Exponential-Size Circuit Complexity is Comeager in Symmetric Exponential Time

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 $S_2^E$ contains problems requiring circuits of size $\frac{2^n}{n}$. In this note, we extend resource-bounded category to $S_2^E$ by defining meagerness relative to single-valued $F{S}_2^P$ strategies in the Banach-Mazur game. We show that Li’s $F{S}_2^P$ algorithm for the Range Avoidance problem yields a winning strategy, proving that SIZE($\frac{2^n}{n}$) is meager in $S_2^E$. Consequently, languages requiring exponential-size circuits are comeager in $S_2^E$: they are typical with respect to resource-bounded category.