Document Open Access Logo

Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

Authors Ninad Rajgopal, Rahul Santhanam, Srikanth Srinivasan

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2018.78.pdf
  • Filesize: 0.56 MB
  • 15 pages

Document Identifiers

Author Details

Ninad Rajgopal
  • Department of Computer Science, University of Oxford, Oxford, United Kingdom
Rahul Santhanam
  • Department of Computer Science, University of Oxford, Oxford, United Kingdom
Srikanth Srinivasan
  • Department of Mathematics, IIT Bombay, Mumbai, India

Funding

  • Rajgopal, Ninad: 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.
  • Santhanam, Rahul: 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.

Cite AsGet BibTex

Ninad Rajgopal, Rahul Santhanam, and Srikanth Srinivasan. Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 78:1-78:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.78

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • circuit satisfiability
  • circuit lower bounds
  • polynomial method
  • derandomization

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Miklos Ajtai. Σ ¹₁-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. Google Scholar
  2. 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. Google Scholar
  3. Richard Beigel and Jun Tarui. On ACC. Computational Complexity, 4:350-366, 1994. Google Scholar
  4. 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. Google Scholar
  5. 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. Google Scholar
  6. 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. Google Scholar
  7. 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.
  8. 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. Google Scholar
  9. Merrick Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. Google Scholar
  10. David Gillman. A Chernoff bound for random walks on expander graphs. SIAM Journal on Computing, 27(4):1203-1220, 1998. Google Scholar
  11. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Symposium on Theory of Computing (STOC), pages 6-20, 1986. Google Scholar
  12. Alexander D Healy. Randomness-efficient sampling within NC. Computational Complexity, 17(1):3-37, 2008. Google Scholar
  13. 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.
  14. 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. Google Scholar
  15. 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. Google Scholar
  16. 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.
  17. Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput., 22(4):838-856, 1993. Google Scholar
  18. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. Google Scholar
  19. Ryan O'Donnell. Analysis of Boolean functions. Cambridge University Press, 2014. Google Scholar
  20. 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. Google Scholar
  21. 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. Google Scholar
  22. 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. Google Scholar
  23. 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. Google Scholar
  24. 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.
  25. 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.
  26. Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput., 42(3):1218-1244, 2013. Google Scholar
  27. 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. Google Scholar
  28. Ryan Williams. Nonuniform ACC circuit lower bounds. Journal of the ACM (JACM), 61(1):2, 2014. Google Scholar
  29. 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. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact