The probabilistic degree of a Boolean function f : {0, 1} n → {0, 1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions -specifically symmetric Boolean functions -have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems.In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

No abstract

Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least n hyperplanes.  We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for polynomial covers of weight-determined sets of strictly unimodal uniform (SU2) grids.   As a central tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) closures of symmetric sets of the Boolean cube. In fact, we obtain a characterization that concerns, more generally, weight-determined sets of SU2 grids. However, in this generality, our characterization is not of the Z-closures but of a new variant of Z-closures defined exclusively for weight-determined sets, which coincides with the Z-closures in the Boolean cube setting, for symmetric sets. This characterization admits a linear time algorithm, and may also be of independent interest.  Indeed, as further applications, we (i) give an alternate proof of a lemma by Alon et al. (IEEE Trans. Inform. Theory, 1988), and (ii) characterize the certifying degrees of weight-determined sets.   In the Boolean cube setting, our above characterization can also be derived using a result of Bernasconi and Egidi (Inf. Comput., 1999) that determines the affine Hilbert functions of symmetric sets. However, our proof is independent of this result, works for all SU2 grids, and could be regarded as being more combinatorial.   We also introduce another new variant of Z-closures to understand better the difference between the hyperplane and polynomial covering problems over uniform grids. Finally, we conclude by introducing a third variant of our covering problems and conjecturing its solution in the Boolean cube setting.

In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC 0 [⊕]. In particular, we show that for any fixed d, the class C d,k of functions that have uniform AC 0 [⊕] formulas of depth d and size n k form an infinite hierarchy. We show this by exibiting the first class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC 0 [⊕] 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 AC 0 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 AC 0 [⊕] formulas (which are also allowed NOT and Parity gates): formally, we show that any AC 0 [⊕] 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 AC 0 [⊕], and a result of Cohen, Ganor and Raz (APPROX-RANDOM 2014), who show a exp(Ω((1/δ) 1/(d−1) )) lower bound for the smaller class AC 0 .The upper bound is a derandomization involving a use of Janson's inequality (from probabilistic combinatorics) and classical combinatorial designs; as far as we know, this is the first such use of Janson's inequality. For the lower bound, we prove an optimal (up to a constant factor) degree lower bound for multivariate polynomials over F 2 solving the δ-coin problem, which may be of independent interest.

No abstract