An Average-Case Depth Hierarchy Theorem for Boolean Circuits

Benjamin Rossman, Rocco A. Servedio, Li-Yang Tan

Abstract: We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d ≥ 2, there is an explicit n-variable Boolean function f, computed by a linear-size depth-d formula, which is such that any depth-(d - 1) circuit that agrees with f on (1/2 + o <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (1)) fraction of all inputs must have size exp(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ω(1/d)</sup> ). This answers an open question posed by Hastad in his Ph.D. thesis [Has86b]. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad [Has86a], Cai [Cai86], and Babai [Bab87]. We also use our result to show that there is no “approximate converse” to the results of Linial, Mansour, Nisan [LMN93] and Boppana [Bop97] on the total influence of constant-depth circuits, thus answering a question posed by Kalai [Kal12] and Hatami [Hat14]. A key ingredient in our proof is a notion of random projections which generalize random restrictions.