A Kleene Theorem for Infinite Trace Languages

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 Schutzenberger, to infinite words by Buchi 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 Metivier 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.

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