Abstract
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.
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Autebert, J.-M., Boasson, L.: Transductions rationelles, Masson (1988)
Berstel, J.: Transductions and context-free languages, Teubner (1979)
Blumensath, A., Grädel, E.: Automatic Structures. In: Proceedings of 15th IEEE Symposium on Logic in Computer Science LICS 2000, pp. 51–62 (2000)
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: ICLMPS, pp. 1–11. Stanford University press (1960)
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)
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)
Courcelle, B.: Handbook of theoretical computer science. In: Graph rewriting: an algebraic and logic approach. Elsevier, Amsterdam (1990)
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)
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)
Ebbinghaus, H.D., Flum, J.: Finite model theory. Springer, Heidelberg (1995)
Eilenberg, S.: Automata, languages and machines, vol. A. Academic Press, London (1974)
Hodgson, B.R.: Décidabilité par automate fini. Ann. Sci. Math. Québec 7, 39–57 (1983)
Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)
Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Logic 6(4), 675–700 (2005)
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)
Morvan, C.: On rational graphs. In: Tiuryn, J. (ed.) ETAPS 2000. LNCS, vol. 1784, pp. 252–266. Springer, Heidelberg (2000)
Les graphes rationnels, Thése de doctorat, Université de Rennes 1 (2001)
Muller, D., Schupp, P.: The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science 37, 51–75 (1985)
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)
Pélecq, L.: Isomorphismes et automorphismes des graphes context-free, équationnels et automatiques, Ph.D. thesis, Université de Bordeau I (1997)
Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. soc. 141, 1–35 (1969)
Rabinovich, A.: Composition theorem for generalized sum, Personal communication (2006)
Rispal, C.: Synchronized graphs trace the context-sensitive languages. In: Mayr, R., Kucera, A. (eds.) Infinity 2002, ENTCS, vol. 68(6) (2002)
Shelah, S.: The monadic theory of order. Ann. Math. 102, 379–419 (1975)
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)
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)
Zeitman, R.S.: The composition method, Phd thesis, Wayne State University, Michigan (1994)
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Carayol, A., Morvan, C. (2006). On Rational Trees. In: Ésik, Z. (eds) Computer Science Logic. CSL 2006. Lecture Notes in Computer Science, vol 4207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11874683_15
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DOI: https://doi.org/10.1007/11874683_15
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