Presentation of Rewriting and Typing 13 2 Abstract Rewriting Systems 15 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.2 Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.3 Morphisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 2.4 Properties of Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : 18 2.5 Strategies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.6 Criteria : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 2.7 Conclusions and Related Work : : : : : : : : : : : : : : : : : : : : : : : : : 24 3 Topology 27 3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.2 Topology : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 3.3 Equivalence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 3.4 Topological Characte...

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and -calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in -calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) -calculus. Note: This article is based on a lecture given by Jan Willem Klop at RTA '98 held in Tsukuba, Japan. Contents 1 Introduction 2 2 Early views on descendants 3 3 Preliminaries 5 3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Reduction . . . . . . . . . . . . . . . . . . . . . ....

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 ...

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.

For some typed λ-calculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. This paper presents a new technique to infer strong normalization of a notion of reduction in a typed λ-calculus from weak normalization of the same notion of reduction. The technique is demonstrated to work on some well-known systems including second-order λ-calculus and the system of positive, recursive types. It gives hope for a positive answer to the Barendregt-Geuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing. The paper also analyzes the relationship between the techniques mentioned above, and reviews, in less detail, other techniques for proving strong normalization.

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.

this paper is to formalize the two

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and -calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in -calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) -calculus. Note: This article is based on a lecture given by Jan Willem Klop at RTA '98 held in Tsukuba, Japan. Contents 1 Introduction 2 2 Early views on descendants 3 3 Preliminaries 5 3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Reduction . . . . . . . . . . . . . . . . . . . . . ....

Using a characterisation of strongly normalising -terms, we give new and simple proofs of the following: 1. all developments and superdevelopments are finite, 2. a certain rewrite strategy is perpetual, 3. a certain rewrite strategy is maximal and thus perpetual, 4. simply typed -calculus is strongly normalising. AMS Subject Classification (1991): 03B40, 03D70. CR Subject Classification (1991): F.3.3, F.4.1. Keywords & Phrases: -calculus, normalisation, perpetual strategies. Note: The research of the first author is supported by NWO/SION project 612-316-606. This report is also available as Computing Science Report 95/20, Eindhoven University of Technology. 1. Introduction 2 1. Introduction This paper represents an effort to shed some more light on various results concerning normalisation in -calculus. We deal with -calculus with only fi-reduction. As a first step towards a better understanding we characterise both the set of weakly normalising terms and the set of strongly norm...

proofs in the λ-calculus