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Torus Polynomials: An Algebraic Approach to ACC Lower Bounds

Authors Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, Sankeerth Rao

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Author Details

Abhishek Bhrushundi
  • Rutgers University, New Brunswick, USA
Kaave Hosseini
  • University of California, San Diego, USA
Shachar Lovett
  • University of California, San Diego, USA
Sankeerth Rao
  • University of California, San Diego, USA

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Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao. Torus Polynomials: An Algebraic Approach to ACC Lower Bounds. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Circuit complexity
  • ACC
  • lower bounds
  • polynomials


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  1. 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. Google Scholar
  2. 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. Google Scholar
  3. 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. Google Scholar
  4. 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. Google Scholar
  5. 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. Google Scholar
  6. Johan Håstad. Computational Limitations of Small-depth Circuits. MIT Press, Cambridge, MA, USA, 1987. Google Scholar
  7. Johan Håstad. The shrinkage exponent of De Morgan formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998. Google Scholar
  8. 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. Google Scholar
  9. 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. Google Scholar
  10. 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. Google Scholar
  11. Alexander A Razborov and Steven Rudich. Natural Proofs. Journal of Computer and System Sciences, 55(1):24-35, 1997. Google Scholar
  12. 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. Google Scholar
  13. Terence Tao. Some notes on “non-classical” polynomials in finite characteristic, 2008. Google Scholar
  14. 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. Google Scholar
  15. Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991. Google Scholar
  16. Ryan Williams. Nonuniform ACC circuit lower bounds. Journal of the ACM (JACM), 61(1):2, 2014. Google Scholar
  17. 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. Google Scholar
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