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Rapport Année : 2012

Quasi-recognizable vs MSO definable languages of one-dimensional overlapping tiles

David Janin

Résumé

It has been shown [6] that, within McAlister inverse monoid, whose element can be seen as overlapping one-dimensional tiles, the class of languages recognizable by finite monoids is fairly smaller than the class of languages definable in Monadic Second Order Logic (MSO). In this paper, we aim at capturing the expressive power of MSO definability of languages of tiles by means of a weakening of the notion of algebraic recognizability we call quasi-recognizability. For that purpose, since the collapse of algebraic recognizability is intrinsically linked with the notion of monoid homomorphisms itself, we consider instead prehomomorphisms. Introduced by McAlister and Reilly [10] these are monotonic mappings over ordered monoids that are only required to be sub-multiplicative with respect to the monoid product, i.e. mapping φ such that for all x and y, φ(xy) ≤ φ(x)φ(y). Doing so, we indeed obtain, with some extra but rather natural closure conditions, the expected quasi-algebraic characterization of MSO definable languages of positive tiles. This result is achieved via a non trivial embedding of any (finite) monoid S into a (sufficiently well-behaved) ordered monoid Q(S).
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Dates et versions

hal-00671917 , version 1 (19-02-2012)
hal-00671917 , version 2 (21-02-2012)
hal-00671917 , version 3 (23-04-2012)

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  • HAL Id : hal-00671917 , version 2

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David Janin. Quasi-recognizable vs MSO definable languages of one-dimensional overlapping tiles. 2012. ⟨hal-00671917v2⟩
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