A Model of Intuitionistic Affine Logic from Stable Domain Theory (Revised and Expanded Version)

Girard worked with the category of coherence spaces and continuous stable maps and observed that the functor that forgets the linearity of linear stable maps has a left adjoint. This fundamental observation gave rise to the discovery of Linear Logic. Since then, the category of coherence spaces and linear stable maps, with the comonad induced by the adjunction, has been considered a canonical model of Linear Logic. Now, the same phenomenon is present if we consider the category of pre dI domains and continuous stable maps, and the category of dI domains and linear stable maps; the functor that forgets the linearity has a left adjoint. This gives an alternative model of Intuitionistic Linear Logic. It turns out that this adjunction can be factored in two adjunctions yielding a model of Intuitionistic Affine Logic; the category of pre dI domains and affine stable functions. It is the goal of this paper to show that this category is actually a model of Intuitionistic Affine Logic, and to show that this category moreover has properties which make it possible to use it to model convergence/divergence behaviour and recursion

On dialogue games and coherent strategies

We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature

Sequentiality vs. Concurrency in Games and Logic

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.Comment: 35 pages, appeared in Mathematical Structures in Computer Scienc

Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of $A$-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart.Comment: Statement of Theorem 4.7 and notation in Section 4.3 correcte

Typing with Leftovers - A mechanization of Intuitionistic Multiplicative-Additive Linear Logic

We start from an untyped, well-scoped lambda-calculus and introduce a bidirectional typing relation corresponding to a Multiplicative-Additive Intuitionistic Linear Logic. We depart from typical presentations to adopt one that is well-suited to the intensional setting of Martin-Löf Type Theory. This relation is based on the idea that a linear term consumes some of the resources available in its context whilst leaving behind leftovers which can then be fed to another program. Concretely, this means that typing derivations have both an input and an output context. This leads to a notion of weakening (the extra resources added to the input context come out unchanged in the output one), a rather direct proof of stability under substitution, an analogue of the frame rule of separation logic showing that the state of unused resources can be safely ignored, and a proof that typechecking is decidable. Finally, we demonstrate that this alternative formalization is sound and complete with respect to a more traditional representation of Intuitionistic Linear Logic. The work has been fully formalised in Agda, commented source files are provided as additional material available at https://github.com/gallais/typing-with-leftovers

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

Substructural logics naturally support a quantitative interpretation of formulas, as they are seen as consumable resources. Distances are the quantitative counterpart of equivalence relations: they measure how much two objects are similar, rather than just saying whether they are equivalent or not. Hence, they provide the natural choice for modelling equality in a substructural setting. In this paper, we develop this idea, using the categorical language of Lawvere's doctrines. We work in a minimal fragment of Linear Logic enriched by graded modalities, which are needed to write a resource sensitive substitution rule for equality, enabling its quantitative interpretation as a distance. We introduce both a deductive calculus and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. The study of 2-categorical properties of Lipschitz doctrines provides us with a universal construction, which generates examples based for instance on metric spaces and quantitative realisability. Finally, we show how to smoothly extend our results to richer substructural logics, up to full Linear Logic with quantifiers