Abstract

We extend to the exponential connectives of linear logic the study initiated in Bucciarelli and Ehrhard 247). We define an indexed version of propositional linear logic and provide a sequent calculus for this system. To a formula A of indexed linear logic, we associate an underlying formula of linear logic, and a family A of elements of , the interpretation of in the category of sets and relations. Then A is provable in indexed linear logic iff the family A is contained in the interpretation of some proof of . We extend to this setting the product phase semantics of indexed multiplicative additive linear logic introduced in Bucciarelli and Ehrhard , defining the symmetric product phase spaces. We prove a soundness result for this truth-value semantics and show how a denotational model of linear logic can be associated to any symmetric product phase space. Considering a particular symmetric product phase space, we obtain a new coherence space model of linear logic, which is non-uniform in the sense that the interpretation of a proof of !A−B contains informations about the behavior of this proof when applied to “chimeric” arguments of type A . In this coherence semantics, an element of a web can be strictly coherent with itself, or two distinct elements can be “neutral”