This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.

. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evy-equivalent (or permutation-equivalent) to a given, finite or infinite, regular (or semi-linear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evy-equivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable -term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...

This paper surveys a part of the theory of fi-reduction in -calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from -terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in -terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in -calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in type-free -calculus [4, 6, 7, 15, 38, 44, 81]---see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...

We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the -calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...

. We show how higher-order rewriting may be encoded into

This paper surveys a part of the theory of fi-reduction in λ-calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λ-terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λ-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λ-calculus and type theory.

Abstract. The λ-calculus has been much used to study the theory of substitution in logical systems and programming languages. However, with explicit substitutions, it is possible to get finer properties with respect to gradual implementations of substitutions as effectively done in runtimes of programming languages. But the theory of explicit substitutions has some defects such as non-confluence or the non-termination of the typed case. In this paper, we stress on the sub-theory of weak substitutions, which is sufficient to analyze most of the properties of programming languages, and which preserves many of the nice theorems of the λ-calculus. 1

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

A rewrite system has standardization i for any rewrite sequence there is an equivalent one which contracts the redexes in a standard order. Standardization is extremely useful for finding normalizing strategies and proving that a rewrite system for a programming language is sound with respect to the language's operational semantics. Although for some rewrite systems the standard-order can be simple, e.g., left-to-right or outermost-first, many systems need a more delicate order. There are abstract notions of standard order which always apply, but proofs (often quite dicult) are required that the rewrite system satis es a number of axioms and not much guidance is provided for finding a concrete order that satisfies the abstract definition. This paper gives a framework based on combinatory reduction systems (CRS's) which is general enough to handle many programming languages. If the CRS is orthogonal and fully extended and a good redex ordering can be found, then a standard order is obtain...

Abstract. The last few years have seen the development of the rewriting calculus (also called rho-calculus or ρ-calculus) that uniformly integrates first-order term rewriting and λ-calculus. The combination of these two latter formalisms has been already handled either by enriching first-order rewriting with higher-order capabilities, like in the Combinatory Reduction Systems (crs), or by adding to λ-calculus algebraic features. In a previous work, the authors showed how the semantics of crs can be expressed in terms of the ρ-calculus. The converse issue is adressed here: rewriting calculus derivations are simulated by Combinatory Reduction Systems derivations. As a consequence of this result, important properties, like standardisation, are deduced for the rewriting calculus.