ERRATUM: the polynomial time hierarchy collapses if the boolean hierarchy collapses
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It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that ${\text{co-NP}} \subseteq {\text{ NP}}^S $, and therefore the polynomial time hierarchy (PH) col...Abstract:
It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that ${\text{co-NP}} \subseteq {\text{ NP}}^S $, and therefore the polynomial time hierarchy (PH) col...read more
Citations
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Book ChapterDOI
A catalog of complexity classes
TL;DR: This chapter discusses the concepts needed for defining the complexity classes, a set of problems of related resource-based complexity that can be solved by an abstract machine M using O(f(n) of resource R, where n is the size of the input.
Journal ArticleDOI
Bounded query classes
TL;DR: The Boolean hierarchy is generalized in such a way that it is possible to characterize P and O in terms of the generalization, and the class $P^{\text{NP}}[O(\log n)]$ can be characterized in very different ways.
Journal ArticleDOI
Propositional circumscription and extended closed-world reasoning are P p 2 -complete
Thomas Eiter,Georg Gottlob +1 more
TL;DR: This work proves that the deduction problem for arbitrary propositional theories under the extended closed-world assumption or under circumscription is Π P 2 -complete, i.e., complete for a class of the second level of the polynomial hierarchy.
Journal ArticleDOI
Notes on polynomially bounded arithmetic
TL;DR: In this article, the collapse of Buss' bounded arithmetic is characterized in terms of the provable collapse of the polynomial time hierarchy, and a general model-theoretical investigation on fragments of bounded arithmetic are presented.
Proceedings Article
P NP [log n] and sparse turing complete sets for NP.
TL;DR: In this article, it was shown that if there exists a sparse set S ϵ NP such that co-NP ( NPs) is contained in PNP[O(log n), then the polynomial hierarchy (PH) of PNP(n) = DP.
References
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Proceedings Article
P NP [log n] and sparse turing complete sets for NP.
TL;DR: In this article, it was shown that if there exists a sparse set S ϵ NP such that co-NP ( NPs) is contained in PNP[O(log n), then the polynomial hierarchy (PH) of PNP(n) = DP.
Journal ArticleDOI
Graph minimal uncolorability is D P -complete
Jin-Yi Cai,Gabriele E. Meyer +1 more
TL;DR: It is shown that minimal-3-uncolorability is D-p-complete, for all fixed $k \geq 3$, and can be modified by using ``sensitive'' gadgets to resolve the planar case, bounded vertex degree case and their combination.
Book ChapterDOI
Collapsing Oracle Hierarchies, Census Functions and Logarithmically Many Queries
Uwe Schöning,Klaus W. Wagner +1 more
TL;DR: Using a uniform method based on a technique by Kadin, many oracle and alternation hierarchies can be shown to collapse to their level Δ2, and this improves in different respects earlier such collapse results.
Journal ArticleDOI
A note on sparse oracles for NP
TL;DR: A single construction yields Mahaney's result for sparse ⩽TP-complete sets for NP and also the corresponding result for co-sparse⩽ TP-complete set for NP, which strengthen the earlier work of Karp, Lipton, and Sipser.