License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2023.51
URN: urn:nbn:de:0030-drops-185856
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Göös, Mika ; Guan, Ziyi ; Mosnoi, Tiberiu

Depth-3 Circuits for Inner Product

LIPIcs-MFCS-2023-51.pdf (0.8 MB)


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.

BibTeX - Entry

  author =	{G\"{o}\"{o}s, Mika and Guan, Ziyi and Mosnoi, Tiberiu},
  title =	{{Depth-3 Circuits for Inner Product}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{51:1--51:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-185856},
  doi =		{10.4230/LIPIcs.MFCS.2023.51},
  annote =	{Keywords: Circuit complexity, inner product}

Keywords: Circuit complexity, inner product
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023

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