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TR23-144 | 22nd September 2023 17:46

Symmetric Exponential Time Requires Near-Maximum Circuit Size


Authors: Lijie Chen, Shuichi Hirahara, Hanlin Ren
Publication: 22nd September 2023 18:01
Downloads: 1074


We show that there is a language in $\mathrm{S}_2\mathrm{E}/_1$ (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 $\Sigma_2\mathrm{E}$, $(\Sigma_2\mathrm{E}\cap\Pi_2\mathrm{E})/_1$, and $\mathrm{ZPE}^{\mathrm{NP}}/_1$. 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 $\Delta_3\mathrm{E} = \mathrm{E}^{\Sigma_2\mathrm{P}}$ (Miltersen, Vinodchandran, and Watanabe COCOON'99).

Our circuit lower bounds are corollaries of an unconditional zero-error pseudodeterministic algorithm with an $\mathrm{NP}$ oracle and one bit of advice ($\mathrm{FZPP}^{\mathrm{NP}}/_1$) that solves the range avoidance problem infinitely often. This algorithm also implies unconditional infinitely-often pseudodeterministic $\mathrm{FZPP}^{\mathrm{NP}}/_1$ 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 $\mathrm{P}^{\mathrm{NP}}$ reduction from the range avoidance problem to constructing hard truth tables (FOCS'21), which was in turn inspired by a result of Je?√°bek 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).

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