Coherent Interaction Graphs

Abstract

We introduce the notion of coherent graphs, and show how those can be used to define dynamic semantics for Multiplicative Linear Logic (MLL) extended with non-determinism. Thanks to the use of a coherence relation rather than mere formal sums of non-deterministic possibilities, our model enjoys some finiteness and sparsity properties.We also show how studying the semantic types generated by a single "test" within our model of MLL naturally gives rise to a notion of proof net, which turns out to be exactly Retoré's R&Bcographs. This revisits the old idea that multplicative proof net correctness is interactive, with a twist: types are characterized not by a set of counter-proofs but by a single non-deterministic counter-proof. IntroductionDynamic semantics of proofs originate in Girard's Geometry of Interaction (GoI) program [7], whose aim was to provide a semantic account for the process of cut-elimination. Indeed, while the proofs-asprograms correspondence expresses that β -reduction of lambda-terms corresponds to cut-elimination, its extension to categorical models (the so-called Curry-Howard-Lambek correspondence) arguably fails to fully reflect these dynamics, as it represents those operations as a simple equality.Since then, GoI has been developed into many directions to give various accounts of cut-elimination without syntactic rewriting. One aspect which has received less attention is the construction of models of linear logic where morphisms are generalized programs on which a GoI-style execution procedure can be defined, and serves to compose morphisms. Girard's successive papers on GoI all investigate such models, using operator algebras as the spaces of generalized programs; recently, Seiller has simplified Girard's models using graphings (a measure-theoretic extension of graphs) instead of operators [24]. Note that discrete graphs suffice to obtain a model of linear logic without exponentials or quantifiers [21,23].The aforementioned models all start from an untyped universe of programs, and interpret formulae as specifications for the behavior of programs: they can be seen as realizability models with operators/graphs as realizers. Futhermore, these specifications will be given as batteries of "tests" or "counterproofs" which are themselves generalized programs.This idea comes from the theory of multiplicative proof nets, where correct proofs have to be distinguished out of a set of "proof structures" by means of a correctness criterion. Girard observed that by representing a proof structure as a permutation σ , he could express the correctness criterion as the cyclicity of σ τ i for all i, where {τ i | i ∈ I} is a set of permutations depending on the formula being proved. This led to a prototypical GoI model [6], limited to Multiplicative Linear Logic (MLL), with permutations as * Part of this work was carried out when the first author was a student at École normale supérieure de Paris.

Exponentially Handsome Proof Nets and Their Normalization

“…The RGB-cographs can be interpreted as a coherent interaction graphs [33] for MLL ∪ {w!p}. We foresee a further application of exponentially handsome proof nets to explore the geometry of interaction of MELL [16] using the same approach from [14], where handsome proof nets are employed to investigate coherent spaces of MLL.…”