Computability An Introduction to Recursive Function Theory

ISBN-10: 0521294657
ISBN-13: 9780521294652
Edition: 1980
Authors: N. J. Cutland
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Description: 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  More...

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Book details

List price: $64.99
Copyright year: 1980
Publisher: Cambridge University Press
Publication date: 6/19/1980
Binding: Paperback
Pages: 264
Size: 6.25" wide x 9.50" long x 0.75" tall
Weight: 1.012
Language: English

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.

Preface
Prologue. Prerequisites and notation
Sets
Functions
Relations and predicates
Logical notation
References
Computable functions
Algorithms, or effective procedures
The unlimited register machine
URM-computable functions
Decidable predicates and problems
Computability on other domains
Generating computable functions
The basic functions
Joining programs together
Substitution
Recursion
Minimalisation
Other approaches to computability: Church's thesis
Other approaches to computability
Partial recursive functions (Godel-Kleene)
A digression: the primitive recursive functions
Turing-computability
Symbol manipulation systems of Post and Markov
Computability on domains other than N
Church's thesis
Numbering computable functions
Numbering programs
Numbering computable functions
Discussion: the diagonal method
The s-m-n theorem
Universal programs
Universal functions and universal programs
Two applications of the universal program
Effective operations on computable functions
Computability of the function [sigma subscript n]
Decidability, undecidability and partial decidability
Undecidable problems in computability
The word problem for groups
Diophantine equations
Sturm's algorithm
Mathematical logic
Partially decidable predicates
Recursive and recursively enumerable sets
Recursive sets
Recursively enumerable sets
Productive and creative sets
Simple sets
Arithmetic and Godel's incompleteness theorem
Formal arithmetic
Incompleteness
Godel's incompleteness theorem
Undecidability
Reducibility and degrees
Many-one reducibility
Degrees
m-complete r.e. sets
Relative computability
Turing reducibility and Turing degrees
Effective operations on partial functions
Recursive operators
Effective operations on computable functions
The first Recursion theorem
An application to the semantics of programming languages
The second Recursion theorem
The second Recursion theorem
Discussion
Myhill's theorem
Complexity of computation
Complexity and complexity measures
The Speed-up theorem
Complexity classes
The elementary functions
Further study
Bibliography
Index of notation
Subject Index

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