Regularity Conditions for Iterated Shuffle on Commutative Regular Languages

Abstract

We introduce the class [Formula: see text] of commutative regular languages that is a positive variety closed under binary shuffle and iterated shuffle (also called shuffle closure). This class arises out of the known positive variety [Formula: see text] by superalphabet closure, an operation on positive varieties we introduce and describe in the present work. We state alternative characterizations for both classes, that the shuffle of any language (resp. any commutative language) with a language from [Formula: see text] gives a regular language [Formula: see text]resp. a language from [Formula: see text] and that [Formula: see text] is also closed for iterated shuffle. Then we introduce the wider class [Formula: see text] that is also closed under iterated shuffle, but fails to be closed for binary shuffle and is not a positive variety. Furthermore, we give an automata-theoretical characterization for the regularity of the iterated shuffle of a regular commutative language. We use this result to show that, for a fixed alphabet, it is decidable in polynomial time whether the iterated shuffle of a commutative regular language given by a deterministic automaton is regular. Lastly, we state some normal form results for the aperiodic, or star-free, commutative languages and the commutative group languages.