Computability Theory: An Introduction

166 Pages·1973·6.79 MB·English

by  Neil D. Jones

ACM MONOGRAPH SERIES
Published under the auspices of the Association for
Computing Machinery Inc.
Editor ROBERT L. ASHENHURST The University of Chicago
A. FINERMAN (Ed.) University Education in Computing Science, 1968
A. GINZBURG Algebraic Theory of Automata, 1968
E. F. CODD Cellular Automata, 1968
G. ERNST AND A. NEWELL GPS: A Case Study in Generality and
Problem Solving, 1969
M. A. GAVRILOV AND A. D. ZAKREVSKII (Eds.) LYaPAS: A Programming
Language for Logic and Coding Algorithms, 1969
THEODOR D. STERLING, EDGAR A. BERING, JR., SEYMOUR V. POLLACK,
AND HERBERT VAUGHAN, JR. (Eds. ) Visual Prosthesis :
The Interdisciplinary Dialogue, 1971
JOHN R. RICE (Ed. ) Mathematical Software, 1971
ELLIOTT I. ORGANICK Computer System Organization: The B5700/B6700
Series, 1973
NEIL D. JONES Computability Theory: An Introduction, 1973
In preparation
ARTO SALOMAA Formal Languages
HARVEY ABRAMSON Theory and Application of a Bottom-Up Syntax-
Directed Translator
Previously published and available from The Macmillan Company,
New York City
V. KRYLOV Approximate Calculation of Integrals (Translated by A. H.
Stroud), 1962
COMPUTABILITY
T H E O RY
AN INTRODUCTION
NEIL D. JONES
Computer Science Department
The Pennsylvania State University
University Park, Pennsylvania
ACADEMIC PRESS New York and London 1973
COPYRIGHT © 1973, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-9331
AMS (MOS) 1970 Subject Classifications: 02F05, 02F15, 02F25, 68A10, 68A25
PRINTED IN THE UNITED STATES OF AMERICA

PREFACE
The purpose of this text is to introduce the major concepts, con-
structions, and theorems of the elementary theory of computability of
recursive functions. The concept of "effective process" is introduced
and emphasized early, so that a clear intuitive understanding of the
effective computability of partial and total functions and the effective
enumerability and decidability of sets is obtained before proceeding to
the more rigorous aspects of the theory. This is followed by a formal
development of the equivalence of Turing machine computability,
enumerability, and decidability with other formulations of these con-
cepts as defined by formulas of the predicate calculus, systems of recur-
sion equations, and Post's production systems.
In a certain sense this text has just two purposes: to establish the
fundamental properties of the partial recursive functions and the recur-
sive and recursively enumerable sets; and to give mathematical evidence
for the validity of the Church-Turing thesis.
This text is suitable for a one-semester advanced undergraduate or
beginning graduate course. Alternately, Chapters I and II can be used
on a stand-alone basis as an informal introduction to the various
concepts related to effectiveness, as a part (say 20 to 30 percent) of an
introductory course in mathematical machine theory. Such a course
might also contain material on finite automata, push-down automata,
and context-free languages, with a primary emphasis on the former.
Vll
Vlll Preface
The student is expected to have some informal experience with
algorithms (such as that obtained in an introductory programming
course), and to be generally comfortable with elementary discrete
mathematics (basic logic, sets, relations, functions, etc.); it is not
required that he be knowledgeable in higher-level mathematical
subjects such as mathematical logic, abstract algebra, or number theory.
The basic approach is to prove the equivalence of effectiveness as
defined by Turing machines with effectiveness as defined by S-rudi-
mentary and other predicate calculus formulas. On the one hand, it is
shown that the S-rudimentary formulas (e.g. Smullyan [SI]) can accu-
rately describe the structure and computations of Turing machines; on
the other hand, we show that Turing machines may be used to evaluate
S-rudimentary and other formulas. As a consequence the equivalence of
two very different formulations of computability is established, as well as
normal form theorems (e.g., as in Kleene [Kl]). The fundamental
theorems concerning computability, enumerability, etc. are then derived
from these results.
The Turing machine model used is similar to the Wang variant
[Wl], in that it has a program as a control unit.
Several factors were involved in the choice of the S-rudimentary
predicates for Turing machine description, as opposed to the more
customary choice of the primitive recursive functions. The author has
observed that many of the difficulties encountered in teaching a first
course in computability (e.g. from Davis [Dl] and Hermes [HI]) seem
to derive from arithmetization techniques which use various devices
from number theory, such as properties of prime numbers and the
Chinese remainder theorem. In addition the representation of an essen-
tially nonnumeric object such as a Turing machine by means of a number
often causes conceptual problems.
In this text the entire approach is syntactic rather than numeric;
the basic objects are words rather than numbers (although of course
numbers can be represented by words in a natural way). Thus the
arithmetization of Turing machines, computations, etc. is replaced
by simple encoding techniques, most of which merely involve replace-
ment of symbols by strings of symbols, or the use of auxiliary symbols
used as special markers.
The elegant methods of Smullyan (SI) are very well adapted to this
approach; in fact we use many of his basic techniques, including the
S-rudimentary predicates. As a side effect we get normal form theorems
which are stronger than usual, because the function U and predicate
Preface ix
T are simpler than primitive recursive (in fact S-rudimentary). Another
n
side eifect which is of interest because of recent work in computational
complexity is that U and T are computable and decidable (respectively)
n
in time, which is a polynomial function of the length of their input
words.
The true power of the syntactic approach is evidenced in Chapter VI
in which the same tools are used to give straightforward proofs that
systems of recursion equations on one hand, and Post canonical systems
on the other, are both exactly equivalent to Turing machines.
Teaching experience with this approach to computability to date has
indicated that it does in fact make computability theory more accessible
to many students, particularly those in computer science.
LIST OF SPECIAL SYMBOLS
First
Symbol Meaning Reference
1. iff if and only if 15
2. 1 marks the end of the definition or
proof 22
3. xeS x is an element of S 6
4. χφΞ x is not an element of S 6
5. S<=T S is a subset of T 6
6. S$T S is a proper subset of T 6
7. 0 the empty set 7
8. #S the number of elements in S 7
9. {x\P(x)} the set of x such that P(x) is true 7
10. SuT the union of S and T 7
11. SnT the intersection of S and T 7
12. S\T the difference of S and T 7
13. (a,b) an ordered pair 8
xi
xii List of Special Symbols
First
Symbol Meaning Reference
14. Ol, ···,*„) an ordered «tuple 9
15. abbreviation for the sequence
Xn
9
16. SxT the cartesian product of S and T 8
17. S" the «fold cartesian product of S
with itself 9
18. Oj X 02 X * * * X *3„ the cartesian product of
9
19. λ the empty word 12
20. 1*1 the number of symbols in a word x 12
21. Λ,Ρ alphabets, that is, finite nonempty
sets 11
22. *y the catenation of word x with
word y 12
23. xn the catenation of« copies of word x 12
24. A* the set of all words over alphabet A 12
25. P(x ...,x ) an «ary predicate 13
u n
26. -^P(Xn) "not P," the negation of P(x„) 14
27. P(x„) Λ Q(xn) "P and Ö," the conjunction of
P(x„) and Q(x ) 14
n
28. P(X ) V £(*„) "P or Ö," the disjunction of P(x„)
n
and Q(x ) 14
n
29. P(x )^Q(x ) "P implies ζ?" 14
n n
30. P(xn)oQ(xn) "P equivalent to β" 14
31. P(t) an explicit transform of P(x„) 15
32. iyP(z ,y) " for some y, P(x„, y) is true," the
n
existential quantification of P(x„, >>) 15
33. VyP(w) " for all y, P(xn, y) is true," the
universal quantification of P(x„, y) 16
34. μy Q(xn ·> y) the least >> such that Q(xn, >>) is 19
true