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Intersection suffices for Boolean hierarchy equivalence

  • Lane A. Hemaspaandra
  • Jörg Rothe
Session 7B: Complexity Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)

Abstract

It is known that for any class C closed under union and intersection, the Boolean closure of C, the Boolean hierarchy over C, and the symmetric difference hierarchy over C all are equal. We prove that these equalities hold for any complexity class closed under intersection.

Keywords

Complexity Class 13th IEEE Symposium Polynomial Hierarchy Polynomial Time Hierarchy SlAM Journal 
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. [BBJ+89]
    A. Bertoni, D. Bruschi, D. Joseph, M. Sitharam, and P. Young. Generalized Boolean hierarchies and Boolean hierarchies over RP. In Proceedings of the 7th Conference on Fundamentals of Computation Theory, pages 35–46. Springer-Verlag Lecture Notes in Computer Science #380, August 1989.Google Scholar
  2. [BCO93]
    R. Beigel, R. Chang, and M. Ogiwara. A relationship between difference hierarchies and relativized polynomial hierarchies. Mathematical Systems Theory, 26:293–310, 1993.CrossRefGoogle Scholar
  3. [BG82]
    A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80–88, 1982.CrossRefGoogle Scholar
  4. [BG94]
    R. Beigel and J. Goldsmith. Downward separation fails catastrophically for limited nondeterminism classes. In Proceedings of the 9th Structure in Complexity Theory Conference, pages 134–138. IEEE Computer Society Press, June/July 1994.Google Scholar
  5. [CGH+88]
    J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The Boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6): 1232–1252, 1988.CrossRefGoogle Scholar
  6. [CGH+89]
    J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The Boolean hierarchy II: Applications. SIAM Journal on Computing, 18(1):95–111, 1989.CrossRefGoogle Scholar
  7. [CH85]
    J. Cai and L. Hemachandra. The Boolean hierarchy: Hardware over NP. Technical Report 85-724, Cornell University, Department of Computer Science, Ithaca, NY, December 1985.Google Scholar
  8. [Cha91]
    R. Chang. On the Structure of NP Computations under Boolean Operators. PhD thesis, Cornell University, Ithaca, NY, 1991.Google Scholar
  9. [CK90a]
    R. Chang and J. Kadin. The Boolean hierarchy and the polynomial hierarchy: A closer connection. In Proceedings of the 5th Structure in Complexity Theory Conference, pages 169–178. IEEE Computer Society Press, July 1990.Google Scholar
  10. [CK90b]
    R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report TR 90-1118, Department of Computer Science, Cornell University, Ithaca, NY, May 1990.Google Scholar
  11. [CM87]
    J. Cai and G. Meyer. Graph minimal uncolorability is DP-complete. SIAM Journal on Computing, 16(2):259–277, 1987.CrossRefGoogle Scholar
  12. [Coo71]
    S. Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on Theory of Computing, pages 151–158, 1971.Google Scholar
  13. [Hau14]
    F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.Google Scholar
  14. [HH88]
    J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.CrossRefGoogle Scholar
  15. [HJ93]
    L. Hemachandra and S. Jha. Defying upward and downward separation. In Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science, pages 185–195. Springer-Verlag Lecture Notes in Computer Science #665, February 1993.Google Scholar
  16. [HJV93]
    L. Hemaspaandra, S. Jain, and N. Vereshchagin. Banishing robust Turing completeness. International Journal of Foundations of Computer Science, 4(3):245–265, 1993.CrossRefGoogle Scholar
  17. [HR92]
    L. Hemachandra and R. Rubinstein. Separating complexity classes with tally oracles. Theoretical Computer Science, 92(2):309–318, 1992.CrossRefGoogle Scholar
  18. [Kad88]
    J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263–1282, 1988. Erratum appears in the same journal, 20(2):404.CrossRefGoogle Scholar
  19. [KSW87]
    J. Köbler, U. Schöning, and K. Wagner. The difference and truth-table hierarchies for NP. R.A.I.R.O. Informatique théorique et Applications, 21:419–435, 1987.Google Scholar
  20. [Lev73]
    L. Levin. Universal sorting problems. Problems of Information Transmission, 9:265–266, 1973.Google Scholar
  21. [MS72]
    A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pages 125–129, 1972.Google Scholar
  22. [PY84]
    C. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2):244–259, 1984.CrossRefGoogle Scholar
  23. [Reg89]
    K. Regan. Provable complexity properties and constructive reasoning. Manuscript, April 1989.Google Scholar
  24. [Sto77]
    L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.CrossRefGoogle Scholar
  25. [Val76]
    L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23, 1976.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Jörg Rothe
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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