A Fixed-Depth Size-Hierarchy Theorem for $\mathrm{AC}^0[\oplus]$ via the Coin Problem.

In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC0[⊕]. In particular, we show that for any fixed d, the class Cd,k of functions that have uniform AC0[⊕] formulas of depth d and size nk form an infinite hierarchy. We show this by exhibiting the first class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC0[⊕] formulas. The explicit functions are derived from the δ-Coin Problem, which is the computational problem of distinguishing between coins that are heads with probability (1+δ)/2 or (1−δ)/2, where δ is a parameter that is going to 0. We study the complexity of this problem and make progress on both upper bound and lower bound fronts. Upper bounds. For any constant d≥ 2, we show that there are explicit monotone AC0 formulas (i.e. made up of AND and OR gates only) solving the δ-coin problem that have depth d, size exp(O(d(1/δ)1/(d−1))), and sample complexity (i.e. number of inputs) poly(1/δ). This matches previous upper bounds of O’Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size (which is optimal) and improves the sample complexity from exp(O(d(1/δ)1/(d−1))) to poly(1/δ). Lower bounds. We show that the above upper bounds are nearly tight (in terms of size) even for the significantly stronger model of AC0[⊕] formulas (which are also allowed NOT and Parity gates): formally, we show that any AC0[⊕] formula solving the δ-coin problem must have size exp(Ω(d(1/δ)1/(d−1))). This strengthens a result of Shaltiel and Viola (SICOMP 2010), who prove a exp(Ω((1/δ)1/(d+2))) lower bound for AC0[⊕], and a lower bound of exp(Ω((1/δ)1/(d−1))) shown by Cohen, Ganor and Raz (APPROX-RANDOM 2014) for the class 0. The upper bound is a derandomization involving a use of Janson’s inequality and classical combinatorial designs. The lower bound involves proving an optimal degree lower bound for polynomials over 2 solving the δ-coin problem.