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 $\mathsf{ZPE}^{\mathsf{NP}}/_1$. 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 $\Delta_3\mathsf{E} = \mathsf{E}^{\Sigma_2\mathsf{P}}$ (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 ($\mathsf{FZPP}^{\mathsf{NP}}/_1$) 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).

翻译：我们证明，存在一个属于 $\mathsf{S}_2\mathsf{E}/_1$（带一位建议的对称指数时间）的语言，其电路复杂度至少为 $2^n/n$。特别地，上述结果也意味着对于类 $\Sigma_2\mathsf{E}$、$(\Sigma_2\mathsf{E}\cap\Pi_2\mathsf{E})/_1$ 和 $\mathsf{ZPE}^{\mathsf{NP}}/_1$ 具有相同的接近最大电路下界。此前，对于这些复杂度类仅已知“半指数”电路下界，而需要指数电路复杂度的最小复杂度类为 $\Delta_3\mathsf{E} = \mathsf{E}^{\Sigma_2\mathsf{P}}$（Miltersen、Vinodchandran 和 Watanabe，COCOON'99）。我们的电路下界是一个带 $\mathsf{NP}$ 预言机和一位建议的无条件零误差伪确定算法（$\mathsf{FZPP}^{\mathsf{NP}}/_1$）的推论，该算法能无限常地解决范围避免问题。该算法还无条件地实现了对于拉姆齐图、刚性矩阵、双源提取器、线性码和 $\mathrm{K}^{\mathrm{poly}}$ 随机串的无限常伪确定 $\mathsf{FZPP}^{\mathsf{NP}}/_1$ 构造，且参数接近最优。我们的证明具有相对化性质。两个主要技术要素是：（1）Korten 将范围避免问题归约到构造真值表的 $\mathsf{P}^{\mathsf{NP}}$ 归约（FOCS'21），该归约本身受 Jeřábek 关于有界算术可证明性的结果启发（Ann. Pure Appl. Log. 2004）；（2）Chen、Lu、Oliveira、Ren 和 Santhanam（FOCS'23）最近的迭代双赢范式。