Randomness Buys Depth for Approximate Counting

Abstract

We show that the promise problem of distinguishing n-bit strings of relative Hamming weight $${1/2 + \Omega(1/{\rm lg}^{d-1} n)}$$ from strings of weight $${1/2 - \Omega(1/{\rm \lg}^{d - 1} n)}$$ can be solved by explicit, randomized (unbounded fan-in) poly(n)-size depth-d circuits with error $${\leq 1/3}$$ , but cannot be solved by deterministic poly(n)-size depth-(d+1) circuits, for every $${d \geq 2}$$ ; and the depth of both is tight. Our bounds match Ajtai’s simulation of randomized depth-d circuits by deterministic depth-(d + 2) circuits (Ann. Pure Appl. Logic; ’83) and provide an example where randomization buys resources. To rule out deterministic circuits, we combine Håstad’s switching lemma with an earlier depth-3 lower bound by the author (Computational Complexity 2009). To exhibit randomized circuits, we combine recent analyses by Amano (ICALP ’09) and Brody and Verbin (FOCS ’10) with derandomization. To make these circuits explicit, we construct a new, simple pseudorandom generator that fools tests $${A_1 \times A_2 \times \cdots \times A_{{\rm lg}{n}}}$$ for $${A_i \subseteq [n], |A_{i}| = n/2}$$ with error 1/n and seed length O(lg n), improving on the seed length $${\Omega({\rm lg}\, n\, {\rm lg}\, {\rm lg}\, n)}$$ of previous constructions.