Logical Foundations of Proof Complexity

Stephen Cook and Phuong Nguyen

Year: 2010
496 pages. Hardcover.
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This book treats bounded arithmetic and propositional proof complexity from the point of view of computational complexity. The first seven chapters include the necessary logical background for the material and are suitable for a graduate course. Associated with each of many complexity classes are both a two-sorted predicate calculus theory, with induction restricted to concepts in the class, and a propositional proof system. The result is a uniform treatment of many systems in the literature, including Buss�s theories for the polynomial hierarchy and many disparate systems for complexity classes such as AC0, AC0(m), TC0, NC1, L, NL, NC, and P.

Table of Contents

  1. Introduction
  2. The predicate calculus and the system LK
  3. Peano arithmetic and its subsystems
  4. Two-sorted logic and complexity classes
  5. The theory V0 and AC0
  6. The theory V1 and polynomial time
  7. Propositional translations
  8. Theories for polynomial time and beyond
  9. Theories for small classes
  10. Proof systems and the reflection principle
  11. Computation models