On the Structure of Bounded Queries to Arbitrary NP Sets

Kadin [6] showed that if the Polynomial Hierarchy (PH) has infinitely many levels, then for all k, PSAT[k] ⊂ PSAT[k+1]. This paper extends Kadin’s technique and shows that a proper query hierarchy is not an exclusive property of NP complete sets. In fact, for any A ∈ NP − l̂ow3, if PH has infinitely many levels, then PA[k] ⊂ PA[k+1]. Moreover, for the case of parallel queries, PA‖[k+1] is not even contained in PSAT‖[k]. These same techniques can be used to explore some other questions about query hierarchies. For example, if there exists any A such that PA[2] = PSAT[1], then PH collapses to ∆3 .