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 : {0,1} n→{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) n∈ 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.
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