Nondeterministic Quasi-Polynomial Time is Average-Case Hard for [math] Circuits

SIAM Journal on Computing, Ahead of Print.
Abstract. Following the seminal work of [R. R. Williams, J. ACM, 61 (2014)], in a recent breakthrough, [C. D. Murray and R. R. Williams, STOC 2018] proved that [math] (nondeterministic quasi-polynomial time) does not have polynomial-size [math] circuits (constant depth circuits consisting of [math]/[math]/[math] gates for a fixed constant [math], a frontier class in circuit complexity). We strengthen the above lower bound to an average-case one, by proving that for all constants [math], there is a language in [math] that cannot be [math]-approximated by polynomial-size [math] circuits. Our work also improves the average-case lower bound for [math] against polynomial-size [math] circuits by [R. Chen, I. C. Oliveira, and R. Santhanam, LATIN 2018, pp. 317–330]. Our new lower bound builds on several interesting components, including the following: 1. Barrington’s theorem and the existence of an [math]-complete language that is random self-reducible. 2. The subexponential witness-size lower bound for [math] against [math] and the conditional nondeterministic pseudorandom generator (PRG) construction in [R. R. Williams, SIAM J. Comput., 45 (2016), pp. 497–529]. 3. An “almost” almost-everywhere [math] average-case lower bound (which strengthens the corresponding worst-case lower bound in [C. D. Murray and R. R. Williams, STOC 2018]). 4. A [math]-complete language that is downward self-reducible, same-length checkable, error-correctable, and paddable. Moreover, all its reducibility properties have corresponding low-depth nonadaptive oracle circuits. Our construction builds on [L. Trevisan and S. P. Vadhan, Comput. Complexity, 16 (2007), pp. 331–364]. Like other lower bounds proved via the “algorithmic approach,” the only property of [math] exploited by us is the existence of a nontrivial [math] algorithm for [math] [R. R. Williams, J. ACM, 61 (2014)]. Therefore, for any typical circuit class [math], our results apply to [math] as well if a nontrivial [math] (in fact, [math]) algorithm for [math] is discovered.