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)


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.


Inverse Semigroup Maximal Element Regular Language Closure Property Free Monoid 
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  1. 1.
    Berthaut, F., Janin, D., Martin, B.: Advanced synchronization of audio or symbolic musical patterns. Technical Report RR1461-12, LaBRI, Université de Bordeaux (2012)Google Scholar
  2. 2.
    Birget, J.-C.: Concatenation of inputs in a two-way automaton. Theoretical Computer Science 63(2), 141–156 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Diekert, V.: The Book of Traces. World Scientific Publishing Co., Inc., River Edge (1995)CrossRefGoogle Scholar
  4. 4.
    Gould, V.: Restriction and Ehresmann semigroups. In: Proceedings of the International Conference on Algebra 2010. World Scientific (2010)Google Scholar
  5. 5.
    Hollings, C.D.: From right PP monoids to restriction semigroups: a survey. European Journal of Pure and Applied Mathematics 2(1), 21–57 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Janin, D.: On languages of one-dimensional overlapping tiles. Technical Report RR-1457-12, LaBRI, Université de Bordeaux (2012)Google Scholar
  7. 7.
    Janin, D.: Quasi-inverse monoids. Technical Report RR-1459-12, LaBRI, Université de Bordeaux (2012)Google Scholar
  8. 8.
    Janin, D.: Vers une modélisation combinatoire des structures rythmiques simples de la musique. Revue Francophone d’Informatique Musicale 2 (to appear, 2012)Google Scholar
  9. 9.
    Lawson, M.V.: Semigroups and ordered categories. i. the reduced case. Journal of Algebra 141(2), 422–462 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lawson, M.V.: McAlister semigroups. Journal of Algebra 202(1), 276–294 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Margolis, S.W., Pin, J.-E.: Languages and Inverse Semigroups. In: Paredaens, J. (ed.) ICALP 1984. LNCS, vol. 172, pp. 337–346. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  12. 12.
    McAlister, D.B., Reilly, N.R.: E-unitary covers for inverse semigroups. Pacific Journal of Mathematics 68, 178–206 (1977)MathSciNetGoogle Scholar
  13. 13.
    Nambooripad, K.S.S.: The natural partial order on a regular semigroup. Proc. Edinburgh Math. Soc. 23, 249–260 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Pécuchet, J.-P.: Automates boustrophedon, semi-groupe de birget et monoide inversif libre. ITA 19(1), 71–100 (1985)zbMATHGoogle Scholar
  15. 15.
    Pin, J.-E.: Mathematical foundations of automata theory. Lecture Notes (2011)Google Scholar
  16. 16.
    Rhodes, J., Birget, J.-C.: Almost finite expansions of arbitrary semigroups. J. Pure and Appl. Algebra 32, 239–287 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Silva, P.V.: On free inverse monoid languages. ITA 30(4), 349–378 (1996)zbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

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