These 2011 nostalgia autists didn't expect that: Structure and Interpretation of Computer Programs
second edition
Harold Abelson and Gerald Jay Sussman with Julie Sussman
foreword by Alan J. Perlis
The MIT Press Cambridge, Massachusetts London, England
McGraw-Hill Book Company New York St. Louis San Francisco Montreal Toronto
This book is one of a series of texts written by faculty of the Electrical Engineering and Computer Science Department at the Massachusetts Institute of Technology. It was edited and produced by The MIT Press under a joint production-distribution arrangement with the McGraw-Hill Book Company.
Ordering Information:
North America Text orders should be addressed to the McGraw-Hill Book Company. All other orders should be addressed to The MIT Press.
Outside North America All orders should be addressed to The MIT Press or its local distributor.
Creative Commons License Structure and Interpretation of Computer Programs by Harold Abelson and Gerald Jay Sussman with Julie Sussman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License by the MIT Press.
This book was set by the authors using the LATEX typesetting system and was printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Abelson, Harold Structure and interpretation of computer programs / Harold Abelson and Gerald Jay Sussman, with Julie Sussman. -- 2nd ed. p. cm. -- (Electrical engineering and computer science series) Includes bibliographical references and index. ISBN 0-262-01153-0 (MIT Press hardcover) ISBN 0-262-51087-1 (MIT Press paperback) ISBN 0-07-000484-6 (McGraw-Hill hardcover) 1. Electronic digital computers -- Programming. 2. LISP (Computer program language) I. Sussman, Gerald Jay. II. Sussman, Julie. III. Title. IV. Series: MIT electrical engineering and computer science series. QA76.6.A255 1996 005.13'3 -- dc20 96-17756
Fourth printing, 1999
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)
Contents
Foreword
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
1 Building Abstractions with Procedures 1.1 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 1.2 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 1.3 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
2 Building Abstractions with Data 2.1 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 2.2 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 2.3 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 2.4 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 2.5 Systems with Generic Operations 2.5.1 Generic Arithmetic Operations 2.5.2 Combining Data of Different Types 2.5.3 Example: Symbolic Algebra
3 Modularity, Objects, and State 3.1 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 3.2 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 3.3 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 3.4 Concurrency: Time Is of the Essence 3.4.1 The Nature of Time in Concurrent Systems 3.4.2 Mechanisms for Controlling Concurrency 3.5 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
4 Metalinguistic Abstraction 4.1 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 4.2 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 4.3 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 4.4 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
5 Computing with Register Machines 5.1 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 5.2 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 5.3 Storage Allocation and Garbage Collection 5.3.1 Memory as Vectors 5.3.2 Maintaining the Illusion of Infinite Memory 5.4 The Explicit-Control Evaluator 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 5.5 Compilation 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
warmly awaiting the rest of this, perhaps it will be easier to read if I consider it a shitpost than a textbook. I’m a 2011 nostalgia autist
Name:
Anonymous2021-03-16 8:50
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 1 1/2 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
Name:
Anonymous2021-03-16 8:51
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 4.4), we have students use the register-machine simulator but we do not cover its implementation (section 5.2), and we give only a cursory overview of the compiler (section 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 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.
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.
Name:
Anonymous2021-03-16 8:52
Acknowledgments
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 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 Moyé 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 André 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 R4RS, 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 support from Hewlett-Packard, made possible by Ira Goldstein and Joel Birnbaum, and support from DARPA, made possible by Bob Kahn.
Name:
Anonymous2021-03-16 8:52
Chapter 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.
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).
Name:
Anonymous2021-03-16 8:53
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.4 In later chapters we will see that these same rules allow us to build procedures to manipulate compound data as well.
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 printing5
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:
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.6
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.7
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.8 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).9
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.10
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 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.
Name:
Anonymous2021-03-16 8:54
Figure 1.1: Tree representation, showing the value of each subcombination.
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 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 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.11
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.12
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.13 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.14 The <name> and the <formal parameters> are grouped within parentheses, just as they would be in an actual call to the procedure being defined.
Name:
Anonymous2021-03-16 8:55
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, x2 + y2 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.
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 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 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 chapters 3 and 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 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 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.15
Name:
Anonymous2021-03-16 8:55
Applicative order versus normal order
According to the description of evaluation given in section 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 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 chapters 3 and 4.16
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
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.17
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 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 =.18 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.
Name:
Anonymous2021-03-16 8:56
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.19
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))
Name:
Anonymous2021-03-16 8:56
Exercise 1.2. Translate the following expression into prefix form
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.)
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
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.20
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.21 For example, we can compute the square root of 2 as follows. Suppose our initial guess is 1:
Continuing this process, we obtain better and better approximations to the square root.
Name:
Anonymous2021-03-16 8:57
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:
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 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):22
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.24
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:
What happens when Alyssa attempts to use this to compute square roots? Explain.
Name:
Anonymous2021-03-16 8:58
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
Use this formula to implement a cube-root procedure analogous to the square-root procedure. (In section 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.)
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 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 figure 1.2) that mirrors the decomposition of the problem into subproblems.
Figure 1.2: Procedural decomposition of the sqrt program.
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.25
(define (square x) (* x x))
(define (square x) (exp (double (log x))))
(define (double x) (+ x x))
Name:
Anonymous2021-03-16 8:58
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:
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.26 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.
Name:
Anonymous2021-03-16 8:59
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:
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
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.27
We will use block structure extensively to help us break up large programs into tractable pieces.28 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.
4 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.
5 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.
6 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.
7 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.''
8 In this book, we do not show the interpreter's response to evaluating definitions, since this is highly implementation-dependent.
9 Chapter 3 will show that this notion of environment is crucial, both for understanding how the interpreter works and for implementing interpreters.
10 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.
11 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.''
12 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 1.3.2.
13 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.
14 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.
15 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.
16 In 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 4.2 we will modify the Scheme interpreter to produce a normal-order variant of Scheme.
17 ``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.
18 Abs also uses the ``minus'' operator -, which, when used with a single operand, as in (- x), indicates negation.
19 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.
20 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 chapter 4.
Name:
Anonymous2021-03-16 9:00
21 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 1.3.4.
22 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.
23 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 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.
24 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 1.2.1.
25 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.
26 The concept of consistent renaming is actually subtle and difficult to define formally. Famous logicians have made embarrassing errors here.
27 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.
28 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.
Name:
Anonymous2021-03-16 9:01
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.
1.2.1 Linear Recursion and Iteration
Figure 1.3: A linear recursive process for computing 6!.
We begin by considering the factorial function, defined by
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:
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 1.1.5 to watch this procedure in action computing 6!, as shown in 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!.
Once again, we can recast our description as a procedure for computing factorials:29
As before, we can use the substitution model to visualize the process of computing 6!, as shown in 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 figure 1.3. The expansion occurs as the process builds up a chain of 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.30
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.
Name:
Anonymous2021-03-16 9:01
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 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.31
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 5n2.
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:
In general, the Fibonacci numbers can be defined by the rule
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).
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 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 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 exercise 1.13), Fib(n) is the closest integer to n /5, where
is the golden ratio, which satisfies the equation
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
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.32 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:33
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:
(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.34
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.
Name:
Anonymous2021-03-16 9:02
Exercise 1.12. The following pattern of numbers is called Pascal's triangle.
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.35 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 n/5, where = (1 + 5)/2. Hint: Let = (1 - 5)/2. Use induction and the definition of the Fibonacci numbers (see section 1.2.2) to prove that Fib(n) = (n - n)/5.
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 (f(n)), written R(n) = (f(n)) (pronounced ``theta of f(n)''), if there are positive constants k1 and k2 independent of n such that
for any sufficiently large value of n. (In other words, for large n, the value R(n) is sandwiched between k1f(n) and k2f(n).)
For instance, with the linear recursive process for computing factorial described in section 1.2.1 the number of steps grows proportionally to the input n. Thus, the steps required for this process grows as (n). We also saw that the space required grows as (n). For the iterative factorial, the number of steps is still (n) but the space is (1) -- that is, constant.36 The tree-recursive Fibonacci computation requires (n) steps and space (n), where is the golden ratio described in section 1.2.2.
Orders of growth provide only a crude description of the behavior of a process. For example, a process requiring n2 steps and a process requiring 1000n2 steps and a process requiring 3n2 + 10n + 17 steps all have (n2) 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 (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 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 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 x x if x is sufficiently small, and the trigonometric identity
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:
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?
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 bn. One way to do this is via the recursive definition
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 (n) steps and (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 (n) steps and (1) space.
We can compute exponentials in fewer steps by using successive squaring. For instance, rather than computing b8 as
we can compute it using three multiplications:
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
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 b2n using fast-expt requires only one more multiplication than computing bn. 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 (log n) growth.37
The difference between (log n) growth and (n) growth becomes striking as n becomes large. For example, fast-expt for n = 1000 requires only 14 multiplications.38 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 exercise 1.16), although, as is often the case with iterative algorithms, this is not written down so straightforwardly as the recursive algorithm.39
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 (bn/2)2 = (b2)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 bn 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 exercises 1.16 and 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.40
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 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 Tn, 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 Tpq, where Tpq transforms the pair (a,b) according to a bq + aq + ap and b bp + aq. Show that if we apply such a transformation Tpq twice, the effect is the same as using a single transformation Tp'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 Tn 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:41
(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)))))
Name:
Anonymous2021-03-16 9:03
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 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
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,
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 Euclid's Algorithm.42
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:
Lamé'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.43
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) k/5. Therefore the number of steps k grows as the logarithm (to the base ) of n. Hence, the order of growth is (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 1.1.5. (The normal-order-evaluation rule for if is described in 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?
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 (n), and a ``probabilistic'' algorithm with order of growth (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 n.44 This means that the algorithm need only test divisors between 1 and n. Consequently, the number of steps required to identify n as prime will have order of growth (n).
The Fermat test
The (log n) primality test is based on a result from number theory known as Fermat's Little Theorem.45
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 an 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 1.2.4. It uses successive squaring, so that the number of steps grows logarithmically with the exponent.46
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)))
Name:
Anonymous2021-03-16 9:04
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 an 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.47 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 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.48
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.
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 (n), you should expect that testing for primes around 10,000 should take about 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 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 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 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 (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 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 (log n) process into a (n) process.'' Explain.
Exercise 1.27. Demonstrate that the Carmichael numbers listed in footnote 47 really do fool the Fermat test. That is, write a procedure that takes an integer n and tests whether an 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 an-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.
Name:
Anonymous2021-03-16 9:05
29 In a real program we would probably use the block structure introduced in the last section to hide the definition of fact-iter:
We avoided doing this here so as to minimize the number of things to think about at once.
30 When we discuss the implementation of procedures on register machines in 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.
31 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 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.
32 An example of this was hinted at in section 1.1.3: The interpreter itself evaluates expressions using a tree-recursive process.
33 For example, work through in detail how the reduction rule applies to the problem of making change for 10 cents using pennies and nickels.
34 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 exercise 3.27.
35 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, Traité du triangle arithmétique. According to Knuth (1973), the same pattern appears in the Szu-yuen Yü-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 Bháscara Áchárya.
36 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.
37 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 k1 and k2 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 (log n).
38 You may wonder why anyone would care about raising numbers to the 1000th power. See section 1.2.6.
39 This iterative algorithm is ancient. It appears in the Chandah-sutra by Áchárya 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.
40 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.
41 This exercise was suggested to us by Joe Stoy, based on an example in Kaldewaij 1990.
42 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 (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.
43 This theorem was proved in 1845 by Gabriel Lamé, a French mathematician and engineer known chiefly for his contributions to mathematical physics. To prove the theorem, we consider pairs (ak ,bk), where ak> bk, for which Euclid's Algorithm terminates in k steps. The proof is based on the claim that, if (ak+1, bk+1) (ak, bk) (ak-1, bk-1) are three successive pairs in the reduction process, then we must have bk+1> bk + bk-1. To verify the claim, consider that a reduction step is defined by applying the transformation ak-1 = bk, bk-1 = remainder of ak divided by bk. The second equation means that ak = qbk + bk-1 for some positive integer q. And since q must be at least 1 we have ak = qbk + bk-1 > bk + bk-1. But in the previous reduction step we have bk+1 = ak. Therefore, bk+1 = ak> bk + bk-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 (ak+1, bk+1) (ak, bk) (ak-1, bk-1) be successive pairs in the reduction process. By our induction hypotheses, we have bk-1> Fib(k - 1) and bk> Fib(k). Thus, applying the claim we just proved together with the definition of the Fibonacci numbers gives bk+1 > bk + bk-1> Fib(k) + Fib(k - 1) = Fib(k + 1), which completes the proof of Lamé's Theorem.
44 If d is a divisor of n, then so is n/d. But d and n/d cannot both be greater than n.
45 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.
46 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 be/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 exercise 1.25.)
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.
48 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.
Name:
Anonymous2021-03-16 9:06
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.
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
which converges to /8 (very slowly):49
(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
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:50
(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 :
(* 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
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
where h = (b - a)/n, for some even integer n, and yk = 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.
Name:
Anonymous2021-03-16 9:06
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.51 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 using the formula52
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 (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 (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).
1.3.2 Constructing Procedures Using Lambda
In using sum as in section 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.53
Name:
Anonymous2021-03-16 9:07
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
which we could also express as
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> <expn>)) <body>)
which can be thought of as saying
let <var1> have the value <exp1> and <var2> have the value <exp2> and
<varn> have the value <expn> 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 (<var1> ...<varn>) <body>) <exp1>
<expn>)
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.54
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.
Name:
Anonymous2021-03-16 9:07
1.3.3 Procedures as General Methods
We introduced compound procedures in section 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 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 (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:
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 1.1.7 for computing square roots:55
(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.56
(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 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 x3 - 2x - 3 = 0 between 1 and 2:
(half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3)) 1.0 2.0) 1.89306640625
Name:
Anonymous2021-03-16 9:08
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,
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:
The fixed-point process is reminiscent of the process we used for finding square roots in section 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 y2 = x. Putting this equation into the equivalent form y = x/y, we recognize that we are looking for a fixed point of the function58 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 y1. The next guess is y2 = x/y1 and the next guess is y3 = x/y2 = x/(x/y1) = y1. This results in an infinite loop in which the two guesses y1 and y2 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 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 (section 1.2.2) is a fixed point of the transformation x 1 + 1/x, and use this fact to compute 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 exercise 1.22. Then find a solution to xx = 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
As an example, one can show that the infinite continued fraction expansion with the Ni and the Di all equal to 1 produces 1/, where is the golden ratio (described in section 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 k-term finite continued fraction -- has the form
Suppose that n and d are procedures of one argument (the term index i) that return the Ni and Di 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/ 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.
Name:
Anonymous2021-03-16 9:08
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 Ni are all 1, and the Di are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, .... Write a program that uses your cont-frac procedure from 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:
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 exercise 1.37.
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 1.3.3. We formulated a new version of the square-root procedure as a fixed-point search, starting with the observation that 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 x2. Applying this resulting procedure to 10 returns the average of 10 and 100, or 55:59
((average-damp square) 10) 55
Using average-damp, we can reformulate the square-root procedure as follows:
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 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/y2, so we can immediately generalize our square-root procedure to one that extracts cube roots:60
When we first introduced the square-root procedure, in section 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
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.61 For many functions g and for sufficiently good initial guesses for x, Newton's method converges very rapidly to a solution of g(x) = 0.62
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 x3 is the function x 3x2. 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
Thus, we can express the idea of derivative (taking dx to be, say, 0.00001) as the procedure
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 x3 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:
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 y2 - x starting with an initial guess of 1.63 This provides yet another form of the square-root procedure:
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:
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 y2 - x) as
We began section 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:64
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.65
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.66
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 x3 + ax2 + 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 2nth 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 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 exercise 1.43.
Exercise 1.45. We saw in section 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/y2. 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/y3 converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of y x/y3) 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/yn-1. Use this to implement a simple procedure for computing nth roots using fixed-point, average-damp, and the repeated procedure of 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 1.1.7 and the fixed-point procedure of section 1.3.3 in terms of iterative-improve.
Name:
Anonymous2021-03-16 9:10
49 This series, usually written in the equivalent form (/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 3.5.3.
50 Notice that we have used block structure (section 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 1.3.2.
51 The intent of exercises 1.31-1.33 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 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.
52 This formula was discovered by the seventeenth-century English mathematician John Wallis.
53 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 calculus, a mathematical formalism introduced by the mathematical logician Alonzo Church (1941). Church developed the calculus to provide a rigorous foundation for studying the notions of function and function application. The calculus has become a basic tool for mathematical investigations of the semantics of programming languages.
54 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 4.1.6, after we learn more about evaluation.
55 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.
56 This can be accomplished using error, which takes as arguments a number of items that are printed as error messages.
57 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.
58 (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.
59 Observe that this is a combination whose operator is itself a combination. 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.
60 See exercise 1.45 for a further generalization.
61 Elementary calculus books usually describe Newton's method in terms of the sequence of approximations xn+1 = xn - g(xn)/Dg(xn). Having language for talking about processes and using the idea of fixed points simplifies the description of the method.
62 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.
63 For finding square roots, Newton's method converges rapidly to the correct solution from any starting point.
64 The notion of first-class status of programming-language elements is due to the British computer scientist Christopher Strachey (1916-1975).
65 We'll see examples of this after we introduce data structures in chapter 2.
66 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 4.1, these variables are stored in the procedure's environment.
Name:
Anonymous2021-03-16 9:10
Chapter 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 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 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 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 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 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.
1 The ability to directly manipulate procedures provides an analogous increase in the expressive power of a programming language. For example, in section 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.
Name:
Anonymous2021-03-16 9:11
2.1 Introduction to Data Abstraction
In section 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.
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 the rational 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:
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.2 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 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:3
(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:4
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 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.
Name:
Anonymous2021-03-16 9:11
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 2.1. The horizontal lines represent 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.
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:
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:
Exercise 2.3. Implement a representation for rectangles in a plane. (Hint: You may want to make use of 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?
2.1.3 What Is Meant by Data?
We began the rational-number implementation in section 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
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.5
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 chapter 3 when we address the issues of modeling and simulation.
Name:
Anonymous2021-03-16 9:12
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 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 2a 3b. 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
This representation is known as Church numerals, after its inventor, Alonzo Church, the logician who invented the 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).
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 Rp of two resistors R1 and R2 using the formula
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:
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.)
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:
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:
and
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 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.)
2 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.''
3 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.
4 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.
5 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 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).
Name:
Anonymous2021-03-16 9:13
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. 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.
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 exercises 2.2 and 2.3.) As a consequence, pairs provide a universal building block from which we can construct all sorts of data structures. Figure 2.3 shows two ways to use pairs to combine the numbers 1, 2, 3, and 4.
Figure 2.2: Box-and-pointer representation of (cons 1 2).
Figure 2.3: Two ways to combine 1, 2, 3, and 4 using pairs.
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.6 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 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.7
2.2.1 Representing Sequences
Figure 2.4: The sequence 1, 2, 3, 4 represented as a chain of pairs.
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 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 conses, is called a list, and Scheme provides a primitive called list to help in constructing lists.8 The above sequence could be produced by (list 1 2 3 4). In general,
Lisp systems conventionally print lists by printing the sequence of elements, enclosed in parentheses. Thus, the data object in 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.9 The constructor cons makes a list like the original one, but with an additional item at the beginning.
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.''10
List operations
The use of pairs to represent sequences of elements as lists is accompanied by conventional programming techniques for manipulating lists by successively ``cdring 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.
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:
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 cdring 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:
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 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:
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 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?
Name:
Anonymous2021-03-16 9:13
Exercise 2.20. The procedures +, *, and list take arbitrary numbers of arguments. One way to define such procedures is to use define with 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).11
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:
We can abstract this general idea and capture it as a common pattern expressed as a higher-order procedure, just as in section 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:12
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 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 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:
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,
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.
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. 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)).
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. Figure 2.6 shows the structure in figure 2.5 viewed as a tree.
Figure 2.6: The list structure in figure 2.5 viewed as a tree.
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 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 cars 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:13
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 figure 2.6).
Exercise 2.25. Give combinations of cars and cdrs 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 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:
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?
Name:
Anonymous2021-03-16 9:14
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 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:
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:
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:
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 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 2.2.2, which takes a tree as argument and computes the sum of the squares of the leaves that are odd:
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 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.
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 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
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:
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:
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:
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:
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
These examples give just a hint of the vast range of operations that can be expressed as sequence operations.15
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 3.5, when we generalize the sequence-processing paradigm to admit infinite sequences.
Name:
Anonymous2021-03-16 9:16
Exercise 2.33. Fill in the missing expressions to complete the following definitions of some basic list-manipulation operations as accumulations:
Exercise 2.34. Evaluating a polynomial in x at a given value of x can be formulated as an accumulation. We evaluate the polynomial
using a well-known algorithm called Horner's rule, which structures the computation as
In other words, we start with an, multiply by x, add an-1, multiply by x, and so on, until we reach a0.16 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 a0 through an.
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 = (vi) as sequences of numbers, and matrices m = (mij) as sequences of vectors (the rows of the matrix). For example, the matrix
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:
We can define the dot product as17
(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 exercise 2.36.)
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))
We can extend the sequence paradigm to include many computations that are commonly expressed using nested loops.18 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:
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:19
(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:
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
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:
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,20 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:21
(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:
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.
Exercise 2.42.
Figure 2.8: A solution to the eight-queens puzzle.
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 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 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 exercise 2.42 solves the puzzle in time T.
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 figure 2.9, which are composed of repeated elements that are shifted and scaled.22 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.
Name:
Anonymous2021-03-16 9:16
The picture language
When we began our study of programming in section 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 figure 2.10. The actual shape of the drawing depends on the frame -- all four images in 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 figure 2.11.23 The four images in figure 2.11 are drawn with respect to the same four frames as the wave images in 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.
Figure 2.11: Images of William Barton Rogers, founder and first president of MIT, painted with respect to the same four frames as in figure 2.10 (original image reprinted with the permission of the MIT Museum).
Figure 2.12 shows the drawing of a painter called wave4 that is built up in two stages starting from wave:
Figure 2.12: Creating a complex figure, starting from the wave painter of figure 2.10.
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
Figure 2.13: Recursive plans for right-split and corner-split.
We can produce balanced patterns by branching upwards as well as towards the right (see exercise 2.44 and figures 2.13 and 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))))))
(right-split wave 4) (right-split rogers 4)
(corner-split wave 4) (corner-split rogers 4)
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 figure 2.9.
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 figure 2.9:
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.
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