Infinitary Lambda Calculi from a Linear Perspective

Abstract

We introduce a linear infinitary λ-calculus, called Λ∞, 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 Λ∞, 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 Λ built around the principles of SLL and 4LL. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.

Finitary Simulation of Infinitary $\beta$-Reduction via Taylor Expansion, and Applications

“…Definition 2.1 can be unfolded using a mixed formal system (in such a system, simple bars denote inductive rules and double bars denote coinductive rules). This reformulation, inspired by [Dal16], provides a graphical description of terms in Λ 001 ∞ . Definition 2.2 (001-infinitary terms, using a mixed formal system).…”

“…Definition 2.10 provides an inductive-coinductive presentation of the notion of strongly convergent reduction sequences defined by [Ken+97], in the specific setting of Λ 001 ∞ : the only coinductive step occurs in argument position in the application rule, which is the position where depth 001 is incremented. In that we follow Dal Lago [Dal16], whereas the fully coinductive approach of Endrullis and Polonsky [EP13] is limited to Λ 111 ∞ .…”