Computability

An Introduction to Recursive Function Theory

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Paperback | 18 August 1980

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264 Pages

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What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gildel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest.

Industry Reviews

"Dr. Cutland has produced here an excellent and much needed textbook which will undoubtedly help to establish recursion theory as a more widely taught branch of mainstream mathematics." Mathematics & Physics

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Preface	p. viii
Prologue. Prerequisites and notation	p. 1
Sets	p. 1
Functions	p. 2
Relations and predicates	p. 4
Logical notation	p. 4
References	p. 5
Computable functions	p. 7
Algorithms, or effective procedures	p. 7
The unlimited register machine	p. 9
URM-computable functions	p. 16
Decidable predicates and problems	p. 22
Computability on other domains	p. 23
Generating computable functions	p. 25
The basic functions	p. 25
Joining programs together	p. 25
Substitution	p. 29
Recursion	p. 32
Minimalisation	p. 42
Other approaches to computability: Church's thesis	p. 48
Other approaches to computability	p. 48
Partial recursive functions (Godel-Kleene)	p. 49
A digression: the primitive recursive functions	p. 51
Turing-computability	p. 52
Symbol manipulation systems of Post and Markov	p. 57
Computability on domains other than N	p. 65
Church's thesis	p. 67
Numbering computable functions	p. 72
Numbering programs	p. 72
Numbering computable functions	p. 76
Discussion: the diagonal method	p. 79
The s-m-n theorem	p. 81
Universal programs	p. 85
Universal functions and universal programs	p. 85
Two applications of the universal program	p. 90
Effective operations on computable functions	p. 93
Computability of the function [sigma subscript n]	p. 95
Decidability, undecidability and partial decidability	p. 100
Undecidable problems in computability	p. 101
The word problem for groups	p. 106
Diophantine equations	p. 107
Sturm's algorithm	p. 108
Mathematical logic	p. 109
Partially decidable predicates	p. 112
Recursive and recursively enumerable sets	p. 121
Recursive sets	p. 121
Recursively enumerable sets	p. 123
Productive and creative sets	p. 133
Simple sets	p. 140
Arithmetic and Godel's incompleteness theorem	p. 143
Formal arithmetic	p. 143
Incompleteness	p. 146
Godel's incompleteness theorem	p. 149
Undecidability	p. 155
Reducibility and degrees	p. 157
Many-one reducibility	p. 158
Degrees	p. 161
m-complete r.e. sets	p. 165
Relative computability	p. 167
Turing reducibility and Turing degrees	p. 174
Effective operations on partial functions	p. 182
Recursive operators	p. 182
Effective operations on computable functions	p. 189
The first Recursion theorem	p. 192
An application to the semantics of programming languages	p. 196
The second Recursion theorem	p. 200
The second Recursion theorem	p. 200
Discussion	p. 207
Myhill's theorem	p. 210
Complexity of computation	p. 212
Complexity and complexity measures	p. 213
The Speed-up theorem	p. 218
Complexity classes	p. 223
The elementary functions	p. 225
Further study	p. 236
Bibliography	p. 239
Index of notation	p. 241
Subject Index	p. 246
Table of Contents provided by Syndetics. All Rights Reserved.

Computability

An Introduction to Recursive Function Theory

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Paperback

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