When Does Partial Commutative Closure Preserve Regularity?

The closure of a regular language under commutation or partial commutation has been extensively studied [1, 11, 12, 13], notably in connection with regular model checking [2, 3, 7] or in the study of Mazurkiewicz traces, one of the models of parallelism [14, 15, 16, 22]. We refer the reader to the survey [10, 9] or to the recent articles of Ochmański [17, 18, 19] for further references. In this paper, we present new advances on two problems of this area. The first problem is well-known and has a very precise statement. The second problem is more elusive, since it relies on the somewhat imprecise notion of robust class. By a robust class, we mean a class of regular languages closed under some of the usual operations on languages, such as Boolean operations, product, star, shuffle, morphisms, inverses of morphisms, residuals, etc. For instance, regular languages form a very robust class, commutative languages (languages whose syntactic monoid is commutative) also form a robust class. Finally, group languages (languages whose syntactic monoid is a finite group) form a semi-robust class: they are closed under Boolean operation, residuals and inverses of morphisms, but not under product, shuffle, morphisms or star. Here are the two problems: