Logical Foundations of Proof Complexity

Abstract

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 complexity classes range from AC0 for the weakest theory up to the polynomial hierarchy. Each bounded theorem in a theory translates into a family of (quantified) propositional tautologies with polynomial size proofs in the corresponding proof system. The theory proves the soundness of the associated 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.

Formalizing Randomized Matching Algorithms

“…Basic bounded arithmetic. The theory VPV for polynomial time reasoning used here is a two-sorted theory described by Cook and Nguyen [7]. The two-sorted language has variables x, y, z, .…”

“…Similarly, a number function is Σ B 0 -definable from L if it is p-bounded and its graph is represented by a Σ B 0 (L)-formula. The theory V 0 for AC 0 is the basis to develop theories for small complexity classes within P in [7]. The theory V 0 consists of the vocabulary L 2 A and axiomatized by the sets of 2-BASIC axioms as given in Figure 1, which express basic properties of symbols in L 2 A , …”

“…A universally-axiomatized conservative extension V 0 of V 0 was also obtained by introducing function symbols and their defining axioms for all FAC 0 functions. In [7,Chapter 9], Cook and Nguyen showed how to associate a theory VC to each complexity class C ⊆ P, where VC extends V 0 with an additional axiom asserting the existence of a solution to a complete problem for C. General techniques are also presented for defining a universally-axiomatized conservative extension VC of VC which has function symbols and defining axioms for every function in FC , and VC admits induction on open formulas in this enriched vocabulary. It follows from Herbrand's Theorem that the provably-total functions in VC (and hence in VC ) are precisely the functions in FC .…”

“…In this paper, we are particularly interested in the theory VPV for polytime reasoning [7,Chapter 8.2] since we will use it to formalize all of our theorems. The universal theory VPV is based on Cook's single-sorted theory PV [8], which was historically the first theory designed to capture polytime reasoning.…”

“…There is a substantial literature on theories such as PV, S 1 2 , VPV, V 1 which capture polynomial time reasoning [8,4,17,7]. These theories prove the existence of polynomial time functions, and in many cases they can prove properties of these functions that assert the correctness of the algorithms computing these functions.…”