On Boolean Lowness and Boolean Highness

  • Steffen Reith
  • Klaus W. Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1449)


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 2 p . 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 2 p (k − 1) ⊕ NP(k).


Computational complexity lowness highness boolean lowness boolean highness boolean hierarchy polynomial-time hierarchy hard/easy advice collapse 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Steffen Reith
    • 1
  • Klaus W. Wagner
    • 1
  1. 1.Lehrstuhl für Theoretische InformatikUniversität WürzburgWürzburgGermany

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