A decisive characterization of BPP

The complexity class BPP (defined by Gill) contains problems that can be solved in polynomial time with bounded error probability. A new and simple characterization of BPP is given. It is shown that a language L is in BPP iff (x ∈ L → ∃⁺y∀zP(x, y, z)) ∧ (x ∉ L → ∀y∃⁺ z ¬ P(x, y, z)) for a polynomial-time predicate P and for |y|, |z| ⩽ poly (|x|). The formula ∃⁺ yP(y) with the random quantifier ∃⁺ means that the probability Pr({y|P(y)}) ⩾+ ɛ for a fixed ɛ. This characterization allows a simple proof that BPP ⊆ ZPP^NP, which strengthens the result of (Lautemann, Inform. Process. Lett. 17 (1983), 215–217; Sipser, in “Proceedings, 15th Annu. ACM Sympos. Theory of Comput.,” 1983, pp. 330–335) that BPP ⊆ Σ₂^p ∩ Π₂^p. Several other results about probabilistic classes can be proved using similar techniques, e.g., NP^R ⊆ ZPP^NP and Σ₂^p,BPP = Σ₂^p.