Advertisement

Journal of Automated Reasoning

, Volume 6, Issue 1, pp 39–49 | Cite as

On the recursive decomposition ordering with lexicographical status and other related orderings

  • Pierre Lescanne
Article

Abstract

This paper studies three orderings, useful in theorem proving, especially for proving termination of term rewriting systems: the recursive decomposition ordering with status, the recursive path ordering with status and the closure ordering. It proves the transitivity of the recursive path ordering, the strict inclusion of the recursive path ordering in the recursive decomposition ordering, the totality of the recursive path ordering — therefore of the recursive decomposition ordering — the strict inclusion of the recursive decomposition ordering in the closure ordering and the stability of the closure ordering by instantiation.

Key words

Well-foundedness rewrite systems termination 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dershowitz, N., ‘Orderings of term-rewriting systems’, Theor. Comp. Sci. 17, 279–301 (1982).Google Scholar
  2. 2.
    Dershowitz, N., ‘Termination of rewriting’, J. Symb. Comp. 3(1 & 2), 69–116 (1987).Google Scholar
  3. 3.
    Forgaard, R., ‘A program for generating and analyzing term rewriting systems’, Tech. Report 343, Laboratory for Computer Science, Massachusetts Institute of Technology (1984). Master's Thesis.Google Scholar
  4. 4.
    Forgaard, R. and Detlefs, D., ‘An incremental algorithm for proving termination of term rewriting systems’, in J.-P. Jouannaud (ed.), Proc. 1st Int. Conf. on Rewriting Techniques and Applications, pp. 255–270, Springer-Verlag (1985).Google Scholar
  5. 5.
    Harel, D., ‘On folk theorems’, Commun. Assoc. Comp. Mach. 23(7), 379–389 (1980).Google Scholar
  6. 6.
    Higman, G., ‘Ordering by divisibility in abstract algebra’, Proc. London Math. Soc. 3(2), (1952).Google Scholar
  7. 7.
    Hsiang, J. and Rusinowitch, M., ‘On word problem in equational theories’, in Th. Ottmann (ed.), Proc. 14th Int. Colloq. on Automata, Languages and Programming, Karlsruhe (West Germany), Springer-Verlag (July 1987). Lecture Notes in Computer Science, vol. 267.Google Scholar
  8. 8.
    Huet, G., ‘Formal structures for computation and deduction’, Tech. Report, INRIA (May 1986).Google Scholar
  9. 9.
    Jouannaud, J. P. and Lescanne, P., ‘La réécriture’, Techniques et Sciences Informatiques 5(6), 433–452 (1987).Google Scholar
  10. 10.
    Jouannaud, J. P. and Lescanne, P., ‘Rewriting systems’, Technology and Sciences of Informatics 6(3), 180–199 (June 1987). Translated from ref. [9].Google Scholar
  11. 11.
    Jouannaud, J. P., Lescanne, P., and Reinig, F., ‘Recursive decomposition ordering’, in D.Bjørner, (ed.), Formal Description of Programming Concepts 2, pp. 331–348, North Holland, Garmisch-Partenkirchen, RFA (1982).Google Scholar
  12. 12.
    Kamin, S. and Lévy, J.-J., ‘Two generalizations of the recursive path ordering’ (1980). Unpublished manuscript.Google Scholar
  13. 13.
    Kapur, D., Narendran, P., and Sivakumar, G., ‘A path ordering for proving termination of term rewriting systems’, in H. Ehrig, C. Floyd, M. Nivat, and J. Thatcher (eds.), Proc. 6th Conf. on Automata, Algebra and Programming, Springer-Verlag (1985).Google Scholar
  14. 14.
    Kapur, D. and Sivakumar, G., ‘Experiments with an architecture of RRL, a rewrite rule laboratory’, in Proc. NSF Workshop on the Rewrite Rule Laboratory, pp. 33–56 (1983).Google Scholar
  15. 15.
    Lescanne, P., ‘Computer experiments with the REVE term rewriting systems generator’, in Proc. 10th ACM Symp. on Principles of Programming Languages, ACM (1983).Google Scholar
  16. 16.
    Lescanne, P., ‘Uniform termination of term rewriting systems recursive decomposition ordering with status’, in B.Courcelle (ed.), Proc. 9th Colloque les Arbres en Algebre et en Programmation, pp. 182–194, Cambridge University Press, Bordeaux, France (1984).Google Scholar
  17. 17.
    Lévy, J.-J., ‘Dershowitzeries’ (1981). Unpublished manuscript.Google Scholar
  18. 18.
    Plaisted, D., ‘A recursively defined ordering for proving termination of term rewriting systems’, Tech. Report R-78-943, Univ. of Illinois, Dept. of Computer Science (1978).Google Scholar
  19. 19.
    Puel, L., ‘Bon préordres sur les arbres associés à des ensembles inévitables et preuves de terminaison de systèmes de réécriture’, Thèse d'Etat (September 1987), Université Paris VII.Google Scholar
  20. 20.
    Rusinowitch, M., ‘Path of subterms ordering and recursive decomposition ordering revisited’, J. Symb. Comp. 3(1 & 2), 117–132 (1987).Google Scholar
  21. 21.
    Sakai, K., ‘An ordering method for term rewriting systems’, in Proc. First Int. Conf. on Fifth Generation Computer Systems, Tokyo, Japan (November 1984).Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Pierre Lescanne
    • 1
  1. 1.Centre de Recherche en Informatique de NancyINRIA-LorraineVandoeuvre-les-NancyFrance

Personalised recommendations