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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.
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.
Strong Normalization for Nonstructural Subtyping via Saturated Sets
, 1996
"... We show that the standard method of saturated sets for proving strong normalization of fireduction in the simply typed and second order polymorphic lambda calculus incorporates nonstructural subtyping systems in a natural way. This shows that strong normalization for nonstructural subtyping pr ..."
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Cited by 2 (0 self)
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We show that the standard method of saturated sets for proving strong normalization of fireduction in the simply typed and second order polymorphic lambda calculus incorporates nonstructural subtyping systems in a natural way. This shows that strong normalization for nonstructural subtyping proved by Wand, O'Keefe and Palsberg [24] via coercion interpretations can be obtained in a straightforward extension of the standard method. The proof presented here is compared to other proofs of strong normalization for subtyping systems.
A Formalised FirstOrder . . .
, 2002
"... We present the titular proof development that has been verified in Isabelle/HOL. As a first, the proof is conducted exclusively by the primitive proof principles of the standard syntax and of the considered reduction relations: the naive way, so to speak. Curiously, the Barendregt Variable Conventio ..."
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We present the titular proof development that has been verified in Isabelle/HOL. As a first, the proof is conducted exclusively by the primitive proof principles of the standard syntax and of the considered reduction relations: the naive way, so to speak. Curiously, the Barendregt Variable Convention takes on a central technical role in the proof. We also show (i) that our presentation of the λcalculus coincides with Curry’s and Hindley’s when terms are considered equal up to αequivalence and (ii) that the confluence properties of all considered systems are equivalent.
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
Normal Forms for the Algebraic LambdaCalculus
"... We study the problem of defining normal forms of terms for the algebraic λcalculus, an extension of the pure λcalculus where linear combinations of terms are firstclass entities: the set of terms is enriched with a structure of vector space, or module, over a fixed semiring. Towards a solution to ..."
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We study the problem of defining normal forms of terms for the algebraic λcalculus, an extension of the pure λcalculus where linear combinations of terms are firstclass entities: the set of terms is enriched with a structure of vector space, or module, over a fixed semiring. Towards a solution to the problem, we propose a variant of the original reduction notion of terms which avoids annoying behaviours affecting the original version, but we find it not even locally confluent. Finally, we consider reduction of linear combinations of terms over the semiring of polynomials with nonnegative integer coefficients: terms coefficients are replaced by indeterminates and then, after reduction has taken placed, restored back to their original value by an evaluation function. Such a special setting permits us to talk about normal forms of terms and, via an evaluation function, to define such notion for any semiring. 1.