Relational type-checking for MELL proof-structures. Part 1: Multiplicatives

Relational semantics for linear logic is a form of non-idempotent intersection type system, from which several informations on the execution of a proof-structure can be recovered. An element of the relational interpretation of a proof-structure R with conclusion $Γ$ acts thus as a type (refining $Γ$) having R as an inhabitant. We are interested in the following type-checking question: given a proof-structure R, a list of formulae $Γ$, and a point x in the relational interpretation of $Γ$, is x in the interpretation of R? This question is decidable. We present here an algorithm that decides it in time linear in the size of R, if R is a proof-structure in the multiplicative fragment of linear logic. This algorithm can be extended to larger fragments of multiplicative-exponential linear logic containing $λ$-calculus.

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