Pseudorandom generators hard for k-DNF resolution and polynomial calculus resolution

Abstract

A pseudorandom generator Gn: {0,1}n → {0,1}m is hard for a propositional proof system P if (roughly speaking) P cannot efficiently prove the statement Gn(x1,., xn) ≠ b for any string b 2 {0,1}m. We present a function (m ≥ 2nΩ(1)) generator which is hard for Res(ε 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(ε 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 log N) (N ~ 2n is the overall number of propositional variables) does not possess efficient proofs of NP ⊈ 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 eld is different from 2. As a byproduct, we also improve on the small restriction switching lemma due to Segerlind, Buss and Impagliazzo by removing a square root from the nal bound. This in particular implies that the (moderately) weak pigeonhole principle PHP2nn is hard for Res(ε log n/log log n).