Symmetric Groups and Quotient Complexity of Boolean Operations

Abstract

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L′ are binary regular languages with quotient complexities m and n, and that the subgroups of permutations in the transition semigroups of the minimal deterministic automata accepting L and L′ are the symmetric groups S _m and S _n of degrees m and n, respectively. Denote by ∘ any binary boolean operation that is not a constant and not a function of one argument only. For m,n ≥ 2 with $(m,n)\not \in \{(2,2),(3,4),(4,3),(4,4)\}$ we prove that the quotient complexity of L ∘ L′ is mn if and only either (a) $m\not= n$ or (b) m = n and the bases (ordered pairs of generators) of S _m and S _n are not conjugate. For (m,n) ∈ {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.

For a complete version of this work see http://arxiv.org/abs/1310.1841.