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Mathematical systems theory

, Volume 26, Issue 3, pp 293–310 | Cite as

A relationship between difference hierarchies and relativized polynomial hierarchies

  • Richard Beigel
  • Richard Chang
  • Mitsunori Ogiwara
Article

Abstract

Chang and Kadin have shown that if the difference hierarchy over NP collapses to levelk, then the polynomial hierarchy (PH) is equal to thekth level of the difference hierarchy over Σ 2 p . We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to levelk, then PH collapses to (P (k−1) NP )NP, the class of sets recognized in polynomial time withk − 1 nonadaptive queries to a set in NPNP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any classC that has ≤ m p -complete sets and is closed under ≤ conj p -and ≤ m NP -reductions (alternatively, closed under ≤ disj p -and ≤ m co-NP -reductions), if the difference hierarchy overC collapses to levelk, then PH C = (P (k−1)−tt NP ) C . Then we show that the exact counting class C_P is closed under ≤ disj p - and ≤ m co-NP -reductions. Consequently, if the difference hierarchy over C_P collapses to levelk, then PHPP(= PHC_P) is equal to (P (k−1)−tt NP )PP. In contrast, the difference hierarchy over the closely related class PP is known to collapse.

Finally we consider two ways of relativizing the bounded query class P k−tt NP : the restricted relativization P k−tt NP C and the full relativization (P k−tt NP ) C . IfC is NP-hard, then we show that the two relativizations are different unless PH C collapses.

Keywords

Maximal Sequence Polynomial Hierarchy Full Relativization Complete Language Boolean Hierarchy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • Richard Beigel
    • 1
  • Richard Chang
    • 2
  • Mitsunori Ogiwara
    • 3
  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA
  2. 2.Department of Computer ScienceUpson Hall, Cornell UniversityIthacaUSA
  3. 3.Department of Computer ScienceUniversity of Electro-CommunicationsTokyoJapan

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