We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of “finitary ” subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics...). In particular, the standard fix-point operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete version of the Köthe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard’s quantitative semantics with coefficients in the field). We obtain in that way a new model of the recently introduced differential lambda-calculus. Notations. If S is a set, we denote by M(S) = N S the set of all multi-sets over S. If µ ∈ M(S), |µ| denotes the support of µ which is the set of all a ∈ S such that µ(a) ̸ = 0. A multi-set is finite if it has a finite support. If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multi-set over S. The usual operations on natural numbers are extended to multi-sets pointwise. If (Si)i∈I are sets, we denote by πi the i-th projection πi: ∏ j∈I Sj → Si.

We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

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 ( , 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 ( , 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.

In the refinement calculus, monotonic predicate transformers are used to model specifications for (imperative) programs. Together with a natural notion of simulation, they form a category enjoying many algebraic properties. We build on this structure to make predicate transformers into a denotational model of full linear logic: all the logical constructions have a natural interpretation in terms of predicate transformers (i.e. in terms of specifications). We then interpret proofs of a formula by a safety property for the corresponding specification.

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 show that the extensional collapse of the relational model of linear logic is the model of prime-algebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambda-calculus.

We propose a translation of a finitary (that is, replication-free) version of the pi-calculus into promotionfree differential interaction net structures, a linear logic version of the differential lambda-calculus (or, more precisely, of a resource lambda-calculus). For the sake of simplicity only, we restrict our attention to a monadic version of the pi-calculus, so that the differential interaction net structures we consider need only to have exponential cells. We prove that the nets obtained by this translation satisfy an acyclicity criterion weaker than the standard Girard (or Danos-Regnier) acyclicity criterion, and we compare the operational semantics of the pi-calculus, presented by means of an environment machine, and the reduction of differential interaction nets. Differential interaction net structures being of a logical nature, this work provides a Curry-Howard interpretation of processes.

In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to dene new denotational models of linear logic using symmetric product phase models (truthvalue models) of indexed linear logic. We present here a sequent calculus of indexed linear logic which strictly extends the system LL(I) presented in [BE99] and for which the symmetric product phase spaces provide a complete semantics. We study the connection between this new system and LL(I).

In this paper, we extend the non-uniform denotational semantics de ned by Bucciarelli and Ehrhard in [BuEh,00] to second-order linear logic.

Abstract. We present an extension to second order of the indexed linear logic system that was once introduced by Bucciarelli and Ehrhard in 1999-2000. After a brief reminder on the relational model of second order indexed linear logic (defined by the author in [Br,00]), we will present the system LL 2 (I) and show how it is possible to express through this system definability properties in the relational model of second order indexed linear logic.