Associative path orderings
In this paper we introduce a new class of orderings — associative path orderings — for proving termination of associative commutative term rewriting systems. These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E-congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. The associative path ordering is similar to another termination ordering for proving AC termination, described in Dershowitz, et al. (83), which is also based on the idea of transforming terms. Our ordering is conceptually simpler, however, since any term is transformed into a single term, whereas in Dershowitz, et al. (83) the transform of a term is a multiset of terms. More important yet, we show how to lift our ordering to non-ground terms, which is essential for applications of the Knuth-Bendix completion method but was not possible with the previous ordering.
Associative path orderings require less expertise than polynomial orderings. They are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition, the associative pair condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings, boolean algebra, etc.) and, in addition, point out ways of dealing with equational theories for which the associative pair condition does not hold.
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