skip to main content
research-article

Polynomial-Time Random Oracles and Separating Complexity Classes

Authors Info & Claims
Published:21 January 2021Publication History
Skip Abstract Section

Abstract

Bennett and Gill [1981] showed that PA ≠ NPA ≠ coNPA for a random oracle A, with probability 1. We investigate whether this result extends to individual polynomial-time random oracles. We consider two notions of random oracles: p-random oracles in the sense of martingales and resource-bounded measure [Lutz 1992; Ambos-Spies et al. 1997], and p-betting-game random oracles using the betting games generalization of resource-bounded measure [Buhrman et al. 2000]. Every p-betting-game random oracle is also p-random; whether the two notions are equivalent is an open problem.

(1) We first show that PA ≠ NPA for every oracle A that is p-betting-game random.

Ideally, we would extend (1) to p-random oracles. We show that answering this either way would imply an unrelativized complexity class separation:

(2) If PA ≠ NPA relative to every p-random oracle A, then BPP ≠ EXP.

(3) If PA ≠ NPA relative to some p-random oracle A, then P ≠ PSPACE.

Rossman, Servedio, and Tan [2015] showed that the polynomial-time hierarchy is infinite relative to a random oracle, solving a longstanding open problem. We consider whether we can extend (1) to show that PHA is infinite relative to oracles A that are p-betting-game random. Showing that PHA separates at even its first level would also imply an unrelativized complexity class separation:

(4) If NPA ≠ coNPA for a p-betting-game measure 1 class of oracles A, then NP ≠ EXP.

(5) If PHA is infinite relative to every p-random oracle A, then PH ≠ EXP.

We also consider random oracles for time versus space, for example:

(6) LA ≠ PA relative to every oracle A that is p-betting-game random.

References

  1. E. Allender and M. Strauss. 1994. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science. IEEE Computer Society, 807--818 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. K. Ambos-Spies and E. Mayordomo. 1997. Resource-bounded measure and randomness. In Complexity, Logic and Recursion Theory, A. Sorbi (Ed.). Marcel Dekker, New York, NY, 1--47.Google ScholarGoogle Scholar
  3. L. Babai. 1987. Random oracles separate PSPACE from the polynomial-time hierarchy. Information Processing Letters 26 (1987), 51--53. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. R. Beigel. 1989. On the relativized power of additional accepting paths. In Proceedings of the 4th Annual Structure in Complexity Theory Conference. IEEE Computer Society, 216--224.Google ScholarGoogle ScholarCross RefCross Ref
  5. C. H. Bennett and J. Gill. 1981. Relative to a random oracle A, PA ≠ NPA ≠ co-NPA with Probability 1. SIAM Journal on Computing 10 (1981), 96--113.Google ScholarGoogle ScholarCross RefCross Ref
  6. R. V. Book. 1974. Tally languages and complexity classes. Information and Control 26 (1974), 186--193.Google ScholarGoogle ScholarCross RefCross Ref
  7. R. V. Book, J. H. Lutz, and K. W. Wagner. 1994. An observation on probability versus randomness with applications to complexity classes. Mathematical Systems Theory 27 (1994), 201--209. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. H. Buhrman, D. van Melkebeek, K. W. Regan, D. Sivakumar, and M. Strauss. 2001. A generalization of resource-bounded measure, with application to the BPP vs. EXP problem. SIAM Journal on Computing 30, 2 (2001), 576--601. http://pages.cs.wisc.edu/ dieter/Research/m-betting.html. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. J. Cai. 1989. With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy. Journal of Computer and System Sciences 38 (1989), 68--85. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. R. Chang, B. Chor, O. Goldreich, J. Hartmanis, J. Håstad, D. Ranjan, and R. Rohatgi. 1994. The random oracle hypothesis is false. Journal of Computer and System Sciences 49, 1 (1994), 24--39. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. R. C. Harkins and J. M. Hitchcock. 2013. Exact learning algorithms, betting games, and circuit lower bounds. ACM Transactions on Computation Theory 5, 4 (2013), Article 18. DOI:https://doi.org/10.1145/2539126.2539130 Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. J. Håstad. 1986. Computational Limitations for Small-Depth Circuits. The MIT Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. J. M. Hitchcock. 2006. Hausdorff dimension and oracle constructions. Theoretical Computer Science 355, 3 (2006), 382--388. http://www.cs.uwyo.edu/ jhitchco/papers/hdoc.shtml. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. 2005. The fractal geometry of complexity classes. SIGACT News 36, 3 (September 2005), 24--38. http://www.cs.uwyo.edu/ jhitchco/papers/fgcc.shtml.Google ScholarGoogle Scholar
  15. D. W. Juedes and J. H. Lutz. 1995. Weak completeness in E and E2. Theoretical Computer Science 143, 1 (1995), 149--158. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. J. H. Lutz. 1992. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44, 2 (1992), 220--258. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. J. H. Lutz. 1997. The quantitative structure of exponential time. In Complexity Theory Retrospective II, L. A. Hemaspaandra and A. L. Selman (Eds.). Springer-Verlag, 225--254. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. J. H. Lutz and W. J. Schmidt. 1993. Circuit size relative to pseudorandom oracles. Theoretical Computer Science 107, 1 (March 1993), 95--120. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. P. Martin-Löf. 1966. The definition of random sequences. Information and Control 9 (1966), 602--619.Google ScholarGoogle Scholar
  20. W. Merkle, J. S. Miller, A. Nies, J. Reimann, and F. Stephan. 2006. Kolmogorov-Loveland randomness and stochasticity. Annals of Pure and Applied Logic 138, 1--3 (2006), 183--210.Google ScholarGoogle ScholarCross RefCross Ref
  21. A. A. Muchnik, A. L. Semenov, and V. A. Uspensky. 1998. Mathematical metaphysics of randomness. Theoretical Computer Science 207, 2 (1998), 263--317. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. N. Nisan and A. Wigderson. 1994. Hardness vs. randomness. Journal of Computer and System Sciences 49, 2 (1994), 149--167. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. B. Rossman, R. A. Servedio, and L.-Y. Tan. 2015. An average-case depth hierarchy theorem for Boolean circuits. In Proceedings of the 56th Symposium on Foundations of Computer Science. IEEE Computer Society, 1030--1048. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Polynomial-Time Random Oracles and Separating Complexity Classes

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 13, Issue 1
      March 2021
      143 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3447816
      Issue’s Table of Contents

      Copyright © 2021 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 21 January 2021
      • Revised: 1 September 2020
      • Accepted: 1 September 2020
      • Received: 1 May 2019
      Published in toct Volume 13, Issue 1

      Permissions

      Request permissions about this article.

      Request Permissions

      Check for updates

      Qualifiers

      • research-article
      • Research
      • Refereed
    • Article Metrics

      • Downloads (Last 12 months)14
      • Downloads (Last 6 weeks)2

      Other Metrics

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    HTML Format

    View this article in HTML Format .

    View HTML Format
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!