Synchronous Games, Simulations and λ-calculus

We refine a model for linear logic based on two well-known ingredients: games and simulations. We have already shown that usual simulation relations form a sound notion of morphism between games; and that we can interpret all linear logic in this way. One particularly interesting point is that we interpret multiplicative connectives by synchronous operations on games. We refine this work by giving computational contents to our simulation relations. To achieve that, we need to restrict to intuitionistic linear logic. This allows to work in a constructive setting, thus keeping a computational content to the proofs. We then extend it by showing how to interpret some of the additional structure of the exponentials. To be more precise, we first give a model for the typed λ-calculus; and then give a model for the differential λ-calculus of Ehrhard and Regnier. Both this models are proved correct constructively.