Interaction Graphs: Multiplicatives

We introduce a graph-theoretical representation of proofs of multiplicative linear logic which yields both a denotational semantics and a notion of truth. For this, we use a locative approach (in the sense of ludics [Gir01]) related to game semantics [HO00, AJM94] and the Danos-Regnier interpretation of GoI operators as paths in proof nets [ADLR94, DR95]. We show how we can retrieve from this locative framework both a categorical semantics for MLL with distinct units and a notion of truth. Moreover, we show how a restricted version of our model can be reformulated in the exact same terms as Girard’s geometry of interaction [Gir08]. This shows that this restriction of our framework gives a combinatorial approach to J.-Y. Girard’s geometry of interaction in the hyperfinite factor, while using only graphtheoretical notions.