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The Shrinking Property for NP and coNP

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Logic and Theory of Algorithms (CiE 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5028))

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Abstract

We study the shrinking and separation properties (two notions well-known in descriptive set theory) for and and show that under reasonable complexity-theoretic assumptions, both properties do not hold for and the shrinking property does not hold for . In particular we obtain the following results.

  1. 1

    and do not have the shrinking property, unless is finite. In general, and do not have the shrinking property, unless is finite. This solves an open question from [25].

  2. 1

    The separation property does not hold for , unless .

  3. 1

    The shrinking property does not hold for , unless there exist -hard disjoint -pairs (existence of such pairs would contradict a conjecture by Even, Selman, and Yacobi [6]).

  4. 1

    The shrinking property does not hold for , unless there exist complete disjoint -pairs.

Moreover, we prove that the assumption is too weak to refute the shrinking property for in a relativizable way. For this we construct an oracle relative to which , , and has the shrinking property. This solves an open question by Blass and Gurevich [2] who explicitly ask for such an oracle.

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Authors and Affiliations

  1. Julius-Maximilians-Universität Würzburg, Germany

    Christian Glaßer

  2. Siberian Division of the Russian Academy of Sciences, A.P. Ershov Institute of Informatics Systems, Russia

    Christian Reitwießner & Victor Selivanov

Authors
  1. Christian Glaßer
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  2. Christian Reitwießner
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  3. Victor Selivanov
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Arnold Beckmann Costas Dimitracopoulos Benedikt Löwe

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Glaßer, C., Reitwießner, C., Selivanov, V. (2008). The Shrinking Property for NP and coNP. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_25

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  • DOI: https://doi.org/10.1007/978-3-540-69407-6_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69405-2

  • Online ISBN: 978-3-540-69407-6

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