2026-07-22T12:29:06Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15634 2024-03-06T10:55:49Z ddc:004 open_access

Smaller ACC0 Circuits for Symmetric Functions Chapman, Brynmor Williams, R. Ryan ACC CC circuit complexity symmetric functions Chinese Remainder Theorem 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Brynmor Chapman and R. Ryan Williams 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.38 urn:nbn:de:0030-drops-156342 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.38 eng https://creativecommons.org/licenses/by/4.0/legalcode