Labelled calculi of resources

Lévy’s labelled λ-calculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λ-evaluators: labels relate to the multiplicative information of paths. In this paper, we generalise the structure of labels, and the underlying term structure, in order to keep track of exponential information too. We first define two labelled calculi with explicit substitutions and resource management, where labels are in close correspondence with paths in call-by-value and call-by-name translations of the λ-calculus into linear logic proof nets, respectively. We observe a tight relationship between labels and the dynamics of substitutions; this will then guide us through the design of a third calculus that combines the advantages of the previous two, where labels fully reflect the dynamics of substitutions.