We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critica...

Length-two string rewriting systems are length preserving string rewriting systems that consist of length-two rules. This paper shows that confluence, termination, left-most termination and right-most termination are undecidable properties for length-two string rewriting systems. This results mean that these properties are undecidable for the class of linear term rewriting systems in which dept...

We show that, contrary to the situation in first-order term rewriting, almostnone of the usual properties of rewriting are modular for higher-order rewriting, irre-spective of the higher-order rewriting format. For the particular format of simply typedapplicative term rewriting systems we prove that modularity of confluence, normalization,and termination can be recovered by impo...

Term rewriting systems provide a versatile model of computation. An important property which allows to abstract from potential nondeterminism of parallel execution of the modelled program is confluence. In this paper we prove that confluence of a fairly large class of systems, namely right ground term rewriting systems, is decidable. We introduce a labelling of variables with colours and constr...

We define infinitary combinatory reduction systems (iCRSs). This provides the first extension of infinitary rewriting to higher-order rewriting. We lift two well-known results from infinitary term rewriting systems and infinitary λ-calculus to iCRSs: 1. every reduction sequence in a fully-extended left-linear iCRS is compressible to a reduction sequence of length at most ω, and 2. every complet...

The last open problem regarding the modularity of the fundamental properties of Term Rewriting Systems concerns the property of uniqueness of normal forms w r t reduction UN In this article we solve this open problem showing that UN is modular for left linear Term Rewriting Systems The novel pile and delete technique here introduced allows for quite a short proof and is of independent interest ...

We present a new proof of Chew's theorem, which states that normal forms are unique up to conversion in compatible term rewriting systems. We apply the technique of left-right separated conditional term rewriting systems (LRCTRSs), in which the unique normal form property of a term rewriting system is reduced to the Church-Rosser property of its conditional linearization. In contrast to traditi...

For the whole class of linear term rewriting systems, we define bottomup rewriting which is a restriction of the usual notion of rewriting. We show that bottom-up rewriting effectively inverse-preserves recognizability and analyze the complexity of the underlying construction. The BottomUp class (BU) is, by definition, the set of linear systems for which every derivation can be replaced by a bo...