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Canonical Forms in Finitely Presented Algebras

  • Philippe Le Chenadec
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 170)

Abstract

This paper is an overview of rewriting systems as a tool to solve word problems in usual algebras. A successful completion of an equational theory, defining a variety of algebras, induces the existence of a completion procedure for the finite presentations in this variety.

The common background of these algorithms implies a unified vision of several well-known algorithms: Thue systems, abelian group decomposition, Dehn systems for small cancellation groups, Buchberger and Bergman’s algorithms, while experiments on many classical groups proove their practical efficiency despite negative decidability results.

Keywords

Finitely Presented Algebras Word Problem Rewriting Systems Completion Procedures 

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References

  1. Ba179a.
    A.M. Ballantyne and D.S. Lankford, New Decision Algorithms for Finitely Presented Commutative Semigroups, Report MTP-4, Department of Mathematics, Louisiana Tech. U (May 1979).Google Scholar
  2. Ber78a.
    George M. Bergman, The Diamond Lemma for Ring Theory, Advances in Mathematics, 29,2 (pp. 178–218) (Aug. 1978).Google Scholar
  3. Boo82b.
    Ronald V. Book, Confluent and Other Types of Thue Systems, Journal of the ACM, Vol 89, No. 1 (January 1982).Google Scholar
  4. Boo82a.
    Ronald V. Book, Matthias Jantzen, and Celia Wrathall, Monadic Thue Systems, Theoretical Computer Science 19, p. 231–251. North-Holland Publishing (1982).MathSciNetCrossRefGoogle Scholar
  5. Buc81a.
    B. Buchberger, “H-Bases and Grobner-Bases for Polynomial ideals,” CAMP Publ. No 81-2.0, Johannes Kepler Universitat, Austria (Fevrier 1981).Google Scholar
  6. Buc82a.
    B. Buchberger, “Miscellaneous results on Grobner-Bases for polynomial ideals II,” CAMP Publ. No 83-23.0, Johannes Kepler Universitat, Austria (Juin 1982).Google Scholar
  7. Buc79a.
    Hans Bucken, Reduction Systems and Small Cancellation Theory, Proc. Fourth Workshop on Automated Deduction, pp. 53–59 (1979).Google Scholar
  8. Che83a.
    Philippe Le Chenadec, Formes canoniques dans les algèbres finiment présentées, Thèse de 3ème cycle, Univ. d’Orsay (Juin 1983).Google Scholar
  9. Cli67a.
    A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, Vol II, Amer. Math. Soc. Providence, Rhode Island (1967).zbMATHGoogle Scholar
  10. Coc76a.
    Y. Cochet, Church-Rosser congruences on free semigroups, Colloq. Math. Soc. Janos Bolyai: Algeb. theory of Semigroups. 20, 51–80 (1976).MathSciNetzbMATHGoogle Scholar
  11. Cox72a.
    H.S.M. Coxeter and W.O.J Moser, Generators aM Relations for Discrete Groups, Springer-Verlag (1972).Google Scholar
  12. Deh12a.
    M. Dehn, Transformation der Kurve auf zweiseitigen Flache, Math Ann. 72, 413–420 (1912).MathSciNetCrossRefzbMATHGoogle Scholar
  13. Der82a.
    Nachum Dershowitz, Orderings for Term-rewriting Systems, Theoretical Computer Science 17, pp. 279–301 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fag84a.
    F. Fages, Associative-Commutative Unification, 7th Conference on Automated Deduction, Napa Valley California (May 1984).Google Scholar
  15. GreS0a.
    M. Greendlinger, Dehn’s Algorithm for the Word Problem, Communications on Pure and Applied Mathematics, 13, pp. 67–83 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hue80a.
    G. Huet, Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems., JACM 27,4, pp. 797–821 (Oct. 1980).Google Scholar
  17. Hue81a.
    G. Huet, A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm, JCSS 23,1, pp. 11–21 (Aug. 1981).Google Scholar
  18. Hue78a.
    G. Huet and D.S. Lankford, On the Uniform Halting Problem for Term Rewriting Systems, Rapport Laboria 283, IRIA (Mars 1978).Google Scholar
  19. Hue80b.
    G. Huet and D. Oppen, Equations and Rewriting Rules: a Survey, In Formal languages: Perspectives and Open Problems, Ed. Book R., Academic Press (1980).Google Scholar
  20. Hul80a.
    J.M. Hullot, Compilation de Formes Canoniques dans les Théories Equationnelles, Thèse de 3ème cycle, U. de Paris Sud (Nov. 80).Google Scholar
  21. Jou83a.
    J.P Jouannaud, H. Kirchner, and J.L. Remy, Church-Rosser properties of weakly terminating equational term rewriting systems, Centre de Recherche en Informatique de Nancy, R-004 (1983).Google Scholar
  22. Jou81a.
    Jean-Pierre Jouannaud, Pierre Lescanne, and Fernand Reinig, Recursive decomposition ordering, Rapport GRECO No 2 (1981).Google Scholar
  23. Knu70a.
    D. Knuth and P. Bendix, Simple Word Problems in Universal Algebras, In Computational Problems in Abstract Algebra Ed. Leech J., Pergamon Press, pp. 263–297. (1970).Google Scholar
  24. Liv76a.
    M. Livesey and J. Siekmann, Unification of Bags and Sets, Internal Report 3/76, Institut fur Informatik I, U. Karlsru∼he (1976).Google Scholar
  25. Lyn77a.
    R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer-Verlag (1977).Google Scholar
  26. Met3.a.
    Yves Metivier, Systèmes de réécriture de termes et de mots, Thèse de 3ème cycle d’enseignement supérieur, No d’ordre 1841, Université de Bordeaux I (26 Mai 1983.).Google Scholar
  27. Nov55a.
    P.S. Novikov, On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44, 143 (1955).Google Scholar
  28. Pet82a.
    Gerald E. Peterson and Mark E. Stickel, “Complete systems of reductions using associative and/or commutative unification,” Technical note 269, SRI International (October 1982).Google Scholar
  29. Smi66a.
    David A. Smith, A basis algorithm for finitely generated abelian groups, Mathematical Algorithms Vol I,1 (Jan. 1966).Google Scholar
  30. Sti81a.
    M.E. Stickel, A Complete Unification Algorithm for Associative-Commutative Functions, JACM 28,3 pp 423–434 (1981).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Philippe Le Chenadec
    • 1
  1. 1.INRIADomaine de VoluceauLe Chesnay CedexFrance

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