Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems

TL;DR

This paper establishes a downward separation in circuit complexity for certain classes, demonstrating that non-deterministic circuit lower bounds imply stronger lower bounds for larger classes, and also provides weak derandomization results for Arthur-Merlin protocols.

Contribution

It introduces new downward separation results for circuit classes and extends derandomization techniques to promise Arthur-Merlin protocols using Williams' hitting set method.

Findings

Proves that non-existence of polynomial size circuits for E implies the same for SubEXP.

Shows augmented Arthur-Merlin protocols with one bit advice lack fixed polynomial size non-deterministic circuits.

Provides a weak unconditional derandomization of certain promise Arthur-Merlin protocols.

Abstract

We prove a downward separation for $\mathsf{\Sigma}_2$-time classes. Specifically, we prove that if $\Sigma_2$E does not have polynomial size non-deterministic circuits, then $\Sigma_2$SubEXP does not have \textit{fixed} polynomial size non-deterministic circuits. To achieve this result, we use Santhanam's technique on augmented Arthur-Merlin protocols defined by Aydinlio\u{g}lu and van Melkebeek. We show that augmented Arthur-Merlin protocols with one bit of advice do not have fixed polynomial size non-deterministic circuits. We also prove a weak unconditional derandomization of a certain type of promise Arthur-Merlin protocols. Using Williams' easy hitting set technique, we show that $\Sigma_2$-promise AM problems can be decided in $\Sigma_2$SubEXP with $n^c$ advice, for some fixed constant $c$.