Proving termination of (conditional) rewrite systems

In this paper an important problem in the domain of term rewriting, the termination of (conditional) rewrite systems, is dealt with. We show that in many applications, well-founded orderings on terms which only make use of syntactic information of a rewrite systemR, do not suffice for proving termination ofR. Indeed sometimes semantic information is needed to orient a rewrite rule. Therefore we integrate a semantic interpretation of rewrite systems and terms into a well-founded ordering on terms: the notion ofsemantic ordering is the first main contribution of this paper. The use and usefulness of the semantic ordering in proving termination is illustrated by means of some realistic examples.Furthermore the concept of semantic information induces a novel approach for proving termination inconditional rewrite systems. The idea is to employ not only semantic information contained in the terms that are to be compared, but also extra (semantic) information contained in the premiss of the conditional equation in which the terms appear. This leads to our second contribution in the termination problem area: the notion ofcontextual ordering andcontextual semantic ordering. Thecontextual approach allows to prove termination of conditional rewrite systems where all classical partial orderings would fail.