LATIN 1998: LATIN'98: Theoretical Informatics pp 53-64 | Cite as
An Eilenberg theorem for words on countable ordinals
Abstract
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.
Preview
Unable to display preview. Download preview PDF.
References
- 1.J. Almeida. Finite semigroups and universal algebra, volume 3 of Series in algebra. World Scientific, 1994.Google Scholar
- 2.A. Arnold. A syntactic congruence for rational Ω-languages. Theoretical Computer Science, 39:333–335, 1985.MATHCrossRefMathSciNetGoogle Scholar
- 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.N. Bedon. Star-free sets of words on ordinals. IGM report 97-8, submitted to Information and Computation.Google Scholar
- 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.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.Y. Choueka. Finite automata, definable sets, and regular expressions over Ωn-tapes. J. Comp. Syst. Sci., 17:81–97, 1978.MATHCrossRefMathSciNetGoogle Scholar
- 8.S. Eilenberg. Automata, languages and machines, volume B. Academic Press, 1976.Google Scholar
- 9.J.-P. Pécuchet. Etude syntaxique des parties reconnaissables de mots infinis. Lecture Notes in Computer Science, 226:294–303, 1986.MATHGoogle Scholar
- 10.J.-P. Pécuchet. Variétés de semigroupes et mots infinis. Lecture Notes in Computer Science, 210:180–191, 1986.MATHGoogle Scholar
- 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.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.J.-E. Pin. Handbook of formal languages, volume 1, chapter Syntactic semigroups, pages 679–746. Springer, 1997.Google Scholar
- 14.S. Rohde. Alternating automata and the temporal logic of ordinals. PhD thesis, University of Illinois, Urbana-Champaign, 1997.Google Scholar
- 15.J. G. Rosenstein. Linear ordering. Academic Press, New York, 1982.Google Scholar
- 16.M. P. Schützenberger. On finite monoids having only trivial subgroups. Information and Control, 8:190–194, 1965.MATHCrossRefMathSciNetGoogle Scholar
- 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.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.J. Wojciechowski. Finite automata on transfinite sequences and regular expressions. Fundamenta information, 8(3–4):379–396, 1985.MATHMathSciNetGoogle Scholar