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15
The Differential LambdaCalculus
 Theoretical Computer Science
, 2001
"... We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual ..."
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Cited by 44 (9 self)
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We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus.
Böhm trees, Krivine machine and the Taylor expansion of ordinary lambdaterms
, 2005
"... We show that, given an ordinary lambdaterm and a normal resource lambdaterm which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambdaterm reducing to this normal resource term can be obtained by running a version of the Krivi ..."
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Cited by 17 (5 self)
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We show that, given an ordinary lambdaterm and a normal resource lambdaterm which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambdaterm reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine.
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 3 (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.
A Linearization of the LambdaCalculus and Consequences
, 2000
"... We embed the standard #calculus, denoted #, into two larger #calculi, denoted # # and &# # . The standard notion of #reduction for # corresponds to two new notions of reduction, # # for # # and &# # for &# # . A distinctive feature of our new calculus # # (resp., &# # ) is that, in every function ..."
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Cited by 2 (0 self)
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We embed the standard #calculus, denoted #, into two larger #calculi, denoted # # and &# # . The standard notion of #reduction for # corresponds to two new notions of reduction, # # for # # and &# # for &# # . A distinctive feature of our new calculus # # (resp., &# # ) is that, in every function application, an argument is used at most once (resp., exactly once) in the body of the function. We establish various connections between the three notions of reduction, #, # # and &# # . As a consequence, we provide an alternative framework to study the relationship between #weak normalization and #strong normalization, and give a new proof of the oftmentioned equivalence between #strong normalization of standard #terms and typability in a system of "intersection types".
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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Cited by 2 (1 self)
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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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Cited by 2 (0 self)
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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
Filter models: nonidempotent intersection types, orthogonality and polymorphism
"... This paper revisits models of typed λcalculus based on filters of intersection types: By using nonidempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Nonidempotent intersections provide a decreasing measure proving a key ..."
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Cited by 1 (1 self)
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This paper revisits models of typed λcalculus based on filters of intersection types: By using nonidempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Nonidempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two termmodels (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Currystyle System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types.
Presentation
, 2003
"... We present an extension of the lambdacalculus with di erential constructions. We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus. Keywords. Lambda ..."
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We present an extension of the lambdacalculus with di erential constructions. We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus. Keywords. Lambdacalculus, linear logic, denotational semantics, linear head reduction. Prerequisites. This paper assumes from the reader some basic knowledge in lambdacalculus and an elementary (but not technical) knowledge of di erential calculus. Notations. Following [Kri93], we denote by (s)t the lambdacalculus application of s to t. The expression (s)t1... tn denotes the term ( · · · (s)t1 · · ·)tn when n ≥ 1, and s when n = 0. Accordingly, if A1,..., An and A are types, both expressions A1,..., An → A and A1 → · · · → An → A denote the type A1 → ( · · · (An → A) · · ·). If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multiset over S. If x and y are variables, δx,y is equal to 1 if x = y and to 0 otherwise. We denote by N + the set of positive integers {1, 2,...}.
What is a differential partial combinatory algebra?
, 2011
"... In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure wh ..."
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In this thesis we combine Turing categories with Cartesian left additive restriction categories and again with differential restriction categories. The result of the first combination is a new structure which models nondeterministic computation. The result of the second combination is a structure which models the notion of linear resource consumption. We also study the structural background required to understand what new features Turing structure should have in light of addition and differentiation – most crucial to this development is the way in which idempotents split. For the combination of Turing categories with Cartesian left additive restriction categories we will also provide a model.