Bounded Queries and the NP Machine Hypothesis

Abstract

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] .

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

“…However, this collapse is tantalizingly close to a complete upward collapse of PH down to P SAT [1] . Indeed, Chang and Purini [10] showed that under the NP Machine Hypothesis, P SAT [1] = P SAT [2] implies that PH = P SAT [1] = NP. This gives us some hope of proving a complete upward collapse without the additional assumption of the NP Machine Hypothesis.…”