International audienceWe give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped MELL net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut-link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus

The multiset based relational model of linear logic induces a semantics of the type free λ-calculus, which corresponds to a non-idempotent intersection type system, System R. We prove that, in System R, the size of the type derivations and the size of the types are closely related to the execution time of λ-terms in a particular environment machine, Krivine's machine.

We prove a completeness result for Multiplicative Exponential Linear Logic (MELL): we show that the relational model is injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the relational model is exactly axiomatized by cut-elimination.