Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

2026-07-10

AI Summary

This work investigates circuit lower bounds for the exponential-time complexity class $\mathsf{E}^{\mathsf{prMA}}_1$, which consists of languages decidable in deterministic exponential time with access to a promise Merlin-Arthur oracle and one bit of advice. By integrating an iterative win-win strategy, a reduction from the Range Avoidance problem to circuit lower bounds, the PCP theorem, and a refined analysis of adaptive $\mathsf{P}^{\mathsf{NP}}$ query structures with bounded rounds and witness length, the authors establish— for the first time in this setting—a near-optimal circuit lower bound of $2^n / n$. This result substantially advances the understanding of circuit complexity for exponential-time classes augmented with oracles and provides new techniques for characterizing the computational power of restricted interactive proof systems.

Abstract

We prove a near-maximum ($2^n / n$) circuit lower bound for the complexity class $\mathsf{E}^{\mathrm{pr}\mathsf{MA}}/_1$, corresponding to exponential time with access to a promise-$\mathsf{MA}$ oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen--Lu--Oliveira--Ren--Santhanam, FOCS'23), the reduction from the Range Avoidance problem to circuit lower bounds (Jeřábek, Ann. Pure Appl. Log. '04; Korten, FOCS'21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class $\mathsf{P}^\mathsf{NP}[{\textsf{#rounds}}=r, {\textsf{length}}=s]$, which is $\mathsf{P}^\mathsf{NP}$ with $r(n)$ adaptive rounds of $\mathsf{NP}$ queries, where each $\mathsf{NP}$ query has witness length $s(n)$.

Problem

• circuit lower bounds
• exponential time
• Merlin-Arthur
• promise-MA
• advice

Innovation

• circuit lower bounds
• promise-MA oracle
• iterative win-win paradigm
• Range Avoidance
• PCP theorem