Classic text considers general theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.
Author(s): Martin Davis
Series: Mcgraw-Hill Series in Information Processing and Computers
Edition: 1
Publisher: Dover Publications
Year: 1985
Language: English
Commentary: Front cover, OCR, 3 level bookmarks, paginated.
Pages: 237
Introduction
1. Heuristic Remarks on Decision Problems
2. Suggestions to the Reader
3. Notational Conventions
PART 1 - THE GENERAL THEORY OF COMPUTABILITY
Ch 1. Computable Functions
1. Turing Machines
2. Computable Functions and Partially Computable Functions
3. Some Examples
4. Relatively Computable Functions
Ch 2. Operations on Computable Functions
1. Preliminary Lemmas
2. Composition and Minimalization
Ch 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
Ch 4. Turing Machines Self-applied
1. Arithmetization of the Theory of Turing Machines
2. Computability and Recursiveness
3. A Universal Turing Machine
Ch 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
Ch 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
Ch 7. Diophantine Equations
1. Hilbert's Tenth Problem
2. Arithmetical and Diophantine Predicates
3. Arithmetical Representation of Semicomputable Predicates
Ch 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
Ch 9. The Kleene Hierarchy
1. The Iteration 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 2A
7. Post's Representation Theorem
Ch 10. Computable Functionals
1. Functionals
2. Completely Computable Functionals
3. Normal Form Theorems
4. Partially Computable and Computable Functionals
5. Functionals and Relative Recursiveness
6. Decision Problems
7. The Recursion Theorems
Ch 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. Some Results from the Elementary Theory of Numbers
