Advertisement

On Higher Arthur-Merlin Classes

  • Jin-Yi Cai
  • Denis Charles
  • A. Pavan
  • Samik Sengupta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2387)

Abstract

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

Keywords

Turing Machine Query Answer Interactive Proof System Operator Machinery Oracle Turing Machine 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Arvind and J. Köbler, On Pseudorandomness and Resource-Bounded Measure, Proc. 17th FST and TCS, Springer-Verlag, LNCS 1346, 235–249, 1997.Google Scholar
  2. 2.
    V. Arvind and J. Köbler, Graph isomorphism is low for ZPP NP and other lowness results, STACS 2000.Google Scholar
  3. 3.
    S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, Proof verification and hardness of approximation problems. Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, 14–23, 1992.Google Scholar
  4. 4.
    S. Arora and S. Safra, Approximating clique is NP-complete., Proceedings of the 33rd IEEE Symposium on Foundations on Computer Science, 2–13, 1992.Google Scholar
  5. 5.
    M. Agrawal and T. Thierauf, The Boolean isomorphism problem, Proc. 37th Annual Symposium on Foundations of Computer Science, 422–430.Google Scholar
  6. 6.
    L. Babai, Trading group theory for randomness, STOC 17:421–429(85).Google Scholar
  7. 7.
    L. Babai and S. Moran, Arthur-Merlin Games: a randomized proof system, and a hierarchy of complexity classes, Journal of Computer and System Sciences, 36:254–276, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. L. Balcázar, J. Díaz, J. Gabarró, Structural Complexity II, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.Google Scholar
  9. 9.
    R. Boppana, J. Hastad and S. Zachos, Does co-NP have short interactive proofs?, Information Processing Letters, 25:127–132, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. Bshouty, R. Cleve, S. Kannan and C. Tamon, Oracles and Queries that are sufficient for Exact Learning, Proceedings of the 17th Annual ACM conference on Computational Learning Theory, 130–19 (1994).Google Scholar
  11. 11.
    Jin-Yi Cai, S2P ⊆ ZPPNP, ECCC Tech-report TR-02-30, also to appear in FOCS 2001.Google Scholar
  12. 12.
    S. Goldwasser, S. Micali and C. Rackoff, The Knowledge Complexity of Interactive Proofs, Proc. 17th ACM Symp. om Computing, Providence, RI, 1985, pp. 291–304.Google Scholar
  13. 13.
    S. Goldwasser and M. Sipser, Private coins versus public coins in interactive proof systems, STOC 18:59–68(1986).Google Scholar
  14. 14.
    O. Goldreich and D. Zuckerman, Another Proof that BPP ⊆ PH (and more), ECCC, TR97-045, October 1997.Google Scholar
  15. 15.
    J. Köbler and O. Watanabe, New collapse consequences of NP having small circuits ICALP, LNCS 944:196–207(1995).Google Scholar
  16. 16.
    C. Lund, L. Fortnow, H. Karloff and N. Nisan, Algebraic Methods for Interactive Proof Systems, Journal of the ACM, 39(4):859–868, October 1992.Google Scholar
  17. 17.
    A. Shamir, IP = PSPACE, Journal of the ACM, 39(4):869–877, October 1992.Google Scholar
  18. 18.
    S. Toda, PP is as hard as polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865–877, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    H. Vollmer and K. Wagner, The complexity of finding middle elements. International Journal of Foundations of Computer Science, 4:293–307, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    O. Watanabe and S. Toda, Polynomial time 1-Turing reductions from #PH to #P. Theoritical Computer Science, 100(1):205–221, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    S. Zachos and H. Heller, A Decisive characterization of BPP, Information and Control, 69:125–135(1986).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S. Zachos and M. Fürer, Probabilistic quantifiers vs Distrustful adversaries, FSTTCS 1987, LNCS-287:449–455.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Denis Charles
    • 2
  • A. Pavan
    • 3
  • Samik Sengupta
    • 4
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadison
  2. 2.Computer Sciences DepartmentUniversity of WisconsinMadison
  3. 3.NEC Research InstitutePrinceton
  4. 4.Department of Computer Science and EngineeringUniversity at BuffaloBuffalo

Personalised recommendations