Document Open Access Logo

Symmetric Formulas for Products of Permutations

Authors William He, Benjamin Rossman

Thumbnail PDF


  • Filesize: 0.84 MB
  • 23 pages

Document Identifiers

Author Details

William He
  • Duke University, Durham, NC, USA
Benjamin Rossman
  • Duke University, Durham, NC, USA

Cite AsGet BibTex

William He and Benjamin Rossman. Symmetric Formulas for Products of Permutations. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 68:1-68:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study the formula complexity of the word problem Word_{S_n,k} : {0,1}^{kn²} → {0,1}: given n-by-n permutation matrices M₁,… ,M_k, compute the (1,1)-entry of the matrix product M₁⋯ M_k. An important feature of this function is that it is invariant under action of S_n^{k-1} given by (π₁,… ,π_{k-1})(M₁,… ,M_k) = (M₁π₁^{-1},π₁M₂π₂^{-1},… ,π_{k-2}M_{k-1}π_{k-1}^{-1},π_{k-1}M_k). This symmetry is also exhibited in the smallest known unbounded fan-in {and,or,not}-formulas for Word_{S_n,k}, which have size n^O(log k). In this paper we prove a matching n^{Ω(log k)} lower bound for S_n^{k-1}-invariant formulas computing Word_{S_n,k}. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes NC¹ and Logspace. Our more general main theorem gives a nearly tight n^d(k^{1/d}-1) lower bound on the G^{k-1}-invariant depth-d {maj,and,or,not}-formula size of Word_{G,k} for any finite simple group G whose minimum permutation representation has degree n. We also give nearly tight lower bounds on the G^{k-1}-invariant depth-d {and,or,not}-formula size in the case where G is an abelian group.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • circuit complexity
  • group-invariant formulas


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


  1. Boris Alexeev, Michael A Forbes, and Jacob Tsimerman. Tensor rank: Some lower and upper bounds. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 283-291. IEEE, 2011. Google Scholar
  2. Eric Allender and Michal Kouckỳ. Amplifying lower bounds by means of self-reducibility. Journal of the ACM (JACM), 57(3):1-36, 2010. Google Scholar
  3. Matthew Anderson and Anuj Dawar. On symmetric circuits and fixed-point logics. Theory of Computing Systems, 60(3):521-551, July 2016. URL:
  4. László Babai, Albert J Goodman, and László Pyber. On faithful permutation representations of small degree. Communications in Algebra, 21(5):1587-1602, 1993. Google Scholar
  5. David A. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in nc1. Journal of Computer and System Sciences, 38(1):150-164, 1989. Google Scholar
  6. Laurent Bartholdi, Michael Figelius, Markus Lohrey, and Armin Weiß. Groups with ALOGTIME-hard word problems and PSPACE-complete circuit value problems. In Proceedings of the 35th Computational Complexity Conference, CCC '20, pages 1-29, Dagstuhl, DEU, July 2020. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  7. Paul Beame, Russell Impagliazzo, and Toniann Pitassi. Improved depth lower bounds for small distance connectivity. Computational Complexity, 7(4):325-345, December 1998. URL:
  8. Lijie Chen and Roei Tell. Bootstrapping results for threshold circuits “just beyond” known lower bounds. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 34-41, 2019. Google Scholar
  9. Xi Chen, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan. Near-optimal small-depth lower bounds for small distance connectivity. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing. ACM, June 2016. URL:
  10. Stephen A. Cook and Pierre McKenzie. Problems complete for deterministic logarithmic space. Journal of Algorithms, 8(3):385-394, 1987. Google Scholar
  11. Max Dehn. Über unendliche diskontinuierliche gruppen. Mathematische Annalen, 71(1):116-144, March 1911. URL:
  12. Larry Denenberg, Yuri Gurevich, and Saharon Shelah. Definability by constant-depth polynomial-size circuits. Information and Control, 70(2-3):216-240, August 1986. URL:
  13. Édouard Goursat. Sur les substitutions orthogonales et les divisions régulières de lquotesingleespace. Annales scientifiques de lquotesingleÉcole normale supérieure, 6:9-102, 1889. URL:
  14. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC quotesingle86. ACM Press, 1986. URL:
  15. Russell Impagliazzo, Ramamohan Paturi, and Michael E Saks. Size-depth tradeoffs for threshold circuits. SIAM Journal on Computing, 26(3):693-707, 1997. Google Scholar
  16. David Lawrence Johnson. Minimal permutation representations of finite groups. American Journal of Mathematics, 93(4):857-866, 1971. Google Scholar
  17. Valeriy Mihailovich Khrapchenko. Complexity of the realization of a linear function in the class of ii-circuits. Mathematical Notes of the Academy of Sciences of the USSR, 9(1):21-23, 1971. Google Scholar
  18. Alexei Miasnikov, Svetla Vassileva, and Armin Weiß. The conjugacy problem in free solvable groups and wreath products of abelian groups is in TC⁰. Theory of Computing Systems, 63(4):809-832, February 2018. URL:
  19. Alexei Myasnikov and Armin Weiß. TC⁰ circuits for algorithmic problems in nilpotent groups. In 42nd International Symposium on Mathematical Foundations of Computer Science. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany, 2017. URL:
  20. Benjamin Rossman. The average sensitivity of bounded-depth formulas. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 424-430, Berkeley, CA, USA, 2015. IEEE. URL:
  21. Benjamin Rossman. Formulas versus circuits for small distance connectivity. SIAM Journal on Computing, 47(5):1986-2028, January 2018. URL:
  22. Benjamin Rossman. Subspace-invariant AC⁰ formulas. Logical Methods in Computer Science, Volume 15, Issue 3 (July 24, 2019) lmcs:5641, June 2018. URL:
  23. Robert A. Wilson. The Finite Simple Groups. Springer London, 2009. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail