We investigate one of the classical problems of the theory ofterm rewriting, namely termination. We present an ordering for compar-ing higher-order terms that can be utilized for testing termination anddecreasingness of higher-order conditional term rewriting systems. Theordering relies on a first-order interpretation of higher-order terms anda suitable extension of the RPO.
Various methods for proving the termination of term rewriting systems havebeen suggested. Most of them are based on the notion of simplification ordering.In this paper, the theoretical time complexities (of the worst cases) of a collectionof well-known simplification orderings will be presented.
An important property and also a crucial point ofa term rewriting system is its termination. Transformation or-derings, developed by Bellegarde & Lescanne strongly based on awork of Bachmair & Dershowitz, represent a general technique forextending orderings. The main characteristics of this method aretwo rewriting relations, one for transforming terms and the otherfor ensuring the well-foundedness of the ordering. The centralproblem of this approach concerns the choice of the two relationssuch that the termination of a given term rewriting system can beproved. In this communication, we present a heuristic-based al-gorithm that partially solves this problem. Furthermore, we showhow to simulate well-known orderings on strings by transformationorderings.
Orderings on polynomial interpretations of operators represent a powerful technique for proving thetermination of rewriting systems. One of the main problems of polynomial orderings concerns thechoice of the right interpretation for a given rewriting system. It is very difficult to develop techniquesfor solving this problem. Here, we present three new heuristic approaches: (i) guidelines for dealingwith special classes of rewriting systems, (ii) an algorithm for choosing appropriate special polynomialsas well as (iii) an extension of the original polynomial ordering which supports the generation ofsuitable interpretations. All these heuristics will be applied to examples in order to illustrate theirpractical relevance.