Abstract
The low and high hierarchies within NP were introduced by Schöning in order to classify sets in NP. It is not known whether the low and high hierarchies include all sets in NP. In this paper, we provide evidence that the class UP may fall in the gap between the low and high hierarchies. Using the circuit lower bound techniques of Håstad and Ko, we construct an oracle set relative to which UP is not in any level of the low and high hierarchies. Since it is known that UPA is low for PPA for all sets A, our result also shows that the interaction between UP and PP is crucial for the lowness of UP for PP; changing the base class to any level of the polynomial-time hierarchy destroys the lowness of UP.
This work was supported in part by NSF Grant CCR-8909071
Part of this work was done while both authors were visitors in the the Department of Computer Science at New Mexico State University.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
D. Angluin. On counting problems and the polynomial-time hierarchy. Theoretical Computer Science, 12:161–173, 1980.
M. Fürst, 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.
J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM J. Comput., 11(2), April 1988.
J. D. Hastad. Computational limitations for small-depth circuits. PhD thesis, Massachusetts Institute of Technology, 1987.
J. Hartmanis and L. Hemachandra. One-way functions, robustness, and non-isomorphism of NP-complete sets. In Proc. Second Annual IEEE Structure in Complexity Theory Conference, pages 160–173, 1987.
K. Ko. Relativized polynomial time hierarchies having exactly k levels. SIAM J. Cornput., 18(2):392–408, April 1989.
K. Ko. Separating the low and high hierarchies by oracles. Information and Computation, 90(2):156–177, Feburary 1991.
J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. In Proc. 4th Structure in Complexity Theory Conference, IEEE, 1989.
U. Schöning. A low and a high hierarchy within NP. J. Comput. System Sci., 27:14–28, 1983.
M. Sheu and T. Long. The extended low hierarchy is an infinite hierarchy. In Proc. 9th Annual Symposium on Theoretical Aspects of Computer Sceince, pages 187–198, 1992.
L. Stockmeyer. The polynomial-time hierarchy. Theor. Comput. Sci., 3:1–22, 1976.
C. Wrathall. Complete sets and the polynomial hierarchy. Theor. Comput. Sci., 3:23–33, 1976.
A. Yao. Separating the polynomial-time hierarchy by oracles. In Proc. 26th IEEE Symp. on Foundations of Computer Science, pages 1–10, 1985.
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sheu, MJ., Long, T.J. (1992). UP and the low and high hierarchies: A relativized separation. In: Kuich, W. (eds) Automata, Languages and Programming. ICALP 1992. Lecture Notes in Computer Science, vol 623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55719-9_73
Download citation
DOI: https://doi.org/10.1007/3-540-55719-9_73
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55719-7
Online ISBN: 978-3-540-47278-0
eBook Packages: Springer Book Archive