An Exact Characterization of Symmetric Functions in qAC0[2]

qAC0[2] is the class of languages computable by circuits of constant depth and quasi-polynomial \((2^{\log ^{o(1)_{ n} } } )\)size with unbounded fan-in AND, OR, and PARITY gates. Symmetric functions are those functions that are invariant under permutations of the input variables. Thus a symmetric function fn: 0, 1n #x2192; 0, 1 can also be seen as a function fn: 0, 1, ..., n} → 0, 1. We give the following characterization of symmetric functions in qAC0[2], according to how fn(x) changes as x grows from 0 to n. A symmetric function f = (fn) is in qAC0[2] if and only if fn has period 2t(n) = logO(1)n except within both ends of length logO(1)n.