# An Exact Characterization of Symmetric Functions in _{q}*AC*^{0}[2]

## Abstract

_{q}*AC*^{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, 1^{n} #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 _{q}*AC*^{0}[2], according to how *f*_{n}(*x*) changes as *x* grows from 0 to *n*. A symmetric function *f* = (*f*_{n}) is in _{q}*AC*^{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*.

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