An average-case depth hierarchy theorem for Boolean circuits

TL;DR

This paper establishes an average-case depth hierarchy theorem for Boolean circuits, demonstrating explicit functions that require exponentially larger size for lower-depth circuits to approximate, and confirms related conjectures about the polynomial hierarchy and influence bounds.

Contribution

It proves an average-case depth hierarchy theorem for Boolean circuits, answering an open question and confirming the infinitude of the polynomial hierarchy relative to a random oracle.

Findings

Existence of explicit functions requiring exponential size for lower-depth circuits

Confirmation that the polynomial hierarchy is infinite relative to a random oracle

No approximate converse to influence bounds for small-depth circuits

Abstract

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, 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 $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{\Omega(1/d)}}).$ This answers an open question posed by H{\aa}stad in his Ph.D. thesis. 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 H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions.