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The Power of Natural Properties as Oracles

Authors Russell Impagliazzo, Valentine Kabanets, Ilya Volkovich



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Author Details

Russell Impagliazzo
  • Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Ilya Volkovich
  • Department of EECS, CSE Division, University of Michigan, Ann Arbor, MI, USA

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Russell Impagliazzo, Valentine Kabanets, and Ilya Volkovich. The Power of Natural Properties as Oracles. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 7:1-7:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CCC.2018.7

Abstract

We study the power of randomized complexity classes that are given oracle access to a natural property of Razborov and Rudich (JCSS, 1997) or its special case, the Minimal Circuit Size Problem (MCSP). We show that in a number of complexity-theoretic results that use the SAT oracle, one can use the MCSP oracle instead. For example, we show that ZPEXP^{MCSP} !subseteq P/poly, which should be contrasted with the previously known circuit lower bound ZPEXP^{NP} !subseteq P/poly. We also show that, assuming the existence of Indistinguishability Obfuscators (IO), SAT and MCSP are equivalent in the sense that one has a ZPP algorithm if and only the other one does. We interpret our results as providing some evidence that MCSP may be NP-hard under randomized polynomial-time reductions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • natural properties
  • Minimal Circuit Size Problem (MCSP)
  • circuit lower bounds
  • hardness of MCSP
  • learning algorithms
  • obfuscation
  • Indistinguishability Obfuscators (IO)

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References

  1. L. M. Adleman. Two theorems on random polynomial time. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 75-83, 1978. Google Scholar
  2. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: http://dx.doi.org/10.1137/050628994.
  3. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, volume 8635 of Lecture Notes in Computer Science, pages 25-32. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_3.
  4. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing disjunctive normal form formulas and ac^0 circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. URL: http://dx.doi.org/10.1137/060664537.
  5. Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 21-33. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21.
  6. S. Arora and B. Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  7. Vikraman Arvind and Johannes Köbler. On pseudorandomness and resource-bounded measure. Theor. Comput. Sci., 255(1-2):205-221, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(99)00164-4.
  8. L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. Google Scholar
  9. L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993. Google Scholar
  10. B. Barak. A probabilistic-time hierarchy theorem for "slightly non-uniform" algorithms. In RANDOM, pages 194-208, 2002. Google Scholar
  11. B. Barak, O. Goldreich, R. Impagliazzo, S. Rudich, A. Sahai, S. P. Vadhan, and K. Yang. On the (im)possibility of obfuscating programs. In Advances in Cryptology - CRYPTO 2001, 21st Annual International Cryptology Conference, pages 1-18, 2001. Google Scholar
  12. Donald Beaver and Joan Feigenbaum. Hiding instances in multioracle queries. In Christian Choffrut and Thomas Lengauer, editors, STACS 90, 7th Annual Symposium on Theoretical Aspects of Computer Science, Rouen, France, February 22-24, 1990, Proceedings, volume 415 of Lecture Notes in Computer Science, pages 37-48. Springer, 1990. URL: http://dx.doi.org/10.1007/3-540-52282-4_30.
  13. Nader H. Bshouty, Richard Cleve, Ricard Gavaldà, Sampath Kannan, and Christino Tamon. Oracles and queries that are sufficient for exact learning. J. Comput. Syst. Sci., 52(3):421-433, 1996. URL: http://dx.doi.org/10.1006/jcss.1996.0032.
  14. H. Buhrman, L. Fortnow, and T. Thierauf. Nonrelativizing separations. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity (CCC), pages 8-12, 1998. Google Scholar
  15. Harry Buhrman and Steven Homer. Superpolynomial circuits, almost sparse oracles and the exponential hierarchy. In R. K. Shyamasundar, editor, Foundations of Software Technology and Theoretical Computer Science, 12th Conference, New Delhi, India, December 18-20, 1992, Proceedings, volume 652 of Lecture Notes in Computer Science, pages 116-127. Springer, 1992. URL: http://dx.doi.org/10.1007/3-540-56287-7_99.
  16. Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 10:1-10:24. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.10.
  17. J. Feigenbaum and L. Fortnow. Random-self-reducibility of complete sets. SIAM J. on Computing, 22(5):994-1005, 1993. Google Scholar
  18. L. Fortnow and A. R. Klivans. Efficient learning algorithms yield circuit lower bounds. J. Comput. Syst. Sci., 75(1):27-36, 2009. Google Scholar
  19. L. Fortnow and R. Santhanam. Hierarchy theorems for probabilistic polynomial time. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 316-324, 2004. Google Scholar
  20. O. Goldreich and D. Zuckerman. Another proof that BPP ⊆ PH (and more). Studies in Complexity and Cryptography, pages 40-53, 2011. Google Scholar
  21. S. Goldwasser and G. N. Rothblum. On best-possible obfuscation. In Theory of Cryptography, 4th Theory of Cryptography Conference, TCC, pages 194-213, 2007. Google Scholar
  22. Hans Heller. On relativized exponential and probabilistic complexity classes. Information and Control, 71(3):231-243, 1986. URL: http://dx.doi.org/10.1016/S0019-9958(86)80012-2.
  23. Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 18:1-18:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.18.
  24. John M. Hitchcock and Aduri Pavan. On the np-completeness of the minimum circuit size problem. In Prahladh Harsha and G. Ramalingam, editors, 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, volume 45 of LIPIcs, pages 236-245. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.236.
  25. R. Impagliazzo, V. Kabanets, and A. Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. J. of Computer and System Sciences, 65(4):672-694, 2002. Google Scholar
  26. R. Impagliazzo and A. Wigderson. P=BPP unless E has subexponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pages 220-229, 1997. Google Scholar
  27. R. Impagliazzo and A. Wigderson. Randomness vs. time: De-randomization under a uniform assumption. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 734-743, 1998. Google Scholar
  28. V. Kabanets and J.-Y. Cai. Circuit minimization problem. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 73-79, 2000. Google Scholar
  29. Ravi Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55(1-3):40-56, 1982. URL: http://dx.doi.org/10.1016/S0019-9958(82)90382-5.
  30. Richard M. Karp. Turing award lecture. In Bill Healy and Judith D. Schlesinger, editors, Proceedings of the 1985 ACM annual conference on The range of computing: mid-80’s perspective: mid-80’s perspective, Denver, Colorado, USA, October 14-16, 1985, page 193. ACM, 1985. URL: http://dx.doi.org/10.1145/320435.320497.
  31. Richard M. Karp and Richard J. Lipton. Some connections between nonuniform and uniform complexity classes. In Raymond E. Miller, Seymour Ginsburg, Walter A. Burkhard, and Richard J. Lipton, editors, Proceedings of the 12th Annual ACM Symposium on Theory of Computing, April 28-30, 1980, Los Angeles, California, USA, pages 302-309. ACM, 1980. URL: http://dx.doi.org/10.1145/800141.804678.
  32. A. Klivans, P. Kothari, and I. Oliveira. Constructing hard functions from learning algorithms. In Proceedings of the 28th Annual IEEE Conference on Computational Complexity (CCC), pages 86-97, 2013. Google Scholar
  33. Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput., 31(5):1501-1526, 2002. URL: http://dx.doi.org/10.1137/S0097539700389652.
  34. Johannes Köbler and Osamu Watanabe. New collapse consequences of NP having small circuits. SIAM J. Comput., 28(1):311-324, 1998. URL: http://dx.doi.org/10.1137/S0097539795296206.
  35. Ilan Komargodski, Tal Moran, Moni Naor, Rafael Pass, Alon Rosen, and Eylon Yogev. One-way functions and (im)perfect obfuscation. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 374-383. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.47.
  36. C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. JACM, 39(4):859-868, 1992. Google Scholar
  37. D. van Melkebeek and K. Pervyshev. A generic time hierarchy with one bit of advice. Computational Complexity, 16(2):139-179, 2007. Google Scholar
  38. Cody D. Murray and Richard Ryan Williams. On the (non) np-hardness of computing circuit complexity. In David Zuckerman, editor, 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, volume 33 of LIPIcs, pages 365-380. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365.
  39. N. Nisan and A. Wigderson. Hardness vs. randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. Google Scholar
  40. I. C. Oliveira and R. Santhanam. Conspiracies between learning algorithms, circuit lower bounds and pseudorandomness. CoRR, abs/1611.01190, 2016. URL: http://arxiv.org/abs/1611.01190.
  41. A. A. Razborov and S. Rudich. Natural proofs. J. of Computer and System Sciences, 55(1):24-35, 1997. Google Scholar
  42. R. Santhanam. Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009. Google Scholar
  43. S. Toda. PP is as hard as the polynomial time hierarchy. SIAM J. on Computing, 20(5):865-877, 1991. Google Scholar
  44. Boris A. Trakhtenbrot. A survey of russian approaches to perebor (brute-force searches) algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984. URL: http://dx.doi.org/10.1109/MAHC.1984.10036.
  45. L. Trevisan and S. P. Vadhan. Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity, 16(4):331-364, 2007. Google Scholar
  46. C. Umans. Pseudo-random generators for all hardnesses. J. of Computer and System Sciences, 67(2):419-440, 2003. Google Scholar
  47. L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979. Google Scholar
  48. L. G. Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, 1984. Google Scholar
  49. I. Volkovich. On learning, lower bounds and (un)keeping promises. In Proceedings of the 41st ICALP, pages 1027-1038, 2014. Google Scholar
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