# Computing symmetric functions with *AND/OR* circuits and a single *MAJORITY* gate

## Abstract

Fagin et al. characterized those symmetric Boolean functions which can be computed by small *AND/OR* circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for *d-perceptrons* — *AND/OR* circuits of constant depth and unbounded fan-in with a single *MAJORITY* gate at the output. We show that a symmetric function has small (quasipolynomial, or \(2^{\log ^{O(1)n} }\) size) *d*-perceptrons *iff* it has only poly-log many *sign changes* (i.e., it changes value log^{O(1)}*n* times as the number of positive inputs varies from zero to *n*). A consequence of the lower bound is that a recent construction of Beigel is optimal. He showed how to convert a constant-depth unbounded fan-in *AND/OR* circuit with poly-log many *MAJORITY* gates into an equivalent *d*-perceptron — we show that more than poly-log *MAJORITY* gates cannot in general be converted to one.

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