Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, ,which we allow to be infinite, are unique, in contrast to co-normal forms. Strongly converging fair reductions result in normal forms. In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for B6hm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by _L) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows, The top-terminating OTRSs of Dershowitz c.s. are examples of nonunifiable OTRSs.

Currying is a transformation of term rewrite systems which may contain symbols of arbitrary arity into systems which contain only nullary symbols, together with a single binary symbol called application. We show that for all term rewrite systems (whether orthogonal or not) the following properties are preserved by this transformation: strong normalization, weak normalization, weak Church-Rosser, completeness, semi-completeness, and the non-convertibility of distinct normal forms. Under the condition of leftlinearity we show preservation of the properties NF (if a term is reducible to a normal form, then its reducts are all reducible to the same normal form) and UN → (a term is reducible to at most one normal form). We exhibit counterexamples to the preservation of NF and UN → for non-left-linear systems. The results extend to partial currying (where some subset of the symbols are curried), and imply some modularity properties for unions of applicative systems.

A simple calculus (the Director String Calculus-DSC) for expressing abstractions is introduced, which captures the essence of the "long reach" combinators introduced by Turner. We present abstraction rules that preserve the applicative structure of the original lambda term, and that cannot increase the number of subterms in the translation.A translated lambda term can be reduced according to the evaluation rules of DSC. If this terminates with a DSC normal form, this can be translated into a lambda term using rules presetid below. We call this process of abstracting a lambda term, reducing to normal form in the space of DSC terms, and translating back to a lambda term an impZ.ementation.We show that our implementation of the lambda calculus is correct: For lambda terms with a normal form that contains no lambdas (ground term), the implementation is shown to yield a lambda calculus normal form. For lambda terms whose normal forms represent functions, it is shown that the implementation yields lambda terms that are beta-convertible in zero or more steps to the normal form of the original lambda term. In this sense, our implementation involves weak reduction according to Hindley et al. [9].

Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also been a significant influence on the design and implementation of real programming languages, most notably the functional and logic programming families of languages. For a theoretical perspective on the place of rewriting in Computer Science, see for example [14]. For a programming language perspective, see for example [16]

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