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Computability : An Introduction to Recursive Function Theory - Nigel J. Cutland


An Introduction to Recursive Function Theory


Published: 18th August 1980
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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.

"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

Prefacep. viii
Prologue. Prerequisites and notationp. 1
Setsp. 1
Functionsp. 2
Relations and predicatesp. 4
Logical notationp. 4
Referencesp. 5
Computable functionsp. 7
Algorithms, or effective proceduresp. 7
The unlimited register machinep. 9
URM-computable functionsp. 16
Decidable predicates and problemsp. 22
Computability on other domainsp. 23
Generating computable functionsp. 25
The basic functionsp. 25
Joining programs togetherp. 25
Substitutionp. 29
Recursionp. 32
Minimalisationp. 42
Other approaches to computability: Church's thesisp. 48
Other approaches to computabilityp. 48
Partial recursive functions (Godel-Kleene)p. 49
A digression: the primitive recursive functionsp. 51
Turing-computabilityp. 52
Symbol manipulation systems of Post and Markovp. 57
Computability on domains other than Np. 65
Church's thesisp. 67
Numbering computable functionsp. 72
Numbering programsp. 72
Numbering computable functionsp. 76
Discussion: the diagonal methodp. 79
The s-m-n theoremp. 81
Universal programsp. 85
Universal functions and universal programsp. 85
Two applications of the universal programp. 90
Effective operations on computable functionsp. 93
Computability of the function [sigma subscript n]p. 95
Decidability, undecidability and partial decidabilityp. 100
Undecidable problems in computabilityp. 101
The word problem for groupsp. 106
Diophantine equationsp. 107
Sturm's algorithmp. 108
Mathematical logicp. 109
Partially decidable predicatesp. 112
Recursive and recursively enumerable setsp. 121
Recursive setsp. 121
Recursively enumerable setsp. 123
Productive and creative setsp. 133
Simple setsp. 140
Arithmetic and Godel's incompleteness theoremp. 143
Formal arithmeticp. 143
Incompletenessp. 146
Godel's incompleteness theoremp. 149
Undecidabilityp. 155
Reducibility and degreesp. 157
Many-one reducibilityp. 158
Degreesp. 161
m-complete r.e. setsp. 165
Relative computabilityp. 167
Turing reducibility and Turing degreesp. 174
Effective operations on partial functionsp. 182
Recursive operatorsp. 182
Effective operations on computable functionsp. 189
The first Recursion theoremp. 192
An application to the semantics of programming languagesp. 196
The second Recursion theoremp. 200
The second Recursion theoremp. 200
Discussionp. 207
Myhill's theoremp. 210
Complexity of computationp. 212
Complexity and complexity measuresp. 213
The Speed-up theoremp. 218
Complexity classesp. 223
The elementary functionsp. 225
Further studyp. 236
Bibliographyp. 239
Index of notationp. 241
Subject Indexp. 246
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521294652
ISBN-10: 0521294657
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 264
Published: 18th August 1980
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.7
Weight (kg): 0.41