Closure of regular languages under semi-commutations

The closure of a regular language under a [partial, semi-] 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, semi-] commutation still regular? (2) Are there any robust classes of languages closed under [partial, semi-] 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) and is decidable. 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. Finally, we prove a few results on semi-commutations. 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, partial commutation or semi-commutation has been extensively studied [35, 24, 1, 16, 17, 18], notably in connection with regular model checking [2, 3, 8, 9] or in the study of Mazurkiewicz traces, one of the models of parallelism [20, 21, 25, 36]. We refer the reader to the survey [15, 14] or to the recent articles of Ochmański [26, 27, 28] 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 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 authors acknowledge support from the AutoMathA programme of the European Science Foundation. The first author was supported by the project Técnicas de Inferencia Gramatical y aplicación al procesamiento de biosecuencias (TIN2007-60769) supported by the Spanish Ministery of Education and Sciences. The third author was supported by the project ANR 2010 BLAN 0202 02 FREC.