Template games and differential linear logic

Abstract

We extend our template game model of multiplicative additive linear logic (MALL) with an exponential modality of linear logic (LL) derived from the standard categorical construction Sym of the free symmetric monoidal category. We obtain in this way the first game semantics of differential linear logic (DiLL). Its formulation relies on a careful and healthy comparison with the model of distributors and generalised species designed ten years ago by Fiore, Gambino, Hyland and Winskel. Besides the resolution of an old open problem of game semantics, the study reveals an unexpected and promising convergence between linear logic and homotopy theory.

Asynchronous Template Games and the Gray Tensor Product of 2-Categories

“…defined by the multiplication of the asynchronous graph [O, P ] described in (11) which associates to each polarity O or P of the domain the same polarity O or P of the codomain. This definition of the tensor product A, B → A⊗B of two asynchronous games is reminiscent of a very similar construction in the template game model of differential linear logic [23], [24] which we briefly recall now.…”

“…This excess of synchronization in the interpretation of A ⊗ B should be seen as a defect (not as a feature!) of the original concurrent template game semantics formulated in [23], [24].…”

“…The functor − associates to every asynchronous graph (G, ) the 2-category G, whose objects are the vertices of G, whose morphisms are the paths of G and whose 2-dimensional cells are the reshufflings (or reschedulings) between paths of the graph G. A detailed definition of reshuffling appears in §V-A but let us already mention that the main purpose of reshufflings is to identify different combinations of permutation tiles such as ∼ = which deserve to be considered as equivalent in the asynchronous graph G. The fact that the functor − comes equipped with natural isomorphisms of 2-categories: Asynchronous template games: We have accumulated enough material in order to define the notion of asynchronous template game. There is still one fundamental obstruction which we need to resolve however: the notion of template formulated in [23], [24] is defined as an internal category…”

“…in the category S with finite limits, where [0] = game is the object of objects and [1] = strat is the object of morphisms. For that reason, the formalism of template games originally developed in [23], [24] does not work any more when the category S = Cat with finite limits is replaced by the monoidal category S = 2-Cat equipped with the Gray tensor product.…”

“…On the practical side, the tensor product A, B → A ⊗ B of asynchronous template games is defined in Games( asynch ) using the Gray tensor product and thus avoids the defect discussed in (25) and (28). On the conceptual side, the construction extends in a very natural way the original framework of template games [23], [24] to the more general situation where the template = asynch defines an internal category in a monoidal category S such as S = (2-Cat, , 1) -see Aguiar [3] and §VI, §VII for a definition -instead of an internal category in a category S with limits such as S = Cat.…”