## The differential λµ-calculus (2007)

Venue: | Theor. Comput. Sci |

Citations: | 6 - 2 self |

### BibTeX

@ARTICLE{Vaux07thedifferential,

author = {Lionel Vaux},

title = {The differential λµ-calculus},

journal = {Theor. Comput. Sci},

year = {2007},

pages = {166--209}

}

### OpenURL

### Abstract

We define a differential λµ-calculus which is an extension of both Parigot’s λµ-calculus and Ehrhard-Régnier’s differential λ-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing. Contents 1

### Citations

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
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- 1992
(Show Context)
Citation Context ...onal semantics of λ-calculus. Costructural constructs allow to differentiate morphisms: they form the semantic basis of the differential part of the calculus. Michel Parigot introduced λµ-calculus in =-=[Par92]-=-: this extension of λ-calculus lifts the Curry-Howard correspondence from intuitionistic logic to classical logic. In this setting, proofs in classical natural deduction (also introduced in [Par92]) a... |

122 |
Lambda Calcul : types et modèles
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- 1990
(Show Context)
Citation Context ... that typed terms are strongly normalizing, under the aforementioned conditions on R. The structure of the proof is borrowed from [ER03, Section 4], using the Tait reducibility method as presented in =-=[Kri90]-=-. In subsection 5.1, we prove that the set of strongly normalizing terms is exactly the module generated by strongly normalizing simple terms: this uses the conditions we introduced on R and is essent... |

51 | Classical logic, continuation semantics and abstract machines - Streicher, Reus - 1998 |

44 | The differential lambda-calculus
- Ehrhard, Regnier
- 2003
(Show Context)
Citation Context ...al λ-calculus. We prove some basic properties of the system: reduction enjoys ChurchRosser and simply typed terms are strongly normalizing. 1 Introduction Thomas Ehrhard and Laurent Régnier showed in =-=[ER03]-=- how to extend λ-calculus by means of formal derivatives of λ-terms, following the well-known rules of usual differential calculus. This differential λ-calculus involves a strong relationship between ... |

34 |
Proofs of Strong Normalisation for Second Order Classical Natural Deduction
- Parigot
- 1997
(Show Context)
Citation Context ...vely, these are sets closed under backwards reduction. In section 5.3, we interpret types into some particular saturated subsets of the set of strongly normalizing terms. We adapt ideas by Parigot in =-=[Par97]-=- in order to extend that notion of reducibility to the types associated with names. In section 5.4, we prove that typed terms lie in the interpretation of their types: this entails strong normalizatio... |

30 | On Köthe sequence spaces and linear logic
- Ehrhard
- 2001
(Show Context)
Citation Context ...explicit by the interaction between the structural rules of linear logic (contraction, dereliction and weakening), and some new semantic co-structural constructions that were introduced by Ehrhard in =-=[Ehr01]-=- and [Ehr04]. Structural constructs are used to manage arguments in the denotational semantics of λ-calculus. Costructural constructs allow to differentiate morphisms: they form the semantic basis of ... |

17 | Böhm trees, Krivine’s machine and the Taylor expansion of λ-terms. CiE
- Ehrhard, Regnier
- 2006
(Show Context)
Citation Context ...tural rules. From a computer scientist’s point of view, derivatives allow for a precise investigation of the use of arguments by λ-terms, i.e. purely functional programs. In particular, in [ER04] and =-=[ER05]-=-, Ehrhard and Régnier obtain very deep results relating the summands of the Taylor expansion of a pure λ-term with executions of this term in a Krivine machine. One may want to extend these results to... |

17 | Polarized proof-nets and λµ-calculus - Laurent |

14 |
deduction: An analysis of "computations" in classical logic
- Parigot, “Free
- 1991
(Show Context)
Citation Context ...us in [Par92]. It is an extension of λ-calculus, lifting the Curry-Howard correspondence from intuitionistic logic to classical logic. The associated logical system is a restriction of free deduction =-=[Par91]-=-, called classical natural deduction, also introduced in [Par92]. As such, classical natural deduction enjoys an internal notion of cut, similar to that of intuitionistic natural deduction, although i... |

10 | An environment machine for the lambda-mu-calculus - Groote - 1998 |

9 | Uniformity and the Taylor expansion of ordinary lambda-terms - Ehrhard, Regnier - 2008 |

7 |
W.: Lambda-mu-calculus and Böhm’s theorem
- David, Py
- 2001
(Show Context)
Citation Context ...us and this alternative syntax, which he calls Λµ-calculus, are distinct calculi. In particular, he proves that the separation property holds in Λµ-calculus, whereas it fails in λµ-calculus (see also =-=[DP01]-=-). For this reason, we think it better to avoid confusion by denoting differently terms and named terms. Reducing terms with several outputs The reduction of λµ-calculus is given by the usual βreducti... |

7 |
Separation with streams in the λµ-calculus
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- 2005
(Show Context)
Citation Context ... in [Par92]), terms and named terms are denoted by the same symbols. Moreover, there is a variant of λµ-calculus with only one syntactic group: one can form µα s and [α] s whatever the shape of s. In =-=[Sau05]-=-, however, Alexis Saurin shows that Parigot’s λµ-calculus and this alternative syntax, which he calls Λµ-calculus, are distinct calculi. In particular, he proves that the separation property holds in ... |

4 |
Available at http: //iml.univ-mrs.fr/ ∼ girard/linear.pdf
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- 1987
(Show Context)
Citation Context ...atics, linearity is a fundamental notion of algebra. In λ-calculus, however, as in proof theory in general, linearity is a completely different concept at first sight. Jean-Yves Girard’s linear logic =-=[Gir87]-=-, by decomposing intuitionistic implication, made this syntactic concept of linearity prominent. As we stated informally before, a term is said to be linear if it uses its argument exactly once. This ... |

3 |
Liguoro and Adolfo Piperno. Nondeterministic extensions of untyped λ-calculus
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- 1995
(Show Context)
Citation Context ...in the function and not in the argument, in accordance with the computational notion of linearity. Although it is argued, in the introduction of [ER03], that non-deterministic choice (in the sense of =-=[dLP95]-=-) provides a possible computational interpretation of the sum, this is not explicitly introduced in the calculus, if only because it would break confluence. We will rather allow the formation of forma... |

2 |
Finiteness spaces. Available at http://iml.univ-mrs.fr/ ∼ ehrhard/pub/ finite.ps.gz
- Ehrhard
- 2004
(Show Context)
Citation Context ...the interaction between the structural rules of linear logic (contraction, dereliction and weakening), and some new semantic co-structural constructions that were introduced by Ehrhard in [Ehr01] and =-=[Ehr04]-=-. Structural constructs are used to manage arguments in the denotational semantics of λ-calculus. Costructural constructs allow to differentiate morphisms: they form the semantic basis of the differen... |

2 |
Réseaux de preuve polarisés : des réseaux pour LC. Mémoire de D.E.A. de logique et fondements de l’informatique
- Laurent
- 1998
(Show Context)
Citation Context ...malization in the typed case can be found in [Par97]. Like λ-calculus, λµ-calculus can be given a denotational semantics in a variant of linear logic. This variant is Laurent’s polarized linear logic =-=[Lau98]-=-, i.e. linear logic in which all formulas are polarized, and contraction and weakening are allowed on every negative formula. 1.2 Differential λ-calculus Linearity in λ-calculus In the mainstream math... |

2 |
λ-calculus in an algebraic setting. Available at http://iml.univ-mrs.fr
- Vaux
(Show Context)
Citation Context ...present in sections 2 and 3 is already quite complex, and we think it better not to add to that complexity (which does not mean we needn’t be cautious when dealing with sums). The reader may refer to =-=[Vau06]-=- for a presentation of λ-terms with sums handled inside the calculus. The derivative of a term In the abovementioned models of linear logic by Ehrhard, all functions are differentiable. The differenti... |

1 |
Finiteness spaces. http://iml.univ-mrs.fr/%7Eehrhard/pub. html
- Ehrhard
- 2004
(Show Context)
Citation Context ...the interaction between the structural rules of linear logic (contraction, dereliction and weakening), and some new semantic co-structural constructions that were introduced by Ehrhard in [Ehr01] and =-=[Ehr04]-=-. Structural constructs are used to manage arguments in the denotational semantics of λ-calculus. Co-structural constructs allow to derivate morphisms: they form the semantic basis of the differential... |

1 |
2 Of course, the set of simple terms ∆ is not the same when we write R〈∆〉 and Pm
- Lambda-Calculus, Masson, et al.
- 1993
(Show Context)
Citation Context ...〈N 〉, then (t)π ∈ N. 1 We defined simultaneous substitution (as in s[T1, . . . , Tn/x1, . . . , xn]) using one-variable substitution, and not as a primitive operation, as it is done, e.g., in Krivine =-=[Kri93]-=-. Thus, s [T1, . . . , Tn/x1, . . . , xn] is well defined only if no xi is free in any Tj. This enforces a constrained formulation of the latter adequation lemma (theorem 4.32), in the proof of which ... |