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Extending Downward Collapse from 1-versus-2 Queries to j-versus-j + 1 Queries

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STACS 99 (STACS 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1563))

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Abstract

The above figure shows some classes from the boolean and (truth-table) bounded-query hierarchies. It is well-known that if either collapses at a given level, then all higher levels 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 downwards has proven vastly more challenging. In particular, with regard to the figure above, consider the following claim:

$$ P_{m - tt}^{\sum _k^p } = P_{m + 1 - tt}^{\sum _k^p } \Rightarrow DIFF_m (\sum _k^p ) = coDIFF_m (\sum _k^p ) = BH(\sum _k^p ).(**) $$

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 (**) ever held, except in the trivial m = 0 case. Then Hemaspaandra et al. [15] proved that (**) holds for all m, whenever k > 2. For the case k = 2, Buhrman and Fortnow [5] then showed that (**) holds when m = 1. In this paper, we prove that for the case k = 2, (**) holds for all values of m. As Buhrman and Fortnow showed that no relativizable technique can prove “for k = 1, (**) holds for all m,” our achievement of the k = 2 case is unlikely to be strengthened to k = 1 any time in the foreseeable future. The new downward translation we obtain tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded-query hierarchy of the second level of the polynomial hierarchy.

Supported in part by grant NSF-INT-9513368/DAAD-315-PRO-fo-ab. Work done in part while visiting Friedrich-Schiller-Universität Jena.

Supported in part by grants NSF-CCR-9322513 and NSF-INT-9513368/DAAD-315-PRO-fo-ab. Work done in part while visiting Friedrich-Schiller-Universität Jena.

Supported in part by grant NSF-INT-9513368/DAAD-315-PRO-fo-ab. Work done in part while visiting Le Moyne College.

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Author information

Authors and Affiliations

  1. Dept. of Comp. Sci., Rochester Institute of Technology, Rochester, NY, 14623, USA

    Edith Hemaspaandra

  2. Dept. of Computer Science, University of Rochester, Rochester, NY, 14627, USA

    Lane. A. Hemaspaandra

  3. Inst. für Informatik, Friedrich-Schiller-Universität Jena, 07743, Jena, Germany

    Harald Hempel

Authors
  1. Edith Hemaspaandra
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  2. Lane. A. Hemaspaandra
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  3. Harald Hempel
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Editor information

Editors and Affiliations

  1. FB IV - Informatik, Universität Trier, D-54286, Trier, Germany

    Christoph Meinel

  2. LIFL, Université de Lille I, Bătiment 3, F-59655, Villeneuve d’Ascq Cedex, France

    Sophie Tison

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© 1999 Springer-Verlag Berlin Heidelberg

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Hemaspaandra, E., Hemaspaandra, L.A., Hempel, H. (1999). Extending Downward Collapse from 1-versus-2 Queries to j-versus-j + 1 Queries. In: Meinel, C., Tison, S. (eds) STACS 99. STACS 1999. Lecture Notes in Computer Science, vol 1563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49116-3_25

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  • DOI: https://doi.org/10.1007/3-540-49116-3_25

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