A Programming Approach to Computability

258 Pages·1982·5.58 MB·English

by  Michael A. Arbib, A. J. Kfoury, Robert N. Moll

Texts and Monographs in Computer Science
Editor
David Gries
Advisory Board
F. L. Bauer
K.S.Fu
J. J. Horning
R. Reddy
D. C. Tsichritzis
W. M. Waite
The AKM Series in
Theoretical Computer Science
A Subseries of Texts and Monographs in Computer Science
A Basis for Theoretical Computer Science
by M. A. Arbib, A. J. Kfoury, and R. N. Moll
A Programming Approach to Computability
by A. J. Kfoury, R. N. Moll, and M. A. Arbib
An Introduction to Formal Language Theory
by R. N. Moll, M. A. Arbib, and A. J. Kfoury
A Programming
Approach to
Computability
A. J. Kfoury
Robert N. Moll
Michael A. Arbib
I
Springer-Verlag
New York Heidelberg Berlin
A. J. Kfoury Robert N. Moll
Department of Mathematics Department of Computer and
Boston University Information Science
Boston, MA 02215 University of Massachusetts
U.S.A. Amherst, MA 01003
U.S.A.
Michael A. Arbib Series Editor
Department of Computer and
Information Science
David Gries
University of Massachusetts
Department of Computer Science
Amherst, MA 01003
Cornell University
U.S.A.
Upson Hall
Ithaca, NY 14853
U.S.A.
On the front cover is a likeness (based on a postcard kindly supplied by Professor S.
C. Kleene) of the ninth century mathematician Abu lafar Muhammad Ibn Musa
AI-Khowarizmi. The word "algorithm" is derived from the last part of his name.
With 36 Figures
Library of Congress Cataloging in Publication Data
Kfoury, A. 1.
A programming approach to computability.
(The AKM series in theoretical computer science)
(Texts and monographs in computer science)
"February 1982."
Bibliography: p.
Includes index.
I. Computable functions. 2. Programming
(Electronic computers) I. Moll, Robert N.
II. Arbib, Michael A. III. Title. IV. Series.
V. Series: Texts and monographs in computer science.
QA9.59.K46 1982 519.4 82-10488
All rights reserved.
No part of this book may be translated or reproduced in any form without written
permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010,
U.S.A.
© 1982 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1982
9 8 7 6 5 432 I
ISBN-13 :978-1-4612-5751-6 e-ISBN-13:978-1-4612-5749-3
DOl: 10.1007/978-1-4612-5749-3
Preface
Computability theory is at the heart of theoretical computer science. Yet,
ironically, many of its basic results were discovered by mathematical
logicians prior to the development of the first stored-program computer. As
a result, many texts on computability theory strike today's computer
science students as far removed from their concerns. To remedy this, we
base our approach to computability on the language of while-programs, a
lean subset of PASCAL, and postpone consideration of such classic models as
Turing machines, string-rewriting systems, and p.-recursive functions till the
final chapter. Moreover, we balance the presentation of un solvability
results such as the unsolvability of the Halting Problem with a presentation
of the positive results of modern programming methodology, including the
use of proof rules, and the denotational semantics of programs.
Computer science seeks to provide a scientific basis for the study of
information processing, the solution of problems by algorithms, and the
design and programming of computers. The last 40 years have seen
increasing sophistication in the science, in the microelectronics which has
made machines of staggering complexity economically feasible, in the
advances in programming methodology which allow immense programs to
be designed with increasing speed and reduced error, and in the develop-
ment of mathematical techniques to allow the rigorous specification of
program, process, and machine. The present volume is one of a series, the
AKM Series in Theoretical Computer Science, designed to make key
mathematical developments in computer science readily accessible to un-
dergraduate and beginning graduate students. The book is essentially
self-contained-what little background is required may be found in the
AKM volume A Basis for Theoretical Computer Science.
v
vi Preface
The book is organized as follows. Chapter I provides a preview of the
techniques to be used throughout the book in determining whether or not a
function is computable by some algorithm.
Chapters 2 and 3 establish our framework for the study of computabil-
ity, using the lean PASCAL subset we call "while-programs". In these two
chapters we take a close look at the intuitive idea of an algorithm as
formulated in this simple programming language. We also introduce the
idea of a universal program or interpreter which can simulate any program
P when given P's data together with an encoding of P as an additional
input.
In Chapter 4 we develop basic techniques of computability theory,
which we then use to examine the computational status of various natural
problems.
Chapter 5 provides an introduction to program correctness and to
alternative theories of semantics of programming languages.
In Chapter 6 we consider the role of self-reference in computability
theory as embodied in a result known as the Recursion Theorem. We also
study conditions under which the theory can be made model independent.
In Chapters 7 and 8 our emphasis changes from functions to sets.
Recursive sets, whose characteristic functions are computable, are com-
pared with recursively enumerable sets whose elements are listable by
algorithms. As we shall see, these two concepts do not coincide.
Finally, in Chapter 9 we examine alternative formulations of comput-
ability theory, and we show how one of these formulations (the Turing
machine model) can be used to embed the results of computability theory
in a large variety of other symbol-processing systems, including formal
languages.
More detailed outlines may be found at the beginning of each chapter.
In keeping with the style of the AKM series, we have resisted the
temptation to include more material than could possibly be covered in a
one-semester course. Nevertheless, in moderating the pace for a single
term's work, the instructor may find it appropriate to omit material from
one or more of Sections 5.3, 6.2, 7.3, 8.2, 9.2, and 9.3 (the starred sections in
the table of contents).
The book grew out of our teaching classes at the University of Massa-
chusetts at Amherst and at Boston University over several years. We owe
special thanks to Ernest Manes for letting us use material from the
forthcoming text Formal Semantics of Programming Languages by M. A.
Arbib and E. Manes in the preparation of Section 5.1. We thank our
students for helping us integrate computability theory into the "world
view" of modern computer science, and we thank Susan Parker for all her
efforts in typing the manuscript.
February 1982 A. J. Kfoury
R. N. Moll
M. A. Arbib
Contents
CHAPTER 1
Introduction
1.1 Partial Functions and Algorithms I
1.2 An Invitation to Computability Theory 9
1.3 Diagonalization and the Halting Problem 13
CHAPTER 2
The Syntax and Semantics of while-Programs 17
2.1 The Language of while-Programs 17
2.2 Macro Statements 25
2.3 The Computable Functions 34
CHAPTER 3
Enumeration and Universality of the Computable Functions 45
3.1 The Effective Enumeration of while-Programs 45
3.2 Universal Functions and Interpreters 51
3.3 String-Processing Functions 61
3.4 Pairing Functions 67
CHAPTER 4
Techniques of Elementary Computability Theory 75
4.1 Algorithmic Specifications 76
4.2 The s-m-n Theorem 80
4.3 Undecidable Problems 86
vii
viii Contents
CHAPTER 5
Program Methodology 95
5.1 An Invitation to Denotational Semantics 96
5.2 Recursive Programs 110
5.3* Proof Rules for Program Properties 124
CHAPTER 6
The Recursion Theorem and Properties of Enumerations 138
6.1 The Recursion Theorem 138
6.2* Model-Independent Properties of Enumerations 147
CHAPTER 7
Computable Properties of Sets (Part 1) 152
7.1 Recursive and Recursively Enumerable Sets 153
7.2 Indexing the Recursively Enumerable Sets 163
7.3* G6del's Incompleteness Theorem 170
CHAPTER 8
Computable Properties of Sets (Part 2) 176
8.1 Rice's Theorem and Related Results 177
8.2* A Classification of Sets 184
CHAPTER 9
Alternative Approaches to Computability 196
9.1 The Turing Characterization 196
9.2* The Kleene Characterization 205
9.3* Symbol-Manipulation Systems and Formal Languages 224
References 235
Notation Index 239
Author Index 243
Subject Index 245
*These sections may be omitted with no loss of continuity.
CHAPTER 1
Introduction
1.1 Partial Functions and Algorithms
1.2 An Invitation to Computability Theory
1.3 Diagonalization and the Halting Problem
Computability theory looks at the functions computed by algorithms, such
as computer programs. Since a program may not halt for a given input, the
functions so computed will be partial functions. In Section 1.1 we discuss
partial functions and their computability by different families of algo-
rithms. In Section 1.2 we give an example of an unsolvability or noncom-
putability result-showing that no program can compute for an arbitrary
number n whether or not the PASCAL program with ASCII code n will halt
when also given n as input data. Section 1.3 shows that the method of proof
used to show the noncomputability of this halting problem is akin to
Cantor's diagonal argument that the real numbers are uncountable. This
chapter thus gives the reader a taste for a number of techniques to be used
again and again throughout this volume.
1.1 Partial Functions and Algorithms
A function is computable if it is determined by some algorithm, that is, by
some well-specified program or set of rules. In this section we introduce the
notion of a partial function-the proper functional concept for computabil-
ity theory-and then look closely at the relationship between algorithms
and functions.
1
2 1 Introduction
FUNCTIONS
We assume that the reader already knows what the word "function" means.
(The necessary background for this section is provided in the companion
volume, A Basis for Theoretical Computer Science, M. A. Arbib, A. J.
Kfoury, and R. N. Moll, Springer-Verlag, New York, 1981, Chapter 1.)
Recall that in conventional mathematics, a function f: A ~ B is some rule
according to which each element x of A (the domain of arguments) is
associated with an element f(x) in another set B (the codomain of values).
This rule may be a precise algorithm, or merely a hypothetical rule of
correspondence. Most common functions of everyday arithmetic, e.g., the
multiplication function g(x, y) = x * y, have familiar algorithmic descrip-
tions, and are routinely classified as computable functions. However, the
status of a function specified by a hypothetical rule of correspondence may
be less clear. As an example, consider the function f given below mapping
domain N, the set of non-negative integers, to codomain {O, I}.
f(x) = { 1, if Fermat's last theorem 1 is true;
0, if Fermat's last theorem is false.
Now mathematicians have not yet established the validity of Fermat's last
theorem. So it is not known if f(x) is the constant function identically equal
to zero, or the constant function identically equal to 1. In either case,
however, f can be calculated by a trivial algorithm-either "output 0 on all
arguments," or "output 1 on all arguments." Hence, f is considered to be
computable even though the algorithm for calculating its value has not yet
been identified.
As every programmer knows, a poorly designed but runnable algorithm
will often fail to terminate for some input values, and our theory must take
this phenomenon into account. Consider, for example, the function com-
putable by the algorithm shown in the flow chart of Figure 1, and which is
expressed as follows in the computer language PASCAL2:
begin
while Xmod2 = 0 do
X:= X + 2
end
I Fermat's last theorem is the as yet unproved assertion that for any n > 2, the equation
x' + y' = z· has no integer solutions.
2 Our convention for writing variables throughout the book is this: We use lower case letters,
mostly from the end of the Roman alphabet, e.g., XI' x2' y, Y12' z, etc.-except when these
variables appear in the text of programs, in which case we use the corresponding upper case
letters, e.g., X I, X2, Y, Y 12, Z, etc.