Abstract
We consider words indexed by linear orderings. These extend finite, (bi-)infinite words and words on ordinals. We introduce automata and rational expressions for words on linear orderings. We prove that for countable scattered linear orderings they are equivalent. This result extends Kleene’s theorem. The proofs are effective.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Asarin. Equations on timed languages. In T. Henzinger and S. Sastry, editors, Hybrid Systems: Computation and Control, number 1386 in Lect. Notes in Comput. Sci., pages 1–12, 1998.
E. Asarin, P. Caspi, and O. Maler. A Kleene theorem for timed automata. In Proceedings, Twelth Annual IEEE Symposium on Logic in Computer Science, pages 160–171, 1997.
N. Bedon and O. Carton. An Eilenberg theorem for words on countable ordinals. In Cláudio L. Lucchesi and Arnaldo V. Moura, editors, Latin’98: Theoretical Informatics, volume 1380 of Lect. Notes in Comput. Sci., pages 53–64. Springer-Verlag, 1998.
B. Bérard and C. Picaronny. Accepting Zeno words without making time stand still. In Mathematical Foundations of Computer Science 1997, volume 1295 of Lect. Notes in Comput. Sci., pages 149–158, 1997.
P. Bouyer and A. Petit. A Kleene/Büchi-like theorem for clock languages. J. of Automata, Languages and Combinatorics, 2001. To appear.
J. R. Büchi. Weak second-order arithmetic and finite automata. Z. Math. Logik und grundl. Math., 6:66–92, 1960.
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.
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.
Y. Choueka. Finite automata, definable sets, and regular expressions over ωn-tapes. J. Comput. System Sci., 17(1):81–97, 1978.
B. Courcelle. Frontiers of infinite trees. RAIRO Theoretical Informatics, 12(4):319–337, 1978.
D. Girault-Beauquier. Bilimites de langages reconnaissables. Theoret. Comput. Sci., 33(2-3):335–342, 1984.
M. R. Hansen, P. K. Pandya, and Z. Chaochen. Finite divergence. Theoret. Comput. Sci., 138(1):113–139, 1995.
S. Heilbrunner. An algorithm for the solution of fixed-point equations for infinite words. RAIRO Theoretical Informatics, 14(2):131–141, 1980.
S. C. Kleene. Representation of events in nerve nets and finite automata. In C. E. Shannon, editor, Automata studies, pages 3–41. Princeton University Press, Princeton, 1956.
M. Nivat and D. Perrin. Ensembles reconnaissables de mots bi-infinis. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pages 47–59, 1982.
D. Perrin. Finite automata. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 1, pages 1–57. Elsevier, 1990.
J. G. Rosenstein. Linear ordering. Academic Press, New York, 1982.
S. Shelah. The monadic theory of order. Annals of Mathematics, 102:379–419, 1975.
W. Thomas. On frontiers of regular sets. RAIRO Theoretical Informatics, 20:371–381, 1986.
W. Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 4, pages 133–191. Elsevier, 1990.
W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, number 1261 in Lect. Notes in Comput. Sci., pages 118–143. Springer-Verlag, 1997.
J. Wojciechowski. Finite automata on transfinite sequences and regular expressions. Fundamenta Informaticæ, 8(3-4):379–396, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bruyère, V., Carton, O. (2001). Automata on Linear Orderings. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_21
Download citation
DOI: https://doi.org/10.1007/3-540-44683-4_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42496-3
Online ISBN: 978-3-540-44683-5
eBook Packages: Springer Book Archive