Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound.

We give an explicit function h : {0, l}(n) -> {0, 1} such that every de Morgan formula of sizen(3) o(1)/r(2)agrees with h on at most a fraction of 1/2 + 2 (Omega)(r) of the inputs. Our technical contributions include a theorem that shows that the "expected shrinkage" result of HAstad [SIAM T. Comput., 27 (1998), pp. 48-64] actually holds with very high probability (where the restrictions are chosen from a certain distribution that takes into account the structure of the formula), using ideas of Impagliazzo, Meka, and Zuckerman [Proceedings of FOCS, 2012, pp. 111-119].