Two Queries

Abstract

We consider the question whether two queries SAT are as powerful as one query. We show that if

Amplifying ZPP^SAT[1] and the Two Queries Problem

“…The two queries problem has been studied extensively, beginning with Kadin [14] who showed that P SAT [1] = P SAT [2] implies that the Polynomial Hierarchy (PH) collapses to the Σ P 3 level. Subsequent results [1,8,17,18] brought the collapse further down, to within ∆ P 3 .…”

“…A breakthrough in the proof techniques came when Hemaspaandra, Hemaspaandra and Hempel [13] showed that if the queries were made to a Σ P 3 oracle (instead of SAT), then PH ⊆ Σ P 3 which is a downward collapse. Buhrman and Fortnow [2] improved this technique and made it work for queries to a Σ P 2 oracle. Fortnow, Pavan and Sengupta [11] then showed that P SAT [1] = P SAT [2] =⇒ PH ⊆ S P 2 † Address: Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA.…”

“…Buhrman and Fortnow [2] improved this technique and made it work for queries to a Σ P 2 oracle. Fortnow, Pavan and Sengupta [11] then showed that P SAT [1] = P SAT [2] =⇒ PH ⊆ S P 2 † Address: Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA. Email: chang@umbc.edu, suresh1@umbc.edu.…”

“…we have the collapse P SAT [1] = P SAT [2] =⇒ P SAT [1] = P SAT ⊆ ZPP SAT [1] ⊆ ZPP SAT [2] = S P 2 = ZPP SAT = PH.…”

“…Note that P SAT [1] = P SAT [2] does not immediately imply that ZPP SAT [1] = ZPP SAT [2] because a ZPP machine can accept, reject or output "don't know". It takes two queries to SAT to cover all three possibilities.…”