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/sup /spl Omega/(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/sup 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}/sup n/ /spl rarr/ {0, 1} that is very hard on aver-age for constant-depth circuits of size n/sup /spl epsi//spl middot/log n/ with /spl epsi/log/sup 2/ 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).