Maximum Circuit Lower Bounds for Exponential-Time Arthur Merlin
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3717823.3718224.pdf
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Author(s)
Chen, Lijie
Li, Jiatu
Liang, Jingxun
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Lijie Chen, Jiatu Li, and Jingxun Liang. 2025. Maximum Circuit Lower Bounds for Exponential-Time Arthur Merlin. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1348–1358.
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Final published version
Abstract
We show that the complexity class of exponential-time Arthur Merlin with sub-exponential advice (AMEXP/2nε) requires circuit complexity at least 2n/n. Previously, the best known such near-maximum lower bounds were for symmetric exponential time by Chen, Hirahara, and Ren (STOC’24) and Li (STOC’24), or randomized exponential time with MCSP oracle and sub-exponential advice by Hirahara, Lu, and Ren (CCC’23).
Our result is proved by combining the recent iterative win-win paradigm of Chen, Lu, Oliveira, Ren, and Santhanam (FOCS’23) together with the uniform hardness-vs-randomness connection for Arthur-Merlin protocols by Shaltiel-Umans (STOC’07) and van Melkebeek-Sdroievski (CCC’23). We also provide a conceptually different proof using a novel ”critical win-win” argument that extends a technique of Lu, Oliveira, and Santhanam (STOC’21).
Indeed, our circuit lower bound is a corollary of a new explicit construction for properties in coAM. We show that for every dense property P ∈ coAM, there is a quasi-polynomial-time Arthur-Merlin protocol with short advice such that the following holds for infinitely many n: There exists a canonical string wn ∈ P ∩ {0,1}n so that (1) there is a strategy of Merlin such that Arthur outputs wn with probability 1 and (2) for any strategy of Merlin, with probability 2/3, Arthur outputs either wn or a failure symbol ⊥. As a direct consequence of this new explicit construction, our circuit lower bound also generalizes to circuits with an AM ∩ coAM oracle. To our knowledge, this is the first unconditional lower bound against a strong non-uniform class using a hard language that is only ”quantitatively harder”.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3717823.3718224
