Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e ${\bf AC}^{0} \circ {\bf MOD}_m$ circuits. We show that the following holds for several types of gates $\mathcal{G}$: by adding a gate of type $\mathcal{G}$ at the output, it is possible to obtain an equivalent probabilistic depth 2 circuit of quasipolynomial size consisting of a gate of type $\mathcal{G}$ at the output and a layer of modular counting gates, i.e $\mathcal{G} \circ {\bf MOD}_m$ circuits. The types of gates $\mathcal{G}$ we consider are modular counting gates and threshold-style gates. For all of these, strong lower bounds are known for (deterministic) $\mathcal{G} \circ {\bf MOD}_m$ circuits.

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