Baire Categories on Small Complexity Classes and Meager-Comeager Laws

TL;DR

This paper introduces resource-bounded Baire category notions for small complexity classes, providing new characterizations and laws that distinguish them from measure-based approaches, with implications for class properties and completeness notions.

Contribution

It defines two resource-bounded Baire categories, offers alternative characterizations, and establishes meager-comeager laws and nonexistence of certain weak-completeness notions in small complexity classes.

Findings

Almost all subexponentially computable languages satisfy P(A)=BPP(A).

SPARSE is meager in P under locally-computable Baire categories.

No weak-completeness notion exists in P based on locally-computable Baire categories.

Abstract

We introduce two resource-bounded Baire category notions on small complexity classes such as P, SUBEXP, and PSPACE and on probabilistic classes such as BPP, which differ on how the corresponding finite extension strategies are computed. We give an alternative characterization of small sets via resource-bounded Banach-Mazur games. As an application of the first notion, we show that for almost every language A (i.e. all except a meager class) computable in subexponential time, P(A)=BPP(A). We also show that almost all languages in PSPACE do not have small nonuniform complexity. We then switch to the second Baire category notion (called locally-computable), and show that the class SPARSE is meager in P. We show that in contrast to the resource-bounded measure case, meager-comeager laws can be obtained for many standard complexity classes, relative to locally-computable Baire category on BPP and PSPACE. Another topic where locally-computable Baire categories differ from resource-bounded measure is regarding weak-completeness: we show that there is no weak-completeness notion in P based on locally-computable Baire categories, i.e. every P-weakly-complete set is complete for P. We also prove that the class of complete sets for P under Turing-logspace reductions is meager in P, if P is not equal to DSPACE(log n), and that the same holds unconditionally for quasi-poly time. Finally we observe that locally-computable Baire categories are incomparable with all existing resource-bounded measure notions on small complexity classes, which might explain why those two settings seem to differ so fundamentally.