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Lower Bounds and Separations for Torus Polynomials

Authors Vaibhav Krishan , Sundar Vishwanathan

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LIPIcs.ITCS.2026.88.pdf
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ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Circuit complexity
  • ACC
  • lower bounds
  • polynomials

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Abstract

The class ACC⁰ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with AND, NOT and MOD_m gates, where m is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that MAJORITY does not belong to ACC⁰.
A few years ago, Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach towards this conjecture. Torus polynomials approximate Boolean functions when the fractional part of their value on Boolean points is close to half the value of the function. They reduced the conjecture that MAJORITY ∉ ACC⁰ to a conjecture concerning the non-existence of low degree torus polynomials that approximate MAJORITY.
We reduce the non-existence problem further, to a statement about finding feasible solutions for an infinite family of linear programs. The main advantage of this statement is that it allows for incremental progress, which means finding feasible solutions for successively larger collections of these programs. As an immediate first step, we find feasible solutions for a large class of these linear programs, leaving only a finite set for further consideration. Our method is inspired by the method of dual polynomials, which is used to study the approximate degree of Boolean functions. Using our method, we also propose a way to progress further.
We prove several additional key results with the same method, which include:  
- A lower bound on the degree of symmetric torus polynomials that approximate the AND function. As a consequence, we get a separation that symmetric torus polynomials are weaker than their asymmetric counterparts. 
- An error-degree trade-off for symmetric torus polynomials approximating the MAJORITY function, strengthening the corresponding result of Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019). 
- The first lower bounds against torus polynomials approximating AND, showcasing the power of the machinery we develop. This lower bound nearly matches the corresponding upper bound. Hence, we get an almost complete characterization of the torus polynomial approximation degree of AND. 
- Lower bounds against asymmetric torus polynomials approximating MAJORITY, or AND, in the very low error regime. This partially answers a question posed in Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) about error-reduction for torus polynomials.

Cite As Get BibTex

Vaibhav Krishan and Sundar Vishwanathan. Lower Bounds and Separations for Torus Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 88:1-88:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.88

Author Details

Vaibhav Krishan
  • The Institute of Mathematical Sciences, Chennai, India
Sundar Vishwanathan
  • Indian Institute of Technology Bombay, Mumbai, India

Acknowledgements

We thank Shachar Lovett for helpful feedback on an earlier draft. We also thank anonymous referees for their detailed feedback.

References

  1. Scott Aaronson. The Polynomial Method in Quantum and Classical Computing. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, USA, October 25-28, 2008, page 3. IEEE, IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.91.
  2. David A. Mix Barrington. Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹. J. Comput. Syst. Sci., 38(1):150-164, 1989. URL: https://doi.org/10.1016/0022-0000(89)90037-8.
  3. Richard Beigel and Jun Tarui. On ACC. Comput. Complex., 4:350-366, 1994. URL: https://doi.org/10.1007/BF01263423.
  4. Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao. Torus Polynomials: An Algebraic Approach to ACC Lower Bounds. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 13:1-13:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.13.
  5. Enrico Bombieri and Jeffrey D. Vaaler. On Siegel’s lemma. Inventiones mathematicae, 73:11-32, 1983. URL: https://api.semanticscholar.org/CorpusID:121274024.
  6. Harry Buhrman, Richard Cleve, Ronald de Wolf, and Christof Zalka. Bounds for Small-Error and Zero-Error Quantum Algorithms. In 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, New York, NY, USA, October 17-18, 1999, pages 358-368. IEEE, IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814607.
  7. Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. Inf. Comput., 243:2-25, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.003.
  8. Mark Bun and Justin Thaler. Approximate Degree in Classical and Quantum Computing. Found. Trends Theor. Comput. Sci., 15(3-4):229-423, 2022. URL: https://doi.org/10.1561/0400000107.
  9. Lijie Chen. Non-deterministic Quasi-Polynomial Time is Average-Case Hard for ACC Circuits. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 1281-1304. IEEE, IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00079.
  10. Lijie Chen. New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 34:1-34:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.34.
  11. Lijie Chen, Zhenjian Lu, Xin Lyu, and Igor C. Oliveira. Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC¹. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 51:1-51:20. Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.51.
  12. Lijie Chen, Xin Lyu, and R. Ryan Williams. Almost-Everywhere Circuit Lower Bounds from Non-Trivial Derandomization. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1-12. IEEE, IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00009.
  13. Ruiwen Chen, Igor C. Oliveira, and Rahul Santhanam. An Average-Case Lower Bound Against ACC⁰. In Michael A. Bender, Martin Farach-Colton, and Miguel A. Mosteiro, editors, LATIN 2018: Theoretical Informatics - 13th Latin American Symposium, Buenos Aires, Argentina, April 16-19, 2018, Proceedings, volume 10807 of Lecture Notes in Computer Science, pages 317-330. Springer, Springer, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_24.
  14. Frederic Green, Johannes Köbler, Kenneth W. Regan, Thomas Schwentick, and Jacobo Torán. The Power of the Middle Bit of a #P Function. J. Comput. Syst. Sci., 50(3):456-467, 1995. URL: https://doi.org/10.1006/jcss.1995.1036.
  15. Johan Håstad. Almost Optimal Lower Bounds for Small Depth Circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL: https://doi.org/10.1145/12130.12132.
  16. Johan Håstad and Mikael Goldmann. On the Power of Small-Depth Threshold Circuits. Comput. Complex., 1(2):113-129, 1991. URL: https://doi.org/10.1007/BF01272517.
  17. Vaibhav Krishan. Upper Bound for Torus Polynomials. In Rahul Santhanam and Daniil Musatov, editors, Computer Science - Theory and Applications - 16th International Computer Science Symposium in Russia, CSR 2021, Sochi, Russia, June 28 - July 2, 2021, Proceedings, volume 12730 of Lecture Notes in Computer Science, pages 257-263. Springer, Springer, 2021. URL: https://doi.org/10.1007/978-3-030-79416-3_15.
  18. Vaibhav Krishan and Sundar Vishwanathan. Degree Lower Bounds for Torus Polynomials and MAJORITY vs ACC⁰. Electron. Colloquium Comput. Complex., TR24-074, 2024. URL: https://eccc.weizmann.ac.il/report/2024/074/#revision1.
  19. Hermann Minkowski. Geometrie der zahlen. BG Teubner, 1910. Google Scholar
  20. Cody D. Murray and R. Ryan Williams. Circuit Lower Bounds for Nondeterministic Quasi-polytime from a New Easy Witness Lemma. SIAM J. Comput., 49(5):STOC18-300, 2020. URL: https://doi.org/10.1137/18M1195887.
  21. Noam Nisan and Mario Szegedy. On the Degree of Boolean Functions as Real Polynomials. Comput. Complex., 4:301-313, 1994. URL: https://doi.org/10.1007/BF01263419.
  22. Alexander A. Razborov and Avi Wigderson. n^Ω(log n) Lower Bounds on the Size of Depth-3 Threshold Circuits with AND Gates at the Bottom. Inf. Process. Lett., 45(6):303-307, 1993. URL: https://doi.org/10.1016/0020-0190(93)90041-7.
  23. Alexander Alexandrovich Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Matematicheskie Zametki, 41(4):598-607, 1987. Google Scholar
  24. Roman Smolensky. Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL: https://doi.org/10.1145/28395.28404.
  25. Ryan Williams. Nonuniform ACC Circuit Lower Bounds. J. ACM, 61(1):2:1-2:32, 2014. URL: https://doi.org/10.1145/2559903.
  26. Andrew Chi-Chih Yao. On ACC and Threshold Circuits. In 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, October 22-24, 1990, Volume II, pages 619-627. IEEE, IEEE Computer Society, 1990. URL: https://doi.org/10.1109/FSCS.1990.89583.
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