Modular termination of r-consistent and left-linear term rewriting systems

Abstract

A modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems, iff it holds for every involved term rewriting system. A term rewriting system is r-consistent, iff there is no term that can be rewritten to two different variables. We show that the subclass of left-linear and r-consistent term rewriting systems has the modular termination property. This subclass may also contain nonconfluent term rewriting systems. Since confluence implies r-consistency, this constitutes a generalisation of the theorem of Toyama, Klop, and Barendregt on the modularity of termination for confluent and left-linear term rewriting systems.

Modular proofs for completeness of hierarchical term rewriting systems

“…Toyama et al [28] established the modularity of completeness for left-linear term rewriting systems. Marchiori [16] and Schmidt-Schauss and Pintz [25] independently provided a simpler proof of this result.…”

Modular termination of context-sensitive rewriting

“…4 Further relevant works on modularity not mentioned elsewhere include (this list is highly incomplete): [22,23,24], [20], [21], [34,35,36], [42], [46] 5 Typically, the inverse implication is trivial.…”