The Geometry of Concurrent Interaction: Handling Multiple Ports by Way of Multiple Tokens (Long Version)

We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $\pi$-calculus. The introduced model is based on token machines in which not one but multiple tokens are allowed to traverse the underlying net at the same time. We prove soundness and adequacy of the introduced model. The former is proved as a simulation result between the token machines one obtains along any reduction sequence. The latter is obtained by a fine analysis of convergence, both in nets and in token machines

A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry-Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal lambda-calculus by Pfenning and Davies for staged computation

The Geometry of Synchronization (Long Version)

We graft synchronization onto Girard's Geometry of Interaction in its most concrete form, namely token machines. This is realized by introducing proof-nets for SMLL, an extension of multiplicative linear logic with a specific construct modeling synchronization points, and of a multi-token abstract machine model for it. Interestingly, the correctness criterion ensures the absence of deadlocks along reduction and in the underlying machine, this way linking logical and operational properties.Comment: 26 page

Differential Logical Relations, Part I: The Simply-Typed Case

We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The novelty of differential logical relations consists in measuring the distance between terms not (necessarily) by a numerical value, but by a mathematical object which somehow reflects the interactive complexity, i.e. the type, of the compared terms. We exemplify this concept in the simply-typed lambda-calculus, and show a form of soundness theorem. We also see how ordinary logical relations and metric relations can be seen as instances of differential logical relations. Finally, we show that differential logical relations can be organised in a cartesian closed category, contrarily to metric relations, which are well-known not to have such a structure, but only that of a monoidal closed category

A Higher-arity Sequent Calculus for Model Linear Logic (Proof theory and proving)

We propose a cut-free sequent calculus for multiplicative exponential linear logic with S4 necessity and possibility modalities. The calculus has the so-calld "higher-arity" judgment to formalize the exponentials and S4 modalities neatly. An the properties follow from the model of proofs for a higher-arity sequent calculus of classical modal logic S4, which is also proposed and studied here

Study on improvement of fishing port for the sea food hygiene control and sustainable utilization of fishery resources

This paper presents the methods of the development of fishing ports for the supplying of safe seafood products, based on the some rules on food hygiene control and on the eco-label system which is useful for fishery resource management in Japan. First, we analyzed the issue of the hygiene control for fresh fish on the distribution process from the landing places to the consumers and proposed the method to estimate the hygiene level with the two factors; the water quality, control and the sanitary condition of the working areas at the fishing ports, landing places and the auction halls on the distribution process. Secondly, we proposed the method to select whether fishery products contribute to being sustainable or not, so as to secure the validity of the sea food hygiene control at the fishing ports

The Geometry of Synchronization

International audienceWe graft synchronization onto Girard's Geometry of Interaction in its most concrete form, namely token machines. This is realized by introducing proof-nets for SMLL, an extension of multiplica-tive linear logic with a specific construct modeling synchronization points, and of a multi-token abstract machine model for it. Inter-estingly, the correctness criterion ensures the absence of deadlocks along reduction and in the underlying machine, this way linking logical and operational properties

Parallelism and Synchronization in an Infinitary Context

International audienceWe study multitoken interaction machines in the context of a very expressive linear logical system with expo-nentials, fixpoints and synchronization. The advantage of such machines is to provide models in the style of the Geometry of Interaction, i.e., an interactive semantics which is close to low-level implementation. On the one hand, we prove that despite the inherent complexity of the framework, interaction is guaranteed to be deadlock-free. On the other hand, the resulting logical system is powerful enough to embed PCF and to adequately model its behaviour, both when call-by-name and when call-by-value evaluation are considered. This is not the case for single-token stateless interactive machines