The Classical Decision Problem

by Egon Bïrger, Erich Grïdel, Yuri Gurevich

Overview

This book is addressed to all those - logicians, computer scientists, mathe­ maticians, philosophers of science as well as the students in all these disci­ plines - who may be interested in the development and current status of one of the major themes of mathematical logic in the twentieth century, namely the classical decision problem known also as Hilbert's Entscheidungsproblem. The text provides a comprehensive modern treatment of the subject, includ­ ing complexity theoretic analysis. We have made an effort to combine the features of a research monograph and a textbook. Only the basic knowledge of the language of first-order logic is required for understanding of the main parts of the book, and we use standard terminology. The chapters are written in such a way that various combinations of them can be used for introductory or advanced courses on undecidability, decidability and complexity of logical decision problems. This explains a few intended redundancies and repetitions in some of the chapters. The annotated bibliography, the historical remarks at the end of the chapters and the index allow the reader to use the text also for quick reference purposes.

Product Details

ISBN-13:	9783540423249
Publisher:	Springer Berlin Heidelberg
Publication date:	10/16/2001
Series:	Universitext
Edition description:	Softcover reprint of the original 1st ed. 1997
Pages:	482
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1. Introduction: The Classical Decision Problem.- 1.1 The Original Problem.- 1.2 The Transformation of the Classical Decision Problem.- 1.3 What Is and What Isn’t in this Book.- I. Undecidable Classes.- 2. Reductions.- 3. Undecidable Standard Classes for Pure Predicate Logic.- 4. Undecidable Standard Classes with Functions or Equality.- 5. Other Undecidable Cases.- II. Decidable Classes and Their Complexity.- 6. Standard Classes with the Finite Model Property.- 7. Monadic Theories and Decidable Standard Classes with Infinity Axioms.- 8. Other Decidable Cases.- A. Appendix: Tiling Problems.- A.1 Introduction.- A.2 The Origin Constrained Domino Problem.- A.3 Robinson’s Aperiodic Tile Set.- A.4 The Unconstrained Domino Problem.- A.5 The Periodic Problem and the Inseparability Result.- Annotated Bibliography.

Editorial Reviews

From the reviews of the first edition:

"The authors … describe their effort as that of ‘combining the features of a research monograph and a textbook.’ They suggest that the book - or selected chapters of it - might be used for an introductory course on decision problems, undecidability, and the complexity of decision procedures. … So there is usually a lot to think about in making sense of the author’s arguments. This is part of what makes this book so enjoyable." (R. Gregory Taylor, The Review of Modern Logic, Vol. 9 (3-4), 2004)

"This is the most comprehensive treatment available in book form of the classical decision problem of mathematical logic and of the role of the classical decision problem in modern computer science. A revealing analysis of the natural order of decidable and undecidable cases is given. ... Many cases are treated here for the first time, and a great number of simple proofs and exercises have been included." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)

"The book is dedicated to a comprehensive presentation of the classical decision problem of first-order logic. … This book is an essential reference for any researcher in logic, complexity, and artificial intelligence. … Historical references that are placed at the end of each chapter are very enjoyable and help the reader follow the literature and gain a perspective of the field. … an excellent reference book for researchers in the field, and for advanced doctoral students in theoretical computer science and logic." (Dan A. Simovici, SIGACT News, Vol. 35 (1), 2004)

From the Publisher