Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

Author Prerona Chatterjee

Thumbnail PDF


  • Filesize: 0.83 MB
  • 24 pages

Document Identifiers

Author Details

Prerona Chatterjee
  • Tata Institute of Fundamental Research, Mumbai, India


We are thankful to Ramprasad Saptharishi, Mrinal Kumar, C. Ramya and especially Anamay Tengse for the discussions at various stages of this work. We would also like to thank Ramprasad Saptharishi, Anamay Tengse and Kshitij Gajjar for helping with the presentation of the paper. Finally, we would like to thank the anonymous reviewers for their valuable comments that have helped in improving the paper.

Cite AsGet BibTex

Prerona Chatterjee. Separating ABPs and Some Structured Formulas in the Non-Commutative Setting. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The motivating question for this work is a long standing open problem, posed by Nisan [Noam Nisan, 1991], regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question remains open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011]) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards resolving the VF_{nc} vs VBP_{nc} question, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n²-variate degree d abecedarian polynomial f_{n,d}(𝐱) such that - f_{n, d}(𝐱) can be computed by an abecedarian ABP of size O(nd); - any abecedarian formula computing f_{n, log n}(𝐱) must have size at least n^{Ω(log log n)}. We also show that a super-polynomial lower bound against abecedarian formulas for f_{log n, n}(𝐱) would separate the powers of formulas and ABPs in the non-commutative setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Non-Commutative Formulas
  • Lower Bound
  • Separating ABPs and Formulas


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


  1. V. Arvind, P. S. Joglekar, and S. Raja. Noncommutative valiant’s classes: Structure and complete problems. ACM Trans. Comput. Theory, 9(1), 2016. URL:
  2. Vikraman Arvind and Srikanth Srinivasan. On the hardness of the noncommutative determinant. Comput. Complex., 27(1):1-29, 2018. URL:
  3. Michael Ben-Or and Richard Cleve. Computing algebraic formulas using a constant number of registers. SIAM J. Comput., 21(1):54-58, 1992. URL:
  4. Richard P. Brent. The parallel evaluation of general arithmetic expressions. Journal of the ACM, 21(2):201-206, 1974. URL:
  5. 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, volume 102 of LIPIcs, pages 12:1-12:16, 2018. URL:
  6. Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. A quadratic lower bound for algebraic branching programs and formulas. CoRR, 1911.11793v2, 2019. URL:
  7. Zeev Dvir, Guillaume Malod, Sylvain Perifel, and Amir Yehudayoff. Separating multilinear branching programs and formulas. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 615-624. ACM, 2012. URL:
  8. Nathanaël Fijalkow, Guillaume Lagarde, Pierre Ohlmann, and Olivier Serre. Lower bounds for arithmetic circuits via the hankel matrix. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS, volume 154 of LIPIcs, pages 24:1-24:17, 2020. URL:
  9. Pavel Hrubes and Avi Wigderson. Non-commutative arithmetic circuits with division. Theory Comput., 11:357-393, 2015. URL:
  10. Pavel Hrubes, Avi Wigderson, and Amir Yehudayoff. Relationless completeness and separations. In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC, pages 280-290. IEEE Computer Society, 2010. URL:
  11. Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011. URL:
  12. Pavel Hrubes and Amir Yehudayoff. Homogeneous formulas and symmetric polynomials. Comput. Complex., 20(3):559-578, 2011. URL:
  13. L. Hyafil. The power of commutativity. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 171-174, 1977. URL:
  14. K. Kalorkoti. A lower bound for the formula size of rational functions. SIAM J. Comput., 14(3):678-687, 1985. URL:
  15. Guillaume Lagarde, Nutan Limaye, and Srikanth Srinivasan. Lower bounds and PIT for non-commutative arithmetic circuits with restricted parse trees. Comput. Complex., 28(3):471-542, 2019. URL:
  16. Guillaume Lagarde, Guillaume Malod, and Sylvain Perifel. Non-commutative computations: lower bounds and polynomial identity testing. Chic. J. Theor. Comput. Sci., 2019, 2019. URL:
  17. Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for non-commutative skew circuits. Theory of Computing, 12(1):1-38, 2016. URL:
  18. Merriam and Webster. Definition of abecedarian. Word of the Day at, 2019. URL:
  19. Eduard Ivanovich Nechiporuk. On a boolean function. Dokl. Akad. Nauk SSSR, 169:765-766, 1966. URL:
  20. Noam Nisan. Lower bounds for non-commutative computation (extended abstract). In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 410-418. ACM, 1991. URL:
  21. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. J. ACM, 60(6):40:1-40:15, 2013. URL:
  22. Ramprasad Saptharishi and Anamay Tengse. Quasipolynomial hitting sets for circuits with restricted parse trees. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, volume 122 of LIPIcs, pages 6:1-6:19, 2018. URL:
  23. Leslie G. Valiant. Completeness classes in algebra. In Proceedings of the 11h Annual ACM Symposium on Theory of Computing, pages 249-261. ACM, 1979. 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