A Downward Collapse within the Polynomial Hierarchy

Abstract

Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k > 2, if PWe extend this to obtain a more general downward collapse result.

Two Queries

“…Then we will have either the first or second column accepting respectively. ¾ Hemaspaandra, Hemaspaandra and Hempel [HHH99a] give a more general version of Theorem 3.1 for the boolean hierarchy over ¦ Ô for ¾. In a later paper, Hemaspaandra, Hemaspaandra and Hempel [HHH99b] show that our Theorem 3.2 similarly extends to the boolean hierarchy over ¦ Ô ¾ .…”

“…Easy-III are formulas with a short proof of nonsatisfiability given that they failed to be Easy-II. Easy-IV strings are formulas with a short proof of nonsatisfiability using techniques from Hemaspaandra, Hemaspaandra and Hempel [HHH99a] discussed in Section 3.…”

“…One goal might be to show that P NP ½℄ P NP ¾℄ implies NP coNP. Hemaspaandra, Hemaspaandra and Hempel [HHH99a] made a step into that direction. They showed that for ¾, if P ¦ Ô ½℄ P ¦ Ô ¾℄ then ¦ Ô ¥ Ô .…”

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

“…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.…”

Bounded Queries and the NP Machine Hypothesis

“…For technical reasons that we will not go into here, the case of one query versus two queries is a special case. By considering whether the input string itself might be a hard string, downward collapses can be achieved for queries to Σ P i for i ≥ 2 [10,6]: P Σ P i [1] = P Σ P i [2] ⇐⇒ PH ⊆ Σ P i .…”