An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras

  • Abdelilah Kandri-Rody
  • Deepak Kapur
  • Paliath Narendran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 202)


A new approach based on computing the Gröbner basis of polynomial ideals is developed for solving word problems and unification problems for finitely presented commutative algebras. This approach is simpler and more efficient than the approaches based on generalizations of the Knuth-Bendix completion procedure to handle associative and commutative operators. It is shown that (i) the word problem over a finitely presented commutative ring with unity is equivalent to the polynomial equivalence problem modulo a polynomial ideal over the integers, (ii) the unification problem for linear forms is decidable for finitely presented commutative rings with unity, (iii) the word problem and unification problem for finitely presented boolean polynomial rings are co-NP-complete and co-NP-hard respectively, and (iv) the set of all unifiers of two forms over a finitely presented abelian group can be computed in polynomial time. Examples and results of algorithms based on the Gröbner basis computation are also reported.

Key Words

Word Problem Unification Problem Finitely Presented Algebras Commutative Algebras Gröbner Basis Polynomial Ideals Term Rewriting Knuth-Bendix Completion Procedure 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Abdelilah Kandri-Rody
    • 1
  • Deepak Kapur
    • 2
  • Paliath Narendran
    • 2
  1. 1.Department des Mathematiques Faculte des SciencesUniversity Cadi AyyadMarrakechMorocco
  2. 2.Computer Science Branch Corporate Research and DevelopmentGeneral Electric CompanySchenectady

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