Computability and Unsolvability: (Dover Books on Computer Science)

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.

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

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