Bounded Arithmetic, Propositional Logic and Complexity Theory

by Jan Krajicek

Overview

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. Then more advanced topics are treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, simple independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the approximation method and the method of Boolean valuations, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find his comprehensive treatment an excellent guide to this expanding interdisciplinary area.

Product Details

ISBN-13:	9780521452052
Publisher:	Cambridge University Press
Publication date:	11/24/1995
Series:	Encyclopedia of Mathematics and its Applications , #60
Pages:	360
Product dimensions:	6.38(w) x 9.53(h) x 1.06(d)

Table of Contents

1. Introduction; 2. Preliminaries; 3. Basic complexity theory; 4. Basic propositional logic; 5. Basic bounded arithmetic; 6. Definability of computations; 7. Witnessing theorems; 8. Definability and witnessing in second order theories; 9. Translations of arithmetic formulas; 10. Finite axiomatizability problem; 11. Direct independence proofs; 12. Bounds for constant-depth Frege systems; 13. Bounds for Frege and extended Frege systems; 14. Hard tautologies and optimal proof systems; 15. Strength of bounded arithmetic; References; Index.

Editorial Reviews

"...the author has produced a very good piece of work, which can be strongly recommended to anyone wishing to study the relations between bounded arithmetic and complexity of propositional logic." Constantine Dimitracopoulos

"...a mathematical masterpiece...presents students and researchers with the most important tools, general background and literature references to begin research in this exciting new area in the intersection of mathematical logic, theoretical computer science, and finite combinatorics...a major accomplishment...will play a substantial role in the training of future researchers." Journal of Symbolic Logic

From the Publisher

Synthesizes recent research in bounded arithmetic and the complexity of propositional logic, emphasizing independence proofs and lower bound proofs. Also discusses the deep connections between logic and complexity theory and lists several intriguing problems waiting for answers. Introduces the basics of logic and complexity theory, then reports results in propositional proof systems and systems of bounded arithmetic. More advanced topics include polynomial simulations and conservativity results, and various witnessing theorems. Addressed to research mathematicians and computer scientists with no previous knowledge of the subject. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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