Computability and Unsolvability

by Martin D. Davis

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.

Genres: Computer Science, Reference, Mathematics, Science, Philosophy

248 pages, Kindle Edition

First published January 1, 1958

About the author

Martin David Davis (born 1928) is Professor Emeritus at New York University's Computer Science Department.

Reviews

[Roberto Rigolin F Lopes · Rating 4 out of 5]

We are in 1958, Davis is writing from the border between mathematics and computer science. He assumes that you know a great deal about Turing machines, Godel's numbers + incompleteness theorems, and is familiar with their original notations. Non-mathematicians like myself might get scared with the notation (e.g. while wrestling with the "arithmetization theory of Turing machines", what?!); it may even make you cry. Then he goes incrementally showing operations with computable functions, recursive functions and difficulties with decision problems. One proof after another. The cross-references among the several theorems in this book will make you behave like a Turing machine going furiously back and forth trying to "compute" this book. A great challenge indeed.

[Ben Meyers · Rating 3 out of 5]

Fascinating but damn dense.