Modelling Coeffects in the Relational Semantics of Linear Logic

Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic, e.g. [6, 2]. The parameters are taken from a semi-ring, and allow to express coeffects – i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.). We show that all these systems can be interpreted in the relational category (Rel) of sets and relations. This is possible because of the notion of multiplicity semi-ring, introduced in [3] and allowing a great variety of exponential comonads in Rel. The interpretation of a particular typing system corresponds then to give a suitable notion of stratification of the exponential comonad associated with the semi-ring parametrising the exponential modality. 1998 ACM Subject Classification F.3.2 Semantics of Programming Languages (Denotational semantics); F.4.1 Mathematical Logic (Lambda calculus and related systems)