Symmetric Exponential Time Requires Near-Maximum Circuit Size

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 $\Sigma2\mathsf{E}$, $(\Sigma2\mathsf{E}\cap\Pi2\mathsf{E})/1$, and $\mathsf{ZPE}^{{\mathsf{NP}}/_1$.} 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 $\Delta3\mathsf{E} = \mathsf{E}^{\Sigma2\mathsf{P}}$ (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 ($\mathsf{FZPP}^{{\mathsf{NP}}/_1$)} that solves the range avoidance problem infinitely often. This algorithm also implies unconditional infinitely-often pseudodeterministic $\mathsf{FZPP}^{{\mathsf{NP}}/_1$} constructions for Ramsey graphs, rigid matrices, two-source extractors, linear codes, and $\mathrm{K}^{{\mathrm{poly}}$-random} strings with nearly optimal parameters. Our proofs relativize. The two main technical ingredients are (1) Korten's $\mathsf{P}^{{\mathsf{NP}}$} 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).