Pseudorandom bits for constant depth circuits with few arbitrary symmetric gates

We exhibit an explicitly computable ´pseudorandom´ generator stretching l bits into m(l) = l^{Ω(log l)} bits that look random to constant-depth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ´93) that achieves the same stretch but only fools circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan´s generator for constant depth circuits (Combinatorica ´91) (but Nisan´s generator has a much bigger stretch). In particular, we conclude that every function computable by uniform poly(n)-size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME (2^no(1)) This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function f : {0, 1}ⁿ → {0, 1} that is very hard on aver-age for constant-depth circuits of size n^{ε·log n} with εlog² n it arbitrary symmetric gates, and plugging it into the Nisan-Wigderson pseudorandom generator construction (FOCS ´88). The proof of the average-case hardness of this function is a modification of arguments by Razborov and Wigderson (IPL ´93), and Hansen and Miltersen (MFCS ´04), and combines Hdstad´s switching lemma (STOC ´86) with a multiparty communication complexity lower bound by Babai, Nisan and Szegedy (STOC ´89).