In this paper we present a novel condensed narrowing-like semantics that contains the minimal information which is needed to describe compositionally all possible rewritings of a term rewriting system. We provide its goal-dependent top-down definition and, more importantly, an equivalent goal-independent bottom-up fixpoint characterization. We prove soundness and completeness w.r.t. the small-s...

One of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the cong...

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...

We consider combinations of term rewriting with built-in operations which compute objects of some external domain from ground terms. Combining a canonical term rewriting system with a computation relation does in general not lead to a canonical simpliication relation. We show how a canonical simpliication system can be constructed for such rewrite systems with built-in operations. Decomposition...

We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on speciic tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of pr...

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with lefthand side a gener...

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to control their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specifi...

Context-sensitive rewriting is a computational restriction of term rewriting used to model non-strict (lazy) evaluation in functional programming. The goal of this paper is the study and development of techniques to analyze the termination behavior of context-sensitive rewrite systems. For that purpose, several methods have been proposed in the literature which transform contextsensitive rewrit...