Amplifying ZPP and the Two Queries Problem

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, ZPP = ZPP =⇒ ZPP = PH. 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 S2 . 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 computations. This paper also shows that in the deterministic case, PSAT[1] = PSAT‖[2] =⇒ PH⊆ ZPP where the ZPP machine achieves a probability of success of 1/2−2−n k .