This paper presents a purely syntactic account of type generativity and sharing – two key mechanisms in the SML module system – and shows its equivalence with the traditional stamp-based description of these mechanisms. This syntactic description recasts the SML module system in a more abstract, type-theoretic framework.

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.

This paper describes a compiling graph reduction system which realizes the reduction semantics of a fully-fledged applied λ-calculus. High-level functional programs are conceptually executed as sequences of program transformations governed by full β-reductions. They may be carried out step-by-step, and intermediate programs may be displayed in high-level notation, rendering the system suitable for interactive program design, high-level debugging, and also for teaching basic programming language concepts and language interpretation. Run-time efficiency for production runs is achieved by means of an abstract stack machine ASM which serves as an intermediate level of code generation. It employs multiple stacks for reasonably fast function calls, optimized tail-end recursions, and earliest possible releases of subgraphs that are no longer needed. The ASM involves an interpreter if and only if potential naming conflicts need to be resolved when reducing partial function applications.