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) related to game semantics and the Danos-Regnier interpretation of GoI operators as paths in proof nets. 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 latest geometry of interaction. 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 graph-theoretical notions.

💡 Research Summary

The paper proposes a novel graph‑theoretic representation of proofs in Multiplicative Linear Logic (MLL) that simultaneously yields a denotational semantics and a truth‑value notion. Building on a locative perspective inspired by ludics, each logical subformula is assigned a unique locus, and the flow of a proof is captured as paths between these loci. These paths are encoded as edges in an “interaction graph” whose vertices correspond to subformulas and whose edges represent logical connectives (tensor ⊗, par ⅋) as well as cut links. By extending the Danos‑Regnier interpretation of Geometry of Interaction (GoI) operators as path sets, the authors compress all possible execution paths into a single graph structure. Normalisation of proofs becomes a purely graph‑based reduction: edges are rewired and contracted, mirroring GoI’s execution formula without recourse to infinite‑dimensional Hilbert spaces or operator algebras.

The authors then show that this graph model forms a *‑autonomous category: objects are interaction graphs, morphisms are graph transformations, composition is graph composition, and tensor/par correspond to categorical tensor and internal hom. Distinct units (1 and ⊥) are treated as explicit null nodes, solving the asymmetry present in earlier GoI models. Consequently, the model provides both a categorical semantics for MLL with separate units and a concrete truth‑value assignment derived from the presence or absence of certain cycles in the interaction graph.

A restricted variant—where graphs are acyclic directed graphs and each edge encodes a single logical operation—is proved to be exactly equivalent to Girard’s recent GoI construction in the hyperfinite factor. This equivalence demonstrates that the sophisticated operator‑theoretic GoI can be reconstructed purely combinatorially, using only elementary graph notions. The result bridges the gap between high‑level GoI theory and low‑level, implementable graph algorithms.

In summary, the paper delivers two major contributions: (1) a locative, graph‑based semantics for MLL that unifies proof representation, normalisation, and truth evaluation; and (2) a proof that, under a natural restriction, this semantics coincides with Girard’s hyperfinite GoI, offering a purely combinatorial perspective on a traditionally analytic construction. The authors suggest future work extending the framework to exponentials, exploring connections with game semantics, and applying the model to concrete programming language implementations.