Document Open Access Logo

Low-Depth Algebraic Circuit Lower Bounds over Any Field

Author Michael A. Forbes

PDF
Thumbnail PDF

File

LIPIcs.CCC.2024.31.pdf
  • Filesize: 0.74 MB
  • 16 pages

Document Identifiers

Author Details

Michael A. Forbes
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Forbes, Michael A.: Supported by NSF CAREER award 2047310.

Cite AsGet BibTex

Michael A. Forbes. Low-Depth Algebraic Circuit Lower Bounds over Any Field. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.31

Abstract

The recent breakthrough of Limaye, Srinivasan and Tavenas [Limaye et al., 2022] (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this result to small-characteristic ([Limaye et al., 2022; Govindasamy et al., 2022; Fournier et al., 2023]), which in particular is relevant for an approach to prove superpolynomial AC⁰[p]-Frege lower bounds ([Govindasamy et al., 2022]). In this work, we prove super-polynomial algebraic circuit lower bounds against low-depth algebraic circuits over any field, with the same parameters as LST (or even matching the improved parameters of Bhargav, Dutta, and Saxena [Bhargav et al., 2022]). We give two proofs. The first is logical, showing that even though the proof of LST naively fails in small characteristic, the proof is sufficiently algebraic that generic transfer results imply the result over characteristic zero implies the result over all fields. Motivated by this indirect proof, we then proceed to give a second constructive proof, replacing the field-dependent set-multilinearization result of LST with a set-multilinearization that works over any field, by using the Binet-Minc identity [Minc, 1979].

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • algebraic circuits
  • lower bounds
  • low-depth circuits
  • positive characteristic

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, and Bhargav Thankey. Low-depth arithmetic circuit lower bounds: Bypassing set-multilinearization. In Proceedings of the 50th International Colloquium on Automata, Languages and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:20, 2023. Full version in the http://eccc.hpi-web.de/report/2022/151/. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.12.
  2. Robert Andrews. Algebraic hardness versus randomness in low characteristic. In Proceedings of the 35th Annual Structure in Complexity Theory Conference (Structures 2020), volume 169 of Leibniz International Proceedings in Informatics (LIPIcs), pages 37:1-37:32, 2020. Full version at http://arxiv.org/abs/2005.10885. URL: https://doi.org/10.4230/LIPICS.CCC.2020.37.
  3. C. S. Bhargav, Sagnik Dutta, and Nitin Saxena. Improved lower bound, and proof barrier, for constant depth algebraic circuits. In Proceedings of the 47th Internationl Symposium on the Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:16, 2022. URL: https://doi.org/10.4230/LIPICS.MFCS.2022.18.
  4. Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan. On the VNP-hardness of some monomial symmetric polynomials. In 42nd International Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022), volume 250 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:14, 2022. Full version in the http://eccc.hpi-web.de/report/2022/139/. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2022.16.
  5. Ismor Fischer. Sums of like powers of multivariate linear forms. Mathematics Magazine, 67(1):59-61, 1994. URL: http://www.jstor.org/stable/2690560.
  6. Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511801655.
  7. Michael A. Forbes, Amir Shpilka, and Ben Lee Volk. Succinct hitting sets and barriers to proving algebraic circuits lower bounds. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017), pages 653-664, 2017. Full version at http://arxiv.org/abs/1701.05328. URL: https://doi.org/10.1145/3055399.3055496.
  8. Hervé Fournier, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the power of homogeneous algebraic formulas. Electronic Colloquium on Computational Complexity (ECCC), TR23-191, 2023. URL: https://eccc.weizmann.ac.il/report/2023/191.
  9. Nashlen Govindasamy, Tuomas Hakoniemi, and Iddo Tzameret. Simple hard instances for low-depth algebraic proofs. In Preliminary version in the 63rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2022), pages 188-199, 2022. Full version at http://arxiv.org/abs/2205.07175. URL: https://doi.org/10.1109/FOCS54457.2022.00025.
  10. Joshua A. Grochow, Mrinal Kumar, Michael E. Saks, and Shubhangi Saraf. Towards an algebraic natural proofs barrier via polynomial identity testing. arXiv, 1701.01717, 2017. URL: http://arxiv.org/abs/1701.01717.
  11. Joshua A. Grochow and Toniann Pitassi. Circuit complexity, proof complexity, and polynomial identity testing. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), pages 110-119, 2014. Full version at http://arxiv.org/abs/1404.3820. URL: https://doi.org/10.1109/FOCS.2014.20.
  12. Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In Proceedings of Innovations in Theoretical Computer Science (ITCS 2013), pages 529-540, 2013. Full version at http://arxiv.org/abs/1208.5413. URL: https://doi.org/10.1145/2422436.2422494.
  13. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth three. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 578-587, 2013. Full version in the http://eccc.hpi-web.de/report/2013/026/. URL: https://doi.org/10.1109/FOCS.2013.68.
  14. Erich L. Kaltofen. Factorization of polynomials given by straight-line programs. In Silvio Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 375-412. JAI Press, Inc., Greenwich, CT, USA, 1989. URL: http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf.
  15. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In Preliminary version in the 62nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2021), pages 804-814, 2022. Full version in the http://eccc.hpi-web.de/report/2021/081/. URL: https://doi.org/10.1109/FOCS52979.2021.00083.
  16. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. bGuest column: Lower bounds against constant-depth algebraic circuits. SIGACT News, 53(2):40-62, 2022. URL: https://doi.org/10.1145/3544979.3544989.
  17. Henryk Minc. Evaluation of permanents. Proc. Edinburgh Math. Soc. (2), 22(1):27-32, 1979. URL: https://doi.org/10.1017/S0013091500027760.
  18. Ran Raz. Elusive functions and lower bounds for arithmetic circuits. Theory of Computing, 6(1):135-177, 2010. 40th Annual ACM Symposium on Theory of Computing (STOC 2008). URL: https://doi.org/10.4086/TOC.2010.V006A007.
  19. Alexander A. Razborov. aLower bounds on the size of bounded depth circuits over a complete basis with logical addition. Matematicheskie Zametki, 41(4):598-607, April 1987. Google Scholar
  20. Amir Shpilka and Avi Wigderson. Depth-3 arithmetic circuits over fields of characteristic zero. Computational Complexity, 10(1):1-27, 2001. Preliminary version in the 14th Annual IEEE Conference on Computational Complexity (CCC 1999). URL: https://doi.org/10.1007/PL00001609.
  21. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC 1987), pages 77-82, 1987. URL: https://doi.org/10.1145/28395.28404.
  22. Ryan Williams. Finding paths of length k in O^*(2^k) time. Inf. Process. Lett., 109(6):315-318, 2009. URL: https://doi.org/10.1016/J.IPL.2008.11.004.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact