Abstrakt
It is well known (cf. Krajicek and Pudlak ['Propositional proof systems, the consistency of first order theories and the complexity of computations', J.
Symbolic Logic 54 (1989) 1063-1079]) that a polynomial time algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1((k)) and outputs a tautology tau(k) of size at least k such that P is not p-bounded on the set of all formulas tau(k).
We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP not equal coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT boolean AND{0, 1}(k).
In Find, given 1((k)) and a tautology alpha of size at most k(0)(c), one should output a size k tautology beta that has no size k(1)(c) P-proof from substitution instances of alpha. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2(O(k)) algorithm.
Using a stronger hypothesis about the proof complexity of the Nisan-Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan-Wigderson generator as a proof system.