Gaps in Bounded Query Hierarchies

Prior results show that most bounded query hierarchies cannot contain finite gaps. For example, it is known that PSAT (m+1)-tt = PSAT m-tt ) PSAT btt = PSAT m-tt and for all sets A FPA(m+1)-tt = FPAm-tt ) FPAbtt = FPAm-tt PA(m+1)-T = PAm-T ) PAbT = PAm-T FPA(m+1)-T = FPAm-T ) FPAbT = FPAm-T where PAm-tt is the set of languages computable by polynomial-time Turing machines that make m nonadaptive queries toA; PAbtt = Sm PAm-tt; PAm-T andPAbT are the analogous adaptive queries classes; and FPAm-tt, FPAbtt, FPAm-T, and FPAbT in turn are the analogous function classes. It was widely expected that these general results would extend to the remaining case — languages computed with nonadaptive queries — yet results remained elusive. The best known was that PA2m-tt = PAm-tt ) PAbtt = PAm-tt: We disprove the conjecture. In fact, PAb 4 3mc-tt = PAm-tt 6) PA(b 43mc+1)-tt = PAb 43mc-tt: Thus there is a PAm-tt hierarchy that contains a finite gap. We also make progress on the 3-tt vs. 2-tt case: PA3-tt = PA2-tt ) PAbtt PA2-tt=poly: Research performed at the University of Maryland, Yale University, and Lehigh University. Supported in part by the National Science Foundation under grants CCR-8958528, CCR-9796317, and CCR-9996021 and by NASA under grant NAG 52895.