For two hierarchies of properties of term rewriting systems related to con uence and termination, respectively, we prove relative undecidability : for implications X ) Y in the hierarchies the property X is undecidable for term rewriting systems satisfying Y .

In this paper we introduce Associative Commutative Distributive Term Rewriting (ACDTR), a rewriting language for rewriting logical formulae. ACDTR extends AC term rewriting by adding distribution of conjunction over other operators. Conjunction is vital for expressive term rewriting systems since it allows us to require that multiple conditions hold for a term rewriting rule to be used. ACDTR u...

We present a novel proof of Toyama’s famous modularity of confluence result for term rewriting systems. Apart from being short and intuitive, the proof is modular itself in that it factors through the decreasing diagrams technique for abstract rewriting systems, is constructive in that it gives a construction for the converging rewrite sequences given a pair of diverging rewrite sequences, and ...

Recently, rewriting induction, which is one of the induction principles for proving inductive theorems in equational theory, has been extended to deal with constrained term rewriting systems. Rewriting induction has been applied to developing a method for proving the equivalence of imperative programs. To prove inductive theorems, there are many cases where appropriate lemmas need to be added. ...

This paper shows that “root-only” rewrite relations with respect to term rewriting systems can be expressed using Kleene star operations in a gs-monoidal Kleene category with converse. In our framework, we can analyze some properties of term rewriting systems by computing rewrite descendants of tree languages. As an application, we consider an infinite state model-checking problem given by a te...

This paper addresses the problem of how to represent free variables and subexpressions involving-bindings. The aim is to apply what is known as higher-order abstract syntax to higher-order term rewriting systems. Directly applying-reduction for the purpose of subterm-replacement is incompatible with the requirements of term-rewriting. A new meta-level representation of subterms is developed tha...

Term rewriting has a large potential for industrial applications, but these applications are always larger than one could ever dream of: huge sets of rewrite rules and gigantic terms to rewrite pose interesting challenges for implementors and theoreti-cians alike. We give a brief overview of the generation of term-rewriting-based tools as done in the Asf+Sdf Meta-Environment and then we sketch ...

Topological dynamics is, roughly, the study of phenomena related to iterations of continuous maps from a metric space to itself. We show how the rewrite relation in term rewriting gives rise to dynamical systems in two distinct, natural ways: (A) One in which any deterministic rewriting strategy induces a dynamical system on the set of finite and infinite terms endowed with the usual metric, an...

A term rewriting system is called complete if it is both confluent and strongly normalising. Barendregt and Klop showed that the disjoint union of complete term rewriting systems does not need to be complete. In other words, completeness is not a modular property of term rewriting systems. Toyama, Klop and Barendregt showed that completeness is a modular property of left-linear term rewriting s...

Based on a simple metric and a simple partial order on term graphs, we develop two infinitary calculi of term graph rewriting. We show that, similarly to infinitary term rewriting, the partial order formalisation yields a conservative extension of the metric formalisation of the calculus. By showing that the resulting calculi simulate the corresponding well-established infinitary calculi of ter...