Dual weak pigeonhole principle, Boolean complexity, and derandomization

Abstract

We study the extension (introduced as BT by Krajı́ček in Fund. Math. 170 (2001) 123) of the theory S 2 1 by instances of the dual (onto) weak pigeonhole principle for p-time functions, dWPHP( PV) x 2 x . We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S 2 1+ dWPHP( PV). We construct a propositional proof system WF (based on a reformulation of Extended Frege in terms of Boolean circuits), which captures the ∀ Π 1 b -consequences of S 2 1+ dWPHP( PV). We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. (Comput. Complexity 6 (1996/1997) 256). We prove that dWPHP( PV) is (over S 2 1) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction (derandomization of probabilistic p-time algorithms) in a conservative extension of S 2 1+ dWPHP( PV).