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.
Similar content being viewed by others
References
Dershowitz, N., ‘Orderings of term-rewriting systems’, Theor. Comp. Sci. 17, 279–301 (1982).
Dershowitz, N., ‘Termination of rewriting’, J. Symb. Comp. 3(1 & 2), 69–116 (1987).
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.
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).
Harel, D., ‘On folk theorems’, Commun. Assoc. Comp. Mach. 23(7), 379–389 (1980).
Higman, G., ‘Ordering by divisibility in abstract algebra’, Proc. London Math. Soc. 3(2), (1952).
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.
Huet, G., ‘Formal structures for computation and deduction’, Tech. Report, INRIA (May 1986).
Jouannaud, J. P. and Lescanne, P., ‘La réécriture’, Techniques et Sciences Informatiques 5(6), 433–452 (1987).
Jouannaud, J. P. and Lescanne, P., ‘Rewriting systems’, Technology and Sciences of Informatics 6(3), 180–199 (June 1987). Translated from ref. [9].
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).
Kamin, S. and Lévy, J.-J., ‘Two generalizations of the recursive path ordering’ (1980). Unpublished manuscript.
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).
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).
Lescanne, P., ‘Computer experiments with the REVE term rewriting systems generator’, in Proc. 10th ACM Symp. on Principles of Programming Languages, ACM (1983).
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).
Lévy, J.-J., ‘Dershowitzeries’ (1981). Unpublished manuscript.
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).
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.
Rusinowitch, M., ‘Path of subterms ordering and recursive decomposition ordering revisited’, J. Symb. Comp. 3(1 & 2), 117–132 (1987).
Sakai, K., ‘An ordering method for term rewriting systems’, in Proc. First Int. Conf. on Fifth Generation Computer Systems, Tokyo, Japan (November 1984).
Author information
Authors and Affiliations
Additional information
Part of this work was done while the author was visiting the Institute for New Generation Computer Technology, Tokyo, Japan.
Rights and permissions
About this article
Cite this article
Lescanne, P. On the recursive decomposition ordering with lexicographical status and other related orderings. J Autom Reasoning 6, 39–49 (1990). https://doi.org/10.1007/BF00302640
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00302640