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 logO(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.