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On how to move mountains ‘associatively and commutatively’

  • Mike Lai
Regular Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 355)

Abstract

In this paper we give another characterization of a set of rules which defines a Church-Rosser reduction on the term algebra specified by some associative and commutative equations. This characterization requires fewer conditions to be satisfied than those previously given in the literature do. As a result, when the required conditions are satisfied, the word problem in the term algebra defined by the set of rules and the set of associative and commutative equations can be solved by successive applications of rewriting to the elements in question.

In addition, what makes this approach different from the others is that notions such as AC-compatibility or coherence modulo AC of reductions induced by sets of rules, which are essential in [Pe-S] or [Jo-Ki] respectively, are not required here. Consequently, a proof of correctness of the completion algorithm (given in [Lai 2]) for constructing a desired set of rules based on this approach can be compared directly with that of Huet in [Hu 2]. In fact, it turns out that all we have to do is to replace terms in [Hu 2] by AC-equivalence classes of terms. The main reason is that all the complications due to AC-compatibility or coherence modulo AC simply are not present here.

Finally, we shall discuss how to minimize the unnecessary computation of some critical pairs during the completion.

Keywords

Equivalence Relation Peak Reduction Word Problem Free Algebra Free Abelian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    L.Bachmair “Proof Methods for Equational Theories” Ph.D. Thesis Univ. of Illinois. (1987)Google Scholar
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    G. Huet “Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems” J.ACM 27 797–821 (1980)Google Scholar
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    M.Lai “A Peak Reduction in Rewritings of Term Algebras ‘Associatively and Commutatively”’ To appear (1988)Google Scholar
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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Mike Lai
    • 1
  1. 1.Department of Computer Science Royal Holloway and Bedford New CollegeUniversity of LondonUK

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