Skip to main content
Springer Nature Link
Log in

On pot, pans and pudding or how to discover generalised critical Pairs

  • Conference paper
  • First Online:
Automated Deduction — CADE-12 (CADE 1994)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 814))

Included in the following conference series:

Abstract

We develop a new critical pair criterion for term completion modulo equational theories. Our criterion relies on computing generalised critical pairs. It is compatible with most known critical pair criteria based on subconnectedness. Therefore our procedure can profit from the additional benefits of other critical pair criteria. A first test implementation has shown the practical usefulness of the new criterion for completion modulo associative and commutative theories.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Leo Bachmair and Nachum Dershowitz. Critical pair criteria for completion. Journal of Symbolic Computation, 6:1–18, 1988.

    Google Scholar 

  2. George M. Bergman. The diamond lemma for ring theory. Advances in Mathematics, 29:178–218, 1978.

    Google Scholar 

  3. [BHK+88] H.-J. Bürckert, A. Herold, D. Kapur, J. H. Siekmann, M. E. Stickel, M. Tepp, and H. Zhang. Opening the ac-unification race. Journal of Automated Reasoning, 4:465–474, 1988.

    Google Scholar 

  4. Bruno Buchberger and Rüdiger Loos. Algebraic simplification. In Computer Algebra, pages 14–43. Springer-Verlag, 1982.

    Google Scholar 

  5. Timothy B. Baird, Gerald E. Peterson, and Ralph W. Wilkerson. Complete sets of reductions modulo associativity, commutativity and identity. In Nachum Dershowitz, editor, Rewriting Techniques and Applications (LNCS355), pages 29–44. Springer-Verlag, 1989. (Proc. RTA'89, Chapel Hill, NC, USA, April 1989).

    Google Scholar 

  6. Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Universität Innsbruck, 1965.

    Google Scholar 

  7. Bruno Buchberger. A criterion for detecting unnecessary reductions in the construction of Gröbner-Bases. In E. Ng, editor, Symbolic and Algebraic Computing (LNCS 72), pages 3–21. Springer-Verlag, 1979. (Proc. EUROSAM'79, Marseille, France).

    Google Scholar 

  8. Reinhard Bündgen. Completion of integral polynomials by AC-term completion. In Stephen M. Watt, editor. International Symposium on Symbolic and Algebraic Computation, pages 70–78, 1991. (Proc. ISSAC'91, Bonn, Germany, July 1991).

    Google Scholar 

  9. Reinhard Bündgen. Term Completion Versus Algebraic Completion. PhD thesis, Universität Tübingen, D-7400 Tübingen, Germany, May 1991. (reprinted as report WSI 91-3).

    Google Scholar 

  10. Reinhard Bündgen. Reduce the redex → ReDuX. In Claude Kirchner, editor, Rewriting Techniques and Applications (LNCS 690), pages 446–450. Springer-Verlag, 1993. (Proc. RTA'93, Montreal, Canada, June 1993).

    Google Scholar 

  11. Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuven, editor, Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, chapter 6. Elsevier, 1990.

    Google Scholar 

  12. François Fages. Associative commutative unification. Journal of Symbolic Computation, 3:257–275, 1987.

    Google Scholar 

  13. Albrecht Fortenbacher. An algebraic approach to unification under associativity and commutativity. Journal of Symbolic Computation, 3:217–229, 1987.

    Google Scholar 

  14. Jean-Pierre Jouannaud and Hélène Kirchner. Completion of a set of rules modulo a set of equations. SIAM J. on Computing, 14(4): 1155–1194, 1986.

    Google Scholar 

  15. Jean-Pierre Jouannaud and Claude Marché. Termination and completion modulo associativity, commutativity and identity. Theoretical Computer Science, 104:29–51, 1992.

    Google Scholar 

  16. Donald E. Knuth and Peter B. Bendix. Simple word problems in universal algebra. In J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon Press, 1970. (Proc. of a conference held in Oxford, England, 1967).

    Google Scholar 

  17. Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Background: Computational Strcutures, volume 2 of Handbook of Logic in Computer Science, chapter 1. Oxford University Press, 1992.

    Google Scholar 

  18. Deepak Kapur, David R. Musser, and Paliath Narendran. Only prime superpositions need be considered in the Knuth-Bendix completion procedure. Journal of Symbolic Computation, 6:19–36, 1988.

    Google Scholar 

  19. Wolfgang Küchlin. An implementation and investigation of the Knuth-Bendix completion algorithm. Master's thesis, Informatik I, Universität Karlsruhe, D-7500 Karlsruhe, W-Germany, 1982. (Reprinted as Report 17/82.).

    Google Scholar 

  20. Wolfgang Küchlin. A confluence criterion based on the generalised Knuth-Bendix algorithm. In B. F. Caviness, editor, Eurocal'85 (LNCS 204), pages 390–399. Springer-Verlag, 1985. (Proc. Eurocal'85, Linz, Austria, April 1985).

    Google Scholar 

  21. Wolfgang Küchlin. Equational Completion by Proof Transformation. PhD thesis, Swiss Federal Institute of Technology (ETH), CH-8092 Zürich, Switzerland, June 1986. (Also as Equational Completion by Proof Simplification, Report 86-02, Mathematics, ETH Zürich, May 1986).

    Google Scholar 

  22. Wolfgang Küchlin. A generalized Knuth-Bendix algorithm. Technical Report 86-01, Mathematics, Swiss Federal Institute of Technology (ETH), CH-8092 Zürich, Switzerland, January 1986.

    Google Scholar 

  23. Mike Lai. On how to move mountains ‘associatively and commutatively'. In Nachum Dershowitz, editor, Rewriting Techniques and Applications (LNCS 355), pages 187–202. Springer-Verlag, 1989. (Proc. RTA'89, Chapel Hill, NC, USA. April 1989).

    Google Scholar 

  24. Dallas Lankford and A. M. Ballantyne. Decision procedures for simple equational theories with commutative-associative axioms: Complete sets of commutative-associative reductions. Technical Report Report ATP-39, Department of Mathematics and Computer Sciences, University of Texas, Austin, August 1977.

    Google Scholar 

  25. Dallas Lankford and A. M. Ballantyne. Blocked permutative narrowing and resolution. In Proc. 4th Workshop on Automated Deduction, pages 168–174, Austin, Texas, February 1979. (Corrigendum of June, 1979, Addendum of January, 1984).

    Google Scholar 

  26. M. Livesey and J. Siekmann. Unification of bags and sets. Technical report, Institut für Informatik I, Universität Karlsruhe, 7500 Karlsruhe, Fed. Rep. of Germany, 1976.

    Google Scholar 

  27. M. H. A. Newman. On theories with a combinatorial definition of “equivalence”. Annals of Mathematics, 43(2):223–243, 1942.

    Google Scholar 

  28. David Plaisted. Equational reasoning and term rewriting systems. In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Logical Foundations, volume I of Handbook of Logic in Artificial Intelligence and Logic Programming, chapter 5. Oxford University Press, 1993.

    Google Scholar 

  29. G. Peterson and M. Stickel. Complete sets of reductions for some equational theories. Journal of the ACM, 28:223–264, 1981.

    Google Scholar 

  30. J. R. Slagle. Automated theorem-proving for theories with simplifiers, commutativity, and associativity. Journal of the ACM, 21(4): 622–642, October 1974.

    Google Scholar 

  31. Mark E. Stickel. A unification algorithm for associative-commutative functions. JACM, 28(3):423–434, July 1981.

    Google Scholar 

  32. Franz Winkler and Bruno Buchberger. A criterion for eliminating unnecessary reductions in the Knuth-Bendix algorithm. In Proc. Colloquium on Algebra, Combinatorics and Logic in Computer Science. J. Bolyai Math. Soc., J. Bolyai Math. Soc. and North-Holland, 1985. (Colloquium Mathematicum Societatis J. Bolyai, Györ, Hungary, 1983).

    Google Scholar 

  33. Franz Winkler. The Church-Rosser Property in Computer Algebra and Special Theorem Proving: An Investigation of Critical-Pair/Completion Algorithms. PhD thesis, Johannes Kepler Universität Linz, May 1984.

    Google Scholar 

  34. Hanatao Zhang and Deepak Kapur. Consider only general superpositions in completion procedures. In Nachum Dershowitz, editor, Rewriting Techniques and Applications (LNCS 355), pages 513–527. Springer-Verlag, 1989. (Proc. RTA'89, Chapel Hill, NC, USA, April 1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wilhelm-Schickard-Institut, Universität Tübingen, D-72076, Tübingen, Fed. Rep. of Germany

    Reinhard Bündgen

Authors
  1. Reinhard Bündgen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Alan Bundy

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bündgen, R. (1994). On pot, pans and pudding or how to discover generalised critical Pairs. In: Bundy, A. (eds) Automated Deduction — CADE-12. CADE 1994. Lecture Notes in Computer Science, vol 814. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58156-1_50

Download citation

  • DOI: https://doi.org/10.1007/3-540-58156-1_50

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58156-7

  • Online ISBN: 978-3-540-48467-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics