Computability Theory

This textbook is designed to introduce undergraduate mathematics and computer science students to computability theory (recursion theory). It is based on teaching experience, and is designed to be accessible to junior and senior students without a previous background in the subject. The book will prepare the students for further study in computational complexity, logic, theoretical computer science, as well as other topics. It focuses on computability theory, and excludes computer-science topics such as automata theory, and context-free languages.Undergraduates can read and understand the materialPointers to more advanced topics, and provides a base for further study of such topicsCovers a strong introductory set of material from a modern mathematical viewpoint

Foreword	p. ix
Preface	p. xi
The Computability Concept	p. 1
The Informal Concept	p. 1
Decidable Sets	p. 1
Calculable Functions	p. 3
Church's Thesis	p. 10
Exercises	p. 11
Formalizations - An Overview	p. 12
Turing Machines	p. 13
Primitive Recursiveness and Search	p. 18
Loop and While Programs	p. 20
Register Machines	p. 22
Definability in Formal Languages	p. 24
Church's Thesis Revisited	p. 26
Exercises	p. 27
General Recursive Functions	p. 29
Primitive Recursive Functions	p. 29
Bounded Search	p. 40
Search Operation	p. 47
Exercises	p. 49
Programs and Machines	p. 53
Register Machines	p. 53
A Universal Program	p. 60
Exercises	p. 71
Register Machines Over Words	p. 72
Exercises	p. 76
Binary Arithmetic	p. 76
Recursive Enumerability	p. 79
Recursively Enumerable Relations	p. 81
Exercises	p. 92
Parameters	p. 93
Exercises	p. 100
Connections to Logic	p. 103
Arithmetical Hierarchy	p. 103
Exercises	p. 111
Definability in Arithmetic	p. 116
The Complexity of Truth	p. 116
Exercises	p. 120
Degrees of Unsolvability	p. 121
Relative Computability	p. 121
Equivalence Relations	p. 127
Preordering Relations	p. 130
Ordering Degrees	p. 131
Exercises	p. 132
Structure of the Degrees	p. 132
Exercises	p. 137
Polynomial-Time Computability	p. 139
Feasible Computability	p. 139
P versus NP	p. 147
Some other Complexity Classes	p. 148
Exercises	p. 149
Mathspeak	p. 151
Countability	p. 155
Decadic Notation	p. 159
References	p. 163
Index	p. 165
Table of Contents provided by Ingram. All Rights Reserved.

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