Synopses & Reviews
"A clearly written, well-presented survey of an intriguing subject." — Scientific American. 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.
Synopsis
In this classic text, Dr. Davis provides a clear introduction to computability, at an advanced undergraduate level, that serves the needs of specialists and non-specialists alike.
In Part One (Chapters 1-5), Professor Davis outlines the general theory of computability, discussing such topics as computable functions, operations on computable functions, recursive functions, Turing machines, self-applied, and unsolvable decision problems. The author has been careful, especially in the first seven chapters, to assume no special mathematical training on the part of the reader.
Part Two (Chapters 6-8) comprises a concise treatment of applications of the general theory, incorporating material on combinatorial problems, Diophantine Equations (including Hilbert's Tenth Problem) and mathematical logic. The final three chapters (Part 3) present further development of the general theory, encompassing the Kleene hierarchy, computable functionals, and the classification of unsolvable decision problems.
When first published in 1958, this work introduced much terminology that has since become standard in theoretical computer science. Indeed, the stature of the book is such that many computer scientists regard it as their theoretical introduction to the topic. This new Dover edition makes this pioneering, widely admired text available in an inexpensive format.
For Dover's edition, Dr. Davis has provided a new Preface and an Appendix, "Hilbert's Tenth Problem Is Unsolvable," an important article he published in The American Mathematical Monthly in 1973, which was awarded prizes by the American Mathematical Society and the Mathematical Association of America. These additions further enhance the value and usefulness of an "unusually clear and stimulating exposition" (Centre National de la Recherche Scientifique, Paris) now available for the first time in paperback.
Synopsis
Classic graduate-level introduction to theory of computability. Discusses 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, more.
About the Author
Martin Davis: Computer Science Pioneer
Dover's publishing relationship with Martin Davis, now retired from NYU and living in Berkeley, goes back to 1985 when we reprinted his classic 1958 book Computability and Unsolvability, widely regarded as a classic of theoretical computer science. A graduate of New York's City College, Davis received his PhD from Princeton in the late 1940s and became one of the first computer programmers in the early 1950s, working on the ORDVAC computer at The University of Illinois. He later settled at NYU where he helped found the Computer Science Department.
Not many books from the infancy of computer science are still alive after several decades, but Computability and Unsolvability is the exception. And The Undecidable is an anthology of fundamental papers on undecidability and unsolvability by major figures in the field including Godel, Church, Turing, Kleene, and Post.
Critical Acclaim for Computability and Unsolvability:
"This book gives an expository account of the theory of recursive functions and some of its applications to logic and mathematics. It is well written and can be recommended to anyone interested in this field. No specific knowledge of other parts of mathematics is presupposed. Though there are no exercises, the book is suitable for use as a textbook." — J. C. E. Dekker, Bulletin of the American Mathematical Society, 1959
Critical Acclaim for The Undecidable:
"A valuable collection both for original source material as well as historical formulations of current problems." — The Review of Metaphysics
"Much more than a mere collection of papers . . . a valuable addition to the literature." — Mathematics of Computation
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 1. 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