Interaction Graphs: Multiplicatives

Thomas Seiller (1, 2)

1 LAMA - Laboratoire de Mathématiques
2 IML - Institut Mathématiques de Luminy

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, Girard, 2001 [11]) related to game semantics (Hyland and Ong, 2000 [17], Abramsky et al., 1994 [2]) and the Danos–Regnier interpretation of GoI operators as paths in proof nets (Asperti et al., 1994 [3], Danos and Regnier, 1995 [4]). We show how we can retrieve from this locative framework both a categorical semantics for Multiplicative Linear Logic (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 Girardʼs geometry of interaction (Girard, 2011 [14]). This shows that this restriction of our framework gives a combinatorial approach to J.-Y. Girardʼs geometry of interaction in the hyperfinite factor, while using only graph-theoretical notions.

Mots clés: Proof theory; Linear logic; Denotational semantics; Operator algebras; Geometry of interaction

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Thomas Seiller. Interaction Graphs: Multiplicatives. Annals of Pure and Applied Logic, 2012, 163 (12), pp.1808-1837. ⟨10.1016/j.apal.2012.04.005⟩. ⟨hal-01159218⟩