Smaller ACC0 Circuits for Symmetric Functions

Abstract

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 superpolynomial-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-two MOD p • MOD m circuits, 2 Ω(n) lower bounds were known [Barrington Straubing Thérien 1990].