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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.
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.
On finiteness spaces and extensional presheaves over the lawvere theory of polynomials
 Journal of Pure and Applied Algebra
"... theory of polynomials ..."
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,...}.