A new characterization of BPP

Abstract

The complexity class BPP contains languages that can be solved in polynomial time with bounded error probability. It is shown that a language L is in BPP iff (x∈L ↔ ∃_my ∀z P(x,y,z)) and (x∉L ↔ ∀y ∃_mz⌍P(x,y,z)) for a polynomial time predicate P and for |y|, |z| ≦ poly(|x|). The formula ∃_myP(y) with the random quantifier ∃_m means that the probability Pr({y| P(y)}) <1/2 + ɛ for a fixed ɛ. Note that the weaker conditions ∃ y ∀zP(x,y,z) and ∀ y ∃ z⌍P(x,y,z) are complementary and thus decide whether x∈L. Some of the consequences of the characterization of BPP are that various probabilistic polynomial time hierarchies collapse as well as that probabilistic oracles do not add anything to the computing power of classes as low as Σ ^P₂ . For example, Σ ^{P, BPP}₂ = Σ ^P₂ , where Σ ^{P, BPP}₂ is the class σ ^P₂ relativized to BPP.