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)


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 B k , C k , and D k , such that PH(B k ) = σ k P (B k ) δ k P (B k ), PE(C k ) = δ k P (C k ) θ k P (C k ), and PH(D k ) = θ k P (D k ) ≠ σ k 1/P(D k )


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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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