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Quasi-recognizable vs MSO Definable Languages of One-Dimensional Overlapping Tiles

(Extended Abstract)
  • David Janin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

Abstract

It has been shown [6] that, within the McAlister inverse monoid [10], whose elements can be seen as overlapping one-dimensional tiles, the class of languages recognizable by finite monoids collapses compared with the class of languages definable in Monadic Second Order Logic (MSO).

This paper aims at capturing the expressive power of the MSO definability of languages of tiles by means of a weakening of the notion of algebraic recognizability which we shall refer to as quasi-recognizability. For that purpose, since the collapse of algebraic recognizability is intrinsically linked with the notion of monoid morphism itself, we propose instead to use premorphisms, monotonic mappings on ordered monoids that are only required to be sub-multiplicative with respect to the monoid product, i.e. mapping φ so that for all x and y, φ(xy) ≤ φ(x) φ(y).

In doing so, we indeed obtain, with additional but relatively natural closure conditions, the expected quasi-algebraic characterization of MSO definable languages of positive tiles. This result is achieved via the axiomatic definition of an original class of well-behaved ordered monoid so that quasi-recognizability implies MSO definability. An original embedding of any (finite) monoid S into a (finite) well-behaved ordered monoid \({\mathcal Q}(S)\) is then used to prove the converse.

Keywords

Inverse Semigroup Maximal Element Regular Language Closure Property Free Monoid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • David Janin
    • 1
  1. 1.LaBRI UMR 5800Université de BordeauxTalenceFrance

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