Labelled Lambda-calculi with Explicit Copy and Erase

We present two rewriting systems that define labelled explicit substitution lambda-calculi. Our work is motivated by the close correspondence between Levy's labelled lambda-calculus and paths in proof-nets, which played an important role in the understanding of the Geometry of Interaction. The structure of the labels in Levy's labelled lambda-calculus relates to the multiplicative information of paths; the novelty of our work is that we design labelled explicit substitution calculi that also keep track of exponential information present in call-by-value and call-by-name translations of the lambda-calculus into linear logic proof-nets

A Fully Labelled Lambda Calculus: Towards Closed Reduction in the Geometry of Interaction Machine

AbstractWe investigate the possibility of performing new reduction strategies with the Geometry of Interaction Machine (GOIm). To this purpose, we appeal to Lévy's labelled lambda calculus whose labels describe: a) the path that the GOIm will follow in the graph of a term and b) the operations that the GOIm requires to compute the multiplicative part from the Multiplicative and Exponential Linear Logic encoding that the machine uses. Our goal is to unveil the missing exponential information in the structure of the labels. This will provide us with a tool to talk about strategies for computing paths with the GOIm

Geometry of Resource Interaction - A Minimalist Approach

International audienceThe Resource λ-calculus is a variation of the λ-calculus where arguments can be superposed and must be linearly used. Hence it is a model for linear and non-deterministic and programming languages, and the target language of Ehrhard-Taylor expansion of λ-terms. In a strictly typed restriction of the Resource λ-calculus, we study the notion of path persistence, and we define a Geometry of Interaction that characterises it. The construction is also invariant under reduction and able to count addends in normal forms

The cost of usage in the λ-calculus

Abstract-A new "inductive" approach to standardization for the λ-calculus has been recently introduced by Xi, allowing him to establish a double-exponential upper bound |M | 2 |σ| for the length of the standard reduction relative to an arbitrary reduction σ originated in M . In this paper we refine Xi's analysis, obtaining much better bounds, especially for computations producing small normal forms. For instance, for terms reducing to a boolean, we are able to prove that the length of the standard reduction is at most a mere factorial of the length of the shortest reduction sequence. The methodological innovation of our approach is that instead to try to count the cost for producing something, as customary, we count the cost of consuming things. The key observation is that the part of a λ-term that is needed to produce the normal form (or an arbitrary rigid prefix) may rapidly augment along a computation, but can only decrease very slowly (actually, linearly)