A MALL Geometry of Interaction Based on Indexed Linear Logic

We construct a geometry of interaction (GoI; dynamic modeling of Gentzen-style cut elimination) for multiplicativeadditive linear logic (MALL) by employing the Bucciarelli–Ehrhard MALL(I) of indexed linear logic to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI I . The indexes are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indexes. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, w.r.t symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The indexes diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of cut elimination via convergence of the execution formulas on the denotations of (cut-free) proofs.