Abstract
We prove that every rational language of words indexed by linear orderings is definable in monadic second-order logic. We also show that the converse is true for the class of languages indexed by countable scattered linear orderings, but false in the general case. As a corollary we prove that the inclusion problem for rational languages of words indexed by countable linear orderings is decidable.
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References
Bedon, N.: Logic over words on denumerable ordinals. Journal of Computer and System Science 63(3), 394–431 (2001)
Bedon, N., Rispal, C.: Schützenberger and Eilenberg theorems for words on linear orderings. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 134–145. Springer, Heidelberg (2005)
Bès, A., Carton, O.: A Kleene theorem for languages of words indexed by linear orderings. Int. J. Found. Comput. Sci. 17(3), 519–542 (2006)
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)
Bruyère, V., Carton, O.: Hierarchy among automata on linear orderings. In: Baeza-Yate, R., Montanari, U., Santoro, N. (eds.) Foundation of Information technology in the era of network and mobile computing, pp. 107–118. Kluwer Academic Publishers, Dordrecht (2002)
Bruyère, V., Carton, O.: Automata on linear orderings. J. Comput. System Sci. 73(1), 1–24 (2007)
Bruyère, V., Carton, O., Sénizergues, G.: Tree automata and automata on linear orderings. In: Harju, T., Karhumäki, J. (eds.) WORDS 2003, pp. 222–231. Turku Center for Computer Science (2003)
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 1960, 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)
Carton, O.: Accessibility in automata on scattered linear orderings. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 155–164. Springer, Heidelberg (2002)
Gurevich, Y.: Monadic second-order theories. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics. Perspectives in Mathematical Logic, pp. 479–506. Springer, Heidelberg (1985)
Ladner, R.E.: Application of model theoretic games to discrete linear orders and finite automata. Inform. Control 33 (1977)
McNaughton, R., Papert, S.: Counter free automata. MIT Press, Cambridge, MA (1971)
Michaux, C., Point, F.: Les ensembles k-reconnaissables sont définissables dans < N, + ,V k > (the k-recognizable sets are definable in < N, + ,V k > ). C. R. Acad. Sci. Paris Sér. I (303), 939–942 (1986)
Perrin, D.: An introduction to automata on infinite words. In: Perrin, D., Nivat, M. (eds.) Automata on Infinite Words. LNCS, vol. 192, pp. 2–17. Springer, Heidelberg (1984)
Perrin, D.: Recent results on automata and infinite words. In: Chytil, M.P., Koubek, V. (eds.) MFCS 1984. LNCS, vol. 176, pp. 134–148. Springer, Heidelberg (1984)
Perrin, D., Pin, J.E.: First order logic and star-free sets. J. Comput. System Sci. 32, 393–406 (1986)
Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 141, 1–35 (1969)
Rispal, C.: Automates sur les ordres linéaires: Complémentation. PhD thesis, University of Marne-la-Vallée, France (2004)
Rispal, C., Carton, O.: Complementation of rational sets on countable scattered linear orderings. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds.) DLT 2004. LNCS, vol. 3340, pp. 381–392. Springer, Heidelberg (2004)
Rosenstein, J.G.: Linear orderings. Academic Press, New York (1982)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Inform. Control 8, 190–194 (1965)
Shelah, S.: The monadic theory of order. Annals of Mathematics 102, 379–419 (1975)
Thomas, W.: Star free regular sets of ω-sequences. Inform. Control 42, 148–156 (1979)
Thomas, W.: Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 118–143. Springer, Heidelberg (1997)
Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 389–455. Springer, Heidelberg (1997)
Wojciechowski, J.: Finite automata on transfinite sequences and regular expressions. Fundamenta informaticæ 8(3-4), 379–396 (1985)
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Bedon, N., Bès, A., Carton, O., Rispal, C. (2008). Logic and Rational Languages of Words Indexed by Linear Orderings. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_11
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DOI: https://doi.org/10.1007/978-3-540-79709-8_11
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