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.
- 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 Scholar
Digital Library
- 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 Scholar
- L. Babai. 1987. Random oracles separate PSPACE from the polynomial-time hierarchy. Information Processing Letters 26 (1987), 51--53. Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Cross Ref
- R. V. Book. 1974. Tally languages and complexity classes. Information and Control 26 (1974), 186--193.Google Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- J. Håstad. 1986. Computational Limitations for Small-Depth Circuits. The MIT Press. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- D. W. Juedes and J. H. Lutz. 1995. Weak completeness in E and E2. Theoretical Computer Science 143, 1 (1995), 149--158. Google Scholar
Digital Library
- J. H. Lutz. 1992. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44, 2 (1992), 220--258. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- J. H. Lutz and W. J. Schmidt. 1993. Circuit size relative to pseudorandom oracles. Theoretical Computer Science 107, 1 (March 1993), 95--120. Google Scholar
Digital Library
- P. Martin-Löf. 1966. The definition of random sequences. Information and Control 9 (1966), 602--619.Google Scholar
- 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 Scholar
Cross Ref
- A. A. Muchnik, A. L. Semenov, and V. A. Uspensky. 1998. Mathematical metaphysics of randomness. Theoretical Computer Science 207, 2 (1998), 263--317. Google Scholar
Digital Library
- N. Nisan and A. Wigderson. 1994. Hardness vs. randomness. Journal of Computer and System Sciences 49, 2 (1994), 149--167. Google Scholar
Digital Library
- 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 Scholar
Digital Library
Index Terms
Polynomial-Time Random Oracles and Separating Complexity Classes
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