An Eilenberg theorem for words on countable ordinals

  • Nicolas Bedon
  • Olivier Carton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)


We present in this paper an algebraic approach to the theory of languages of words on countable ordinals. The algebraic structure used, called an Ω1-semigroup, is an adaptation of the one used in the theory of regular languages of Ω-words. We show that finite Ω1-semigroups are equivalent to automata. In particular, the proof gives a new algorithm for determinizing automata on countable ordinals. As in the cases of finite and Ω-words, a syntactic Ω1-semigroup can effectively be associated with any regular language of words on countable ordinals. This result is used to prove an Eilenberg type theorem. There is a one-to-one correspondence between varieties of Ω1-languages and pseudo-varieties of Ω1-semigroups.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Almeida. Finite semigroups and universal algebra, volume 3 of Series in algebra. World Scientific, 1994.Google Scholar
  2. 2.
    A. Arnold. A syntactic congruence for rational Ω-languages. Theoretical Computer Science, 39:333–335, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Bedon. Automata, semigroups and recognizability of words on ordinals. IGM report 96-5, to appear in International Journal of Algebra and Computation.Google Scholar
  4. 4.
    N. Bedon. Star-free sets of words on ordinals. IGM report 97-8, submitted to Information and Computation.Google Scholar
  5. 5.
    J. R. Büchi. On a decision method in the restricted second-order arithmetic. In Proc. Int. Congress Logic, Methodology and Philosophy of science, Berkeley 1960, pages 1–11. Stanford University Press, 1962.Google Scholar
  6. 6.
    J. R. Büchi. Transfinite automata recursions and weak second order theory of ordinals. In Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Jerusalem 1964, pages 2–23. North-Holland, 1965.Google Scholar
  7. 7.
    Y. Choueka. Finite automata, definable sets, and regular expressions over Ωn-tapes. J. Comp. Syst. Sci., 17:81–97, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Eilenberg. Automata, languages and machines, volume B. Academic Press, 1976.Google Scholar
  9. 9.
    J.-P. Pécuchet. Etude syntaxique des parties reconnaissables de mots infinis. Lecture Notes in Computer Science, 226:294–303, 1986.zbMATHGoogle Scholar
  10. 10.
    J.-P. Pécuchet. Variétés de semigroupes et mots infinis. Lecture Notes in Computer Science, 210:180–191, 1986.zbMATHGoogle Scholar
  11. 11.
    D. Perrin. Recent results on automata and infinite words. In M. P. Chytil and V. Koubek, editors, Mathematical foundations of computer science, volume 176 of Lecture Notes in Computer Science, pages 134–148, Berlin, 1984. Springer.Google Scholar
  12. 12.
    D. Perrin and J.-E. Pin. Semigroups and automata on infinite words. In J. Fountain and V. A. R. Gould, editors, NATO Advanced Study Institute Semigroups, Formal Languages and Groups, pages 49–72. Kluwer academic publishers, 1995.Google Scholar
  13. 13.
    J.-E. Pin. Handbook of formal languages, volume 1, chapter Syntactic semigroups, pages 679–746. Springer, 1997.Google Scholar
  14. 14.
    S. Rohde. Alternating automata and the temporal logic of ordinals. PhD thesis, University of Illinois, Urbana-Champaign, 1997.Google Scholar
  15. 15.
    J. G. Rosenstein. Linear ordering. Academic Press, New York, 1982.Google Scholar
  16. 16.
    M. P. Schützenberger. On finite monoids having only trivial subgroups. Information and Control, 8:190–194, 1965.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D. Thérien and T. Wilke. Temporal logic and semidirect products: An effective characterization of the until hierarchy. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, 1996. To appear.Google Scholar
  18. 18.
    T. Wilke. An Eilenberg theorem for ∞-languages. In Automata, Languages and Programming: Proc. of 18th ICALP Conference, pages 588–599. Springer, 1991.Google Scholar
  19. 19.
    J. Wojciechowski. Finite automata on transfinite sequences and regular expressions. Fundamenta information, 8(3–4):379–396, 1985.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Nicolas Bedon
    • 1
  • Olivier Carton
    • 1
  1. 1.Institut Gaspard MongeUniversité de Marne-la-ValléeNoisy-le-Grand Cedex

Personalised recommendations