Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds
We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.
circuit satisfiability
circuit lower bounds
polynomial method
derandomization
Theory of computation~Computational complexity and cryptography
78:1-78:15
Regular Paper
Ninad
Rajgopal
Ninad Rajgopal
Department of Computer Science, University of Oxford, Oxford, United Kingdom
This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement No. 615075.
Rahul
Santhanam
Rahul Santhanam
Department of Computer Science, University of Oxford, Oxford, United Kingdom
This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement No. 615075.
Srikanth
Srinivasan
Srikanth Srinivasan
Department of Mathematics, IIT Bombay, Mumbai, India
10.4230/LIPIcs.MFCS.2018.78
Miklos Ajtai. Σ ¹₁-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983.
Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple construction of almost k-wise independent random variables. Random Struct. Algorithms, 3(3):289-304, 1992.
Richard Beigel and Jun Tarui. On ACC. Computational Complexity, 4:350-366, 1994.
Eli Ben-Sasson and Emanuele Viola. Short PCPs with projection queries. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 163-173, 2014.
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 10:1-10:24, 2016.
Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1246-1255, 2016.
Shiteng Chen and Periklis A. Papakonstantinou. Depth-reduction for composites. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 99-108. IEEE Computer Society, 2016. URL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7781469, URL: http://dx.doi.org/10.1109/FOCS.2016.20.
http://dx.doi.org/10.1109/FOCS.2016.20
Anindya De, Omid Etesami, Luca Trevisan, and Madhur Tulsiani. Improved pseudorandom generators for depth 2 circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 504-517. Springer, 2010.
Merrick Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984.
David Gillman. A Chernoff bound for random walks on expander graphs. SIAM Journal on Computing, 27(4):1203-1220, 1998.
Johan Håstad. Almost optimal lower bounds for small depth circuits. In Symposium on Theory of Computing (STOC), pages 6-20, 1986.
Alexander D Healy. Randomness-efficient sampling within NC. Computational Complexity, 17(1):3-37, 2008.
Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4):439-561, 2006. URL: http://dx.doi.org/10.1090/S0273-0979-06-01126-8.
http://dx.doi.org/10.1090/S0273-0979-06-01126-8
Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC^0. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 961-972, 2012.
Swastik Kopparty and Srikanth Srinivasan. Certifying polynomials for AC^0[⊕] circuits, with applications. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, pages 36-47, 2012.
Daniel Lokshtanov, Ramamohan Paturi, Suguru Tamaki, R. Ryan Williams, and Huacheng Yu. Beating brute force for systems of polynomial equations over finite fields. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2190-2202. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.143.
http://dx.doi.org/10.1137/1.9781611974782.143
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput., 22(4):838-856, 1993.
Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994.
Ryan O'Donnell. Analysis of Boolean functions. Cambridge University Press, 2014.
Igor Carboni Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 18:1-18:49, 2017.
Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987.
Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. Annals of Mathematics, 155(1):157-187, 2002.
Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 77-82. ACM, 1987.
Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. URL: http://dx.doi.org/10.1137/0220053.
http://dx.doi.org/10.1137/0220053
Ryan Williams. Guest column: a casual tour around a circuit complexity bound. SIGACT News, 42(3):54-76, 2011. URL: http://dx.doi.org/10.1145/2034575.2034591.
http://dx.doi.org/10.1145/2034575.2034591
Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput., 42(3):1218-1244, 2013.
Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 194-202, 2014.
Ryan Williams. Nonuniform ACC circuit lower bounds. Journal of the ACM (JACM), 61(1):2, 2014.
Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles (preliminary version). In Symposium on Foundations of Computer Science (FOCS), pages 1-10, 1985.
Ninad Rajgopal, Rahul Santhanam, and Srikanth Srinivasan
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