On Rational Trees

  • Arnaud Carayol
  • Christophe Morvan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)


Rational graphs are a family of graphs defined using labelled rational transducers. Unlike automatic graphs (defined using synchronized transducers) the first order theory of these graphs is undecidable, there is even a rational graph with an undecidable first order theory. In this paper we consider the family of rational trees, that is rational graphs which are trees. We prove that first order theory is decidable for this family. We also present counter examples showing that this result cannot be significantly extended both in terms of logic and of structure.


Rational Tree Rational Graph Rational Accessibility Complete Binary Tree Rational Subset 
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  1. 1.
    Autebert, J.-M., Boasson, L.: Transductions rationelles, Masson (1988)Google Scholar
  2. 2.
    Berstel, J.: Transductions and context-free languages, Teubner (1979)Google Scholar
  3. 3.
    Blumensath, A., Grädel, E.: Automatic Structures. In: Proceedings of 15th IEEE Symposium on Logic in Computer Science LICS 2000, pp. 51–62 (2000)Google Scholar
  4. 4.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: ICLMPS, pp. 1–11. Stanford University press (1960)Google Scholar
  5. 5.
    Caucal, D.: On transition graphs having a decidable monadic theory. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 194–205. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Caucal, D.: On infinite terms having a decidable monadic theory. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 165–176. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Courcelle, B.: Handbook of theoretical computer science. In: Graph rewriting: an algebraic and logic approach. Elsevier, Amsterdam (1990)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Epstein, D., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston: Word processing in groups. Jones and Barlett publishers (1992)Google Scholar
  10. 10.
    Ebbinghaus, H.D., Flum, J.: Finite model theory. Springer, Heidelberg (1995)MATHGoogle Scholar
  11. 11.
    Eilenberg, S.: Automata, languages and machines, vol. A. Academic Press, London (1974)MATHGoogle Scholar
  12. 12.
    Hodgson, B.R.: Décidabilité par automate fini. Ann. Sci. Math. Québec 7, 39–57 (1983)MATHMathSciNetGoogle Scholar
  13. 13.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Logic 6(4), 675–700 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Lohrey, M.: Automatic structures of bounded degree. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS (LNAI), vol. 2850, pp. 344–358. Springer, Heidelberg (2003)Google Scholar
  16. 16.
    Morvan, C.: On rational graphs. In: Tiuryn, J. (ed.) ETAPS 2000. LNCS, vol. 1784, pp. 252–266. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Les graphes rationnels, Thése de doctorat, Université de Rennes 1 (2001)Google Scholar
  18. 18.
    Muller, D., Schupp, P.: The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science 37, 51–75 (1985)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Morvan, C., Stirling, C.: Rational graphs trace context-sensitive languages. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 548–559. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Pélecq, L.: Isomorphismes et automorphismes des graphes context-free, équationnels et automatiques, Ph.D. thesis, Université de Bordeau I (1997)Google Scholar
  21. 21.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. soc. 141, 1–35 (1969)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Rabinovich, A.: Composition theorem for generalized sum, Personal communication (2006)Google Scholar
  23. 23.
    Rispal, C.: Synchronized graphs trace the context-sensitive languages. In: Mayr, R., Kucera, A. (eds.) Infinity 2002, ENTCS, vol. 68(6) (2002)Google Scholar
  24. 24.
    Shelah, S.: The monadic theory of order. Ann. Math. 102, 379–419 (1975)CrossRefGoogle Scholar
  25. 25.
    Sénizergues, G.: Definability in weak monadic second-order logic of some infinite graphs. In: Dagstuhl seminar on Automata theory: Infinite computations, Warden, Germany, vol. 28, p. 16 (1992)Google Scholar
  26. 26.
    Thomas, W.: A short introduction to infinite automata. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 130–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  27. 27.
    Zeitman, R.S.: The composition method, Phd thesis, Wayne State University, Michigan (1994)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Arnaud Carayol
    • 1
  • Christophe Morvan
    • 1
  1. 1.IRISARennesFrance

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