Abstract
Let F = { f 1, f 2,…} be a family of symmetric Boolean functions, where f n has n Boolean variables, for each n ⩾ 1. Let μ F ( n) be the minimum number of variables of f n that each have to be set to constant values so that the resulting function is a constant function. We show that the growth rate of μ F ( n) completely determines whether or not the family F is ‘good’, that is, can be realized by a family of constant-depth, polynomial-size circuits (with unbounded fan-in). Furthermore, if μ F ( n) ⩽ (log n) k for some k, then the family F is good. However, if μ F ( n) ⩾ n ϵ for some ϵ > 0, then the family is not good.
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