## Differential Linear Logic and Polarization

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@MISC{Vaux_differentiallinear,

author = {Lionel Vaux},

title = {Differential Linear Logic and Polarization},

year = {}

}

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### Abstract

Abstract. We study an extension of Ehrhard–Regnier’s differential linear logic along the lines of Laurent’s polarization. We show that a particular object of the well-known relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semantics of both differential and polarized linear logics: this justifies our choice of cut elimination rules. Then we show this new system models the recently introduced convolution ¯ λµ-calculus, the same as linear logic decomposes λ-calculus. 1

### Citations

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Citation Context ...where cut elimination is guided by the reduction of nets: call differential linear logic (DiLL) this system. Polarization. The notion of polarities, made prominent by Girard’s work on classical logic =-=[6]-=-, led to the definition of polarized linear logic (LLP) by Laurent in his thesis [7]. LLP is actually the extension of a restriction of linear logic. The restriction is on formulas, which have to be p... |

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Citation Context ...las is built from that of their subformulas (basically variables and exponentials). Work by Laurent and Regnier [10] later showed that this construction generalizes: the`-monoids of a Lafont category =-=[11]-=- freely form a model of LLP. Polarized Costructural Rules. In short, DiLL introduces a symmetry on exponential types, with costructural rules, and provides a differential analysis of proofs through a ... |

61 | From proof nets to interaction nets
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Citation Context ... 2 Polarized Differential Nets Polarized differential nets (PDN) are formal finite sums of simple nets, which are particular multiport interaction nets, such as studied by Mazza [15] following Lafont =-=[16]-=-. The cells of simple PDN are actually those of DN, i.e. DIN plus promotion. Mainly, PDN differ from DN when considering typing, which is relaxed by polarization, and cut elimination, which involves n... |

60 |
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Citation Context ...ed by term constructions reflecting polarized structural rules. These enjoy decompositions into LLP, similar to the translation of λ-calculus into linear logic proof nets studied by Danos and Regnier =-=[8,9]-=-. From a semantical point of view, the idea that polarization canonically extends the coalgebraic structure of exponentials to polarized formulas is also valid. For instance Girard’s correlation space... |

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Citation Context ...fragment of their differential λ-calculus [2]. Both DIN and differential λ-calculus originate in the study of particular models of linear logic introduced by Ehrhard, such as the finiteness spaces of =-=[3]-=-. The distinctive attributes of these models are a monoid structure on morphisms, together with the ability to differentiate the interpretations of intuitionistic proofs. This latter feature can be an... |

39 |
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Citation Context ...ogic (DiLL) this system. Polarization. The notion of polarities, made prominent by Girard’s work on classical logic [6], led to the definition of polarized linear logic (LLP) by Laurent in his thesis =-=[7]-=-. LLP is actually the extension of a restriction of linear logic. The restriction is on formulas, which have to be polarized. The extension is on structural rules, which are allowed on negative formul... |

35 |
Differential interaction nets
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(Show Context)
Citation Context ...uced convolution ¯ λµ-calculus, the same as linear logic decomposes λ-calculus. 1 Introduction Differential Linear Logic. Differential interaction nets (DIN) were introduced by Ehrhard and Regnier in =-=[1]-=- to provide a notion of proof nets for the finitary fragment of their differential λ-calculus [2]. Both DIN and differential λ-calculus originate in the study of particular models of linear logic intr... |

26 | Etude de la polarisation en logique - Laurent - 2002 |

19 | Not enough points is enough
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Citation Context ...s to deduce, in a very natural way, the semantics of polarized costructural rules from that of polarized structural rules: just reverse the corresponding relations. The reflexive object introduced in =-=[13]-=- is well suited for this study: it allows to interpret both DiLL and LLP in a pure (i.e. untyped) setting, so that exponential structural and costructural rules are exchanged by symmetry, and polarize... |

16 | Böhm trees, Krivine’s machine and the Taylor expansion of lambda-terms - Ehrhard, Regnier - 2006 |

13 |
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Citation Context ... : !A ⊸ 1. The basis of differentiation in these models is that morphisms f : !A ⊸ B can be seen as defined by power series, in a construction similar to Girard’s quantitative semantics of λ-calculus =-=[4]-=-: then one can introduce a morphism ∂ : A ⊸ !A such that f ◦ ∂ : A ⊸ B is the linear part of f, i.e. its derivative at point 0; together with the convolution product, this defines the derivative at an... |

12 | About translations of classical logic into polarized linear logic
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Citation Context ...semantics of LLP [7]: the coalgebraic structure of the interpretation of polarized formulas is built from that of their subformulas (basically variables and exponentials). Work by Laurent and Regnier =-=[10]-=- later showed that this construction generalizes: the`-monoids of a Lafont category [11] freely form a model of LLP. Polarized Costructural Rules. In short, DiLL introduces a symmetry on exponential t... |

10 | Differential categories
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Citation Context ...hrhard and Regnier, but they prefered to stick to the finitary system, as the proof theoretic properties of DN were yet to be explored. A categorical investigation of a similar system can be found in =-=[5]-=-, focusing on the differentiation operator as a primitive, and not necessarily involving the whole structure of linear logic models. Our work departs form this, since we put the emphasis on costructur... |

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8 |
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Citation Context ...fferential Linear Logic. Differential interaction nets (DIN) were introduced by Ehrhard and Regnier in [1] to provide a notion of proof nets for the finitary fragment of their differential λ-calculus =-=[2]-=-. Both DIN and differential λ-calculus originate in the study of particular models of linear logic introduced by Ehrhard, such as the finiteness spaces of [3]. The distinctive attributes of these mode... |

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6 | The differential λµ-calculus
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(Show Context)
Citation Context ...ostructural rules in the setting of linear logic.srelations entertained by both of these extensions of the Curry–Howard correspondence and its analysis by linear logic. A first result was provided in =-=[12]-=-, where the author introduces a differential λµ-calculus which is a conservative extension of both λµ-calculus and differential λ-calculus, enjoying confluence and strong normalization of typed terms:... |

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3 |
λ-calcul différentiel et logique classique : interactions calculatoires,” Thèse de Doctorat, Université AixMarseille
- Vaux
- 2007
(Show Context)
Citation Context ...es in PDN are those of group p ′ . It is easily checked that the left part of Figure 3, i.e. groups m, r, p ′ and p except p3, define a confluent and terminating system. A full proof is developped in =-=[17]-=-. Local confluence of the whole system, minus d1, is an tiring but easy exercise. Notice however that local confluence of the system including d1 is only verified up-to a Retoré-like equivalence on ne... |

3 | Intuitionistic di erential nets and lambda-calculus - Tranquilli |

2 | Convolution ¯ λµ-calculus
- Vaux
- 2007
(Show Context)
Citation Context ...n easy to derive the computational behaviour of polarized costructural rules from this semantics. The system presented in the current paper can be seen as the end result of this course of thought. In =-=[14]-=-, the author introduced convolution ¯ λµ-calculus based on similar ideas: interpret Herbelin’s ¯ λµ-calculus into the object of [13] through LLP, then investigate the computational couterpart of the m... |

2 | Interaction Nets: Semantics and Concurrent Extensions. Thèse de doctorat - Mazza - 2006 |

1 |
Substitutions explicites, logique et normalisation
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- 2004
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Citation Context ... proved this notion of reduction is confluent. It also is proved in [17] that the simply typed objects of convolution ¯ λµ-calculus are all strongly normalizing: one adapts the proof by Polonowski in =-=[18]-=- for ¯ λµ˜µ-calculus. Theorem 3. The cut elimination procedure of PDN up-to structural equivalence simulates the reduction of convolution ¯ λµ-calculus. · · · T · · · µ · · · · · · µ · · · · · · µ · ·... |

1 | L.: Di erential interaction nets. Theor - Ehrhard, Regnier - 2009 |

1 | The di erential λµ-calculus. Theor - Vaux - 2009 |

1 | λ-calcul di érentiel et logique classique: interactions calculatoires - Vaux - 2007 |

1 | The di erential λµ-calculus. Theor. Comput. Sci. 379(1-2) (2007) 166 209 14 - Vaux - 2007 |