On Term Rewriting Systems Having a Rational Derivation

  • Antoine Meyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2987)

Abstract

Several types of term rewriting systems can be distinguished by the way their rules overlap. In particular, we define the classes of prefix, suffix, bottom-up and top-down systems, which generalize similar classes on words. Our aim is to study the derivation relation of such systems (i.e. the reflexive and transitive closure of their rewriting relation) and, if possible, to provide a finite mechanism characterizing it. Using a notion of rational relations based on finite graph grammars, we show that the derivation of any bottom-up, top-down or suffix systems is rational, while it can be non recursive for prefix systems.

References

  1. 1.
    Abdulla, P., Jonsson, B., Mahata, P., D’Orso, J.: Regular tree model checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 555–568. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 403–418. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Bouajjani, A., Touili, T.: Extrapolating tree transformations. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 539–554. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Calbrix, H., Knapik, T.: A string-rewriting characterization of Muller and Schupp’s context-free graphs. In: Arvind, V., Sarukkai, S. (eds.) FST TCS 1998. LNCS, vol. 1530, pp. 331–342. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Caucal, D.: On word rewriting systems having a rational derivation. In: Tiuryn, J. (ed.) FOSSACS 2000. LNCS, vol. 1784, pp. 48–62. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Caucal, D., Knapik, T.: A Chomsky-like hierarchy of infinite graphs. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 177–187. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Colcombet, T.: On families of graphs having a decidable first order theory with reachability. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 98–109. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Dauchet, M., Tison, S.: Decidability of the confluence of finite ground term rewrite systems. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 80–89. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  9. 9.
    Dauchet, M., Tison, S.: The theory of ground rewrite systems is decidable. In: LICS 90th, pp. 242–248. IEEE, Los Alamitos (1990)Google Scholar
  10. 10.
    Gyenizse, P., Vágvölgyi, S.: Linear generalized semi-monadic rewrite systems effectively preserve recognizability. TCS 194(1-2), 87–122 (1998)MATHCrossRefGoogle Scholar
  11. 11.
    Lodaya, K., Weil, P.: Rationality in algebras with a series operation. Information and Computation 171(2), 269–293 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Löding, C.: Ground tree rewriting graphs of bounded tree width. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 559–570. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Raoult, J.-C.: A survey of tree transductions. Technical Report 1410, Inria (April 1991)Google Scholar
  14. 14.
    Raoult, J.-C.: Rational tree relations. Bulletin of the Belgian Mathematics Society 4, 149–176 (1997)MATHMathSciNetGoogle Scholar
  15. 15.
    Takai, T., Kaji, Y., Seki, H.: Right-linear finite path overlapping term rewriting systems effectively preserve recognizability. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 246–260. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Antoine Meyer
    • 1
  1. 1.Irisa, campus de Beaulieu, Rennes Liafa, Université de Paris 7 

Personalised recommendations