Regular languages and partial commutations

Abstract

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 2 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

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“…This closure problem for ω-regular languages has been studied for partial commutation relations such as Mazurkiewicz traces [Diekert et al 1995;Muscholl 1996;Peled et al 1998]. Other types of closure problems are studied in Bouajjani et al [2001], Cécé et al [2008], and Cano et al [2011]. These works investigate classes C of regular languages (of finite words) such that for each language L ∈ C the closure [ L] is still in C. Cano et al [2011] also give conditions on the semicommutation (resp.…”

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