Topics in Termination

We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 I n t r o d u c t i o n If no infinite sequences of rewrites are possible, a rewrite system is said to have tile termiuation property. In practice, one usually guarantees termination by devising a well-founded (strict partial) ordering ~, such that s :,t whenever s rewrites to t. As suggested in [Manna and Ness, 1970], it is often convenient to separate reduction orderings into a homomorphism from terms to an algebra with a wellfounded ordering. Tile use, in particular, of polynomial interpretations which map terms into the natural nmnbers, was developed by Lankford [1979]. For a survey of termination methods, see [Dershowitz, 1987]. Virtually all orderings used in practice are simplification orderings [Dershowitz, 1982], satisfying the replacement property, that s ~t implies that any term containing s is not less (under ~) than the same term with that occurrence of s replaced by t, and the snbterm property, that any term containing s is greater or equal to s. Simplification orderings cannot be used to prove termination of "self-embedding" systems, that is, when a term t can be derived in one or more steps from a term t ~, and t t can be obtained by repeatedly replacing subterms of t with subterms of those subterms. Knuth and Bendix [1970] designed a particular class of well-orderings which assigns a weight to a term which is the sum of the weights of its constituent function symbols. Terms of equal weight and headed by the same symbol have their subterms compared lexicographieally. Another class of simplification orderiugs, tile path orderings [Dershowitz, 1982], is based on the idea that a term u should be bigger than any term that is built from smaller terms, all held together by a structure of function symbols that are smaller in some precedence ordering than the root symbol of u. The notion of path ordering was extended by Kamin and Ldvy [1980] to compare subterms lexicographically and to allow for a semantic component; see [Dershowitz, 1987]. Ilere, we generalize these orderings and the conditions under which they work. In the appendix, we describe an implementation of the general ordering. * This research was supported in part by the U. S. National Science Foundation under Grants CCR-90-07195 and CCR-90-24271. The first author was also supported by Lady Davis fellowship at the Hebrew University of Jerusalem.