The extended low hierarchy is an infinite hierarchy

  • Ming-Jye Sheu
  • Timothy Juris Long
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)

Abstract

Balcźar, Book, and Schöning introduced the extended low hierarchy based on the σ-levels of the polynomial-time hierarchy as follows: for k≥1, level k of the extended low hierarchy is the set \(EL_k^{P,\sum } = \left\{ {\sum\nolimits_k^P {(A) \subseteq \sum\nolimits_{k - 1}^P {\left( {A \oplus SAT} \right)} } } \right\}\). Allender and Hemachandra and Long and Sheu introduced refinements of the extended low hierarchy based on the δ and θ-levels, respectively, of the polynomial-time hierarchy: for k≥2, \(EL_k^{P,\Delta } = \left\{ {A|\Delta _k^P \left( A \right) \subseteq \Delta _{k - 1}^P \left( {A \oplus SAT} \right)} \right\}\) and \(EL_k^{P,\Theta } = \left\{ {A|\Theta _k^P \left( A \right) \subseteq \Theta _{k - 1}^P \left( {A \oplus SAT} \right)} \right\}\). In this paper we show that the extended low hierarchy is properly infinite by showing, for k≥2, that \(EL_k^{P,\sum } \subset EL_{k + 1}^{P,\Theta } \subset EL_{k + 1}^{P,\Delta } \subset EL_{k + 1}^{P,\sum } \). Our proofs use the circuit lower bound techniques of Hastad and Ko. As corollaries to our constructions, we obtain, for k≥2, oracle sets Bk, Ck, and Dk, such that PH(Bk) = σkP(Bk) δkP(Bk), PE(Ck) = δkP (Ck) θkP(Ck), and PH(Dk) = θkP(Dk) ≠ σk 1/P(Dk)

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References

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Ming-Jye Sheu
    • 1
  • Timothy Juris Long
    • 1
  1. 1.Department of Computer and Information ScienceThe Ohio State UniversityColumbusUSA

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