Abstract
In a preceding paper (Bruyère and Carton, automata on linear orderings, MFCS’01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Büchi. Kleene’s theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation.
This is a preview of subscription content, log in via an institution to check access.
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
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, London (1974)
Bruyère, V., Carton, O.: Automata on linear orderings. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 236–247. Springer, Heidelberg (2001) IGM report 2001-12
Bruyère, V., Carton, O.: Hierarchy among automata on linear orderings. In: IFIP TCS 2002, pp. 107–118 (2002)
Bedon, N.: Finite automata and ordinals. Theoret. Comput. Sci. 156, 119–144 (1996)
Bedon, N.: Automata, semigroups and recognizability of words on ordinals. Int. J. Alg. Comput. 8, 1–21 (1998)
Bedon, N., Carton, O.: An Eilenberg theorem for words on countable ordinals. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 53–64. Springer, Heidelberg (1998)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik und grundl. Math. 6, 66–92 (1960)
Büchi, J.R.: On a decision method in the restricted second-order arithmetic. In: Proc. Int. Congress Logic, Methodology and Philosophy of science, Berkeley, pp. 1–11. Stanford University Press (1962)
Büchi, J.R.: Transfinite automata recursions and weak second order theory of ordinals. In: Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Jerusalem 1964, pp. 2–23. North Holland, Amsterdam (1965)
Choueka, Y.: Finite automata, definable sets, and regular expressions over ω n-tapes. J. Comput. System Sci. 17(1), 81–97 (1978)
Girault-Beauquier, D.: Bilimites de langages reconnaissables. Theoret. Comput. Sci. 33(2-3), 335–342 (1984)
Hausdorff, F.: Set theory. In Chelsea, New York (1957)
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E. (ed.) Automata studies, pp. 3–41. Princeton University Press, Princeton (1956)
Muller, D.: Infinite sequences and finite machines. In: Proc. of Fourth Annual IEEE Symp., editor, Switching Theory and Logical Design, pp. 3–16 (1963)
Nivat, M., Perrin, D.: Ensembles reconnaissables de mots bi-infinis. In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pp. 47–59 (1982)
Perrin, D., Pin, J.E.: Infinite words. In: Elsevier (ed.) Academic Press, London (2003)
Pin, J.-E.: Syntactic semigroups. In: Handbook of formal languages, vol. 1, pp. 679–746. Springer, Heidelberg (1997)
Rabin, M.O.: Decidability of second-order theories and automata on infinite trees 141, 1–35 (1969)
Ramsey, F.D.: On a problem of formal logic. Proc. of the London math. soc. 30, 338–384 (1929)
Rosenstein, J.G.: Linear ordering. Academic Press, New York (1982)
Safra, S.: On the complexity of ω-automata. In: 29th Annual Symposium on Foundations of computer sciences, pp. 24–29 (1988)
Wojciechowski, J.: 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
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carton, O., Rispal, C. (2004). Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_33
Download citation
DOI: https://doi.org/10.1007/978-3-540-24698-5_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21258-4
Online ISBN: 978-3-540-24698-5
eBook Packages: Springer Book Archive