Abstract
We show how Buchberger's algorithm for Q[x1,..., xr] can be simulated using term completion modulo AC. To specify the rational numbers an infinite term rewriting system is needed. However, for the simulation of each particular ideal completion a finite approximation of the infinite rule set is sufficient. This approximation can be constructed during the completion. Then the division operation in Q reduces to a narrowing procedure which becomes part of the critical pair computation process.
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References
Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nultdimensionalen Polynomideal. PhD thesis, Universität Innsbruck, 1965.
Bruno Buchberger. Gröbner bases: an algorithmic method in polynomial ideal theory. In N. K. Böse, editor, Recent Trends in Multidimensional Systems Theory, chapter 6, Reidel, 1985. (Also Report CAMP-83.29, U. Linz, 1983).
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).
Reinhard Bündgen. Simulating Buchberger's algorithm by Knuth-Bendix completion. In Ronald V. Book, editor, Rewriting Techniques and Applications (LNCS 488), Springer-Verlag, 1991. (Proc. RTA'91, Como, Italy, April 1991).
Reinhard Bündgen. Term Completion Versus Algebraic Completion. PhD thesis, Universität Tübingen, D-7400 Tübingen, Germany, May 1991. (reprinted as report WSI 91-3).
Nachum Dershowitz. Completion and its applications. In H. Aït-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, chapter 2, Academic Press, 1989.
Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuven, editor, Formal Models and Semantics, chapter 6, Elsevier, 1990.
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).
A. Kandri-Rody, D. Kapur, and F. Winkler. Knuth-Bendix procedure and Buchberger algorithm —a synthesis. In International Symposium on Symbolic and Algebraic Computation, pages 55–67, 1989. (Proc. ISSAC'89; in the proceedings only Winkler is listed as author).
Philippe Le Chenadec. Canonical Forms in Finitely Presented Algebras. Pitman, London, 1986.
Rüdiger Loos. Term reduction systems and algebraic algorithms. In Jörg H. Siekmann, editor, GWAI-81 (IFB 47), pages 214–234, Springer-Verlag, 1981. (Proc. German Workshop on AI, Bad Honnef, 1981).
Loïc Pottier. Algorithmes de complétion et généralison en logique du premier ordre PhD thesis, Université Nice-Sophia Antipolis, Parc Valrose, Fac. Sciences, F-06034 Nice, France, February 1989.
G. Peterson and M. Stickel. Complete sets of reductions for some equational theories. Journal of the ACM, 28:223–264, 1981.
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© 1992 Springer-Verlag Berlin Heidelberg
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Bündgen, R. (1992). Buchberger's algorithm: The term rewriter's point of view. In: Kuich, W. (eds) Automata, Languages and Programming. ICALP 1992. Lecture Notes in Computer Science, vol 623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55719-9_90
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DOI: https://doi.org/10.1007/3-540-55719-9_90
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