Symmetric Exponential Time Requires Near-Maximum Circuit Size

Abstract: We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds for the classes $\Sigma_2\mathsf{E}$, $(\Sigma_2\mathsf{E}\cap\Pi_2\mathsf{E})/_1$, and $\mathsf{ZPE}<sup>{\mathsf{NP}}/_1$. Previously, only "half-exponential" circuit lower bounds for these complexity classes were known, and the smallest complexity class known to require exponential circuit complexity was $$\Delta_3\mathsf{E} = \mathsf{E}<sup>{\Sigma_2\mathsf{P}}$$ (Miltersen, Vinodchandran, and Watanabe COCOON'99). Our circuit lower bounds are corollaries of an unconditional zero-error pseudodeterministic algorithm with an $\mathsf{NP}$ oracle and one bit of advice ($\mathsf{FZPP}<sup>{\mathsf{NP}}/_1$) that solves the range avoidance problem infinitely often. This algorithm also implies unconditional infinitely-often pseudodeterministic $\mathsf{FZPP}<sup>{\mathsf{NP}}/_1$ constructions for Ramsey graphs, rigid matrices, two-source extractors, linear codes, and $\mathrm{K}<sup>{\mathrm{poly}}$-random strings with nearly optimal parameters. Our proofs relativize. The two main technical ingredients are (1) Korten's $\mathsf{P}<sup>{\mathsf{NP}}$ reduction from the range avoidance problem to constructing hard truth tables (FOCS'21), which was in turn inspired by a result of Je\v{r}\'abek on provability in Bounded Arithmetic (Ann. Pure Appl. Log. 2004); and (2) the recent iterative win-win paradigm of Chen, Lu, Oliveira, Ren, and Santhanam (FOCS'23).