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 2no(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 average for constant-depth circuits of size nɛ·log n with ɛ log 2 n 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 H˚astad’s switching lemma (STOC ’86) with a multiparty communication complexity lower bound by Babai, Nisan an

Short PCPs with projection queries

We construct a PCP for NTIME(2 n) with constant soundness, 2 n poly(n) proof length, and poly(n) queries where the verifier’s computation is simple: the queries are a projection of the input randomness, and the computation on the prover’s answers is a 3CNF. The previous upper bound for these two computations was polynomial-size circuits. Composing this verifier with a proof oracle increases the circuit-depth of the latter by 2. Our PCP is a simple variant of the PCP by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (CCC 2005). We also give a more modular exposition of the latter, separating the combinatorial from the algebraic arguments. If our PCP is taken as a black box, we obtain a more direct proof of the result by Williams, later with Santhanam (CCC 2013) that derandomizing circuits on n bits from a class C in time 2 n /n ω(1) yields that NEXP is not in a related circuit class C ′. Our proof yields a tighter connection: C is an And-Or of circuits from C ′. Along the way we show that the same lower bound follows if the satisfiability of the And of any 3 circuits from C ′ can be solved in time 2 n /n ω(1). ∗The research leading to these results has received funding from the European Community’