Skip to main content
SpringerLink
Log in

Automating the Knuth Bendix ordering

  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract

Knuth and Bendix proposed a very versatile technique for ordering terms, based upon assigning weights to operators and then to terms by adding up the weights of the operators they contain. Our purpose in this paper is as follows. First we give some examples to indicate the flexibility of the method. Then we give a simple and practical algorithm, based on solving systems of linear inequalities, for determining whether or not a set of rules can be ordered by a Knuth Bendix ordering. We also describe how this algorithm may be incorporated in a completion procedure which thus considers all possible choices of weights which orient a given equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bellegarde, F.: Rewriting systems of FP expressions to reduce the number sequences yielded. Sci. Comput. Program.6, 11–34 (1986)

    Google Scholar 

  2. Boyer, R.S., Moore, J.S.: A computational logic. New York: Academic Press 1979

    Google Scholar 

  3. Ben Cherifa, A., Lescanne, P.: Termination of rewriting systems by polynomial interpretations and its implementation. Sci. Comput. Program.9, 137–160 (1987)

    Google Scholar 

  4. Chvatal, V.: Linear programming. New York: Freeman 1983

    Google Scholar 

  5. Dershowitz, N.: Orderings for term rewriting systems. Theor. Comp. Sci.17, 279–301 (1982)

    Google Scholar 

  6. Dershowitz, N.: Termination of rewriting. J. Symbol. Comput.3, 69–116 (1987)

    Google Scholar 

  7. Forgaard, R., Detlefs, D.: A procedure for automatically proving the termination of a set of rewrite rules. Proceedings of the First International Conference on Rewriting Techniques and Applications. (Lect. Notes Comput. Sci., vol. 202) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  8. Dick, A.J.J.: ERIL — Equational reasoning: an interactive laboratory. Rutherford Appleton Laboratory Report RAL-86-010, March 1985

  9. Dick, A.J.J., Kalmus J.R.: ERIL (Equational reasoning: an interactive laboratory). User's manual. Version R1.6, Rutherford Appleton Laboratory Report RAL-88-055, September 1988

  10. Feferman, S.: Systems of predicative analysis II: Representation of ordinals. J. Symbol. Logic33, 193–220 (1968)

    Google Scholar 

  11. Gale, D.: The theory of linear economic models. New York: McGraw Hill 1960

    Google Scholar 

  12. Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. Ass. Comp. Mach.27(4) 797–821 (1980), preliminary version in 18th Symposium on Foundations of Computer Science, IEEE, 1977

    Google Scholar 

  13. Huet, G., Lankford, D.S.: On the uniform halting problem for term rewriting systems. Rapport Laboria 283, INRIA, March 1978

  14. Higman, G.: Ordering by divisibility in abstract algebras. Proc LMS(3)2, 326–336 (1952)

    Google Scholar 

  15. Huet, G., Oppen, D.C.: Equations and rewrite rules: a survey. In: Book, R. (ed.), Formal language theory, perspectives and open problems, pp. 349–405. New York: Academic Press 1980

    Google Scholar 

  16. Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational problems in abstract algebra, pp. 263–297. New York: Pergamon Press 1970

    Google Scholar 

  17. Kreko, B.: Linear programming. London: Pitman 1968

    Google Scholar 

  18. Lankford, D.S.: Canonical algebraic simplification in computational logic. Mem ATP-25. Automatic Theorem Proving Project. University of Texas, Austin TX, May 1975

    Google Scholar 

  19. Lescanne, P.: Computer experiments with the REVE term rewriting system generator, Proc 10th ACM POPL symposium, Austin, TX 1983, pp. 99–108

  20. Lescanne, P.: Divergence of the Knuth Bendix completion procedure and termination orderings. Bull. Eur. Assoc. Theoret. Comput. Sci.30, 80–83 (1986)

    Google Scholar 

  21. Lescanne, P.: Private communication

  22. Manna, Z., Ness, S.: On the termination of Markov algorithms. Proceedings of the Third Hawaii International Conference on System Science, Honolulu, HI, January 1970, pp. 789–792

  23. Martin, U.: How to choose the weights in the Knuth Bendix ordering. Proceedings of the Second International Conference on Rewriting Techniques and Applications (RTA 87), Bordeaux, France, May 1987. (Lect. Notes Comput. Sci., vol. 256, pp. 42–53) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  24. O'Donnell, M.J.: Computing in systems described by equations (Lect Notes Comput. Sci., vol. 58) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  25. Rusinowitch, M.: Plaisted ordering and recursive decomposition ordering revisited. Proceedings of the First International Conference on Rewriting Techniques and Applications. (Lect. Notes Comput. Sci., vol. 202) Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  26. Uzawa, H.: An elementary method for linear programming. In: Arrow, K.J., Hurwicz, L., Uzawa, H. (eds.) Studies in linear and non-linear programming, pp. 179–188. Stanford: Stanford University Press 1958

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Informatics Department, Rutherford Appleton Laboratory, Chilton, OX11 0QX, Didcot, Oxon, UK

    Jeremy Dick & John Kalmus

  2. Department of Computer Science, Royal Holloway and Bedford New College, University of London, Egham Hill, TW20 0EX, Egham, Surrey, UK

    Ursula Martin

Authors
  1. Jeremy Dick
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Kalmus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ursula Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A preliminary version of this paper [Ma 87] appeared in the Proceedings of RTA-87

Supported by the Science and Engineering Research Council

Supported in part by the Science and Engineering Research Council under grant GR/D 81718

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dick, J., Kalmus, J. & Martin, U. Automating the Knuth Bendix ordering. Acta Informatica 28, 95–119 (1990). https://doi.org/10.1007/BF01237233

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01237233

Keywords

Search

Navigation