An Exact Characterization of Symmetric Functions in _qAC⁰[2]

Abstract

_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: 0, 1ⁿ #x2192; 0, 1 can also be seen as a function f _n: 0, 1, ..., n} → 0, 1. 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) 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.