Quantitative Types for the Linear Substitution Calculus

Abstract : We define two non-idempotent intersection type systems for the linear substitution calculus, a calculus with partial substitutions acting at a distance that is a computational interpretation of linear logic proof-nets. The calculus naturally express linear-head reduction, a notion of evaluation of proof nets that is strongly related to abstract machines. We show that our first (resp. second) quantitave type system characterizes linear-head, head and weak (resp. strong) normalizing sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which decreases with respect to each reduction relation considered in the paper.