Combining computer algebra and rule based reasoning

  • Reinhard Bündgen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 958)


We present extended term rewriting systems as a means to describe a simplification relation for an equational specification with a built-in domain of external objects. Even if the extended term rewriting system is canonical, the combined relation including built-in computations of ‘ground terms’ needs neither be terminating nor confluent. We investigate restrictions on the extended term rewriting systems and the built-in domains under which these properties hold. A very important property of extended term rewriting systems is decomposition freedom. Among others decomposition free extended term rewriting systems allow for efficient simplifications. Some interesting algebraic applications of canonical simplification relations are presented.


Simplification Relation Computer Algebra System External Object Computation Relation Critical Pair 
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  1. [AB94]
    J. Avenhaus and K. Becker. Operational specifications with builtins. In P. Enlbert, E. W. Mayr, and K. W. Wagner, editors, STACS 94 (LNCS 775), pages 263–274. Springer-Verlag, 1994. (Proc. STACS'94, Caen, France, February 1994).Google Scholar
  2. [BKR87]
    B. Benninghofen, S. Kemmerich, and M. M. Richter. Systems of Reductions. Springer-Verlag, Berlin, 1987.Google Scholar
  3. [BL82]
    Bruno Buchberger and Rüdiger Loos. Algebraic simplification. In Computer Algebra, pages 14–43. Springer-Verlag, 1982.Google Scholar
  4. [Buc65]
    Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, UniversitÄt Innsbruck, 1965.Google Scholar
  5. [Buc79]
    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. EU-ROSAM'79, Marseille, France).Google Scholar
  6. [Bün91a]
    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
  7. [Bün91b]
    Reinhard Bündgen. Simulating Buchberger's algorithm by Knuth-Bendix completion. In Ronald V. Book, editor, Rewriting Techniques and Applications (LNCS 488), pages 386–397. Springer-Verlag, 1991. (Proc. RTA'91, Como, Italy, April 1991).Google Scholar
  8. [Bün92]
    Reinhard Bündgen. Buchberger's algorithm: The term rewriter's point of view. In G. Kuich, editor, Automata, Languages and Programming (LNCS 623), pages 380–391, 1992. (Proc. ICALP'92, Vienna, Austria, July 1992).Google Scholar
  9. [Bün94a]
    Reinhard Bündgen. On pots, pans and pudding or how to discover generalized critical pairs. In 12th International Conference on Automated Deduction, (LNCS). Springer-Verlag, 1994. (Proc. CADE'94, Nancy, France, July 1994).Google Scholar
  10. [Bün94b]
    Reinhard Bündgen. Preserving confluence for rewrite systems with built-in operations. In Workshop on Conditional and (Typed) Term Rewriting Systems, 1994. (also to appear in LNCS).Google Scholar
  11. [Der87]
    Nachum Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69–115, 1987.Google Scholar
  12. [DJ90]
    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
  13. [Hul79]
    Jean-Marie Hullot. Associative-commutative pattern matching. In Fifth IJCAI, Tokyo, Japan, 1979.Google Scholar
  14. [Hul80]
    Jean-Marie Hullot. Canonical forms and unification. In Proc. Fifth International Conference on Automated Deduction (LNCS 87), pages 318–334. Springer-Verlag, 1980.Google Scholar
  15. [JK86]
    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.CrossRefGoogle Scholar
  16. [KB70]
    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. [KC89]
    Stéphane Kaplan and Christine Choppy. Abstract rewriting with concrete operators. In Nachum Dershowitz, editor, Rewriting Techniques and Applications (LNCS 355), pages 178–186. Springer-Verlag, 1989. (Proc. RTA'89, Chapel Hill, NC, USA, April 1989).Google Scholar
  18. [KR94]
    M. R. K. Krishna Rao. Simple termination of hierarchical combinations of term rewriting systems. In P. Enlbert, E. W. Mayr, and K. W. Wagner, editors, STACS 94 (LNCS 775), pages 203–223. Springer-Verlag, 1994. (Proc. STACS'94, Caen, France, February 1994).Google Scholar
  19. [Küc85]
    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
  20. [New42]
    M. H. A. Newman. On theories with a combinatorial definition of “equivalence”. Annals of Mathematics, 43(2):223–243, 1942.Google Scholar
  21. [Ohl94]
    Enno Ohlebusch. Modular Properties of Composable Term Rewriting Systems. PhD thesis, UniversitÄt Bielefeld, D-33501 Bielefeld, Germany, Mai 1994.Google Scholar
  22. [PS81]
    G. Peterson and M. Stickel. Complete sets of reductions for some equational theories. Journal of the ACM, 28:223–264, 1981.CrossRefGoogle Scholar
  23. [Sti81]
    Mark E. Stickel. A unification algorithm for associative-commutative functions. JACM, 28(3):423–434, July 1981.CrossRefGoogle Scholar
  24. [WB85]
    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
  25. [Wol91]
    Stephen Wolfram. Mathematica: a system for doing mathematics by computer. Addison-Wesley, Redwood City, CA, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Reinhard Bündgen
    • 1
  1. 1.Wilhelm-Schickard-InstitutUniversitÄt TübingenTübingenFed. Rep. of Germany

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