An Exact Characterization of Symmetric Functions in qAC0[2]

qAC 0 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 f n : f0; 1g n ! f0; 1g can also be seen as a function f n : f0; 1; ; ng ! f0; 1g. We give the following characterization of symmetric functions in qAC 0 2], according to how f n (x) changes as x grows from 0 to n. A symmetric function f = (f n) n2N is in qAC 0 2] if and only if f n has period 2 t(n) = log O(1) n except within both ends of length log O(1) n.