## The Differential Lambda-Calculus (2001)

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Venue: | Theoretical Computer Science |

Citations: | 46 - 9 self |

### BibTeX

@ARTICLE{Ehrhard01thedifferential,

author = {Thomas Ehrhard and Laurent Regnier},

title = {The Differential Lambda-Calculus},

journal = {Theoretical Computer Science},

year = {2001},

volume = {309},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

We present an extension of the lambda-calculus 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 lambda-calculus.

### Citations

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Citation Context ...in terms of the comonad structure of the ! functor 10 , is a cartesian closed category, that is, a model of simply typed lambda-calculus: the Kleisli category of the comonad !. See for instance [Bie=-=-=-95]. We nish this short presentation by a word about dierentiation. Due to the fact that nite sums and products coincide, we can build a canonical linear morphism M from !X !X to !X (we apply the ! fu... |

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37 |
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Citation Context ...ng the domains on which functions act as ordered sets with a least element that plays the role of the divergent program. The structure of domains further allows, as in Scott domains (we refer here to =-=[Bar84-=-], as for all the standard lambda-calculus notions and results), to express some continuity properties of functions that account for the nite nature of computation. The main property is that a continu... |

29 | On köthe sequence spaces and linear logic - Ehrhard |

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17 |
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(Show Context)
Citation Context ... very natural notion in lambda-calculus that arises as soon as one wants to precisely evaluate reduction lengths. Linear head reduction has been considered by several authors (starting with De Bruijn =-=[DB87]-=- where it is called minireduction), and in particular, Danos and the second author, who related it to abstract machines (e.g. Krivine's machine [DR03]), game semantics and linear logic [DHR96]. It is ... |

16 | Coherent Banach spaces: a continuous denotational semantics
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(Show Context)
Citation Context ...of linear morphisms between vector spaces. This is reected by the notation and 1 terminology in use: tensor product, linear maps, direct product, etc. This correspondence has been further studied in [=-=Gir99-=-], an attempt which however failed to fully account for the exponential connectives of linear logic. As it turns out, it is possible to dene models of the typed lambda-calculus (or equivalently of ful... |

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12 |
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Citation Context ...inear and continuous functions turns out to be a bijection. This linear category is a model of multiplicative-additive linear logic, that is, a ?-autonomous category with nite sums and products (see [=-=Bar7-=-9]). In particular, all the operations we have dened are functorial. Of course, we need more for getting a model of full linear logic, or of simply typed lambda-calculus. We have to dene an exponentia... |

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7 |
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(Show Context)
Citation Context ...head reduction. Prerequisites. This paper assumes from the reader some basic knowledge in lambda-calculus and an elementary (but not technical) knowledge of dierential calculus. Notations. Following [Kri93], we denote by (s)t the lambda-calculus application of s to t. The expression (s)t 1 : : : t n denotes the term ( (s)t 1 )t n when n 1, and s when n = 0. Accordingly, if A 1 ; : : : ; An a... |

6 |
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- 2004
(Show Context)
Citation Context ...dered by several authors (starting with De Bruijn [DB87] where it is called minireduction), and in particular, Danos and the second author, who related it to abstract machines (e.g. Krivine's machine =-=[DR03-=-]), game semantics and linear logic [DHR96]. It is a kind of hyperlazy reduction strategy where at each step, only the head occurrence may be substituted. As a side eect, only subterms of the initial ... |

5 |
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Citation Context ... as (jX j; F(X) ? )). In this way, the pair (RhXi; RhX ? i) carries a well-behaved duality (each module can be seen as the topological dual of the other for a suitable linear topology in the sense of =-=[Lef42-=-], but this needs not be explained here). A morphism from X to Y (niteness spaces) is a linear function from RhXi to RhY i which is continuous for the topologies mentioned above. But these morphisms a... |

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1 |
Finiteness spaces. Preliminary version accessible from http://iml.univ-mrs.fr/~ehrhard
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- 2003
(Show Context)
Citation Context ...e spaces. We present here shortly a simplied version of this semantics, based on the notion of niteness spaces, a discrete analogue of Kthe spaces. This model will be presented more thoroughly in [Ehr=-=0-=-3]. Given a set I and a subset F of P(I), let us denote by F ? the set of all subsets of I which have a nite intersection with all the elements of F . A niteness space is a pair X = (jX j; F(X)), wher... |

1 | semantics and abstract machines - Kueker, Smith - 1987 |