On the Completeness of Narrowing for E-Unification

The narrowing mechanism has been a major tool in designing complete E-unification procedures for classes of equational theories in the last few years. Meanwhile, it has also been noticed that it is sometimes difficult to build complete procedures using only the narrowing mechanism. This paper analyzes several examples encountered by the authors in the course of research, which show that in certain situations narrowing fails to yield a complete E-unification procedure. Furthermore, we augment the narrowing mechanism with a restricted form of the paramodulation rule. This yields a complete E-unification procedure for all the equational theories that can be described by a confluent term rewriting system. We then present a new procedure in which paramodulation is eliminated by using a set of preconstructed terms, called cover set, which is a substantially reduced set of terms of certain complexity. This provides a degree of completeness, called relative completeness, which is measured by complexity of terms.

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