Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time

TL;DR

This paper demonstrates that within the symmetric exponential time class, problems requiring exponential-size circuits are typical, showing that such complexity is comeager in this resource-bounded setting.

Contribution

It extends resource-bounded category to the class S^E_2 and proves that exponential-size circuit problems are comeager in this class, indicating their typicality.

Findings

SIZE(2^n/n) is meager in S^E_2.

Languages requiring exponential-size circuits are comeager in S^E_2.

Li's FS^P_2 algorithm provides a winning strategy for the Range Avoidance problem.

Abstract

Lutz (1987) introduced resource-bounded category and showed the circuit size class SIZE($\frac{2^n}{n}$) is meager within ESPACE. Li (2024) established that the symmetric alternation class $S^E_2$ contains problems requiring circuits of size $\frac{2^n}{n}$. In this note, we extend resource-bounded category to $S^E_2$ by defining meagerness relative to single-valued $FS^P_2$ strategies in the Banach-Mazur game. We show that Li's $FS^P_2$ algorithm for the Range Avoidance problem yields a winning strategy, proving that SIZE($\frac{2^n}{n}$) is meager in $S^E_2$. Consequently, languages requiring exponential-size circuits are comeager in $S^E_2$: they are typical with respect to resource-bounded category.