STACS 1992: STACS 92 pp 185-198

# 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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