λν, a calculus of explicit substitutions which preserves strong normalisation

Abstract

Explicit substitutions were proposed by Abadi, Cardelli, Curien, Hardin and Lévy to internalise substitutions into λ-calculus and to propose a mechanism for computing on substitutions. λν is another view of the same concept which aims to explain the process of substitution and to decompose it in small steps. It favours simplicity and preservation of strong normalisation. This way, another important property is missed, namely confluence on open terms. In spirit, λν is closely related to another calculus of explicit substitutions proposed by de Bruijn and called CλξΦ. In this paper, we introduce λν, we present CλξΦ in the same framework as λν and we compare both calculi. Moreover, we prove properties of λν; namely λν correctly implements β reduction, λν is confluent on closed terms, i.e. on terms of classical λ-calculus and on all terms that are derived from those terms, and finally λν preserves strong normalisation in the following sense: strongly β normalising terms are strongly λν normalising.

Choices in Representation and Reduction Strategies for Lambda Terms in Intensional Contexts

“…The majority of the calculi that have been proposed do not, in fact, possess this ability. This is true, for example, of the λυ-calculus [BBLRD96], the λζ-calculus [Muñ96], the λs e -calculus [KR97] and the λ ws ocalculus [DG01]. The last two calculi are distinguished from the others in that they contain mechanisms for interchanging the order of substitutions in certain circumstances; such a device is needed to obtain confluence of rewriting under a graftable interpretation of meta variables.…”

“…However, not all explicit substitution calculi have this kind of combining ability. On the one hand, omitting such an ability has potential theoretical benefits: it is known at least in some cases to lead to the preservation of strong normalization properties of the underlying lambda calculus [BBLRD96,DG01]. On the other hand, we believe that an approach that does not use a mechanism for collapsing multiple substitution traversals into a composite one may not be viable in practice.…”

“…A well-known solution to the problem of identifying two lambda terms that differ only in the names chosen for bound variables is, for instance, to transform them into a nameless form using a scheme due to de Bruijn [Bru72]. Similarly, several new notations for the lambda calculus have been described in recent years that have the purpose of making substitutions explicit (e.g., see [ACCL91,BBLRD96,KR97,NW98]). However, the actual manner in which all these devices should be deployed in a practical context is far from clear.…”

Explicit substitutions with de bruijn's levels

“…For instance, Paul-André Mellies [18] has exhibited strongly β normalisable terms that are not strongly λσ normalisable or strongly λσ ⇑ normalisable. On the other hand, in [17,3], we have proved that the calculus of substitutions λυ preserves strong normalisation. A similar proof can be used for proving that λχ preserves strong normalisation.…”

Explicit Substitutions and Programming Languages

“…But it is confluent on closed terms (of explicit substitutions), which are sufficient to represent all λ-terms. Later several calculi were designed with full confluence [11], or with both properties by suppressing some of the operations of explicit substitutions such as the composition of substitutions [4,23,7]. Until very recently no fully expressive calculus existed with both properties of confluence and strong normalization.…”