An explicit formula for the free exponential modality of linear logic

Abstract : The exponential modality of linear logic associates a commutative comonoid !A to every formula A in order to duplicate it. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We then apply this general recipe to two familiar models of linear logic, based on coherence spaces and on Conway games. This algebraic approach enables to unify for the rst time apparently different constructions of the exponential modality in spaces and games. It also sheds light on the subtle duplication policy of linear logic. On the other hand, we explain at the end of the article why the formula does not work in the case of the niteness space model.
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36th International Colloquium on Automata, Languages and Programming, Jul 2009, Rhodes, Greece. Springer, 5556/2009, pp.247-260, 2009, Lecture Notes in Computer Science. 〈10.1007/978-3-642-02930-1〉
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Paul-André Melliès, Nicolas Tabareau, Christine Tasson. An explicit formula for the free exponential modality of linear logic. 36th International Colloquium on Automata, Languages and Programming, Jul 2009, Rhodes, Greece. Springer, 5556/2009, pp.247-260, 2009, Lecture Notes in Computer Science. 〈10.1007/978-3-642-02930-1〉. 〈hal-00391714v2〉

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