An actual implementation of a procedure that mechanically proves termination of rewriting systems based on inequalities between polynomial interpretations

  • Ahlem Ben Cherifa
  • Pierre Lescanne
Term Rewriting Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)


A method based on polynomial interpretations is currently implemented in REVE. Though this termination check procedure is absolutely necessary in order to run some of the examples we know, like Associativity+Endomorphism, we do not think it will replace the current methods based on recursive path ordering [Dersh82] or recursive decomposition ordering [Jouan.etal.82], since they have shown to have a large scope and to be really easy to use in many practical cases [Forg&Det185], and Dershowitz has exhibited examples where polynomial interpretation fail [Dersh83]. So, we think we will keep both methods in REVE and let the user choose the method he or she wants to use. However in the case of associative-commutative operators, the methods based on extensions of the recursive path ordering either fail or are not ready for being incorporated in a rewrite rule laboratory like REVE. So, it is the only method currently available and we are incorporating it into REVE-3 (the general equational rewriting laboratory) as the mechanism for proving termination in the associative-commutative case.

In conclusion, we would like to mention a limit of our criterion. It cannot prove the positiveness of the polynomial x 1 2 +x 2 2 −2x1x2+1=(x1−x2)2+1. Since we never encountered such a polynomial in termination proofs we feel that would not be an obstacle for using it.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bach&Plai85].
    L. Bachmair and D. Plaisted, ”Associative Path Orderings,” in Proc. 1st Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 202, pp. 241–254, Springer Verlag, Dijon (France), 1985.Google Scholar
  2. [Bach&Plai85b].
    L. Bachmair and D.A. Plaisted, ”Termination Orderings For Associative-Commutative Rewriting Systems,” Journal of Symbolic Computation, 1985.Google Scholar
  3. [Belle84].
    F. Bellegarde, ”Rewriting Systems on FP Expressions that Reduce the Number of Sequences they yield,” in Symposium on LISP and Functional Programming, ACM, Austin, USA, 1984.Google Scholar
  4. [Dersh82].
    Nachum Dershowitz, ”Orderings for Term-Rewriting Systems,” Theoretical Computer Science, vol. 17, pp. 279–301, 1982.CrossRefGoogle Scholar
  5. [Dersh83].
    N. Dershowitz, ”Well-Founded Orderings,” ATR-83(8478)-3, Information Science Research Office, The Aerospace Corporation, El Segundo, California (USA), May 1983.Google Scholar
  6. [Dersh85].
    N. Dershowitz, ”Termination,” in Proc. 1rst Conf. Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 202, pp. 180–224, Springer Verlag, Dijon (France), May 1985.Google Scholar
  7. [Forg&Det185].
    R. Forgaard and D. Detlefs, ”An incremental algorithm for proving termination of term rewriting systems,” in Proc. 1rst International Conference on Rewriting Techniques and Applications., Lecture Notes in Computer Science, vol. 202, Springer Verlag, Dijon (France), 1985.Google Scholar
  8. [Huet80].
    G. Huet, ”Confluent reductions: abstract properties and applications to term rewriting systems,” J. of ACM, vol. 27, no. 4, pp. 797–821, Oct. 1980.CrossRefGoogle Scholar
  9. [Hue&Opp80].
    G. Huet and D. Oppen, ”Equations and Rewrite Rules: A Survey,” in Formal Languages: Perspectives And Open Problems, ed. Book R., Academic Press, 1980.Google Scholar
  10. [Jouannaud82].
    [Jouan.etal.82]. J.P. Jouannaud, P. Lescanne, and F. Reinig, ”Recursive Decomposition Ordering,” in Formal Description of Programming Concepts 2, ed. Bjorner D., pp. 331–346, North Holland, Garmish Partenkirschen, RFA, 1982.Google Scholar
  11. [Lankf79].
    D.S. Lankford, ”On Proving Term Rewriting Systems Are Noetherian,” Report Mtp-3, Math. Dept., Louisiana Tech University, May 1979.Google Scholar
  12. [Lesc83].
    P. Lescanne, ”Computer Experiments with the REVE Term Rewriting System Generator,” in 10th ACM Conf. on Principles of Programming Languages, pp. 99–108, Austin Texas, January 1983.Google Scholar
  13. [Man&Nes70].
    Z. Manna and S. Ness, ”On the Termination of Markov Algorithms,” Third Hawaii International Conference on System Sciences, pp. 789–792, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Ahlem Ben Cherifa
    • 1
  • Pierre Lescanne
    • 1
  1. 1.Centre de Recherche en Informatique de NancyVandoeuvreFRANCE

Personalised recommendations