Automata Classes Accepting Languages Whose Commutative Closure is Regular

Abstract

The commutative closure operation, which corresponds to the Parikh image, is a natural operation on formal languages occurring in verification and model-checking. Commutative closures of regular languages correspond to semilinear sets and, by Parikh’s theorem, to the commutative closures of context-free languages. The commutative closure is not regularity-preserving on the class of regular languages, for example already the commutative closure of the simple language (ab)∗ is not regular. Here, we show that the commutative closure of a binary regular language accepted by a circular automaton yields a regular language. Then, we deduce a sufficient condition on the cycles in automata for regularity of the commutative closure. This yields this property, for example, for the following classes of automata: automata with threshold one transformation semigroups, automata with simple idempotents and almost-group automata. The fact that the commutative closure on group languages and polynomials of group languages is regularity-preserving is known in the literature. Polynomials of group languages correspond to level one-half of the group hierarchy. We also show that on the next level in this hierarchy, i.e., level one, this property is lost and the commutative closure is no longer regularity-preserving. Lastly, we give a binary circular automaton not contained in the largest proper positive variety W closed under shuffle and commutative closure.