The direct sum of two term rewriting systems (TRSs) [2,8] R, and R, is confluent if R, and R, are confluent [lo], but the direct sum is not necessarily terminating even if R, and R, are terminating [ll]; that is, termination is not a “modular” [6] property for general term rewriting systems. Recently, it was proved [9,5,12,4] that for some restricted classes of systems the direct sum is termina...

The notion of inductive theorems is well-established in first-order term rewriting. In higherorder term rewriting, in contrast, it is not straightforward to extend this notion because of extensionality (Meinke, 1992). When extending the term rewriting based program transformation of Chiba et al. (2005) to higher-order term rewriting, we need extensibility, a property stating that inductive theo...

We introduce Higher-Order Rewrite Systems (HRS) which extend term rewriting to-terms. HRS can describe computations on arbitrary terms with bound variables. We show that rewriting is closely related to undirected equational reasoning , and extend three connuence results from term rewriting to HRS: the critical pair lemma by Knuth and Bendix, connuence of rewriting modulo equations a la Huet, an...

Critical Pairs and Confluence of Arbitrary Binary Relations Rémy Haemmerlé and François Fages Projet Contraintes – INRIA Rocquencourt – France [email protected] Abstract. In a seminal paper, Huet introduced abstract properties of term rewriting systems, and the confluence analysis of terminating term rewriting systems by critical pairs computation. In this paper, we provide an abstrac...

Term graphs represent functional expressions such that common subexpressions can be shared, making expression evaluation more efficient than with strings or trees. Rewriting of term graphs proceeds by both applications of term rewrite rules and folding steps which enhance the degree of sharing. The present paper introduces critical pairs in term graph rewriting and establishes a Critical Pair L...

For a hierarchy of properties of term rewriting systems related to confluence we prove relative undecidability, i.e., for implications X ⇒ Y in the hierarchy the property X is undecidable for term rewriting systems satisfying Y . For some of the implications either X or ¬X is semi-decidable, for others neither X nor ¬X is semi-decidable. We prove most of these results for linear term rewrite sy...