We introduce Context-sensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed -calculi possibly enriched with pattern-matching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbert-style proof systems, Gentzen-style sequent-calculi, rewrite systems with rule priorities, and the ß-calculus into CERSs. This last encoding is an (important) example of real context-sensitive rewriting. 1 Introduction A term rewriting system is a pair consisting of an alphabet and a set of rewrite rules. The alphabet is used freely to gene...

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...

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. 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 maximal reduction strategy in untyped λ-calculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λ-calculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, recasting the few known results in our framework and giving a number of new results, the most important of which is an effective maximal strategy in fij. We also present a number of applications illustrating the relevance and usefulness of maximal strategies.

. We define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted -calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higher-order) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...

. We define a class of hyperbalanced l-terms by imposing syntactic constraints on the construction of l-terms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded)subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted h-expansion, all simply typable l-terms are hyperbalanced, implying importance of the class of hyperbalanced terms. 1 Introduction The termination of b-reduction for typed terms is one of the most studied topics in l- calculus. After classical proofs of Tait [21] and Girard [8], many interesting proo...

this paper is to formalize the two

We introduce Context-sensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed λ-calculi possibly enriched with pattern-matching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbert-style proof systems, Gentzen-style sequent-calculi, rewrite systems with rule priorities, and the π-calculus into CERSs. This last encoding is an important example of real context-sensitive rewriting. ○c