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23
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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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
Solvability in resource lambdacalculus
 FOSSACS, volume 6014 of LNCS
, 2010
"... Abstract. The resource calculus is an extension of the λcalculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to nondeterministic choices, expressed by a formal sum. ..."
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Abstract. The resource calculus is an extension of the λcalculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to nondeterministic choices, expressed by a formal sum. Using the λcalculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the nondeterminism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability. 1
The differential λµcalculus
 Theor. Comput. Sci
, 2007
"... We define a differential λµcalculus which is an extension of both Parigot’s λµcalculus and EhrhardRégnier’s differential λcalculus. We prove some basic properties of the system: reduction enjoys ChurchRosser and simply typed terms are strongly normalizing. Contents 1 ..."
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We define a differential λµcalculus which is an extension of both Parigot’s λµcalculus and EhrhardRégnier’s differential λcalculus. We prove some basic properties of the system: reduction enjoys ChurchRosser and simply typed terms are strongly normalizing. Contents 1
A finiteness structure on resource terms
 IN LICS
, 2010
"... We study the Taylor expansion of lambdaterms in a nondeterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambdacalculus we introduced recently. This operation is not possible in the general untyped case where redu ..."
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Cited by 7 (1 self)
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We study the Taylor expansion of lambdaterms in a nondeterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambdacalculus 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.
Execution Time of λTerms via Denotational Semantics and Intersection Types. Research Report RR6638
, 2008
"... The multiset based relational model of linear logic induces a semantics of the type free λcalculus, which corresponds to a nonidempotent intersection type system, System R. We prove that, in System R, the size of the type derivations and the size of the types are closely related to the execution t ..."
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The multiset based relational model of linear logic induces a semantics of the type free λcalculus, which corresponds to a nonidempotent intersection type system, System R. We prove that, in System R, the size of the type derivations and the size of the types are closely related to the execution time of λterms in a particular environment machine, Krivine’s machine.
Execution time of lambdaterms via nonuniform semantics and intersection types. Preprint Institut de Mathématique de Luminy 2006
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Full Abstraction for Resource Calculus with Tests
 In CSL, Lecture Notes in Computer Science
, 2011
"... We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrh ..."
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We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. 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 λcalculus. 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 subcalculus where ordinary λcalculus 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
Resource combinatory algebras
"... Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down fou ..."
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Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λcalculus. We also show that the ideal completion of a resource combinatory (resp. lambda, lambdaabstraction) algebra induces a “classical ” combinatory (resp. lambda, lambdaabstraction) algebra, and that any model of the classical λcalculus raising from a resource lambdaalgebra determines a λtheory which equates all terms having the same Böhm tree. 1
Transport of finiteness structures and applications
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
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Differential Linear Logic and Polarization
"... We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semanti ..."
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We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semantics of both differential and polarized linear logics: this justifies our choice of cut elimination rules. Then we show this new system models the recently introduced convolution _*ucalculus, the same as linear logic decomposes calculus.