This paper shows a complete upward collapse in the Polynomial Hierarchy (PH) if for ZPP, two queries to a SAT oracle is equivalent to one query. That is,These ZPP machines are required to succeed with probability at least 1/2 + 1/p(n) on inputs of length n for some polynomial p(n). This result builds upon recent work by Tripathi [16] who showed a collapse of PH to S P 2 . The use of the probability bound of 1/2 + 1/p(n) is justified in part by showing that this bound can be amplified to 1 − 2 −n k for ZPP SAT[1] computations. This paper also shows that in the deterministic case,where the ZPP SAT[1] machine achieves a probability of success of 1/2 − 2 −n k .

The NP machine hypothesis posits the existence of an > 0 and a nondeterministic polynomial-time Turing machine M which accepts the language 0 * but for which no deterministic Turing machine running in 2 n time can output an accepting path infinitely often. This paper shows two applications of the NP machine hypothesis in bounded query complexity. First, if the NP machine hypothesis holds, thenWithout assuming the NP machine hypothesis, the best known collapse of the Polynomial Hierarchy (PH) is to the class S P 2 due to a result of Fortnow, Pavan and Sengupta [9]. The second application is to bounded query function classes. If the NP machine hypothesis holds then for all constants d > 0, there exists a constant k > d such that for all oracles X, PF SAT[n k ] ⊆ PF X[n d ] . In particular, PF SAT[n d ] PF SAT[n k ] . Without the NP machine hypothesis, there are currently no known consequences even if for all k > 1, PF SAT[n k ] ⊆ PF SAT[n] .