TR19-073 Authors: Igor Carboni Oliveira, Rahul Santhanam, Srikanth Srinivasan

Publication: 18th May 2019 00:48

Downloads: 1986

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We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC$^0[\oplus]$ circuits. Razborov (1987) and Smolensky (1987, 1993) showed that Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/2(d-1)})}$. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\widetilde{O}(n^{1/(d-1)})}$. This gap between upper and lower bounds has stood for nearly three decades.

Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large $d$. We show for $d \geq 5$ that any symmetric function can be computed with depth-$d$ AC$^0[\oplus]$ circuits of size $\exp({\widetilde{O} (n^{\frac{2}{3} \cdot \frac{1}{(d - 4)}} )})$. Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for $d \geq 3$ Majority requires depth-$d$ AC$^0[\oplus]$ circuits of size $2^{\Omega(n^{1/(2d-4)})}$. For depths $d \leq 4$, we are able to refine our techniques to get almost-optimal bounds: the depth-$3$ AC$^0[\oplus]$ circuit size of Majority is $2^{\widetilde{\Theta}(n^{1/2})}$, while its depth-$4$ AC$^0[\oplus]$ circuit size is $2^{\widetilde{\Theta}(n^{1/4})}$.