Abstract
We consider context-free languages equipped with the lexicographic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than ω ω. As an application of this result, we obtain that an ordinal is the order type of the lexicographic ordering of a context-free language if and only if it is less than \(\omega^{\omega^\omega}\).
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References
Bedon, N., Bès, A., Carton, O., Rispal, C.: Logic and Rational Languages of Words Indexed by Linear Orderings. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 76–85. Springer, Heidelberg (2008)
Bès, A., Carton, O.: A Kleene Theorem for Languages of Words Indexed by Linear Orderings. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 158–167. Springer, Heidelberg (2005)
Braud, L., Carayol, A.: Linear Orders in the Pushdown Hierarchy. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 88–99. Springer, Heidelberg (2010)
Bloom, S.L., Choffrut, C.: Long words: the theory of concatenation and omega-power. Theoretical Computer Science 259, 533–548 (2001)
Bloom, S.L., Ésik, Z.: Deciding whether the frontier of a regular tree is scattered. Fundamenta Informaticae 55, 1–21 (2003)
Bloom, S.L., Ésik, Z.: The equational theory of regular words. Information and Computation 197, 55–89 (2005)
Bloom, S.L., Ésik, Z.: Regular and Algebraic Words and Ordinals. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 1–15. Springer, Heidelberg (2007)
Bloom, S.L., Ésik, Z.: Algebraic ordinals. Fundamenta Informaticae 99, 383–407 (2010)
Bloom, S.L., Ésik, Z.: Algebraic linear orderings. Int. J. Foundations of Computer Science 22, 491–515 (2011)
Bruyère, V., Carton, O.: Automata on linear orderings. J. Comput. System Sci. 73, 1–24 (2007)
Caucal, D.: On infinite graphs having a decidable monadic theory. Theoretical Computer Science 290, 79–115 (2003)
Courcelle, B.: Frontiers of infinite trees. Theoretical Informatics and Applications 12, 319–337 (1978)
Courcelle, B.: Fundamental properties of infinite trees. Theoretical Computer Science 25, 95–169 (1983)
Damm, W.: Languages Defined by Higher Type Program Schemes. In: Salomaa, A., Steinby, M. (eds.) ICALP 1977. LNCS, vol. 52, pp. 164–179. Springer, Heidelberg (1977)
Damm, W.: An Algebraic Extension of the Chomsky-Hierarchy. In: Becvar, J. (ed.) MFCS 1979. LNCS, vol. 74, pp. 266–276. Springer, Heidelberg (1979)
Delhommé, C.: Automaticité des ordinaux et des graphes homogènes. C. R. Acad. Sci. Paris, Ser. I 339, 5–10 (2004)
Ésik, Z.: Representing small ordinals by finite automata. In: Proc. 12th Workshop Descriptional Complexity of Formal Systems, Saskatoon, Canada. EPTCS, vol. 31, pp. 78–87 (2010)
Ésik, Z.: An undecidable property of context-free linear orders. Information Processing Letters 111, 107–109 (2010)
Ésik, Z.: Scattered Context-Free Linear Orderings. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 216–227. Springer, Heidelberg (2011)
Ésik, Z., Iván, S.: Büchi context-free languages. Theoretical Computer Science 412, 805–821 (2011)
Ésik, Z., Iván, S.: On Müller Context-Free Grammars. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 173–184. Springer, Heidelberg (2010)
Heilbrunner, S.: An algorithm for the solution of fixed-point equations for infinite words. Theoretical Informatics and Applications 14, 131–141 (1980)
Khoussainov, B., Rubin, S., Stephan, F.: On automatic partial orders, in. In: Eighteenth IEEE Symposium on Logic in Computer Science, LICS, pp. 168–177. IEEE Press (2003)
Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Log. 6, 625–700 (2005)
Kuske, D., Liu, J., Lohrey, M.: The isomorphism problem on classes of automatic structures. In: LICS 2010, pp. 160–169. IEEE Computer Society (2010)
Lohrey, M., Mathissen, C.: Isomorphism of Regular Trees and Words. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 210–221. Springer, Heidelberg (2011)
Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge (1997)
Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press (1982)
Thomas, W.: On frontiers of regular trees. Theoretical Informatics and Applications 20, 371–381 (1986)
Toulmin, G.H.: Shuffling ordinals and transfinite dimension. Proc. London Math. Soc. 3, 177–195 (1954)
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Ésik, Z., Iván, S. (2012). Hausdorff Rank of Scattered Context-Free Linear Orders . In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_25
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DOI: https://doi.org/10.1007/978-3-642-29344-3_25
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