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Relativized polynomial hierarchies extending two levels. (English) Zbl 0543.03028

Relativized polynomial hierarchies are studied in the paper. The existence of relativized polynomial hierarchies extending exactly two levels is shown. In these hierarchies two polynomially bounded quantifiers yield more than a single one, but three or more polynomially bounded quantifiers do not yield more than two ones. More precisely, the author constructs oracles X, Y such that \(NP(X)\subsetneqq \Delta_ 2^{P,X}=\Sigma_ 2^{P,X}\) and \(\Delta_ 2^{P,Y}\subsetneqq \Sigma_ 2^{P,Y}=\Pi_ 2^{P,Y}.\) The meaning of the symbols is the following. For an arbitrary set Z, P(Z) and NP(Z) denote the class of languages accepted deterministically or nondeterministically (respectively) with the oracle Z. \(\Delta_ 2^{P,Z}\) is defined as P(NP(Z)), \(\Sigma_ 2^{P,Z}\) as NP(NP(Z)), and \(\Pi_ 2^{P,Z}\) as the class of complements of sets in \(\Sigma_ 2^{P,Z}\).
Reviewer: M.Chytil

MSC:

03D15 Complexity of computation (including implicit computational complexity)
03D55 Hierarchies of computability and definability
68Q25 Analysis of algorithms and problem complexity
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References:

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