A MALL geometry of interaction based on indexed linear logic

Masahiro Hamano

doi:10.1017/s0960129521000062

Abstract

AbstractWe construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) 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 1. The indices 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 indices. 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, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices 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 the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

Highlights

The indexed multiplicative-additive linear logic MALL(I), introduced by Bucciarelli– Ehrhard (2000), is a conservative extension of Girard’s MALL in which all formulas and proofs come equipped with sets of indices
We prove zero convergence, which means that execution formulas converge to zero when two proofs interact with mismatched locations
We prove two main results: (i) (Invariance of the execution formula during MALL normalisation): The execution formula in our dynamic categorical modelling is shown to converge to the denotational interpretation of proofs in the static categorical model

Summary

Introduction

The indexed multiplicative-additive linear logic MALL(I), introduced by Bucciarelli– Ehrhard (2000), is a conservative extension of Girard’s MALL in which all formulas and proofs come equipped with sets of indices. It is well known that the category Rel of sets and relations constitutes a denotational semantics of MALL, that is, the interpretation is invariant, (π[ ], )∗ = (π[ ], )∗, for any reduction π[ ], £ π[ ], of MALL cut elimination. Every MALL proof π[ ], of a sequent [ ] is interpreted as a subset of an associated set of the conclusion (without the cut list),. Every MALL proof π[ ], of a sequent [ ], is interpreted by which is defined inductively and in the same manner as in Definition 2.4, except for the cut rule to make the interpretation differ from the standard (1) in that is visible without performing the relational composition. The fundamental lemma is shown to be preserved under our extended syntax and semantics, designed to accommodate cut formulas in MALL[c](I) and in Rel[c], respectively.

Lifting MALL Reduction over Indices

The Execution formula

Zero convergence of execution formula

Conclusion and Future Work