State Complexity Bounds for the Commutative Closure of Group Languages

In this work we construct an automaton for the commutative closure of a given regular group language. The number of states of the resulting automaton is bounded by the number of states of the original automaton, raised to the power of the alphabet size, times the product of the order of the letters, viewed as permutations of the state set. This gives the asymptotic state bound $O((n\exp(\sqrt{n\ln n}))^{|\Sigma|})$, if the original regular language is accepted by an automaton with $n$ states. Depending on the automaton in question, we label points of $\mathbb N_0^{|\Sigma|}$ by subsets of states and introduce unary automata which decompose the thus labelled grid. Based on these constructions, we give a general regularity condition, which is fulfilled for group languages.