Simulation Buchberger's Algorithm by Knuth-Bendix Completion

We present a canonical term rewriting system whose initial model is isomorphic to GF(q)[x1,...,x n ]. Using this set of rewrite rules and additional ground equations specifying an ideal we can simulate Buchberger's algorithm for polynomials over finite fields using Knuth-Bendix term completion modulo AC. In order to simplify our proofs we exhibit a critical pair criterion which transforms critical pairs into simpler ones.

[1]  Rüdiger Loos,et al.  Term Reduction Systems and Algebraic Algorithms , 1981, GWAI.

[2]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[3]  Franz Winkler,et al.  Knuth-Bendix procedure and Buchberger algorithm: a synthesis , 1989, ISSAC '89.

[4]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[5]  Nachum Dershowitz,et al.  A Note on Simplification Orderings , 1979, Inf. Process. Lett..

[6]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[7]  Philippe le Chenadec.  Canonical forms in finitely presented algebras , 1984, Research notes in theoretical computer science.

[8]  Paliath Narendran,et al.  An Ideal-Theoretic Approach to Work Problems and Unification Problems over Finitely Presented Commutative Algebras , 1985, RTA.

[9]  Mark E. Stickel,et al.  A Unification Algorithm for Associative-Commutative Functions , 1981, JACM.

[10]  Jieh Hsiang,et al.  Topics in Automated Theorem Proving and Program Generation , 1983 .

[11]  Nachum Dershowitz,et al.  Termination of Rewriting , 1987, J. Symb. Comput..

[12]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[13]  Nachum Dershowitz,et al.  Completion and Its Applications , 1989 .

[14]  Nachum Dershowitz,et al.  Critical Pair Criteria for Completion , 1988, J. Symb. Comput..

[15]  Bruno Buchberger,et al.  A criterion for eliminating unnecessary reductions in the Knuth-Bendix algorithm , 1983, SIGS.

[16]  D. McIlroy.  Algebraic Simplification , 1966, CACM.