Amplifying ZPP^SAT[1] 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
^SAT[1]
= ZPP
^SAT||[2]
rArr ZPP
^SAT[1]
= 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 who showed a collapse of PH to S
₂

^P
. 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
^-nk
for ZPP
^SAT[1]
computations. This paper also shows that in the deterministic case, P
^SAT[1]
= P
^SAT||[2]
rArr PH sube ZPP
^SAT[1]
where the ZPP
^SAT[1]
machine achieves a probability of success of 1/2 - 2
^-nk
.