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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 ..."
Abstract

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.
Realizability : a machine for analysis and set theory
, 2006
"... In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this. ..."
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Cited by 4 (4 self)
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In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this.
Functions as proofs as processes
 CoRR
"... Abstract. This paper presents a logical approach to the translation of functional calculi into concurrent process calculi. The starting point is a type system for the πcalculus closely related to linear logic. Decompositions of intuitionistic and classical logics into this system provide typeprese ..."
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Cited by 2 (0 self)
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Abstract. This paper presents a logical approach to the translation of functional calculi into concurrent process calculi. The starting point is a type system for the πcalculus closely related to linear logic. Decompositions of intuitionistic and classical logics into this system provide typepreserving translations of the λ and λµcalculus, both for callbyname and callbyvalue evaluation strategies. Previously known encodings of the λcalculus are shown to correspond to particular cases of this logical embedding. The realisability interpretation of types in the πcalculus provides systematic soundness arguments for these translations and allows for the definition of typesafe extensions of functional calculi. 1
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,...}.
Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction
"... Abstract. Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λterms. Our analysis sheds light on a new condition on λterms, called local sco ..."
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Abstract. Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λterms. Our analysis sheds light on a new condition on λterms, called local scope. We show that the reduction of locally scoped terms matches closely that of bounding skeletons. Exploiting this connection, we give upper bound to the length of linear head reduction for simplytyped locally scoped terms. General terms lose this connection to bounding skeletons. To compensate for that, we show that λlifting allows us to transform any λterm into a locally scoped one. We deduce from that an upper bound to the length of linear head reduction for arbitrary simplytyped λterms. In both cases, we prove the asymptotical optimality of the upper bounds by providing matching lower bounds. 1
DOI: 10.1016/j.apal.2009.07.016 Totality in arena games
, 2009
"... We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We disc ..."
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We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We discuss how this theorem relates to normalization of linear head reduction in simplytyped λcalculus, leading us to a semantic realizability proof à la Kleene of our theorem. We then present another proof of a more combinatorial nature. Finally, we discuss the exact class of strategies to which our theorems apply. 1.