An Average-Case Lower Bound Against $\mathsf {ACC}^0$

Abstract

In a seminal work, Williams [22] showed that $\mathsf {NEXP}$ (non-deterministic exponential time) does not have polynomial-size $\mathsf {ACC}^0$ circuits. Williams’ technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known. We show that there is a language L in $\mathsf {NEXP}$ and a function $\varepsilon (n) = 1/\log (n)^{\omega (1)}$ such that no sequence of polynomial size $\mathsf {ACC}^0$ circuits solves L on more than a $1/2+\varepsilon (n)$ fraction of inputs of length n for all large enough n. Complementing this result, we give a nontrivial pseudo-random generator against polynomial-size $\mathsf {AC}^0[6]$ circuits. We also show that learning algorithms for quasi-polynomial size $\mathsf {ACC}^0$ circuits running in time $2^n/n^\omega (1)$ imply lower bounds for the randomised exponential time classes $\mathsf {RE}$ (randomized time $2^{O(n)}$ with one-sided error) and $\mathsf {ZPE}/1$ (zero-error randomized time $2^{O(n)}$ with 1 bit of advice) against polynomial size $\mathsf {ACC}^0$ circuits. This strengthens results of Oliveira and Santhanam [15].