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Rapport Année : 2012

On languages of one-dimensional overlapping tiles

David Janin

Résumé

A one-dimensional tile with overlaps is a standard finite word that carries some more information that is used to say when the concatenation of two tiles is legal. Known since the mid 70's in the rich mathematical field of inverse monoid theory, this model of tiles with the associated partial product have yet not been much studied in theoretical computer science despite some implicit appearances in studies of two-way automata in the 80's. We aim in this paper at initializing such a systematic computer science flavored theoretical study. For that purpose, after describing the richness of the underlying algebraic structure, we define and study several classical classes of languages of tiles: from recognizable languages definable by morphism into finite monoids up to languages definable in monadic second order logic (MSO). We show that recognizable languages of tiles are tightly linked with covers of periodic bi-infinite words. We also show that the class of MSO definable languages of tiles is both simple: these languages are finite sums of Cartesian products of rational languages, and robust: the class is closed under product, iterated product (star) and shifts (two tiles specific operators). An equivalent notion of regular expression is then provided.
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Dates et versions

hal-00659202 , version 1 (12-01-2012)
hal-00659202 , version 2 (14-01-2012)
hal-00659202 , version 3 (10-04-2012)
hal-00659202 , version 4 (11-07-2012)

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  • HAL Id : hal-00659202 , version 3

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David Janin. On languages of one-dimensional overlapping tiles. 2012. ⟨hal-00659202v3⟩
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