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Computability and Unsolvability
Martin D. Davis
Table of Contents
Preface to the Dover Edition;Preface to the First Edition;
Glossary of special symbols
Introduction
1. Heuristic Remarks on Decision Problems
2. Suggestions to the Reader
3. Notational Conventions
Part I. The general theory of computability
Chapter 1.Computable Functions
1. Turing Machines
2. Computable Functions and Partially Computable functions
3. Some Examples
4. Relatively Computable functions
Chapter 2. Operations on Computable Functions
1. Preliminary Lemmas
2. Composition and Minimalization
Chapter 3. Recursive functions
1. Some Classes of Functions
2. Finite Sequences of Natural Numbers
3. Primitive Recursion
4. Primitive Recursive functions
5. Recursive Sets and Predicates
Chapter 4. Turing Machines Self-applied
1. Arithmetization of the Theory of Turing Machines
2. Computability and Recursiveness
3. A Universal Turing Machine
Chapter 5. Unsolvable Decision Problems
1. Semicomputable Predicates
2.Decision Problems
3.Properties of Semicomputable Predicates
4.Recursively enumerable Sets
5.Two Recursively enumerable Sets
6.A Set Which Is Not Recursively Enumerable
Part 2. Applications of the General Theory
Chapter 6. Combinatorial Problems
1. Combinatorial systems
2. Turing machines and Semi-Thue Systems
3. Thue Systems
4. The Word Problem for Semigroups
5. Normal Systems and Post Systems
Chapter 7. Diophantine Equations
1. Hilbert's Tenth Problem
2. Arithmetical and Diophantine Predicates
3. Arithmetical Representation of Semicomputable Predicates
Chapter 8. Mathematical Logic
1. Logics
2. Incompleteness and Unsolvability Theorems for Logics
3. Arithmetical Logics
4. First-order Logics
5. Partial Propositional Calculi
Part 3. Further Development of the General Theory
Chapter 9. The Kleene Hierarchy
1. The Interation Theorem
2. Some First Applications of the Iteration Theorem
3. Predicates, Sets, and Functions
4. Strong Reducibility
5. Some Classes of Predicates
6. A Representation Theorem for P subscript 2 superscript A
7. Post's Representation Theorem
Chapter 10. Computable Functionals
1. Functionals
2. Complete Computable functionals
3. Normal Form Theorems
4. Partially Computable and Computable Functionals
5. Functionals and Relative Recursiveness
6. Decision Problems
7. The Recursion Theorems
Chapter 11. The Classification of Unsolvable Decision Problems
1. Reducibility and the Kleene Hierarchy
2. Incomparability
3. Creative Sets and Simple Sets
4. Constructive Ordinals
5. Extensions of the Kleene Hierarchy
Appendix 1. Some Results from the Elementary Theory of Numbers
Appendix 2. Hilbert's Tenth Problem is Unsolvable
References; Index