Boolean functions, invariance groups and parallel complexity

The authors study the invariance groups S(f) of Boolean functions f∈B_n on n variables. They give necessary and sufficient conditions for a general permutation group to be of the form S(f), for some f ∈B_n. This leads to an almost optimal algorithm for deciding the representability of an arbitrary permutation group. For cyclic groups G⩽ S_n, an NC-algorithm is given for determining whether the given group is of the form S(f), for some f ∈ B_n . Further, it is proved that asymptotically almost all Boolean functions have trivial invariance groups. Finally, for any formal language L, where L_n is the characteristic function of the set of all strings in L which have length exactly n and S_n(L) is the invariance group of L_n, the classification results on maximal permutation groups are used to show that any language satisfying |S_n:S_n(L)|=n⁰⁽¹⁾ is in NC¹