Limits of Minimum Circuit Size Problem as Oracle
The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-Turing reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP != EXP, which is a major open problem in computational complexity.
In this paper, we provide strong evidence that current techniques cannot establish NP-hardness of MCSP, even under polynomial-time Turing reductions or randomized reductions: Specifically, we introduce the notion of oracle-independent reduction to MCSP, which captures all the currently known reductions. We say that a reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSP^A for any oracle A, where MCSP^A denotes an oracle version of MCSP. We prove that no language outside P is reducible to MCSP via an oracle-independent polynomial-time Turing reduction. We also show that the class of languages reducible to MCSP via an oracle-independent randomized reduction that makes at most one query is contained in AM intersect coAM. Thus, NP-hardness of MCSP cannot be established via such oracle-independent reductions unless the polynomial hierarchy collapses.
We also extend the previous results to the case of more general reductions: We prove that establishing NP-hardness of MCSP via a polynomial-time nonadaptive reduction implies ZPP != EXP, and that establishing NP-hardness of approximating circuit complexity via a polynomial-time Turing reduction also implies ZPP != EXP. Along the way, we prove that approximating Levin's Kolmogorov complexity is provably not EXP-hard under polynomial-time Turing reductions, which is of independent interest.
minimum circuit size problem
NP-completeness
randomized reductions
resource-bounded Kolmogorov complexity
Turing reductions
18:1-18:20
Regular Paper
Shuichi
Hirahara
Shuichi Hirahara
Osamu
Watanabe
Osamu Watanabe
10.4230/LIPIcs.CCC.2016.18
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.
http://dx.doi.org/10.1137/050628994
Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 25-32, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_3.
http://dx.doi.org/10.1007/978-3-662-44465-8_3
Eric Allender, Joshua Grochow, and Cristopher Moore. Graph isomorphism and circuit size. Electronic Colloquium on Computational Complexity (ECCC), 22:162, 2015. URL: http://eccc.hpi-web.de/report/2015/162.
http://eccc.hpi-web.de/report/2015/162
Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In Proceedings of 32nd International Symposium on Theoretical Aspects of Computer Science (STACS), pages 21-33, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.21
Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded kolmogorov complexity in computational complexity theory. J. Comput. Syst. Sci., 77(1):14-40, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.004.
http://dx.doi.org/10.1016/j.jcss.2010.06.004
Andrej Bogdanov and Luca Trevisan. On worst-case to average-case reductions for NP problems. SIAM J. Comput., 36(4):1119-1159, 2006. URL: http://dx.doi.org/10.1137/S0097539705446974.
http://dx.doi.org/10.1137/S0097539705446974
Ravi Boppana, Johan Håstad, and Stathis Zachos. Does co-NP have short interactive proofs? Inf. Process. Lett., 25(2):127-132, 1987. URL: http://dx.doi.org/10.1016/0020-0190(87)90232-8.
http://dx.doi.org/10.1016/0020-0190(87)90232-8
Harry Buhrman and Elvira Mayordomo. An excursion to the Kolmogorov random strings. J. Comput. Syst. Sci., 54(3):393-399, 1997. URL: http://dx.doi.org/10.1006/jcss.1997.1484.
http://dx.doi.org/10.1006/jcss.1997.1484
Lance Fortnow. The complexity of perfect zero-knowledge. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 204-209, 1987. URL: http://dx.doi.org/10.1145/28395.28418.
http://dx.doi.org/10.1145/28395.28418
Shafi Goldwasser and Michael Sipser. Private coins versus public coins in interactive proof systems. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pages 59-68, 1986. URL: http://dx.doi.org/10.1145/12130.12137.
http://dx.doi.org/10.1145/12130.12137
Juris Hartmanis and Richard Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, pages 285-306, 1965.
Johan Håstad, Russell Impagliazzo, Leonid Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. URL: http://dx.doi.org/10.1137/S0097539793244708.
http://dx.doi.org/10.1137/S0097539793244708
Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC), pages 220-229, 1997. URL: http://dx.doi.org/10.1145/258533.258590.
http://dx.doi.org/10.1145/258533.258590
Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 73-79, 2000. URL: http://dx.doi.org/10.1145/335305.335314.
http://dx.doi.org/10.1145/335305.335314
Ker-I Ko. On the complexity of learning minimum time-bounded turing machines. SIAM J. Comput., 20(5):962-986, 1991. URL: http://dx.doi.org/10.1137/0220059.
http://dx.doi.org/10.1137/0220059
Leonid Levin. Randomness conservation inequalities; information and independence in mathematical theories. Information and Control, 61(1):15-37, 1984. URL: http://dx.doi.org/10.1016/S0019-9958(84)80060-1.
http://dx.doi.org/10.1016/S0019-9958(84)80060-1
Cody Murray and Ryan Williams. On the (non) NP-hardness of computing circuit complexity. In Proceedings of 30th Conference on Computational Complexity (CCC), pages 365-380, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.365
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