In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...

In the area of foundations of mathematics and computer science, three related topics dominate. These are -calculus, type theory and logic.

Extending the λ-calculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the - calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fi-reduction of g and the classical -calculus. Furthermore, we study the simply typed versions of s and gs, and show that well-typed terms are strongly normalizing and that other properties,...

The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oe-calculus. In (Kamareddine & R'ios, 1995a), we extended the -calculus with explicit substitutions by turning de Bruijn's meta-operators into objectoperators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the -calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine & R'ios, 1995a), is extended in this paper to a confluent calculus on open terms: the se-caculus. Since the establishment of these results, another calculus, i, came into being in (Mu~noz Hurtado, 1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that se still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi-reduction, whereas i is not. To ...

Postponement of fi K -contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. This then is used to solve an open problem of fi e : the Preservation of Strong Normalisation 1 . Keywords: Generalised fi-reduction, Postponement of K-contractions, Generalised Conservation, Preservation of Strong Normalisation. 1 Introduction 1.1 Background and Motivation In the term (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P . It is also the case that the function starting with y and the argument Q will result in another redex when the first redex is contracted. This idea has been exploited by many researchers and reduction has been extended so that the generalised redex based on the matching y and Q is given the same priority a...

The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oe-calculus. In [KR95a], we extended the -calculus with explicit substitutions by turning de Bruijn's meta-operators into object-operators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the -calculus from an intuitive point of view and, while preserving strong normalisation ([KR95a]), is extended in this paper to a confluent calculus on open terms: the s e -caculus. Since the establishment of the results of this paper 1 , another calculus, i, came into being in [MH95] which preserves strong normalisation and is itself confluent on open terms. However, we believe that s e still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi-reduction, whereas i is not. To prove confluence we introduce a ge...

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

We show the soundness of a -calculus B where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and step-wise reduction and the first clear account of exactly when ff-reduction is needed. Keywords: Explicit Substitution, de Bruijn indices, Variable names, Soundness. 1. Introduction Variables play a very demanding role in the reduction and substitution of the -calculus. This has lead in many cases to using explicit rather than implicit substitution. Implementations of the -calculus provide their own explicit substitution procedures as in Nuprl 9 and Automath 23 . Furthermore, research on theories of explicit substitution has been striving lately 5;12;13;22;4;18 . In this paper, we extend the calculus of [13] (which is influenced by Automath...

We study lambda calculus and refine the notions of fi-reduction and fi-equality. In particular, we define the operation TS (term reshuffling) on -terms which reshuffles a term in such a way that more redexes become visible. Two terms are called shuffleequivalent if they have syntactically equivalent TS-images. The shuffle-equivalence classes are shown to divide the classes of fi-equal terms into smaller classes consisting of terms with similar reduction behaviour. The refinement of fi-reduction from a relation on terms to a relation on shuffle classes, called shuffle-reduction, allows one to make more redexes visible and to contract these newly visible redexes. This enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. Moreover, this gain in reductional breadth is not at the expense of reductional length. We show that the -cube of [Barendregt 92] extended with shuffle...

. We present the ! and !e calculi, the two-sorted (term and substitution) versions of the s (cf. [KR95a]) and se (cf. [KR96a]) calculi, respectively. We establish an isomorphism between the s-calculus and the term restriction of the !-calculus, which extends to an isomorphism between se and the term restriction of !e . Since the ! and !e calculi are given in the style of the oe-calculus (cf. [ACCL91]) they bridge calculi between s and oe and between se and oe and thus we are able to better understand one calculus in terms of the other. We improve our knowledge on the open problem of strong normalisation (SN) of the associated calculus of substitutions se by showing SN for two subcalculi (we use the isomorphism with !e for the proof of SN of one of them). Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !-calculus is a calculus in the oe-style that simulates one step fi-reduction, is confluent ...