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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A fully abstract game semantics for an idealized programming language with local state and higher rank polymorphism -System F extended with general references -is described. It quite concrete, and extends existing games models by a simple development of the existing question/answer labelling to represent "copycat links" between positive and negative occurrences of type variables, using a notion of scoping for question moves. It is effectively presentable, opening the possibility of extending existing model checking techniques to polymorphic types, for example. It is also a novel example of a model of System F with the genericity property. We prove definability of finite elements, and thus a full abstraction result, using a decomposition argument. This also establishes that terms may be approximated up to observational equivalence when instantiation is restricted to tuples of type variables.

Differential categories were introduced by Blute, Cockett and Seely to axiomatize categorically Ehrhard and Regnier's syntactic differential operator. We present an abstract construction that takes a symmetric monoidal category and yields a differential category, and show how this construction may be applied to categories of games. In one instance, we recover the category previously used to give a fully abstract model of a nondeterministic imperative language. The construction exposes the differential structure already present in this model, and shows how the differential combinator may be encoded in the imperative language. The second instance corresponds to a new Cartesian differential category of games. We give a model of a simply-typed resource calculus, Resource PCF, in this category and show that it possesses the finite definability property. Comparison with a semantics based on Bucciarelli, Ehrhard and Manzonetto's relational model reveals that the latter also possesses this property and is fully abstract.

Differential categories were introduced by Blute, Cockett and Seely to axiomatize categorically Ehrhard and Regnier's syntactic differential operator. We present an abstract construction that takes a symmetric monoidal category and yields a differential category, and show how this construction may be applied to categories of games. In one instance, we recover the category previously used to give a fully abstract model of a nondeterministic imperative language. The construction exposes the differential structure already present in this model, and shows how the differential combinator may be encoded in the imperative language. A second instance corresponds to a new cartesian differential category of games. We give a model of a simply-typed resource calculus, Resource PCF, in this category and show that it possesses the finite definability property. Comparison with a semantics based on Bucciarelli, Ehrhard and Manzonetto's relational model reveals that the latter also possesses this property and is fully abstract.