Depth-3 Circuits for Inner Product

Authors Mika Göös, Ziyi Guan, Tiberiu Mosnoi

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Author Details

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


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

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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)


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
  • Circuit complexity
  • inner product


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