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On Boolean Lowness and Boolean Highness

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Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

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Abstract

The concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by Schöning [Sch85]. Informally, a set is low (high, resp.) for a relativizable class K of languages if it does not add (adds maximal, resp.) power to K when used as an oracle. In this paper we introduce the notions of boolean lowness and boolean highness. Informally, a set is boolean low (boolean high, resp.) for a class K of languages if it does not add (adds maximal, resp.) power to K when combined with K by boolean operations. We prove properties of boolean lowness and boolean highness which show a lot of similarities with the notions of lowness and highness. Using Kadin’s technique of hard strings (see [Kad88, Wag87, CK96, BCO93]) we show that the sets which are boolean low for the classes of the boolean hierarchy are low for the boolean closure of Σ p2 . Furthermore, we prove a result on boolean lowness which has as a corollary the best known result (see [BCO93]; in fact even a bit better) on the connection of the collapses of the boolean hierarchy and the polynomial-time hierarchy: If BH = NP(k) then PH = Σ p2 (k − 1) ⊕ NP(k).

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Author information

Authors and Affiliations

  1. Lehrstuhl für Theoretische Informatik, Universität Würzburg, Am Exerzierplatz 3, 97072, Würzburg, Germany

    Steffen Reith & Klaus W. Wagner

Authors
  1. Steffen Reith
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  2. Klaus W. Wagner
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Editor information

Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

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© 1998 Springer-Verlag Berlin Heidelberg

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Reith, S., Wagner, K.W. (1998). On Boolean Lowness and Boolean Highness. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_18

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  • DOI: https://doi.org/10.1007/3-540-68535-9_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

  • eBook Packages: Springer Book Archive

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