On languages of one-dimensional overlapping tiles
Résumé
In the perspective of defining a modeling language for computational music theory [8], this paper deals with languages of one-dimensional discrete overlapping tiles, i.e. triples of words of the form (u,v,w) where v is the root of tile and u and w define compatibility conditions for a (partial) product. This leads to define a monoid called the monoid of positive tiles. Completed with negative tiles, it is shown to be isomorphic with the McAlister (inverse) monoid [15, 13]. Several properties of this monoid are studied. We prove in particular that some canonical left and right residual operations are defined on positive tiles and that they encode within positive tiles product with negative tiles in McAlister monoid. Then we define particular classes of languages of tiles: languages recognizable by finite monoids, rational languages of tiles, and languages definable by monadic second order (MSO) formulae. In contrast with former studies of word languages recognizable by inverse monoids [14, 20], recognizable languages of tiles are shown to be even weaker : they are at most characterized by finite sets of bi- infinite periodic words. Rational languages of tiles are shown to be MSO-definable. For the most general class of MSO-definable languages, we provide a simple though indirect Myhill-Nerode like characterization. It induces a simple characterization of MSO definable languages of positive tiles by means of rational languages of words. This shows that the class of MSO definable languages of tiles is a class of languages as robust and as simple the class of rational languages of words.
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