The category Rel of sets and relations yields one of the simplest denotational semantics of Linear Logic. Rel can be viewed as the biproduct completion of the Boolean ring. We consider the generalization of this construction to arbitrary continuous semirings, producing categories that provide cpo-enriched models of linear logic akin to Rel, and investigate models of PCF in their co-Kleisli categories. These models contain quantitative information, provided by the elements of the semiring R. Specific instances of R allow us to compare programs not only with respect to “what they can do”, but also “in how many steps” or “in how many different ways” (for non-deterministic PCF) or even “with what probability” (for probabilistic PCF).

Joint work with Jim Laird (Bath) and Giulio Manzonetto and Michele Pagani (LIPN, Paris-Nord)