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Lower bounds for the low hierarchy

Extended abstract
  • Eric Allender
  • Lane A. Hemachandra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 372)

Abstract

The low hierarchy in NP [Sc-83] and the extended low hierarchy [BBS-86] have been useful in characterizing the complexity of certain interesting classes of sets. However, until now, there has been no way of judging whether or not a given lowness result is the best possible.

We prove absolute lower bounds on the location of classes is the extended low hierarchy, and relativized lower bounds on the location of classes in the low hierarchy in NP. In some cases, we are able to show that the classes are lower in the hierarchies than was known previously. In almost all cases, we are able to prove that our results are essentially optimal.

We also examine the interrelationships among the levels of the low hierarchies and the classes of sets reducible to or equivalent to sparse and tally sets under different notions of reducibility. We feel that these results clarify the structure underlying the low hierarchies.

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References

  1. [AH-89]
    E. Allender and L. Hemachandra, in preparation.Google Scholar
  2. [AW-88]
    E. Allender and O. Watanabe, Kolmogorov complexity and degrees of tally sets, Proc. 3rd Structure in Complexity Theory Conference, pp. 102–111.Google Scholar
  3. [ABG-88]
    A. Amir, R. Beigel, and W. Gasarch, Cheatable, P-terse, and P-superterse Sets, manuscript.Google Scholar
  4. [AG-87]
    A. Amir and W. Gasarch, 1987 Polynomial Terse Sets, Information and Computation 77, 37–56.CrossRefGoogle Scholar
  5. [BGS-75]
    T. Baker, J. Gill, and R. Solovay, 1975 Relativizations of the P=?NP question, SIAM J. Comput. 4, 431–444.CrossRefGoogle Scholar
  6. [BS-79]
    T. Baker and A. Selman, 1979 A second step toward the polynomial hierarchy, Theoretical Computer Science 8, 177–187.CrossRefGoogle Scholar
  7. [BB-86]
    J. Balcázar and R. Book, 1986 Sets with small generalized Kolmogorov complexity, Acta Informatica 23, 679–688.CrossRefGoogle Scholar
  8. [BBS-86]
    J. Balcázar, R. Book, and U. Schöning, 1986 Sparse sets, lowness, and highness, SIAM J. Comput. 15, 739–747.CrossRefGoogle Scholar
  9. [BDG-88]
    J. Balcázar, J. Díaz, and J. Gabarró, 1988 Structural Complexity I, Springer-Verlag, Berlin/New York.Google Scholar
  10. [Be-89]
    R. Beigel, Bounded queries to SAT and the Boolean hierarchy, to appear in Theoretical Computer Science. Also available as Technical Report 7, Johns Hopkins University, Dept. of Computer Science, 1987.Google Scholar
  11. [BK-88]
    R. Book and K. Ko, 1988 On sets reducible to sparse sets, SIAM J. Comput. 17, 903–919.CrossRefGoogle Scholar
  12. [GJY-87]
    J. Goldsmith, D. Joseph, and P. Young, Self-reducible, P-selective, neartestable, and P-cheatable sets: the effect of internal structure on the complexity of a set, Proc. 2nd Structure in Complexity Theory Conference, pp. 50–59.Google Scholar
  13. [Kä-88]
    J. Kämper, 1988 Non-uniform proof systems: a new framework to describe non-uniform and probabilistic complexity classes, Proceedings, Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science 338, pp. 193–210.Google Scholar
  14. [Ko-82]
    K. Ko, 1982 The maximum value problem and NP real numbers, Journal of Computer and System Sciences 24, 15–35.CrossRefGoogle Scholar
  15. [Ko-83]
    K. Ko, 1983 On the definition of some complexity classes of real numbers, Mathematical Systems Theory 16, 95–109.CrossRefGoogle Scholar
  16. [Ko-86]
    K. Ko, 1986 Applying Techniques of Discrete Complexity Theory to Numerical Computation, in Studies in Complexity Theory, R. V. Book, ed., Wiley and Sons, New York, NY, pp. 1–62.Google Scholar
  17. [Ko-87]
    K. Ko, 1987 On helping by robust oracle machines, Theoretical Computer Science 52, 15–36.CrossRefGoogle Scholar
  18. [Ko-88]
    K. Ko, Distinguishing Bounded Reducibilities by Sparse Sets, Proc. 3rd Structure in Complexity Theory Conference, pp. 181–193.Google Scholar
  19. [Ko-89a]
    K. Ko, Separating the low and high hierarchies by oracles, submitted for publication.Google Scholar
  20. [Ko-89b]
    K. Ko, Constructing oracles by lower bound techniques for circuits, to appear in Proc. International Symposium on Combinatorial Optimization, China, 1988 (Ding-Zhu Du, editor).Google Scholar
  21. [KS-85]
    K. Ko and U. Schöning, 1985 On circuit-size complexity and the low hierarchy in NP, SIAM J. Comput. 14, 41–51.CrossRefGoogle Scholar
  22. [LLS-75]
    R. Ladner, N. Lynch, and A. Selman, 1975 A comparison of polynomial-time reducibilities, Theoretical Computer Science 1, 103–123.CrossRefGoogle Scholar
  23. [Sc-83]
    U. Schöning, 1983 A low and a high hierarchy within NP, Journal of Computer and System Sciences 27, 14–28.CrossRefGoogle Scholar
  24. [Sc-85]
    U. Schöning, Complexity and Structure, Lecture Notes in Computer Science 211.Google Scholar
  25. [Sc-87]
    U. Schöning, 1987 Graph Isomorphism is in the Low Hierarchy, Journal of Computer and System Sciences 37, 312–323.CrossRefGoogle Scholar
  26. [Sc-88]
    U. Schöning, 1988 Robust oracle machines, Proceedings, 13th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 324, pp. 93–106.Google Scholar
  27. [Se-81]
    A. Selman, 1981 Some observations on NP real numbers and P-selective sets, Journal of Computer and System Sciences 23, 326–332.CrossRefGoogle Scholar
  28. [TB-88]
    S. Tang and R. Book, 1988 Separating polynomial-time Turing and truth-table degrees of tally sets, Proc. 15th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science 317, pp. 591–599.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Eric Allender
    • 1
  • Lane A. Hemachandra
    • 2
  1. 1.Department of Computer ScienceRutgers UniversityNew Brunswick
  2. 2.Department of Computer ScienceUniversity of RochesterRochester

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