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Smaller ACC0 Circuits for Symmetric Functions

Authors Brynmor Chapman, R. Ryan Williams

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  • 19 pages

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Author Details

Brynmor Chapman
  • Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA
R. Ryan Williams
  • Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA


We thank Arkadev Chattopadhyay and Kristoffer Arnsfelt Hansen for useful pointers and discussion.

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Brynmor Chapman and R. Ryan Williams. Smaller ACC0 Circuits for Symmetric Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 38:1-38:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


What is the power of constant-depth circuits with MOD_m gates, that can count modulo m? Can they efficiently compute MAJORITY and other symmetric functions? When m is a constant prime power, the answer is well understood. In this regime, Razborov and Smolensky proved in the 1980s that MAJORITY and MOD_m require super-polynomial-size MOD_q circuits, where q is any prime power not dividing m. However, relatively little is known about the power of MOD_m gates when m is not a prime power. For example, it is still open whether every problem decidable in exponential time can be computed by depth-3 circuits of polynomial-size and only MOD_6 gates. In this paper, we shed some light on the difficulty of proving lower bounds for MOD_m circuits, by giving new upper bounds. We show how to construct MOD_m circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC^0[m] circuits when m is a prime power. Furthermore, we observe that our size-depth tradeoff circuits have essentially optimal dependence on m and d in the exponent, under a natural circuit complexity hypothesis. For example, we show that for every ε > 0, every symmetric function can be computed using MOD_m circuits of depth 3 and 2^{n^ε} size, for a constant m depending only on ε > 0. In other words, depth-3 CC^0 circuits can compute any symmetric function in subexponential size. This demonstrates a significant difference in the power of depth-3 CC^0 circuits, compared to other models: for certain symmetric functions, depth-3 AC^0 circuits require 2^{Ω(√n)} size [Håstad 1986], and depth-3 AC^0[p^k] circuits (for fixed prime power p^k) require 2^{Ω(n^{1/6})} size [Smolensky 1987]. Even for depth-2 MOD_p ∘ MOD_m circuits, 2^{Ω(n)} lower bounds were known [Barrington Straubing Thérien 1990].

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • ACC
  • CC
  • circuit complexity
  • symmetric functions
  • Chinese Remainder Theorem


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