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A Kleene theorem for infinite trace languages

  • Formal Languages (Session 6)
  • Conference paper
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Automata, Languages and Programming (ICALP 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 510))

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Abstract

Kleene's theorem is considered as one of the cornerstones of theoretical computer science. It ensures that, for languages of finite words, the family of recognizable languages is equal to the family of rational languages. It has been generalized in various ways, for instance, to formal power series by Schützenberger, to infinite words by Büchi and to finite traces by Ochmanski. Finite traces have been introduced by Mazurkiewicz in order to modelize the behaviours of distributed systems. The family of recognizable trace languages is not closed by Kleene's star but by a concurrent version of this iteration. This leads to the natural definition of co-rational languages obtained as the rational one by simply replacing the Kleene's iteration by the concurrent iteration. Cori, Perrin and Métivier proved, in substance, that any co-rational trace language is recognizable. Independently,Ochmanski generalized Kleene's theorem showing that the recognizable trace languages are exactly the co-rational languages. Besides, infinite traces have been recently introduced as a natural extension of both finite traces and infinite words. In this paper we generalize Kleene's theorem to languages of infinite traces proving that the recognizable languages of finite or infinite traces are exactly the co-rational languages.

This work has been partly supported by the ESPRIT Basic Research Actions No 3166 (ASMICS) and No 3148 (DEMON) and by the PRC Math-Info.

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Author information

Authors and Affiliations

  1. LITP, Institut Blaise Pascal, Université PARIS 6, 4, place Jussieu, 75 252, Paris Cedex 05, France

    Paul Gastin

  2. LRI, URA CNRS 410, Université PARIS SUD, Bât. 490, 91 405, Orsay Cedex, France

    Antoine Petit

  3. LaBRI, Université BORDEAUX 1, 351, Cours de la Libération, 33 405, Talence Cedex, France

    Wieslaw Zielonka

Authors
  1. Paul Gastin
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  2. Antoine Petit
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  3. Wieslaw Zielonka
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Editor information

Javier Leach Albert Burkhard Monien Mario Rodríguez Artalejo

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© 1991 Springer-Verlag Berlin Heidelberg

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Gastin, P., Petit, A., Zielonka, W. (1991). A Kleene theorem for infinite trace languages. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds) Automata, Languages and Programming. ICALP 1991. Lecture Notes in Computer Science, vol 510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54233-7_139

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  • DOI: https://doi.org/10.1007/3-540-54233-7_139

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