Resource-bounded Measure on Probabilistic Classes

We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain notions of resource-bounded measure on probabilistic complexity classes such as BPE and BPEXP. Unlike former attempts, our resource bounded measure notions satisfy all three basic measure properties, that is every singleton {L} has measure zero, the whole space has measure one, and "enumerable infinite unions" of measure zero sets have measure zero

Resource-bounded Measure on Probabilistic Classes

We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain notions of resource-bounded measure on probabilistic complexity classes such as BPE and BPEXP. Unlike former attempts, our resource bounded measure notions satisfy all three basic measure properties, that is every singleton {L} has measure zero, the whole space has measure one, and "enumerable infinite unions" of measure zero sets have measure zero

Baire categories on small complexity classes and meager–comeager laws

We introduce two resource-bounded Baire category notions on small complexity classes such as P, QUASIPOLY, 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, PA = BPPA. 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 QUASIPOLY. 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

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Probabilistic Martingales and BPTIME Classes

We define probabilistic martingales based on randomized approximation schemes, and show that the resulting notion of probabilistic measure has several desirable robustness properties. Probabilistic martingales can simulate the "betting games" of [BMR + 98], and can cover the same class that a "natural proof " diagonalizes against, as implicitly already shown in [RSC95]. The notion would become a full-fledged measure on bounded-error complexity classes such as BPP and BPE if it could be shown to satisfy the "measure conservation" axiom of [Lut92] for these classes. We give a sufficient condition in terms of simulation by "decisive" probabilistic martingales that implies not only measure conservation, but also a much tighter bounded error probabilistic time hierarchy than is currently known. In particular it implies BPTIME[O(n)] 6= BPP, which would stand in contrast to recent claims of an oracle A giving BPTIME A [O(n)] = BPP A . This paper also makes new contributions to the problem of defining (deterministic) measure on P and other sub-exponential classes.