An exact characterization of symmetric functions in qAC0[2]

Abstract

qAC0[2] is the class of languages computable by circuits of constant depth and quasi-polynomial (2logO(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, 1}n → {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.