Randomness Buys Depth for Approximate Counting

We show that the promise problem of distinguishing n-bit strings of hamming weights 1/2 + / - Ω(1/lg^d-1 n) can be solved by explicit, randomized (unbounded-fan-in) poly(n)- size depth-d circuits with error ≤ 1/3, but cannot be solved by deterministic poly(n)-size depth-(d +1) circuits, for every d ≥ 2; and the depth of both is tight. Previous results bounded the depth to within at least an additive 2. Our sharper 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 (provably) buys resources. Techniques: To rule out deterministic circuits we combine the switching lemma with an earlier depth-3 lower bound by the author (Comp. 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 which we find important for the main message of this paper we construct a new pseudorandom generator for certain combinatorial rectangle tests. Based on expander walks, the generator for example fools tests A₁ × A₂ × ... × A_{lg n} for A_i ⊆ [n], |A_i| = n/2 with error 1/n and seed length O(lg n), improving on the seed length Ω(lg n lg lg n) of previous constructions.