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New Lower Bounds Against Homogeneous Non-Commutative Circuits

Authors Prerona Chatterjee , Pavel Hrubeš

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Author Details

Prerona Chatterjee
  • Department of Computer Science, Tel Aviv University, Israel
Pavel Hrubeš
  • Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic


Prerona would like to acknowledge for being such a nice place to work from. Pavel thanks Amir Yehudayoff for useful ideas on this topic which were exchanged in distant and joyous past.

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Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Algebraic circuit complexity
  • Non-Commutative Circuits
  • Homogeneous Computation
  • Lower bounds against algebraic circuits


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  1. Walter Baur and Volker Strassen. The complexity of partial derivatives. Theoretical computer science, 22(3):317-330, 1983. Google Scholar
  2. Peter Bürgisser, Michael Clausen, and M Amin Shokrollahi. Algebraic complexity theory, with the collaboration of thomas lickteig. Grundlehren der Mathematischen Wissenschaften, 315, 1997. Google Scholar
  3. Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin. Hardness amplification for non-commutative arithmetic circuits. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference, CCC 2018, June 22-24, 2018, San Diego, CA, USA, volume 102 of LIPIcs, pages 12:1-12:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  4. Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. Quadratic lower bounds for algebraic branching programs and formulas. Comput. Complex., 31(2):8, 2022. Google Scholar
  5. N.G. de Bruijn. A combinatorial problem. Proceedings of the Section of Sciences of the Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, 49(7):758-764, 1946. Google Scholar
  6. Pavel Hrubes, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. J. Amer. Math. Soc., 24(3):871-898, 2011. Google Scholar
  7. Laurent Hyafil. The power of commutativity. In 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October - 1 November 1977, pages 171-174. IEEE Computer Society, 1977. Google Scholar
  8. K. Kalorkoti. A lower bound for the formula size of rational functions. SIAM J. Comput., 14(3):678-687, 1985. Google Scholar
  9. Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 410-418. ACM, 1991. Google Scholar
  10. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex., 6(3):217-234, 1997. Google Scholar
  11. Joe Sawada, Aaron Williams, and Dennis Wong. Generalizing the classic greedy and necklace constructions of de bruijn sequences and universal cycles. Electron. J. Comb., 23(1):1, 2016. Google Scholar
  12. Arnold Schönhage and Volker Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7(3-4):281-292, 1971. Google Scholar
  13. Volker Strassen. Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten. Numerische Mathematik, 20(3):238-251, 1973. Google Scholar
  14. Sébastien Tavenas, Nutan Limaye, and Srikanth Srinivasan. Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplication. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20-24, 2022, pages 416-425. ACM, 2022. Google Scholar
  15. Leslie G. Valiant. Negation can be exponentially powerful. Theor. Comput. Sci., 12:303-314, 1980. Google Scholar
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