Computing the closure of sets of words under partial commutations

  • Yves Métivier
  • GwénaËl Richomme
  • Pierre-André Wacrenier
Automata and Formal Languages I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 944)


The aim of this paper is the study of a procedure S given in [11, 13]. We prove that this procedure can compute the closure of the star of a closed recognizable set of words if and only if this closure is also recognizable. This necessary and sufficient condition gives a semi algorithm for the Star Problem. As intermediary results, using S, we give new proofs of some known results.

In the last part, we compare the power of S with the rank notion introduced by Hashigushi [9]. Finally, we characterize the recognizability of the closure of star of recognizable closed sets of words using this rank notion.


commutation recognizability rank Star Problem trace monoids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.V. Anisimov et D.E. Knuth, Inhomogeneous Sorting, Int. J. of Computer and Information Sciences, Vol 8, No 4, 1979.Google Scholar
  2. 2.
    P. Cartier et D. Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Math. 85, 1969.Google Scholar
  3. 3.
    R. Cori and D. Perrin, Automates et commutations partielles, RAIRO Inform. Théor. 19, p 21–32, 1985.Google Scholar
  4. 4.
    S. Eilenberg, Automata, languages and machines, Academic Press, New York, 1974.Google Scholar
  5. 5.
    M. Fliess, Matrices de Hankel, J. Math Pures et Appl. 53, p197–224, 1974.Google Scholar
  6. 6.
    P. Gastin, E. Ochmański, A. Petit et B. Rozoy, Decidability of the star problem in A * ×{b} *, Inform. Process. Lett. 44, p65–71, 1992.Google Scholar
  7. 7.
    S. Ginsburg et E. Spanier, Semigroups, Presburger formulas and languages, Pacific journal of mathematics 16, p285–296, 1966.Google Scholar
  8. 8.
    S. Ginsburg et E. Spanier, Bounded regular sets, Proceedings of the AMS, vol. 17(5), p1043–1049, 1966.Google Scholar
  9. 9.
    K. Hashigushi, Recognizable closures and submonoids of free partially commutative monoids, Theoret. Comput. Sci. 86, p233–241, 1991.Google Scholar
  10. 10.
    A. Mazurkiewicz, Concurrent program schemes and their interpretations, Aarhus university, DAIMI rep. PB 78, 1977.Google Scholar
  11. 11.
    Y. Métivier, Contribution à l'étude des monoÏdes de commutations, Thèse d'état, université Bordeaux I, 1987.Google Scholar
  12. 12.
    Y. Métivier, On recognisable subsets of free partially Commutative Monoids, Theoret. Comput. Sci. 58, p201–208, 1988.Google Scholar
  13. 13.
    Y. Métivier et B. Rozoy, On the star operation in free partially commutative monoids, International Journal of Foundations of Computer Science 2, p257–265, 1991.Google Scholar
  14. 14.
    E. Ochmański, Regular behaviour of concurrent systems, Bulletin of EATCS 27, p56–67, 1985.Google Scholar
  15. 15.
    E. Ochmański, P.-A. Wacrenier, On Regular Compatibility of Semi-Commutations, Proceedings of ICALP'93, LNCS 700, p445–456, 1993.Google Scholar
  16. 16.
    G. Richomme, Some trace monoids where both the Star Problem and the Finite Power Property Problem are decidable, Proc. of MFCS'94, LNCS 841, p577–586, 1994.Google Scholar
  17. 17.
    G. Richomme, Equivalence decidability of the Star Problem and the Finite Power Property Problem in trace monoids, LaBRI internal report 835.94, 1994.Google Scholar
  18. 18.
    J. Sakarovitch, The “last” decision problem for rational trace languages, Proceedings of LATIN'92, LNCS 583, p460–473, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Yves Métivier
    • 1
    • 2
  • GwénaËl Richomme
    • 1
    • 2
  • Pierre-André Wacrenier
    • 1
    • 2
  1. 1.LaBRIUniversité Bordeaux ITalenceFrance
  2. 2.Faculté de Mathématiques et d'InformatiqueLAMIFAAmiensFrance

Personalised recommendations