We define a differential λµ-calculus which is an extension of both Parigot’s λµ-calculus and Ehrhard-Régnier’s differential λ-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing. Contents 1

We study the Taylor expansion of lambda-terms in a non-deterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambda-calculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.

We introduce the notion of differential λ-category as an extension of Blute-Cockett-Seely’s differential Cartesian categories. We prove that differential λ-categories can be used to model the simply typed versions of: (i) the differential λ-calculus, a λ-calculus extended with a syntactic derivative operator; (ii) the resource calculus, a non-lazy axiomatisation of Boudol’s λ-calculus with multiplicities. Finally, we provide two

Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on non-idempotent intersection types. We give a principal typing property for this type system. We then prove that the size of the derivations is closely related to the execution time of lambda-terms in a particular environment machine, Krivine’s machine.

We study the semantics of a resource sensitive extension of the λ-calculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λ-calculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrhard and Regnier’s differential extension of Linear Logic and of the λ-calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a “must ” parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary λ-calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. 1998 ACM Subject Classification F.4.1 Lambda calculus and related systems

We study iteration and recursion operators in the denotational semantics of typed λ-calculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard’s finiteness spaces. 1

Abstract. Given a semi-ring with unit which satis es some algebraic conditions, we de ne an exponential functor on the category of sets and relations which allows to de ne a denotational model of di erential linear logic and of the lambda-calculus with resources. We show that, when the semi-ring has an element which is in nite in the sense that it is equal to its successor, this model does not validate the Taylor formula and that it is possible to build, in the associated Kleisli cartesian closed category, a model of the pure lambda-calculus which is not sensible. This is a quantitative analogue of the standard graph model construction in the

Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λ-calculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambda-algebras and resource lambda-abstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λ-calculus. We also show that the ideal completion of a resource combinatory (resp. lambda-, lambda-abstraction) algebra induces a “classical ” combinatory (resp. lambda-, lambda-abstraction) algebra, and that any model of the classical λ-calculus raising from a resource lambda-algebra determines a λ-theory which equates all terms having the same Böhm tree. 1

applications niteness structures and

Abstract. We study iteration and recursion operators in the denotational semantics of typed λ-calculi derived from the multiset relational model of linear logic. Although these operators are de ned as xpoints of typed functionals, we prove them nitary in the sense of Ehrhard's niteness spaces. 1991 Mathematics Subject Classi cation. 03B70, 03D65, 68Q55. 1.