On boolean lowness and boolean highness

The concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by Schoning [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 Σ2p. 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 = Σ2p (k − 1) ⊕ NP(k).

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