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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

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

Library of Congress subject headings for this publication: Recursive functions, Unsolvability (Mathematical logic)Computable functions