Pseudorandom Generators Hard for k-DNF Resolution and Polynomial Calculus Resolution

A pseudorandom generator Gn : {0, 1}n → {0, 1}m is hard for a propositional proof system P if (roughly speaking) P can not efficiently prove the statement Gn(x1, . . . , xn) 6= b for any string b ∈ {0, 1}m. We present a function (m ≥ 2nΩ(1)) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing k-DNFs instead of clauses. As a direct consequence of this result, we show that whenever t ≥ n2, every Res( log t) proof of the principle ¬Circuitt(fn) (asserting that the circuit size of a Boolean function fn in n variables is greater than t) must have size exp(tΩ(1)). In particular, Res(log logN) (N ∼ 2n is the overall number of propositional variables) does not possess efficient proofs of NP 6⊆ P/poly. Similar results hold also for the system PCR (the natural common extension of Polynomial Calculus and Resolution) when the characteristic of the ground field is different from 2. As a by-product, we also improve on the small restriction switching lemma from [SBI04] by removing a square root from the final bound. This in particular implies that the (moderately) weak pigeonhole principle PHP 2n n is hard for Res( log n/ log log n). ∗Institute for Advanced Study, Princeton, US, on leave from Steklov Mathematical Institute, Moscow, Russia. Supported by The State of New Jersey and by the RFBR grant 02-02-01290. Current address: University of Chicago, US, [email protected].