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 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 )

## Keywords

Distinct Level Query String Random Restriction High Hierarchy Infinite 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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## References

1. [AH91]
E. Allender and L. Hemachandra. Lower bounds for the low hierarchy. J. ACM, 1991. to appear.Google Scholar
2. [BBS86]
J. Balcźar, R. Book, and U. Schöming. Sparse sets, lowness, and highness. SIAM J. Comput, 15:739–747, 1986.Google Scholar
3. [Has87]
J. D. Håstad. Computational limitations for small-depth circuits. PhD thesis, Massachusetts Institute of Technology, 1987.Google Scholar
4. [Ko89]
K. Ko. Relativized polynomial time hierarchies having exactly k levels. SIAM J. Cornput., 18(2):392–408, April 1989.Google Scholar
5. [LS91]
T. Long and M. Sheu. A refinement of the low and high hierarchies. Technical Report OSU-CISRC-2/91-TR6, The Ohio State University, 1991.Google Scholar
6. [MFS81]
M. Furst, J. Saxe, and M. Sipser. Pairty, circuits, and the polynomial-time hierarchy. In Proc. 22th Annual IEEE Symposium on Foundations of Computer Science, pages 260–270, 1981.Google Scholar
7. [Sch83]
U. Schöming. A low and a high hierarchy within NP. J. Comput. System Sci., 27:14–28, 1983.Google Scholar
8. [Yao85]
A. Yao. Separating the polynomial-time hierarchy by oracles. In Proc. 26th IEEE Symp. on Foundations of Computer Science, pages 1–10, 1985.Google Scholar

© Springer-Verlag Berlin Heidelberg 1992

## Authors and Affiliations

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