ON THE PROOF COMPLEXITY OF THE NISAN–WIGDERSON GENERATOR BASED ON A HARD NP ∩ coNP FUNCTION

Abstract

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τ(g)b expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement (S) that the tautologies have no polynomial size proofs in any propositional proof system. This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement (S) is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory (properly included in Buss's theory [Formula: see text]) then Razborov's conjecture would follow, and if TPV could be added too then Statement (S) would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems.