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Query order in the polynomial hierarchy

  • Edith Hemaspaandra
  • Lane A. Hemaspaandra
  • Harald Hempel
Technical Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1279)

Abstract

The study of query order was initiated by Hemaspaandra, Hempel, and Wechsung [HHW]. Their goal was to learn whether the order of access to information sources affects the class of problems that can be solved. They showed that in the boolean hierarchy over NP, order matters. In the present paper, we study the power of query order when accessing levels of the polynomial hierarchy, and we show that here order does not matter. In particular, let PC:D denote the class of languages computable by a polynomial-time machine that is allowed one query to C followed by one query to D [HHW]. We prove that the levels of the polynomial hierarchy are order-oblivious:
$$P^{\sum _i^P :\sum _k^P } = P^{\sum _k^P :\sum _j^P } .$$

Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that a wide range of other classes (UP, BPP, ⊕P, PP, etc.) inherit all order-obliviousness results that hold for deterministic polynomialtime transducers.

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References

  1. [ABT96]
    M. Agrawal, R. Beigel, and T. Thierauf. Pinpointing computation with modular queries in the boolean hierarchy. In Proceedings of the 16th Conference on Foundations of Software Technology and Theoretical Computer Science, pages 322–334. Springer-Verlag Lecture Notes in Computer Science #1180, December 1996.Google Scholar
  2. [BC93]
    D. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. Prentice Hall, 1993.Google Scholar
  3. [BC97]
    R. Beigel and R. Chang. Commutative queries. In Proceedings of the 5th Israeli Symposium on Theory of Computing and Systems. IEEE Computer Society Press, June 1997. To appear.Google Scholar
  4. [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
  5. [BDG95]
    J. Balcázar, J. Díaz, and J. Gabarró.Structural Complexity I. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 2nd edition, 1995.Google Scholar
  6. [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
  7. [CK96]
    R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing, 25(2):340–354, 1996.CrossRefGoogle Scholar
  8. [FR91]
    L. Fortnow and N. Reingold. PP is closed under truth-table reductions. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 13–15. IEEE Computer Society Press, June/July 1991.Google Scholar
  9. [Her97]
    U. Hertrampf. Acceptance by transformation monoids (with an application to local self-reductions). In Proceedings of the 12th Annual IEEE Conference on Computational Complexity. IEEE Computer Society Press, June 1997. To appear.Google Scholar
  10. [HHH96]
    E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Query order in the polynomial hierarchy. Technical Report TR-634, University of Rochester, Department of Computer Science, Rochester, NY, September 1996.Google Scholar
  11. [HHH97a]
    E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. A downward translation in the polynomial hierarchy. In Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science, pages 319–328. Springer-Verlag Lecture Notes in Computer Science #1200, February/March 1997.Google Scholar
  12. [HHH97b]
    E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. R1-ttSN(NP) distinguishes robust many-one and Turing completeness. In Proceedings of the 3rd Italian Conference on Algorithms and Complexity, pages 49–60. Springer-Verlag Lecture Notes in Computer Science #1203, March 1997.Google Scholar
  13. [HHH97c]
    E. Hemaspaandra, L. Hemaspaandra, and H. Hempel. Translating equality downwards. Technical Report TR-657, University of Rochester, Department of Computer Science, Rochester, NY, April 1997.Google Scholar
  14. [HHW]
    L. Hemaspaandra, H. Hempel, and G. Wechsung. Query order. SIAM Journal on Computing. To appear.Google Scholar
  15. [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
  16. [KSW87]
    J. Köbler, U. Schöning, and K. Wagner. The difference and truth-table hierarchies for NP. RAIRO Theoretical Informatics and Applications, 21:419–435, 1987.Google Scholar
  17. [Pap94]
    C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google Scholar
  18. [Sel94]
    V. Selivanov. Two refinements of the polynomial hierarchy. In Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science, pages 439–448. Springer-Verlag Lecture Notes in Computer Science #775, February 1994.Google Scholar
  19. [Wag90]
    K. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833–846, 1990.CrossRefGoogle Scholar
  20. [Wag97]
    K. Wagner. A note on parallel queries and the difference hierarchy (draft). Manuscript, February 3, 1997. *** DIRECT SUPPORT *** A0008123 00007Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Edith Hemaspaandra
    • 1
  • Lane A. Hemaspaandra
    • 2
  • Harald Hempel
    • 3
  1. 1.Department of MathematicsLe Moyne CollegeSyracuseUSA
  2. 2.Department of Computer ScienceUniversity of RochesterRochesterUSA
  3. 3.Inst. für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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