Locating P/poly optimally in the extended low hierarchy

  • Johannes Köbler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)

Abstract

The low hierarchy in NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes, and almost all of them could be located optimally therein. However, until now, the exact location of P/poly remained open.

We show that P/poly is contained in the third theta level EL3pΘ of the extended low hierarchy. Since Allender and Hemachandra have shown that there exist sparse sets outside of EL2p, this is optimal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Angluin. Queries and concept learning. Machine Learning, 2:319–342, 1988.Google Scholar
  2. [2]
    E. Allender and L. Hemachandra. Lower bounds for the low hierarchy. Journal of the ACM, 39(1):234–250, 1992.Google Scholar
  3. [3]
    V. Arvind, J. Köbler, M. Mundhenk. Lowness and the complexity of sparse and tally descriptions. To appear in Proceedings 3rd Symposium on Algorithms and Computation, Springer-Verlag, 1992.Google Scholar
  4. [4]
    J.L. Balcázar, R. Book, and U. Schöning. Sparse sets, lowness and highness. SIAM J. Comput., 23:679–688, 1986.Google Scholar
  5. [5]
    J.L. Balcázar, J. Díaz, J. Gabarró. Structural Complexity Theory I + II. Springer-Verlag, 1988 and 1990.Google Scholar
  6. [6]
    L. Berman, J. Hartmanis. On isomorphism and density of NP and other complexity classes. SIAM J. Comput., 6:305–327, 1977.Google Scholar
  7. [7]
    J.L. Carter and M.N. Wegman. Universal classes of hash functions. Journal of Computer and System Sciences 18:143–154, 1979.Google Scholar
  8. [8]
    R. Gavalda. Bounding the complexity of advice functions. In Proceedings of the 7th Structure in Complexity Theory Conference, 249–254. IEEE Computer Society Press, June 1992.Google Scholar
  9. [9]
    L. Hemachandra, M. Ogiwara, and O. Watanabe. How hard are sparse sets. In Proceedings of the 7th Structure in Complexity Theory Conference, 222–238. IEEE Computer Society Press, June 1992.Google Scholar
  10. [10]
    R.M. Karp and R.J. Lipton. Some connections between nonuniform and uniform complexity classes. Proc. 12th ACM Symp. Theory of Comput. Science, 302–309, 1980.Google Scholar
  11. [11]
    K. Ko and U. Schöning. On circuit-size complexity and the low hierarchy in NP. SIAM Journ. Comput. 14:41–51, 1985.Google Scholar
  12. [12]
    J. Köbler. Strukturelle Komplexität von Anzahlproblemen. Doctoral Dissertation. University of Stuttgart, 1989.Google Scholar
  13. [13]
    J. Köbler, U. Schöning, J. Torán. On counting and approximation. Acta Informatica, 26:363–379, 1989.Google Scholar
  14. [14]
    J. Köbler and T. Thierauf. Complexity classes with advice. Proceedings 5th Structure in Complexity Theory Conference, 305–315, IEEE Computer Society, 1990.Google Scholar
  15. [15]
    T.J. Long and M.-J. Sheu. A refinement of the low and high hierarchies. Technical Report OSU-CISRC-2/91-TR6, The Ohio State University, 1991.Google Scholar
  16. [16]
    M.-J. Sheu and T.J. Long. The extended low hierarchy is an infinite hierarchy. Proceedings of 9th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, #577:187–189, Springer-Verlag 1992.Google Scholar
  17. [17]
    N. Pippenger. On simultaneous resource bounds. In Proceedings 20th Symposium on Foundations of Computer Science, 307–311, IEEE Computer Society, 1979.Google Scholar
  18. [18]
    V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing 4 (1975), 214–220.Google Scholar
  19. [19]
    U. Schöning. A low hierarchy within NP. Journal of Computer and System Sciences, 27:14–28, 1983.Google Scholar
  20. [20]
    U. Schöning. Complexity and Structure. Springer-Verlag Lecture Notes in Computer Science 211, 1986.Google Scholar
  21. [21]
    U. Schöning. Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 37:312–323, 1988.Google Scholar
  22. [22]
    M. Sipser. A complexity theoretic approach to randomness. Proc. 15th ACM Symp. Theory of Comput. Science, 330–335, 1983.Google Scholar
  23. [23]
    L.J. Stockmeyer. The polynomial-time hierarchy. Theor. Comput. Science, 3:1–22, 1977.Google Scholar
  24. [24]
    L.J. Stockmeyer. On approximation algorithms for #P. SIAM Journ. Comput. 14:849–861, 1985.Google Scholar
  25. [25]
    K.W. Wagner. Bounded query classes. SIAM Journ. Comput. 19(5):833–846, 1990.Google Scholar
  26. [26]
    C. Wrathall. Complete sets and the polynomial-time hierarchy. Theor. Comput. Science 3:23–33, 1977.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Johannes Köbler
    • 1
  1. 1.Abteilung für Theoretische InformatikUniversität UlmUlmGermany

Personalised recommendations