Termination of term rewriting: interpretation and type elimination

Abstract

We investigate proving termination of term rewriting systems by interpretation of terms in a well-founded monotone algebra. The well-known polynomial interpretations can be considered as a particular case in this framework. A classi cation of types of termination, including simple termination, is proposed based on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of the system SUBST of Hardin and Laville (1986) is given. This system describes explicit substitution incalculus. Another tool for proving termination is based on introduction and removal of type restrictions. A property of many-sorted term rewriting systems is called persistent if it is not a ected by removing the corresponding typing restriction. Persistence turns out to be a generalization of direct sum modularity, but is more powerful for both proving con uence and termination. Termination is proved to be persistent for the class of term rewriting systems for which not both duplicating rules and collapsing rules occur, generalizing a similar result of Rusinowitch for modularity. This result has nice applications, in particular in undecidability proofs.

Proving Infinitary Normalization

“…the property that all rewrite sequences must end eventually in a normal form. Numerous advanced techniques and tools have been developed to prove SN, including interpretations of terms in monotone algebras [7,8] and in weakly monotone algebras [4].…”

Approximating Dependency Graphs Using Tree Automata Techniques

“…Transformation methods which do not attempt to prove termination directly but rather transform the given rewrite system into another rewrite system such that termination of the latter system is easier to prove and implies termination of the former system. Examples include the transformation order of Bellegarde and Lescanne [6], and Zantema's distribution elimination [27] and semantic labelling [28]. Transformations differ in their degree of automation.…”

Type introduction for equational rewriting

“…In this paper we extend the type introduction technique of Zantema [22] for proving properties of rewriting to equational rewriting. More precisely, we show that termination is a persistent property of equational rewrite systems R/E such that R does not contain both collapsing and duplicating rules and E is variable preserving and does not contain collapsing axioms.…”

“…This paper is organized as follows. In the next section we briefly define equational rewriting and we recall the results of Zantema [22] on type introduction.…”

“…In Section 4 the usefulness of the results of Section 3 is illustrated by showing the undecidability of AC-termination for terminating rewrite systems and in Section 5 we address persistence of acyclicity and non-loopingness. Persistence is closely related ( [18,22]) to modularity, a property which has been thoroughly investigated in the term rewriting literature. Along this line we obtain several new modularity results.…”