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...

this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of -terms in . Quite apart from the new light it sheds on fi-reduction, such an analysis turns out to have several other benefits

We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some well-known results and proof techniques. Our proof uses a variant of Klop's extended -calculus, for which it is shown that strong normalization is equivalent to weak normalization. This is

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 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 embed the standard #-calculus, denoted #, into two larger #-calculi, denoted # # and &# # . The standard notion of #-reduction for # corresponds to two new notions of reduction, # # for # # and &# # for &# # . A distinctive feature of our new calculus # # (resp., &# # ) is that, in every function application, an argument is used at most once (resp., exactly once) in the body of the function. We establish various connections between the three notions of reduction, #, # # and &# # . As a consequence, we provide an alternative framework to study the relationship between #-weak normalization and #-strong normalization, and give a new proof of the oft-mentioned equivalence between #-strong normalization of standard #-terms and typability in a system of "intersection types".

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...

Extending the -calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the -calculus. In particular, we show that sg preserves strong normalisation, is sound and it simulates classical fi-reduction. Furthermore, we study the simply typed -calculus extended with both generalised reduction and explicit substitution and show that well-typed terms are strongly normalising and that other properties such as subtyping and subject reduction hold. 1 Introduction 1.1 The -calculus with generalised reduction In (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P which when contracted will turn the function starting with y and Q i...

Postponement of K -contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, that e has the Preservation of Strong Normalisation property. Keywords: Generalised -reduction, Postponement of K-contractions, Generalised Conservation, Preservation of Strong Normalisation. 1 The -calculus with generalized reduction In the term (( x : y :N)P )Q, the abstraction starting with x and the argument P form the redex ( x : y :N)P . When this redex is contracted, the abstraction starting with y and Q will in turn form a redex. It is important to note that Q (or some residual of Q) is the only argument that the abstraction (or some residual of the abstraction) starting with y can ever have. This fact has been exploited by many researchers. Reduction has been ex...

We define an equivalence relation on lambda-terms called shuffle-equivalence which attempts to capture the notion of reductional equivalence on strongly normalizing terms. The aim of reductional equivalence is to characterize the evaluation behavior of programs. The shuffle-equivalence classes are shown to divide the classes of beta-equal strongly normalising terms (programs which lead to the same final value/output) into smaller ones consisting of terms with similar evaluation behavior. We refine beta-reduction from a relation on terms to a relation on shuffle-equivalence classes, called shuffle-reduction, and show that this refinement captures existing generalisations of lambda-reduction. Shuffle-reduction allows one to make more redexes visible and to contract these newly visible redexes. Moreover, it allows 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. This can benefit both programming language...