Star-Free Open Languages and Aperiodic Loops

  • Martin Beaudry
  • François Lemieux
  • Denis Thérien3
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2010)

Abstract

It is known that recognition of regular languages by finite monoids can be generalized to context-free languages and finite groupoids, which are finite sets closed under a binary operation. A loop is a groupoid with a neutral element and in which each element has a left and a right inverse. It has been shown that finite loops recognize exactly those regular languages that are open in the group topology. In this paper, we study the class of aperiodic loops, which are those loops that contain no nontrivial group. We show that this class is stable under various definitions, and we prove some closure properties. We also prove that aperiodic loops recognize only star-free open languages and give some examples. Finally, we show that the wreath product principle can be applied to groupoids, and we use it to prove a decomposition theorem for recognizers of regular open languages.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Beaudry
    • 1
  • François Lemieux
    • 2
  • Denis Thérien3
    • 3
  1. 1.Département de mathématiques et d’informatiqueUniversité de SherbrookeSherbrooke (Qc)Canada
  2. 2.Département d’informatique et de mathématiqueUniversité du Québec à ChicoutimiChicoutimi (Qc)Canada
  3. 3.School of Computer ScienceMcGill UniversityMontréal (Qc)Canada

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