Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A well-known fact is uniform normalisation of orthogonal non-erasing term rewrite systems, e.g. the λI-calculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and non-erasingness to the non-linear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first- and second-order term rewrite systems as well as to a λ-calculus with explicit substitutions. 1

In orthogonal expression reduction systems, a common generalization of term rewriting and #-calculus, we extend the concepts of normalization and needed reduction by considering, instead of the set of normal forms, a set S of "results". When S satisfies some simple axioms, we prove the corresponding generalizations of some fundamental theorems: the existence of needed redexes, that needed reduction is normalizing, the existence of minimal normalizing reductions, and the optimality theorem. 1 Introduction Since a normalizable term in a rewriting system may have an infinite reduction, it is important to have a normalizing strategy which enables one to construct reductions to normal form. It is well known that the leftmost-outermost strategy is normalizing in the #-calculus [10]. Normalization by Needed Reduction For Orthogonal Term Rewriting Systems (OTRSs), a general normalizing strategy, called the needed strategy, was found by Huet and Levy in [18]. The needed strategy always contr...

For a notion of reduction in a #-calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that single-step reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form, or all reduction paths will lead to a normal form.