Regular languages and partial commutations

The closure of a regular language under a [partial] commutation I has been extensively studied. We present new advances on two problems of this area: (1) When is the closure of a regular language under [partial] commutation still regular? (2) Are there any robust classes of languages closed under [partial] commutation? We show that the class Pol(G) of polynomials of group languages is closed under commutation, and under partial commutation when the complement of I in A is a transitive relation. We also give a sufficient graph theoretic condition on I to ensure that the closure of a language of Pol(G) under I-commutation is regular. We exhibit a very robust class of languages W which is closed under commutation. This class contains Pol(G), is decidable and can be defined as the largest positive variety of languages not containing (ab)∗. It is also closed under intersection, union, shuffle, concatenation, quotients, lengthdecreasing morphisms and inverses of morphisms. If I is transitive, we show that the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, finiteness conditions on semigroups and properties of insertion systems. The closure of a regular language under commutation or partial commutation has been extensively studied [37, 25, 1, 17, 18, 19], notably in connection with regular model checking [2, 3, 9, 10] or in the study of Mazurkiewicz traces, one of the models of parallelism [20, 21, 26, 27, 28, 29, 38]. We refer the reader to the book [16] and to the survey [15] 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, quotients, 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 Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, Camino de Vera s/n, P.O. Box: 22012, E-46020 Valencia. LITIS EA 4108, Université de Rouen, BP12, 76801 Saint Etienne du Rouvray, France. LIAFA, Université Paris-Diderot and CNRS, Case 7014, 75205 Paris Cedex 13, France. The first author was supported by the project Autómatas en dispositivos móviles: interfaces de usuario y realidad aumentada (PAID 2019-06-11) supported by Universidad Politécnica de Valencia. The third author was supported by the project ANR 2010 BLAN 0202 02 FREC.