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Relational semantics for linear logic is a form of non-idempotentintersection type system, from which several informations on the execution of aproof-structure can be recovered. An element of the relational interpretationof a proof-structure R with conclusion $\Gamma$ acts thus as a type (refining$\Gamma$) having R as an inhabitant. We are interested in the followingtype-checking question: given a proof-structure R, a list of formulae $\Gamma$,and a point x in the relational interpretation of $\Gamma$, is x in theinterpretation of R? This question is decidable. We present here an algorithmthat decides it in time linear in the size of R, if R is a proof-structure inthe multiplicative fragment of linear logic. This algorithm can be extended tolarger fragments of multiplicative-exponential linear logic containing$\lambda$-calculus.

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