Smaller ACC0 Circuits for Symmetric Functions

What is the power of constant-depth circuits with $MODm$ 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: Razborov and Smolensky proved in the 1980s that MAJORITY and $MODm$ require super-polynomial-size $MODq$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MODm$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomial size and only $MOD6$ gates. We shed some light on the difficulty of proving lower bounds for $MODm$ circuits, by giving new upper bounds. We construct $MODm$ circuits computing symmetric functions with non-prime power $m$, with size-depth tradeoffs that beat the longstanding lower bounds for $AC^0[m]$ circuits for prime power $m$. 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 for every $\varepsilon > 0$ that every symmetric function can be computed with depth-3 $MODm$ circuits of $\exp(O(n^{{\varepsilon}))$} size, for a constant $m$ depending only on $\varepsilon > 0$. That is, depth-$3$ $CC^0$ circuits can compute any symmetric function in \emph{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 $2^{{\Omega(\sqrt{n})}$} size [H{\aa}stad 1986], and depth-$3$ $AC^0[pk]$ circuits (for fixed prime power $p^k$) require $2^{{\Omega(n{1/6})}$} size [Smolensky 1987]. Even for depth-two $MODp \circ MODm$ circuits, $2^{\Omega(n)}$ lower bounds were known [Barrington Straubing Th\'erien 1990].