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14
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 51 (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.
On Köthe sequence spaces and linear logic
 Mathematical Structures in Computer Science
, 2001
"... 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 ..."
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Cited by 30 (9 self)
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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 coKleisli 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
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.
Predicate transformers and Linear Logic  yet another Denotational Model
 In http://jumpstart.anr.mcnc.org
, 2004
"... 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 denotationa ..."
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Cited by 7 (6 self)
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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.
Execution time of lambdaterms via non uniform semantics and intersection types. Research report
, 2006
"... Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on nonidempotent 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 execut ..."
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Cited by 4 (2 self)
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Abstract. The relational semantics for Linear Logic induces a semantics for the type free Lambda Calculus. This one is built on nonidempotent 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 lambdaterms in a particular environment machine, Krivine’s machine.
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.
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.
A Completeness Theorem for Symmetric Product Phase Spaces
, 2000
"... 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 in ..."
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Cited by 1 (0 self)
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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).
On Phase Semantics and Denotational Semantics: The SecondOrder
, 2002
"... In this paper, we extend the nonuniform denotational semantics de ned by Bucciarelli and Ehrhard in [BuEh,00] to secondorder linear logic. ..."
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Cited by 1 (0 self)
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In this paper, we extend the nonuniform denotational semantics de ned by Bucciarelli and Ehrhard in [BuEh,00] to secondorder linear logic.