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Finiteness spaces
 Mathematical Structures in Computer Science
, 1987
"... 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 dif ..."
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Cited by 53 (13 self)
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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 fixpoint 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 lambdacalculus. Notations. If S is a set, we denote by M(S) = N S the set of all multisets 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 multiset 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 multiset over S. The usual operations on natural numbers are extended to multisets pointwise. If (Si)i∈I are sets, we denote by πi the ith projection πi: ∏ j∈I Sj → Si.
Not enough points is enough
 IN: COMPUTER SCIENCE LOGIC. VOLUME 4646 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... Models of the untyped λcalculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as “λmodels”, 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: ..."
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Cited by 19 (9 self)
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Models of the untyped λcalculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as “λmodels”, 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 λmodel 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 λmodel. It is well known that, if C does not have enough points, then the applicative structure C ( , U) is not a λmodel in general. This paper: (i) shows that this mismatch can be avoided by choosing appropriately the carrier set of the λmodel associated with U; (ii) provides an example of an extensional reflexive object D in a ccc without enough points: the Kleislicategory of the comonad “finite multisets ” on Rel; (iii) presents some algebraic properties of the λmodel associated with D by (i) which make it suitable for dealing with nondeterministic extensions of the untyped λcalculus.
Between logic and quantic: a tract
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2003
"... We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of ηconversion, a.k.a, extensionality. ..."
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Cited by 16 (1 self)
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We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of ηconversion, a.k.a, extensionality.
Categorical models for simply typed resource calculi
 ENTCS
"... We introduce the notion of differential λcategory as an extension of BluteCockettSeely’s differential Cartesian categories. We prove that differential λcategories can be used to model the simply typed versions of: (i) the differential λcalculus, a λcalculus extended with a syntactic derivative ..."
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Cited by 7 (3 self)
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We introduce the notion of differential λcategory as an extension of BluteCockettSeely’s differential Cartesian categories. We prove that differential λcategories can be used to model the simply typed versions of: (i) the differential λcalculus, a λcalculus extended with a syntactic derivative operator; (ii) the resource calculus, a nonlazy axiomatisation of Boudol’s λcalculus with multiplicities. Finally, we provide two
What is a Categorical Model of the Differential and the Resource λCalculi?
"... The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitab ..."
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Cited by 5 (1 self)
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The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows to write the full Taylor expansion of a program. Through this expansion every program can be decomposed into an infinite sum (representing nondeterministic choice) of ‘simpler’ programs that are strictly linear. The aim of this paper is to develop an abstract ‘model theory ’ for the untyped differential λcalculus. In particular, we investigate what should be a general categorical definition of denotational model for this calculus. Starting from the work of Blute, Cockett and Seely on differential categories we provide the notion of Cartesian closed differential category and we prove that linear reflexive objects living in such categories constitute sound models of the untyped differential λcalculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This entails that every model living in such categories equates all programs having the same full Taylor expansion. We then
A general class of models of H
 in "Mathematical Foundations of Computer Science (MFCS’09)", Lecture Notes in Computer Science
"... Abstract. We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpoenriched cartesian closed categories to have H ∗ , the maximal cons ..."
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Cited by 3 (1 self)
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Abstract. We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpoenriched cartesian closed categories to have H ∗ , the maximal consistent sensible λtheory, as their equational theory. Finally, we prove that our relational model fulfils these conditions.
The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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Cited by 3 (1 self)
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
Polynomial functors and polynomial monads
, 2009
"... Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship wi ..."
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Cited by 2 (0 self)
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Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.
On Differential Interaction Nets and the Picalculus
 Preuves, Programmes et Systèmes
, 2006
"... We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, w ..."
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Cited by 1 (0 self)
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We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, we restrict our attention to a monadic version of the picalculus, 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 DanosRegnier) acyclicity criterion, and we compare the operational semantics of the picalculus, 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 CurryHoward interpretation of processes.
Exponentials with in nite multiplicities
"... Abstract. Given a semiring 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 lambdacalculus with resources. We show that, when the semiring has ..."
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Abstract. Given a semiring 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 lambdacalculus with resources. We show that, when the semiring 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 lambdacalculus which is not sensible. This is a quantitative analogue of the standard graph model construction in the