Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time

John M. Hitchcock

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arXiv · 2605.03306

Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time

John M. Hitchcock

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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.

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John M. Hitchcock. 2026-05-05. Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time. https://arxiv.org/abs/2605.03306

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