Abstract
What is the power of constantdepth 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 superpolynomialsize 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 depth3 circuits of polynomialsize 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 nonprime power m, with sizedepth tradeoffs that beat the longstanding lower bounds for AC^0[m] circuits when m is a prime power. Furthermore, we observe that our sizedepth 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, depth3 CC^0 circuits can compute any symmetric function in subexponential size. This demonstrates a significant difference in the power of depth3 CC^0 circuits, compared to other models: for certain symmetric functions, depth3 AC^0 circuits require 2^{Ω(√n)} size [Håstad 1986], and depth3 AC^0[p^k] circuits (for fixed prime power p^k) require 2^{Ω(n^{1/6})} size [Smolensky 1987]. Even for depth2 MOD_p ∘ MOD_m circuits, 2^{Ω(n)} lower bounds were known [Barrington Straubing Thérien 1990].
BibTeX  Entry
@InProceedings{chapman_et_al:LIPIcs.ITCS.2022.38,
author = {Chapman, Brynmor and Williams, R. Ryan},
title = {{Smaller ACC0 Circuits for Symmetric Functions}},
booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
pages = {38:138:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772174},
ISSN = {18688969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15634},
URN = {urn:nbn:de:0030drops156342},
doi = {10.4230/LIPIcs.ITCS.2022.38},
annote = {Keywords: ACC, CC, circuit complexity, symmetric functions, Chinese Remainder Theorem}
}
Keywords: 

ACC, CC, circuit complexity, symmetric functions, Chinese Remainder Theorem 
Collection: 

13th Innovations in Theoretical Computer Science Conference (ITCS 2022) 
Issue Date: 

2022 
Date of publication: 

25.01.2022 