Document Open Access Logo

Depth-3 Circuits for Inner Product

Authors Mika Göös, Ziyi Guan, Tiberiu Mosnoi

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2023.51.pdf
  • Filesize: 0.76 MB
  • 12 pages

Document Identifiers

Author Details

Mika Göös
  • EPFL, Lausanne, Switzerland
Ziyi Guan
  • EPFL, Lausanne, Switzerland
Tiberiu Mosnoi
  • EPFL, Lausanne, Switzerland

Funding

This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

We thank the anonymous reviewers for a careful reading of the paper and comments that helped us improve the presentation.

Cite As Get BibTex

Mika Göös, Ziyi Guan, and Tiberiu Mosnoi. Depth-3 Circuits for Inner Product. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.51

Abstract

What is the Σ₃²-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2^{α_n n} for some α_n = Ω(1). We show that the limiting constant α := lim sup α_n satisfies 0.847... ≤ α ≤ 0.965... . Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that α ∈ [0.5,1]. To obtain our improved bounds, we analyse a covering LP that captures the Σ₃²-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Circuit complexity
  • inner product

Metrics

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

References

  1. Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. A Variant of the VC-Dimension with Applications to Depth-3 Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215, pages 72:1-72:19, Dagstuhl, 2022. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.72.
  2. Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit Depth Reductions. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185, pages 24:1-24:20, Dagstuhl, 2021. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.24.
  3. J. Håstad, S. Jukna, and P. Pudlák. Top-down lower bounds for depth-three circuits. Computational Complexity, 5(2):99-112, June 1995. URL: https://doi.org/10.1007/bf01268140.
  4. Suichi Hirahara. A duality between depth-three formulas and approximation by depth-two. Technical report, arXiv, 2017. URL: https://arxiv.org/abs/1705.03588.
  5. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, December 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  6. Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 247-258. IEEE, 2010. URL: https://doi.org/10.1109/CCC.2010.31.
  7. Stasys Jukna. Boolean Function Complexity: Advances and Frontiers, volume 27 of Algorithms and Combinatorics. Springer, 2012. Google Scholar
  8. Mauricio Karchmer, Eyal Kushilevitz, and Noam Nisan. Fractional covers and communication complexity. SIAM Journal on Discrete Mathematics, 8(1):76-92, February 1995. URL: https://doi.org/10.1137/s0895480192238482.
  9. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  10. Troy Lee and Adi Shraibman. Lower Bounds in Communication Complexity, volume 3. Now Publishers, 2007. URL: https://doi.org/10.1561/0400000040.
  11. L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13(4):383-390, 1975. URL: https://doi.org/10.1016/0012-365X(75)90058-8.
  12. R. Paturi, M. E. Saks, and F. Zane. Exponential lower bounds for depth three boolean circuits. computational complexity, 9(1):1-15, 2000. URL: https://doi.org/10.1007/PL00001598.
  13. Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. Journal of the ACM, 52(3):337-364, May 2005. URL: https://doi.org/10.1145/1066100.1066101.
  14. Ramamohan Paturi, Pavel Pudlak, and Francis Zane. Satisfiability coding lemma. Chicago Journal of Theoretical Computer Science, 5(1):1-19, 1999. URL: https://doi.org/10.4086/cjtcs.1999.011.
  15. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Jozef Gruska, editor, Mathematical Foundations of Computer Science 1977, pages 162-176, Berlin, Heidelberg, 1977. Springer Berlin Heidelberg. Google Scholar
  16. Emanuele Viola. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science, 5(1):1-72, 2009. URL: https://doi.org/10.1561/0400000033.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact