Generating polynomial orderings

Abstract

There are various methods of proving termination of term rewriting systems (TRS). Most of these are based on reduction orderings, which are well-founded, compatible with the structure of terms and stable with respect to substitutions. The notion of reduction ordering allows the following characterization of termination of TRS: A TRS 9? terminates iff there exists a reduction ordering + such that I t r for each rule I + r of 9. Polynomial orderings are special reduction orderings, mapping terms into a well-founded set by attaching monotonic polynomials (so-called interpretations) to operators. They have been studied by Manna and Ness [12], Lankford [lo], Dershowitz [3], Huet and Oppen [8] and BenCherifa and Lescanne [l]. In contrast to the others, Dershowitz uses an arbitrary set of functions (polynomials) by requiring them to possess the subterm property, i.e. his ordering is a simplification ordering including the homeomorphic embedding relation. One of the main problems concerning polynomial orderings is the choice of adequate interpretations for a given TRS. The object of this paper is to describe an algorithm (being powerful in practice, see Section 5) based on the Simplex method for finding polynomial interpretations for a given TRS.