Bounded Arithmetic, Propositional Logic and Complexity Theory

Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.

Jan Krajicek (Author)

9780521452052, Cambridge University Press

Hardback, published 24 November 1995

360 pages
24.2 x 16.2 x 2.7 cm, 0.65 kg

'It can be strongly recommended especially to mathematicians and computer scientists working in the field and to graduate students.' European Mathematical Society Newsletter

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 theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then 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, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, 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 this comprehensive treatment an excellent guide to this expanding interdisciplinary area.

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.

Subject Areas: Mathematical theory of computation [UYA], Mathematical foundations [PBC]