On languages of one-dimensional overlapping tiles

Abstract

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. In this paper, after giving an explicit description of McAlister monoid, we define and study several classical classes of languages of tiles: from recognizable languages (REC) definable by morphism into finite monoids up to languages definable in monadic second order logic (MSO). We 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), inverse and projection on context tiles. A equivalent notion of extended regular expression is deduced from these results. The much smaller class of REC recognizable languages of tiles is then studied. We describe few examples and we prove that these languages are tightly linked with covers of periodic bi-infinite words. © 2013 Springer-Verlag Berlin Heidelberg.