2 research outputs found

    Using the No-Search Easy-Hard Technique for Downward Collapse

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    The top part of the preceding figure [figure appears in actual paper] shows some classes from the (truth-table) bounded-query and boolean hierarchies. It is well-known that if either of these hierarchies collapses at a given level, then all higher levels of that hierarchy collapse to that same level. This is a standard ``upward translation of equality'' that has been known for over a decade. The issue of whether these hierarchies can translate equality {\em downwards\/} has proven vastly more challenging. In particular, with regard to the figure above, consider the following claim: Pmβˆ’ttΞ£kp=Pm+1βˆ’ttΞ£kpβ€…β€ŠβŸΉβ€…β€ŠDIFFm(Ξ£kp)coDIFFm(Ξ£kp)=BH(Ξ£kp).(βˆ—)P_{m-tt}^{\Sigma_k^p} = P_{m+1-tt}^{\Sigma_k^p} \implies DIFF_m(\Sigma_k^p) coDIFF_m(\Sigma_k^p) = BH(\Sigma_k^p). (*) This claim, if true, says that equality translates downwards between levels of the bounded-query hierarchy and the boolean hierarchy levels that (before the fact) are immediately below them. Until recently, it was not known whether (*) {\em ever\/} held, except for the degenerate cases m=0m=0 and k=0k=0. Then Hemaspaandra, Hemaspaandra, and Hempel \cite{hem-hem-hem:j:downward-translation} proved that (*) holds for all mm, for k>2k > 2. Buhrman and Fortnow~\cite{buh-for:j:two-queries} then showed that, when k=2k=2, (*) holds for the case m=1m = 1. In this paper, we prove that for the case k=2k=2, (*) holds for all values of mm. Since there is an oracle relative to which ``for k=1k=1, (*) holds for all mm'' fails \cite{buh-for:j:two-queries}, our achievement of the k=2k=2 case cannot to be strengthened to k=1k=1 by any relativizable proof technique. The new downward translation we obtain also tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded-query hierarchy of the second level of the polynomial hierarchy.Comment: 22 pages. Also appears as URCS technical repor

    Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

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    It is known that for any class C closed under union and intersection, the Boolean closure of C, the Boolean hierarchy over C, and the symmetric difference hierarchy over C all are equal. We prove that these equalities hold for any complexity class closed under intersection; in particular, they thus hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove that the Hausdorff hierarchy and the nested difference hierarchy over UP both fail to capture the Boolean closure of UP in some relativized worlds. Karp and Lipton proved that if nondeterministic polynomial time has sparse Turing-complete sets, then the polynomial hierarchy collapses. We establish the first consequences from the assumption that unambiguous polynomial time has sparse Turing-complete sets: (a) UP is in Low_2, where Low_2 is the second level of the low hierarchy, and (b) each level of the unambiguous polynomial hierarchy is contained one level lower in the promise unambiguous polynomial hierarchy than is otherwise known to be the case.Comment: 27 page
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