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Collapsing Recursive Oracles for Relativized Polynomial Hierarchies

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Fundamentals of Computation Theory (FCT 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3623))

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Abstract

Certain recursive oracles can force the polynomial hierarchy to collapse to any fixed level. All collections of such oracles associated with each collapsing level form an infinite hierarchy, called the collapsing recursive oracle polynomial (CROP) hierarchy. This CROP hierarchy is a significant part of the extended low hierarchy (note that the assumption P = NP makes the both hierarchies coincide). We prove that all levels of the CROP hierarchy are distinct by showing “strong” types of separation. First, we prove that each level of the hierarchy contains a set that is immune to its lower level. Second, we show that any two adjacent levels of the CROP hierarchy can be separated by another level of the CROBPP hierarchy—a bounded-error probabilistic analogue of the CROP hierarchy. Our proofs extend the circuit lower-bound techniques of Yao, Håstad, and Ko.

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Computer Science Program, Trent University, Peterborough, Ontario, K9J 7B8, Canada

    Tomoyuki Yamakami

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  1. Tomoyuki Yamakami
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Editors and Affiliations

  1. Institut für Theoretische Informatik, Universität zu Lübeck, Germany

    Maciej Liśkiewicz

  2. Institut für Theoretische Informatik, Universität zu Lübeck, Ratzeburger Allee 160, 23538, Lübeck, Germany

    Rüdiger Reischuk

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© 2005 Springer-Verlag Berlin Heidelberg

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Yamakami, T. (2005). Collapsing Recursive Oracles for Relativized Polynomial Hierarchies. In: Liśkiewicz, M., Reischuk, R. (eds) Fundamentals of Computation Theory. FCT 2005. Lecture Notes in Computer Science, vol 3623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537311_14

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  • DOI: https://doi.org/10.1007/11537311_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28193-1

  • Online ISBN: 978-3-540-31873-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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