On phase semantics and denotational semantics: the exponentials

Abstract

We extend to the exponential connectives of linear logic the study initiated in Bucciarelli and Ehrhard (Ann. Pure. Appl. Logic 102 (3) $\text{(2000)}$ 247). We define an indexed version of propositional linear logic and provide a sequent calculus for this system. To a formula $\text{A}$ of indexed linear logic, we associate an underlying formula $\text{A}\text{̄}$ of linear logic, and a family $\text{〈A〉}$ of elements of $\text{|}\text{A}\text{̄}\text{|}$, the interpretation of $\text{A}\text{̄}$ in the category of sets and relations. Then $\text{A}$ is provable in indexed linear logic iff the family $\text{〈A〉}$ is contained in the interpretation of some proof of $\text{A}\text{̄}$. We extend to this setting the product phase semantics of indexed multiplicative additive linear logic introduced in Bucciarelli and Ehrhard $\text{(2000)}$, 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 $\text{!A−∘B}$ contains informations about the behavior of this proof when applied to “chimeric” arguments of type $\text{A}$ (for instance: booleans whose value can change during the computation). In this coherence semantics, an element of a web can be strictly coherent with itself, or two distinct elements can be “neutral” (that is, neither strictly coherent, nor strictly incoherent).