-coherence: 3-4-1. (line 222)
+* call-by-name <1>: 3-5-1. (line 405)
+* call-by-name: 4-2-2. (line 383)
+* call-by-name thunks: 3-5-1. (line 412)
+* call-by-need <1>: 3-5-1. (line 411)
+* call-by-need: 4-2-2. (line 383)
+* call-by-need thunks: 3-5-1. (line 413)
+* capturing: 1-1-8. (line 123)
+* Carmichael numbers: 1-2-6. (line 305)
+* case analysis: 1-1-6. (line 20)
+* cell: 3-4-2. (line 409)
+* chronological: 4-3-1. (line 86)
+* Church numerals: 2-1-3. (line 126)
+* Church-Turing thesis: 4-1-5. (line 147)
+* clauses: 1-1-6. (line 41)
+* closed world assumption: 4-4-3. (line 153)
+* closure: Chapter 2. (line 130)
+* closure property: 2-2. (line 56)
+* code generator: 5-5-1. (line 21)
+* coerce: 2-5-2. (line 270)
+* coercion: 2-5-2. (line 63)
+* combinations: 1-1-1. (line 44)
+* comments: 2-2-3. (line 671)
+* compacting: 5-3-2. (line 309)
+* compilation: 5-5. (line 45)
+* compile-time environment: 5-5-6. (line 77)
+* composition: 1-3-4. (line 233)
+* compound data: Chapter 2. (line 37)
+* compound data object: Chapter 2. (line 64)
+* compound procedure: 1-1-4. (line 34)
+* computability: 4-1-5. (line 142)
+* computational process: Chapter 1. (line 18)
+* concurrently: 3-4. (line 38)
+* congruent modulo: 1-2-6. (line 56)
+* connectors: 3-3-5. (line 35)
+* consequent: 1-1-6. (line 48)
+* constraint networks: 3-3-5. (line 35)
+* constructors: 2-1. (line 27)
+* continuation procedures: 4-3-3. (line 27)
+* continued fraction: 1-3-3. (line 199)
+* control: 4-4-3. (line 11)
+* controller: 5-1. (line 7)
+* conventional interfaces <1>: Chapter 2. (line 133)
+* conventional interfaces: 2-2-3. (line 11)
+* current time: 3-3-4. (line 491)
+* data <1>: 2-1-3. (line 13)
+* data: Chapter 1. (line 20)
+* data abstraction <1>: Chapter 2. (line 75)
+* data abstraction: 2-1. (line 15)
+* data paths: 5-1. (line 6)
+* data-directed <1>: Chapter 2. (line 156)
+* data-directed <2>: 2-4-3. (line 42)
+* data-directed: 2-4. (line 61)
+* deadlock: 3-4-2. (line 497)
+* deadlock-recovery: 3-4-2. (line 646)
+* debug: Chapter 1. (line 53)
+* deep binding: 4-1-3. (line 227)
+* deferred: 1-2-1. (line 96)
+* delayed: 3-5-1. (line 137)
+* delayed argument: 3-5-4. (line 73)
+* delayed evaluation <1>: Chapter 3. (line 64)
+* delayed evaluation: 3-5. (line 40)
+* dense: 2-5-3. (line 252)
+* dependency-directed backtracking: 4-3-1. (line 215)
+* depth-first search: 4-3-1. (line 86)
+* deque: 3-3-2. (line 253)
+* derived expressions: 4-1-2. (line 230)
+* digital signals: 3-3-4. (line 19)
+* dispatching on type: 2-4-3. (line 7)
+* displacement number: 5-5-6. (line 49)
+* dotted-tail: 2-2-1. (line 239)
+* driver loop: 4-1-4. (line 77)
+* empty list: 2-2-1. (line 90)
+* encapsulated: 3-1-1. (line 300)
+* enclosing: 3-2. (line 26)
+* entry points: 5-1-1. (line 23)
+* enumerator: 2-2-3. (line 71)
+* environment: 1-1-2. (line 55)
+* environment model: Chapter 3. (line 56)
+* environments: 3-2. (line 21)
+* Euclid's: 1-2-5. (line 38)
+* Euclidean ring: 2-5-3. (line 684)
+* evaluating: 1-1-1. (line 9)
+* evaluator: Chapter 4. (line 56)
+* event-driven simulation: 3-3-4. (line 13)
+* evlis tail recursion: 5-4-1. (line 288)
+* execution procedure: 4-1-7. (line 30)
+* explicit-control evaluator: 5-4. (line 11)
+* expression: 1-1-1. (line 8)
+* failure continuation: 4-3-3. (line 31)
+* FIFO: 3-3-2. (line 17)
+* filter <1>: 2-2-3. (line 73)
+* filter: 1-3-1. (line 218)
+* first-class: 1-3-4. (line 200)
+* fixed point: 1-3-3. (line 102)
+* fixed-length: 2-3-4. (line 28)
+* forcing: 4-2-2. (line 22)
+* forwarding address: 5-3-2. (line 126)
+* frame <1>: 4-4-2. (line 47)
+* frame: 5-5-6. (line 48)
+* frame coordinate map: 2-2-4. (line 265)
+* framed-stack: 5-4-1. (line 266)
+* frames: 3-2. (line 23)
+* free: 1-1-8. (line 112)
+* free list: 5-3-1. (line 290)
+* front: 3-3-2. (line 13)
+* full-adder: 3-3-4. (line 127)
+* function boxes: 3-3-4. (line 21)
+* functional programming: 3-1-3. (line 18)
+* functional programming languages: 3-5-5. (line 172)
+* garbage: 5-3-2. (line 12)
+* garbage collection <1>: 5-3. (line 35)
+* garbage collection: 5-3-2. (line 27)
+* garbage collector: 3-3-1. (line 454)
+* garbage-collected: 4-2-2. (line 398)
+* generic operations: Chapter 2. (line 152)
+* generic procedures <1>: 2-3-4. (line 220)
+* generic procedures: 2-4. (line 56)
+* glitches: Chapter 1. (line 38)
+* global <1>: 3-2. (line 28)
+* global: 1-2. (line 35)
+* global environment: 1-1-2. (line 55)
+* golden ratio: 1-2-2. (line 70)
+* grammar: 4-3-2. (line 148)
+* half-adder: 3-3-4. (line 52)
+* half-interval method: 1-3-3. (line 20)
+* Halting Theorem: 4-1-5. (line 185)
+* headed list: 3-3-3. (line 21)
+* hiding principle: 3-1-1. (line 301)
+* hierarchical: 2-2. (line 60)
+* hierarchy of types: 2-5-2. (line 159)
+* higher-order procedures: 1-3. (line 39)
+* Horner's rule: 2-2-3. (line 285)
+* imperative programming: 3-1-3. (line 223)
+* indeterminates: 2-5-3. (line 39)
+* index: 5-3-1. (line 36)
+* indexing: 4-4-2. (line 421)
+* instantiated with: 4-4-1. (line 202)
+* instruction: 5-2-4. (line 80)
+* instruction execution: 5-2-1. (line 119)
+* instruction sequence: 5-5-1. (line 88)
+* instruction tracing: 5-2-4. (line 86)
+* instructions <1>: Chapter 5. (line 37)
+* instructions: 5-1-1. (line 22)
+* integerizing factor: 2-5-3. (line 534)
+* integers: 1-1. (line 54)
+* integrator: 3-5-3. (line 421)
+* interning: 5-3-1. (line 117)
+* interpreter <1>: Chapter 1. (line 74)
+* interpreter: Chapter 4. (line 56)
+* invariant quantity: 1-2-4. (line 95)
+* inverter: 3-3-4. (line 25)
+* iterative improvement: 1-3-4. (line 290)
+* iterative process: 1-2-1. (line 110)
+* k-term: 1-3-3. (line 215)
+* key: 2-3-3. (line 389)
+* labels: 5-1-1. (line 23)
+* lazy: 4-2-1. (line 13)
+* lexical address: 5-5-6. (line 48)
+* lexical addressing: 4-1-3. (line 228)
+* lexical scoping: 1-1-8. (line 192)
+* linear iterative process: 1-2-1. (line 117)
+* linear recursive process: 1-2-1. (line 105)
+* linkage descriptor: 5-5-1. (line 52)
+* list: 2-2-1. (line 404)
+* list structure: 2-2-1. (line 405)
+* list-structured: 2-1-1. (line 121)
+* list-structured memory: 5-3. (line 8)
+* local evolution: 1-2. (line 32)
+* local state variables: 3-1. (line 29)
+* location: 5-3-1. (line 8)
+* logic-programming: Chapter 4. (line 129)
+* logical and: 3-3-4. (line 33)
+* logical deductions: 4-4-1. (line 474)
+* logical or: 3-3-4. (line 37)
+* machine language: 5-5. (line 23)
+* macro: 4-1-2. (line 401)
+* map: 2-2-3. (line 74)
+* mark-sweep: 5-3-2. (line 298)
+* means of abstraction: 1-1. (line 21)
+* means of combination: 1-1. (line 18)
+* Memoization: 3-3-3. (line 258)
+* memoization: 1-2-2. (line 245)
+* memoize: 4-2-2. (line 27)
+* merge: 3-5-5. (line 206)
+* message passing <1>: 2-4-3. (line 359)
+* message passing: 2-1-3. (line 91)
+* message-passing: 3-1-1. (line 187)
+* metacircular: 4-1. (line 12)
+* Metalinguistic abstraction: Chapter 4. (line 52)
+* Miller-Rabin test: 1-2-6. (line 253)
+* modular: Chapter 3. (line 24)
+* modulo: 1-2-6. (line 59)
+* modus ponens: 4-4-3. (line 257)
+* moments in time: 3-4. (line 30)
+* Monte Carlo: 3-1-2. (line 43)
+* Monte Carlo integration: 3-1-2. (line 140)
+* mutable data: 3-3. (line 26)
+* mutators: 3-3. (line 18)
+* mutex: 3-4-2. (line 387)
+* mutual exclusion: 3-4-2. (line 593)
+* n-fold smoothed function: 1-3-4. (line 267)
+* native language: 5-5. (line 22)
+* needed: 5-5-1. (line 162)
+* networks: Chapter 4. (line 147)
+* Newton's method: 1-3-4. (line 67)
+* nil: 2-2-1. (line 90)
+* non-computable: 4-1-5. (line 186)
+* non-strict: 4-2-1. (line 44)
+* nondeterministic: 3-4-1. (line 237)
+* nondeterministic choice point: 4-3-1. (line 59)
+* nondeterministic computing <1>: Chapter 4. (line 120)
+* nondeterministic computing: 4-3. (line 7)
+* normal-order: 4-2-1. (line 9)
+* normal-order evaluation <1>: 1-1-5. (line 125)
+* normal-order evaluation: Chapter 4. (line 116)
+* obarray: 5-3-1. (line 109)
+* object program: 5-5. (line 47)
+* objects: Chapter 3. (line 45)
+* open-code: 5-5-5. (line 383)
+* operands: 1-1-1. (line 45)
+* operator <1>: 4-1-6. (line 302)
+* operator: 1-1-1. (line 45)
+* or-gate: 3-3-4. (line 35)
+* order of growth: 1-2-3. (line 8)
+* ordinary: 2-5-1. (line 25)
+* output prompt: 4-1-4. (line 81)
+* package: 2-4-3. (line 97)
+* painter: 2-2-4. (line 31)
+* pair: 2-1-1. (line 85)
+* parse: 4-3-2. (line 135)
+* Pascal's triangle: 1-2-2. (line 211)
+* pattern: 4-4-1. (line 124)
+* pattern matcher: 4-4-2. (line 37)
+* pattern matching: 4-4-2. (line 23)
+* pattern variable: 4-4-1. (line 128)
+* pipelining: 3-4. (line 73)
+* pointer: 2-2. (line 10)
+* poly: 2-5-3. (line 77)
+* power series: 3-5-2. (line 293)
+* predicate: 1-1-6. (line 41)
+* prefix: 2-3-4. (line 56)
+* prefix code: 2-3-4. (line 57)
+* prefix notation: 1-1-1. (line 51)
+* pretty-printing: 1-1-1. (line 89)
+* primitive constraints: 3-3-5. (line 27)
+* primitive expressions: 1-1. (line 14)
+* probabilistic algorithms: 1-2-6. (line 144)
+* procedural abstraction: 1-1-8. (line 45)
+* procedural epistemology: Preface 1e. (line 66)
+* procedure: 1-2-1. (line 132)
+* procedure definitions: 1-1-4. (line 18)
+* procedures: Chapter 1. (line 119)
+* process: 1-2-1. (line 131)
+* program: Chapter 1. (line 21)
+* programming languages: Chapter 1. (line 31)
+* prompt: 4-1-4. (line 79)
+* pseudo-random: 3-1-2. (line 200)
+* pseudodivision: 2-5-3. (line 538)
+* pseudoremainder: 2-5-3. (line 539)
+* quasiquote: 5-5-2. (line 279)
+* queries: 4-4. (line 99)
+* query language: 4-4. (line 98)
+* queue: 3-3-2. (line 12)
+* quote: 2-3-1. (line 20)
+* Ramanujan numbers: 3-5-3. (line 398)
+* rational: 2-5-3. (line 426)
+* RC circuit: 3-5-3. (line 465)
+* read-eval-print loop: 1-1-1. (line 97)
+* reader macro characters: 4-4-4-7. (line 128)
+* real numbers: 1-1. (line 55)
+* rear: 3-3-2. (line 13)
+* Recursion: 4-1-5. (line 162)
+* recursion equations: Chapter 1. (line 68)
+* recursive <1>: 1-1-8. (line 7)
+* recursive: 1-1-3. (line 23)
+* recursive process: 1-2-1. (line 100)
+* red-black trees: 2-3-3. (line 459)
+* referentially transparent: 3-1-3. (line 156)
+* register: Chapter 5. (line 36)
+* register table: 5-2-1. (line 105)
+* registers: Chapter 5. (line 38)
+* released: 3-4-2. (line 388)
+* remainder of: 1-2-6. (line 58)
+* resolution: 4-4. (line 134)
+* ripple-carry adder: 3-3-4. (line 238)
+* robust: 2-2-4. (line 530)
+* RSA algorithm: 1-2-6. (line 322)
+* rules <1>: 4-4-1. (line 334)
+* rules: 4-4. (line 113)
+* satisfy: 4-4-1. (line 201)
+* scope: 1-1-8. (line 113)
+* selectors: 2-1. (line 27)
+* semaphore: 3-4-2. (line 596)
+* separator code: 2-3-4. (line 53)
+* sequence: 2-2-1. (line 18)
+* sequence accelerator: 3-5-3. (line 97)
+* sequences: 1-3-1. (line 254)
+* serializer: 3-4-2. (line 30)
+* serializers: 3-4-2. (line 91)
+* series RLC circuit: 3-5-4. (line 161)
+* shadow: 3-2. (line 62)
+* shared: 3-3-1. (line 248)
+* side-effect bugs: 3-1-3. (line 320)
+* sieve of: 3-5-2. (line 55)
+* smoothing: 1-3-4. (line 259)
+* source language: 5-5. (line 34)
+* source program: 5-5. (line 39)
+* sparse: 2-5-3. (line 254)
+* special: 1-1-3. (line 96)
+* stack <1>: 5-1-4. (line 82)
+* stack: 1-2-1. (line 233)
+* state variables <1>: 1-2-1. (line 112)
+* state variables: 3-1. (line 11)
+* statements: 5-5-1. (line 167)
+* stop-and-copy: 5-3-2. (line 36)
+* stratified design: 2-2-4. (line 499)
+* stream processing: 1-1-5. (line 157)
+* streams <1>: 3-5. (line 11)
+* streams: Chapter 3. (line 48)
+* strict: 4-2-1. (line 46)
+* subroutine: 5-1-3. (line 75)
+* substitution: 1-1-5. (line 153)
+* substitution model: 1-1-5. (line 58)
+* subtype: 2-5-2. (line 159)
+* success continuation: 4-3-3. (line 29)
+* summation of a series: 1-3-1. (line 50)
+* summer: 3-5-3. (line 422)
+* supertype: 2-5-2. (line 162)
+* symbolic expressions: Chapter 2. (line 139)
+* syntactic sugar: 1-1-3. (line 118)
+* syntax: 4-1. (line 42)
+* systematically search: 4-3-1. (line 75)
+* systems: Chapter 4. (line 148)
+* tableau: 3-5-3. (line 138)
+* tabulation <1>: 1-2-2. (line 245)
+* tabulation: 3-3-3. (line 258)
+* tagged: 5-3-1. (line 274)
+* tail-recursive <1>: 1-2-1. (line 155)
+* tail-recursive: 5-4-2. (line 82)
+* target: 5-5-1. (line 50)
+* thrashing: UTF. (line 22)
+* thunk: 4-2-2. (line 369)
+* thunks: 4-2-2. (line 16)
+* time: 3-4. (line 14)
+* time segments: 3-3-4. (line 473)
+* tower: 2-5-2. (line 165)
+* tree accumulation: 1-1-3. (line 46)
+* tree recursion: 1-2-2. (line 6)
+* trees: 2-2-2. (line 36)
+* truth maintenance: 4-3-1. (line 220)
+* Turing machine: 4-1-5. (line 145)
+* type: 2-4. (line 59)
+* type field: 5-3-1. (line 268)
+* type tag: 2-4-2. (line 25)
+* type-inferencing: 3-5-4. (line 310)
+* typed pointers: 5-3-1. (line 51)
+* unbound: 3-2. (line 32)
+* unification <1>: 4-4-2. (line 24)
+* unification: 4-4. (line 43)
+* unification algorithm: 4-4. (line 133)
+* univariate polynomials: 2-5-3. (line 41)
+* universal: 4-1-5. (line 73)
+* upward-compatible extension: 4-2-2. (line 341)
+* value: 1-1-2. (line 8)
+* value of a variable: 3-2. (line 28)
+* values: 2-3-1. (line 22)
+* variable: 1-1-2. (line 8)
+* variable-length: 2-3-4. (line 30)
+* vector: 5-3-1. (line 19)
+* width: 2-1-4. (line 87)
+* wires: 3-3-4. (line 19)
+* wishful thinking: 2-1-1. (line 24)
+* zero crossings: 3-5-3. (line 506)
+
+
+\1f
+Tag Table:
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+Node: Dedication\7f11431
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+Node: Acknowledgements\7f32802
+Node: Chapter 1\7f39392
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+Ref: Exercise 1-2\7f82482
+Ref: Exercise 1-3\7f82676
+Ref: Exercise 1-4\7f82827
+Ref: Exercise 1-5\7f83111
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+Ref: Exercise 1-11\7f128012
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+Ref: Exercise 1-17\7f139630
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+Ref: 1-2-4-Footnote-2\7f143135
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+Ref: Exercise 1-25\7f159530
+Ref: Exercise 1-26\7f159941
+Ref: Exercise 1-27\7f160942
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+Ref: 1-2-6-Footnote-2\7f162909
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+Ref: Exercise 1-30\7f173401
+Ref: Exercise 1-31\7f173845
+Ref: Exercise 1-32\7f174704
+Ref: Exercise 1-33\7f175791
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+Ref: Exercise 1-39\7f197124
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+Ref: 2-2-1-Footnote-2\7f277830
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+Ref: 2-3-1-Footnote-2\7f355077
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+Ref: Exercise 2-63\7f383461
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+Ref: Figure 3-15\7f608759
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+Ref: Exercise 3-13\7f613349
+Ref: Exercise 3-14\7f613778
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+Ref: Figure 3-17\7f615896
+Ref: Exercise 3-15\7f618802
+Ref: Exercise 3-16\7f618938
+Ref: Exercise 3-17\7f619721
+Ref: Exercise 3-18\7f620047
+Ref: Exercise 3-19\7f620346
+Ref: Exercise 3-20\7f622108
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+Ref: 3-3-1-Footnote-2\7f622615
+Ref: 3-3-1-Footnote-3\7f622917
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+Ref: Figure 3-21\7f632126
+Ref: Exercise 3-21\7f633233
+Ref: Exercise 3-22\7f634543
+Ref: Exercise 3-23\7f635187
+Ref: 3-3-2-Footnote-1\7f635788
+Ref: 3-3-2-Footnote-2\7f636093
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+Ref: Figure 3-23\7f640640
+Ref: Exercise 3-24\7f646213
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+Ref: Exercise 3-27\7f647787
+Ref: 3-3-3-Footnote-1\7f649789
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+Node: 3-3-4\7f650234
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+Ref: Figure 3-25\7f653606
+Ref: Figure 3-26\7f656527
+Ref: Exercise 3-28\7f660018
+Ref: Exercise 3-29\7f660153
+Ref: Exercise 3-30\7f660450
+Ref: Figure 3-27\7f661415
+Ref: Exercise 3-31\7f668906
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+Ref: 3-3-4-Footnote-3\7f675366
+Ref: 3-3-4-Footnote-4\7f675578
+Ref: 3-3-4-Footnote-5\7f675901
+Node: 3-3-5\7f676141
+Ref: Figure 3-28\7f678861
+Ref: Exercise 3-33\7f694633
+Ref: Exercise 3-34\7f694921
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+Ref: 3-3-5-Footnote-3\7f698176
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+Ref: 3-4-1-Footnote-5\7f716709
+Ref: Footnote 39\7f716729
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+Ref: Exercise 3-40\7f723602
+Ref: Exercise 3-41\7f724115
+Ref: Exercise 3-42\7f725311
+Ref: Exercise 3-43\7f730773
+Ref: Exercise 3-44\7f731634
+Ref: Exercise 3-45\7f732622
+Ref: Exercise 3-46\7f737908
+Ref: Exercise 3-47\7f738275
+Ref: Exercise 3-48\7f740140
+Ref: Exercise 3-49\7f740618
+Ref: 3-4-2-Footnote-1\7f743726
+Ref: 3-4-2-Footnote-2\7f744087
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+Ref: 3-4-2-Footnote-7\7f745814
+Ref: 3-4-2-Footnote-8\7f746245
+Ref: 3-4-2-Footnote-9\7f747377
+Ref: 3-4-2-Footnote-10\7f747816
+Ref: 3-4-2-Footnote-11\7f748413
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+Ref: Exercise 3-52\7f766807
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+Ref: 3-5-1-Footnote-2\7f767771
+Ref: 3-5-1-Footnote-3\7f767905
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+Ref: 3-5-1-Footnote-6\7f769102
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+Ref: Exercise 3-54\7f779701
+Ref: Exercise 3-55\7f780097
+Ref: Exercise 3-56\7f780366
+Ref: Exercise 3-57\7f782294
+Ref: Exercise 3-58\7f782733
+Ref: Exercise 3-59\7f783208
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+Ref: Exercise 3-61\7f785997
+Ref: Exercise 3-62\7f786775
+Ref: 3-5-2-Footnote-1\7f787328
+Ref: 3-5-2-Footnote-2\7f787914
+Ref: 3-5-2-Footnote-3\7f788266
+Ref: 3-5-2-Footnote-4\7f788352
+Ref: 3-5-2-Footnote-5\7f788950
+Node: 3-5-3\7f789303
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+Ref: Exercise 3-65\7f797449
+Ref: Exercise 3-66\7f802137
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+Ref: Exercise 3-69\7f803458
+Ref: Exercise 3-70\7f803800
+Ref: Exercise 3-71\7f805240
+Ref: Exercise 3-72\7f805976
+Ref: Figure 3-32\7f807605
+Ref: Exercise 3-73\7f808274
+Ref: Figure 3-33\7f809445
+Ref: Exercise 3-74\7f810281
+Ref: Exercise 3-75\7f812190
+Ref: Exercise 3-76\7f813456
+Ref: 3-5-3-Footnote-1\7f814139
+Ref: 3-5-3-Footnote-2\7f814332
+Ref: 3-5-3-Footnote-3\7f814436
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+Ref: Figure 3-35\7f821315
+Ref: Exercise 3-78\7f822184
+Ref: Exercise 3-79\7f822954
+Ref: Exercise 3-80\7f823151
+Ref: Figure 3-36\7f824328
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+Ref: 3-5-4-Footnote-1\7f828660
+Ref: 3-5-4-Footnote-2\7f828962
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+Ref: 3-5-5-Footnote-1\7f841967
+Ref: 3-5-5-Footnote-2\7f842197
+Ref: 3-5-5-Footnote-3\7f842538
+Ref: 3-5-5-Footnote-4\7f843069
+Node: Chapter 4\7f843642
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+Ref: 4-1-Footnote-1\7f856851
+Ref: 4-1-Footnote-2\7f857273
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+Ref: 4-1-1-Footnote-2\7f869406
+Ref: 4-1-1-Footnote-3\7f869617
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+Ref: Exercise 4-6\7f880989
+Ref: Exercise 4-7\7f881521
+Ref: Exercise 4-8\7f882407
+Ref: Exercise 4-9\7f883284
+Ref: Exercise 4-10\7f883787
+Ref: 4-1-2-Footnote-1\7f884154
+Ref: 4-1-2-Footnote-2\7f884440
+Ref: 4-1-2-Footnote-3\7f884763
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+Ref: 4-1-3-Footnote-2\7f894537
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+Ref: 4-1-4-Footnote-3\7f901728
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+Ref: 4-1-5-Footnote-1\7f908135
+Ref: 4-1-5-Footnote-2\7f909764
+Ref: 4-1-5-Footnote-3\7f910256
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+Ref: Exercise 4-20\7f918981
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+Ref: 4-1-6-Footnote-2\7f924005
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+Ref: 4-1-7-Footnote-2\7f935550
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+Ref: Exercise 4-31\7f955664
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+Ref: 4-2-2-Footnote-2\7f957612
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+Ref: 4-2-3-Footnote-2\7f964719
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+Ref: 4-3-1-Footnote-3\7f978395
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+Ref: Exercise 5-30\7f1306872
+Ref: 5-4-4-Footnote-1\7f1309117
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+Ref: Figure 5-18\7f1381415
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+Ref: Exercise 5-37\7f1384613
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+Ref: 5-5-5-Footnote-2\7f1389198
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+Ref: Exercise 5-49\7f1411445
+Ref: Exercise 5-50\7f1411953
+Ref: Exercise 5-51\7f1412401
+Ref: Exercise 5-52\7f1412748
+Ref: 5-5-7-Footnote-1\7f1413071
+Ref: 5-5-7-Footnote-2\7f1413338
+Ref: 5-5-7-Footnote-3\7f1413719
+Ref: 5-5-7-Footnote-4\7f1414378
+Ref: 5-5-7-Footnote-5\7f1414517
+Ref: 5-5-7-Footnote-6\7f1415891
+Ref: 5-5-7-Footnote-7\7f1416476
+Node: References\7f1416871
+Node: Index\7f1432991
+\1f
+End Tag Table
+++ /dev/null
-The second edition of the book Structure and Interpretation of
-Computer Programs (SICP) in info format.
-
-This library provides the feature `sicp' and does nothing else.
-This allows making the info file available on Melpa. The texi
-file was taken from http://www.neilvandyke.org/sicp-texi. The
-html version can be found at https://mitpress.mit.edu/sicp.
-
-If you want to recreate the info file, you can do so using
-
- makeinfo –no-split sicp.texi -o sicp.info
projects / emacsconfig.git / commitdiff