Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable pseudorandom generator stretching $l$ bits into $m(l) = l^{\Omega(\log l)}$ 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 [Proceedings of the Second Israel Symposium on Theory of Computing Systems, 1993, pp. 18-24] that achieves the same stretch but fools only 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 (but Nisan’s generator has a much bigger stretch) [Combinatorica, 11 (1991), pp. 63-70]. 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 $\mathit{TIME}(2^{n^{o(1)}})$. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function $f : \zo^n \to \zo$ that is very hard on average for constant-depth circuits of size $s(n) = n^{\Omega(\log n)}$ with $\log s(n)$ arbitrary symmetric gates, and plugging it into the Nisan-Wigderson pseudorandom generator construction [J. Comput. System Sci., 49 (1994), pp. 149-167]. The proof of the average-case hardness of this function is a modification of arguments by Razborov and Wigderson [Inform. Process. Lett., 45 (1993), pp. 303-307] and Hansen and Miltersen [Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 3153, Springer-Verlag, Berlin, 2004, pp. 334-345] and combines Harstad’s switching lemma [Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, MA, 1987] with a multiparty communication complexity lower bound by Babai, Nisan, and Szegedy [J. Comput. System Sci., 45 (1992), pp. 204-232].