Torus Polynomials: An Algebraic Approach to ACC Lower Bounds
We propose an algebraic approach to proving circuit lower bounds for ACC^0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC^0 and ACC^0 can be reformulated in this framework, implying that ACC^0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC^0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.
Circuit complexity
ACC
lower bounds
polynomials
Theory of computation~Circuit complexity
13:1-13:16
Regular Paper
Supported by NSF grant CCF-1614023.
https://arxiv.org/abs/1804.08176
Abhishek
Bhrushundi
Abhishek Bhrushundi
Rutgers University, New Brunswick, USA
Part of this work was done when the author was visiting the University of California, San Diego. Research supported in part by Rutgers AAUP-AFT TA-GA Professional Development Fund.
Kaave
Hosseini
Kaave Hosseini
University of California, San Diego, USA
Shachar
Lovett
Shachar Lovett
University of California, San Diego, USA
Sankeerth
Rao
Sankeerth Rao
University of California, San Diego, USA
10.4230/LIPIcs.ITCS.2019.13
Richard Beigel and Jun Tarui. On ACC (circuit complexity). In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, FOCS 1991, pages 783-792. IEEE, 1991.
Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett. Every Locally Characterized Affine-invariant Property is Testable. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC 2013, pages 429-436. ACM, 2013.
Abhishek Bhowmick and Shachar Lovett. Nonclassical polynomials as a barrier to polynomial lower bounds. In Proceedings of the 30th Conference on Computational Complexity, CCC 2015, pages 72-87. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015.
Abhishek Bhrushundi, Prahladh Harsha, and Srikanth Srinivasan. On Polynomial Approximations Over ℤ/2^k ℤ. In Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, pages 12:1-12:12, 2017.
Frederic Green, Johannes Kobler, and Jacobo Toran. The power of the middle bit. In Proceedings of the 7th Annual Structure in Complexity Theory Conference, 1992, pages 111-117. IEEE, 1992.
Johan Håstad. Computational Limitations of Small-depth Circuits. MIT Press, Cambridge, MA, USA, 1987.
Johan Håstad. The shrinkage exponent of De Morgan formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998.
Cody Murray and Ryan Williams. Circuit Lower Bounds for Nondeterministic Quasi-polytime: An Easy Witness Lemma for NP and NQP. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 890-901. ACM, 2018.
Noam Nisan and Mario Szegedy. On the Degree of Boolean Functions As Real Polynomials. In Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC 1992, pages 462-467. ACM, 1992.
Alexander A Razborov. Lower bounds for the size of circuits of bounded depth with basis ∧, ⊕. Math. notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987.
Alexander A Razborov and Steven Rudich. Natural Proofs. Journal of Computer and System Sciences, 55(1):24-35, 1997.
Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of computing, STOC 1987, pages 77-82. ACM, 1987.
Terence Tao. Some notes on “non-classical” polynomials in finite characteristic, 2008.
Terence Tao and Tamar Ziegler. The inverse conjecture for the Gowers norm over finite fields in low characteristic. Annals of Combinatorics, 16(1):121-188, 2012.
Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991.
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. In Proceedings of the 26th Annual Symposium on Foundations of Computer Science, FOCS 1985, pages 1-10. IEEE, 1985.
Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode