Infinitary Lambda Calculi from a Linear Perspective

We introduce a linear infinitary λ-calculus, called ℓΛ
_∞
, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative λ-calculus and is universal for computations over infinite strings. What is particularly interesting about ℓΛ
_∞
, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of ℓΛ built around the principles of SLL and 4LL. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.