On how to move mountains ‘associatively and commutatively’

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.

Supported by the U.K. Science and Engineering Research Council under grant GR/E83634