International audienceThe category Rel of sets and relations yields one of the simplest denotational semantics of Linear Logic (LL). It is known that Rel is the biproduct completion of the Boolean ring. We consider the generalization of this construction to an arbitrary continuous semiring R, producing a cpo-enriched category which is a semantics of LL, and its (co)Kleisli category is an adequate model of an extension of PCF, parametrized by 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)

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† This work was partly funded by the NWO Project 612.000.936 CALMOC (CAtegorical and ALgebraic Models of Computation) and by Digiteo/Île-de-France Project 2009-28HD COLLODI (Complexity and concurrency through ludics and differential linear logic).

Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as "λ-models", or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: given a λ-model A, one may define a ccc in which A (the carrier set) is a reflexive object; conversely, if U is a reflexive object in a ccc C, having enough points, then C(½, U ) may be turned into a λ-model. It is well known that, if C does not have enough points, then the applicative structure C(½, U ) is not a λ-model in general. This paper: (i) shows that this mismatch can be avoided by choosing appropriately the carrier set of the λ-model associated with U ; (ii) provides an example of an extensional reflexive object D in a ccc without enough points: the Kleisli-category of the comonad "finite multisets" on Rel; (iii) presents some algebraic properties of the λ-model associated with D by (i) which make it suitable for dealing with non-deterministic extensions of the untyped λ-calculus.