## Convolution ¯ λµ-calculus (2007)

Venue: | of Lecture Notes in Computer Science |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Vaux07convolution¯,

author = {Lionel Vaux},

title = {Convolution ¯ λµ-calculus},

booktitle = {of Lecture Notes in Computer Science},

year = {2007},

pages = {381--395},

publisher = {Springer}

}

### OpenURL

### Abstract

We define an extension of Herbelin’s ¯ λµ-calculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of Ehrhard-Regnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambda-model in the category of sets and relations. 1

### Citations

51 | Finiteness spaces
- Ehrhard
- 2005
(Show Context)
Citation Context ... happens that original models of linear logic lead to novel extensions of λcalculus, using the Curry-Howard correspondence as a tool to draw semantical properties back into the syntax. In [Ehr01] and =-=[Ehr05]-=-, Ehrhard introduced models of linear logic in which formulas are interpreted by particular vector spaces, and proofs by linear maps between these spaces. Moreover, morphisms with type !A −◦ B corresp... |

46 | The differential lambda-calculus
- Ehrhard, Regnier
(Show Context)
Citation Context ...yped λ-calculus in which λ-terms are interpreted by smooth functions between vector spaces in a very natural way, but also led to the introduction of differential λ-calculus by Ehrhard and Regnier in =-=[ER03]-=-. Differential Nets. Here we briefly outline some features of the differential interaction nets from [ER05], which may be considered as a syntactic presentation of the models from [Ehr01] and [Ehr05].... |

35 |
Differential interaction nets
- Ehrhard, Regnier
(Show Context)
Citation Context ... introduction, we briefly review notions and ideas that led to the definition of convolution ¯ λµ-calculus: it is the pure calculus associated with an extension of Ehrhard-Regnier’s differential nets =-=[ER05]-=-, along the lines of Laurent’s polarization of linear logic proof nets [Lau02]. In section 2 we introduce the objects of convolution ¯ λµ-calculus, and the associated notion of reduction, for which we... |

29 | On köthe sequence spaces and linear logic
- Ehrhard
(Show Context)
Citation Context ...03]. It also happens that original models of linear logic lead to novel extensions of λcalculus, using the Curry-Howard correspondence as a tool to draw semantical properties back into the syntax. In =-=[Ehr01]-=- and [Ehr05], Ehrhard introduced models of linear logic in which formulas are interpreted by particular vector spaces, and proofs by linear maps between these spaces. Moreover, morphisms with type !A ... |

14 |
Sequent calculi for second order logic
- Danos, Joinet, et al.
- 1995
(Show Context)
Citation Context ... [Her95] transposes the Curry-Howard correspondence between classical natural deduction and λµ-calculus to the setting of classical sequent calculus LK (in fact one of its deterministic versions: LKT =-=[DJS95]-=-). In particular, the notion of application, corresponding to the modus ponens of natural deduction, is replaced with the notion of cut between a term and a context. More precisely, ¯ λµ-calculus invo... |

5 | Execution time of lambda-terms via non-uniform semantics and intersection types. Preprint Institut de Mathématique de Luminy 2006
- Carvalho
(Show Context)
Citation Context ...ion 3 we define a reflexive object D in the category of sets and relations, following [BE04]. Then we introduce a type-system, in which types are elements of D, along the lines of Carvalho’s system R =-=[dC06]-=-. We conclude by proving that terms identified by reduction have exactly the same types. 1.1 Classical Logic and Co-structural Rules Denotational semantics of linear logic gives rise to models of simp... |

2 |
An extensional model of the lambda-calculus in the category of sets and relations
- Bucciarelli, Ehrhard
- 2004
(Show Context)
Citation Context ...tion ¯ λµ-calculus, and the associated notion of reduction, for which we prove the Church-Rosser property. In section 3 we define a reflexive object D in the category of sets and relations, following =-=[BE04]-=-. Then we introduce a type-system, in which types are elements of D, along the lines of Carvalho’s system R [dC06]. We conclude by proving that terms identified by reduction have exactly the same type... |

1 |
Intersection and union types in the lambda-bar-mu-mu-tilde-calculus
- Dougherty, Ghilezan, et al.
- 2005
(Show Context)
Citation Context ...: the same as for costructural rules, this amounts to reverse the direction of the corresponding relation. 1.2 Related Work A system of intersection and union types for the ¯ λµ-calculus is presented =-=[DGL05]-=-. This system bears some similarity with the type system we present in section 3: this is underlined by the fact that all weakly normalizing terms are typable. It comes as no 6ssurprise, since our sys... |