Complexity and structure
The Boolean hierarchy: hardware over NP
Proc. of the conference on Structure in complexity theory
The Boolean hierarchy I: structural properties
SIAM Journal on Computing
The polynomial time hierarchy collapses if the Boolean hierarchy collapses
SIAM Journal on Computing
Bounded queries, approximations, and the Boolean hierarchy
Information and Computation
The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection
SIAM Journal on Computing
A Downward Translation in the Polynomial Hierarchy
STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
FCT '85 Fundamentals of Computation Theory
Two Queries
Translating Equality Downwards
Translating Equality Downwards
What''s Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses
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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 Σp2. 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 = Σp2 (k - 1) ⊕ NP(k).