Document Open Access Logo

Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Author Suryajith Chillara

Thumbnail PDF


  • Filesize: 0.79 MB
  • 15 pages

Document Identifiers

Author Details

Suryajith Chillara
  • CSTAR, International Institute of Information Technology, Hyderabad, India


The author is grateful to Nikhil Balaji, Mrinal Kumar, Noga Ron-Zewi, Nitin Saurabh, and Nithin Varma for helpful discussions. The author thanks Nikhil Balaji for telling him more about the Boolean complexity of Iterated Matrix Multiplication. The author thanks Ramprasad Saptharishi for patiently presenting the results in [Michael A. Forbes et al., 2016] while the author visited Tel Aviv University in 2016, hosted by Amir Shpilka.

Cite AsGet BibTex

Suryajith Chillara. Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 14:1-14:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^{O(1)}-variate and degree d polynomial P_{d} ∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to P_d, then C must have size 2^Ω(√dlog{d}). The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC⁰ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC⁰ can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form - sums of powers of polynomials) of 2^(log^O(1) n) size. Thus they argued that a 2^{ω(polylog n)} "functional" lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the "corresponding Boolean function" of Q against non-uniform ACC⁰. In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log²n) ≤ d ≤ n^{0.01}, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O(d/k²) that functionally computes Iterated Matrix Multiplication polynomial IMM_{n,d} (∈ VP) over {0,1}^{n²d} must have size n^Ω(k). Since Iterated Matrix Multiplication IMM_{n,d} over {0,1}^{n²d} is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC⁰ from GapL. For the sake of completeness, we also show a syntactic size lower bound against any Σ∧ΣΠ circuit computing IMM_{n,d} (for the same regime of d) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute IMM_{n,d}, for a slightly larger range of individual degree.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Algebraic complexity theory
  • Functional Lower Bounds
  • Boolean Circuit Lower Bounds
  • Depth Four
  • Connections to Boolean Complexity
  • Iterated Matrix Multiplication


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


  1. Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75. IEEE Computer Society, 2008. URL:
  2. Eric Allender and Vivek Gore. A uniform circuit lower bound for the permanent. SIAM J. Comput., 23(5):1026-1049, 1994. URL:
  3. Vikraman Arvind, Johannes Köbler, Uwe Schöning, and Rainer Schuler. If NP has polynomial-size circuits, then MA=AM. Theor. Comput. Sci., 137(2):279-282, 1995. URL:
  4. Walter Baur and Volker Strassen. The complexity of partial derivatives. Theor. Comput. Sci., 22:317-330, 1983. URL:
  5. Richard Beigel and Jun Tarui. On ACC. Comput. Complex., 4:350-366, 1994. URL:
  6. Peter Bürgisser. Cook’s versus valiant’s hypothesis. Theoretical Computer Science, 235(1):71 - 88, 2000. URL:, URL:
  7. Suryajith Chillara. New exponential size lower bounds against depth four circuits of bounded individual degree. Electronic Colloquium on Computational Complexity (ECCC), 27:33, 2020. URL:
  8. Suryajith Chillara. On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), volume 154 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1-47:16, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  9. Suryajith Chillara. Functional lower bounds for restricted arithmetic circuits of depth four. Electron. Colloquium Comput. Complex., page 105, 2021. URL:
  10. Suryajith Chillara, Christian Engels, Nutan Limaye, and Srikanth Srinivasan. A near-optimal depth-hierarchy theorem for small-depth multilinear circuits. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 934-945. IEEE Computer Society, 2018. URL:
  11. Suryajith Chillara, Nutan Limaye, and Srikanth Srinivasan. Small-depth multilinear formula lower bounds for iterated matrix multiplication with applications. SIAM J. Comput., 48(1):70-92, 2019. URL:
  12. Suryajith Chillara and Partha Mukhopadhyay. Depth-4 lower bounds, determinantal complexity: A unified approach. computational complexity, May 2019. URL:
  13. Ismor Fischer. Sums of like powers of multivariate linear forms. Mathematics Magazine, 67(1):59-61, 1994. URL:
  14. Michael A. Forbes, Mrinal Kumar, and Ramprasad Saptharishi. Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 33:1-33:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL:
  15. Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth-4 formulas computing iterated matrix multiplication. SIAM J. Comput., 44(5):1173-1201, 2015. URL:
  16. Dima Grigoriev and Alexander A. Razborov. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Eng. Commun. Comput., 10(6):465-487, 2000. URL:
  17. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. Journal of the ACM (JACM), 61(6):33, 2014. URL:
  18. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM Journal of Computing, 45(3):1064-1079, 2016. URL:
  19. Nikhil Gupta, Chandan Saha, and Bhargav Thankey. A super-quadratic lower bound for depth four arithmetic circuits. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 23:1-23:31. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  20. Richard M. Karp and Richard J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC '80, pages 302-309, New York, NY, USA, 1980. Association for Computing Machinery. URL:
  21. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 119-127. ACM, 2014. URL:
  22. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An exponential lower bound for homogeneous depth four arithmetic formulas. SIAM J. Comput., 46(1):307-335, 2017. URL:
  23. Neeraj Kayal and Chandan Saha. Multi-k-ic depth three circuit lower bound. Theory Comput. Syst., 61(4):1237-1251, 2017. URL:
  24. Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 146-153. ACM, 2014. URL:
  25. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. On the size of homogeneous and of depth-four formulas with low individual degree. Theory of Computing, 14(16):1-46, 2018. URL:
  26. Mrinal Kumar and Shubhangi Saraf. Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 751-762. Springer, 2014. URL:
  27. Mrinal Kumar and Shubhangi Saraf. The limits of depth reduction for arithmetic formulas: It’s all about the top fan-in. SIAM J. Comput., 44(6):1601-1625, 2015. URL:
  28. Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. SIAM J. Comput., 46(1):336-387, 2017. URL:
  29. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. Electron. Colloquium Comput. Complex., 28:81, 2021. URL:
  30. 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), 2020. URL:
  31. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL:
  32. Ran Raz. Multilinear-NC² ≠ multilinear-NC¹. In proceedings of Foundations of Computer Science (FOCS), pages 344-351, 2004. URL:
  33. Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(1):121-135, 2006. URL:
  34. Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory Comput., 6(1):135-177, 2010. URL:
  35. Ran Raz and Amir Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Computational Complexity, 17(4):515-535, 2008. URL:
  36. Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. URL:
  37. Herbert John Ryser. Combinatorial mathematics, volume 14. American Mathematical Soc., 1963. URL:
  38. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity version 8.0.4. Github survey, 2019. URL:
  39. Victor Shoup and Roman Smolensky. Lower bounds for polynomial evaluation and interpolation problems. Computational Complexity, 6(4):301-311, 1997. URL:
  40. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5(3-4):207-388, 2010. URL:
  41. Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL:
  42. Leslie G. Valiant. Why is Boolean Complexity Theory so Difficult?, pages 84-94. London Mathematical Society Lecture Note Series. Cambridge University Press, 1992. URL:
  43. V. Vinay. Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, Chicago, Illinois, USA, June 30 - July 3, 1991, pages 270-284. IEEE Computer Society, 1991. URL:
  44. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2:1-2:32, 2014. 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