## About Translations of Classical Logic into Polarized Linear Logic (2003)

Venue: | In Proceedings of the eighteenth annual IEEE symposium on Logic In Computer Science |

Citations: | 12 - 0 self |

### BibTeX

@INPROCEEDINGS{Laurent03abouttranslations,

author = {Olivier Laurent and Preuves Programmes Systemes and Laurent Regnier},

title = {About Translations of Classical Logic into Polarized Linear Logic},

booktitle = {In Proceedings of the eighteenth annual IEEE symposium on Logic In Computer Science},

year = {2003},

pages = {11--20},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categorical model of Linear Logic by a construction similar to the co-Kleisli category. Secondly we analyse two standard Continuation-Passing Style (CPS) translations, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL.

### Citations

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...situation is given by the category of coherent spaces with the multiset version of !. We denote by ? the endofunctor U C ? C on C (and by ! its dual). It is standard fact that it is a monad on C (see =-=[19]-=-). Still in accordance with linear logic terminology we will denote der A : A ! ?A and dig A : ??A ! ?A the natural transformations associated to ?. To keep notations light we will also denote der A :... |

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...istic logic: normalization, Church-Rosser, denotational semantics. . . Let us mention Girard's system LC which introduces the important notion of polarities in logic [9], Parigot's lambda-mu-calculus =-=[22]-=- which is a natural extension of lambda-calculus embodying classical principles, and Danos-Joinet-Schellinx (DJS)'s impressive classification of classical constructivizations within the syntactical fr... |

234 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...ns, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL. 1. Introduction Curry-Howard for classical logic. Thanks to the seminal work of Griffin =-=[10]-=- who showed that classical principles could be used to type control operators such as Scheme's call/cc, it is now well known that classical logic (LK), as intuitionistic logic (LJ), fits into the Curr... |

220 |
Call-by-name, call-by-value, and the #-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...struct simply by composing a CPS translation with a translation of LJ into LL, e.g. Girard's translation: LK CPS - LJ !A(B - LL For example if we consider the call by name (CBN) Plotkin's translation =-=[23]-=-, this yields LK !?!A(?!B - LL where the formula on the arrow is the linear formula associated to the implication of LK. As we see the defect of this method is that it makes intensive use of exponenti... |

162 |
A new constructive logic: classical logic
- Girard
- 1991
(Show Context)
Citation Context ...es analogous to those of intuitionistic logic: normalization, Church-Rosser, denotational semantics. . . Let us mention Girard's system LC which introduces the important notion of polarities in logic =-=[9]-=-, Parigot's lambda-mu-calculus [22] which is a natural extension of lambda-calculus embodying classical principles, and Danos-Joinet-Schellinx (DJS)'s impressive classification of classical constructi... |

103 | A new deconstructive logic: Linear logic
- Danos, Joinet, et al.
- 1997
(Show Context)
Citation Context ...tural extension of lambda-calculus embodying classical principles, and Danos-Joinet-Schellinx (DJS)'s impressive classification of classical constructivizations within the syntactical framework LK tq =-=[6, 7]-=-. Decomposing the intuitionistic implication. An important advance in the study of (typed) lambda-calculus, was the discovery by Girard [8] that the intuitionistic implication could be decomposed in l... |

91 |
The linear abstract machine
- Lafont
- 1988
(Show Context)
Citation Context ...otations light we will also denote der A : !A ! A and dig A : !A ! !!A the dual natural transformations associated to the comonad !. The property that U C has a left adjoint makes C a Lafont category =-=[14]-=-. A detailed proof that Lafont's categories are sound models of linear logic has been produced by Bierman (see [3, 4]). It uses some interesting consequences of the existence of an adjunction among wh... |

89 |
Dual intuitionistic linear logic
- Barber
- 1996
(Show Context)
Citation Context ...s on continuations, getting a system called CLC which, as pointed above, is very similar to Berdine et al. and/or Laird's target language, and may also be seen as a subsystem of Barber-Plotkin's DILL =-=[1]-=-. The call by value case. For the sake of simplicity we stick to CBN calculus and translations. Let us point though that the call by value (CBV) case is symmetrical: instead of Girard's translation, u... |

83 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
- Selinger
(Show Context)
Citation Context ...ch is identified with Plotkin's in the linear world. Categorical semantics. The paper begins with the construction of a categorical model of classical logic, a control category as defined by Selinger =-=[25]-=-, by a method very similar to the construction of the co-Kleisli of a monoidal closed category. This construction may be seen as a semantical version of the linear analysis of classical logic by LLP. ... |

49 |
An evaluation semantics for classical proofs
- Murthy
- 1991
(Show Context)
Citation Context ...e's call/cc, it is now well known that classical logic (LK), as intuitionistic logic (LJ), fits into the CurryHoward paradigm. Work on CPS-translations showing that they correspond to ::-translations =-=[21]sa la Gode-=-l, have been used to compile these control operators into lambdacalculus. From the viewpoint of logic, work has been carried out on the so-called "classical constructivization" problem, and ... |

46 |
On intuitionistic linear logic
- Bierman
- 1993
(Show Context)
Citation Context ...to the comonad !. The property that U C has a left adjoint makes C a Lafont category [14]. A detailed proof that Lafont's categories are sound models of linear logic has been produced by Bierman (see =-=[3,-=- 4]). It uses some interesting consequences of the existence of an adjunction among which: for any pair of objects A and B we have ?(A B) _ ' ?A P ?B. Any P-monoid N is a ?-algebra and any P-morphi... |

44 |
Game semantics and abstract machines
- Danos, Herbelin, et al.
- 1996
(Show Context)
Citation Context ...semantics, in particular using the first author game model for LLP [17]. This could yield some similar relation between game semantics for classical logic and abstract machines as the ones studied in =-=[5]-=- for intuitionistic logic. There is also the work by Jim Laird [15] mentioned in the introduction, that introduces a special kind of game models for CPS translations. The question remains open whether... |

35 | H.: LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of the classical implication
- Danos, Joinet, et al.
- 1995
(Show Context)
Citation Context ...tural extension of lambda-calculus embodying classical principles, and Danos-Joinet-Schellinx (DJS)'s impressive classification of classical constructivizations within the syntactical framework LK tq =-=[6, 7]-=-. Decomposing the intuitionistic implication. An important advance in the study of (typed) lambda-calculus, was the discovery by Girard [8] that the intuitionistic implication could be decomposed in l... |

32 |
Étude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille
- Laurent
- 2002
(Show Context)
Citation Context ...are translated by negative formulas. The first author has introduced a proof system LLP (Polarized Linear Logic) for polarized formulas, which relaxes the use of structural rules on negative formulas =-=[16]-=-. This system is better suited than LL or LL pol for encoding classical logic because it allows to design a translation LK !A(B - LLP which is a straightforward extension of Girard's translation. The ... |

23 | Categorical models of linear logic revisited, to appear
- Melliès
- 2009
(Show Context)
Citation Context ...r each pair of objectssA and B there is a morphism m A;B : ?(A P B) ! ?A P ?B which is a P-morphism. As before we will use the same notation for the dual morphism: m A;B : !A !B ! !(A B). 1 Following =-=[20]-=- one may weaken this condition and only suppose that the restriction of U C to a subcategory of P-Mon(C) closed by P and & has a left adjoint. 3.2. Building a control category We will now construct fr... |

22 |
Lambda-Calcul : Types et Modèles. Etudes et Recherches en Informatique
- Krivine
- 1990
(Show Context)
Citation Context ...ion on : 0 - types is satisfied by this translation. It is straightforward to check that the type translation rules are compatible with the term ones. 4.4. Krivine's translation Krivine's translation =-=[12, 13-=-] is essentially a slight amelioration of the former Godel's translation of classical into intuitionistic logic; contrarily to Plotkin's one it is type dependent. A CLC type A is associated to a lamb... |

15 | A general storage theorem for integers in call-by-name lambdacalculus
- Krivine
- 1994
(Show Context)
Citation Context ...ion on : 0 - types is satisfied by this translation. It is straightforward to check that the type translation rules are compatible with the term ones. 4.4. Krivine's translation Krivine's translation =-=[12, 13-=-] is essentially a slight amelioration of the former Godel's translation of classical into intuitionistic logic; contrarily to Plotkin's one it is type dependent. A CLC type A is associated to a lamb... |

11 |
Polarized games (extended abstract), in
- Laurent
- 2002
(Show Context)
Citation Context ...d linear systems if one eventually shows up. An interesting remaining question is to study CPS translations in the framework of game semantics, in particular using the first author game model for LLP =-=[17]-=-. This could yield some similar relation between game semantics for classical logic and abstract machines as the ones studied in [5] for intuitionistic logic. There is also the work by Jim Laird [15] ... |

10 |
T.: Continuation models are universal for lambda-mu-calculus
- Hofmann, Streicher
- 1997
(Show Context)
Citation Context ...slation internal to linear logic which can be used to compare them as we do with Krivine's and Plotkin's translations. Typically the same work could be achieved with the Hofmann-Streicher translation =-=[11]-=- which is identified with Plotkin's in the linear world. Categorical semantics. The paper begins with the construction of a categorical model of classical logic, a control category as defined by Selin... |

7 |
Tortora de Falco, Polarisation des preuves classiques et renversement
- Quatrini, L
- 1996
(Show Context)
Citation Context ...Q being positive, and contain only negative formulas the translations of which begin with a ?. The reversing translation. The second translation, that has been used by Tortora de Falco and Quatrini [24] to refine the DJS embedding of LK into LL, uses a lighter decoration: X = ?X ? (N P M) = N P M (?P ) = ?P and dually for the positive formulas. Sequents ` are translated as ` . Note tha... |

6 | 2002, `A Game Semantics of Linearly Used Continuations
- Laird
(Show Context)
Citation Context ...o be regular lambda-calculus typed with usual types and linear types; this kind of calculus has already been used in work analyzing linearity in CPS translations, e.g. by Berdine et al. [2] and Laird =-=[15]-=-. It allows us to give a clear factorization of DJS translation: LK CPS - CLC H H H H H H !?A(?B j LL !A(B ? Linearity of continuations vs. continuation-passing. The linearity of CBN continuations sho... |

3 |
Polarized proof-nets and -calculus
- Laurent
- 2001
(Show Context)
Citation Context ...LP derivations by exponential connectives in the spirit of decorations of classical logic into linear logic by Danos-Joinet-Schellinx [7]; it has been used to prove the strong normalization of LLP in =-=[18]-=-. Viewed from the positive side, this translation encapsulates every positive formula of a proof into a !-box, thus its name. The goal is to make LL-correct each application of a generalized structura... |

2 |
What is a categorical model of Linear Logic
- Bierman
- 1995
(Show Context)
Citation Context ...to the comonad !. The property that U C has a left adjoint makes C a Lafont category [14]. A detailed proof that Lafont's categories are sound models of linear logic has been produced by Bierman (see =-=[3,-=- 4]). It uses some interesting consequences of the existence of an adjunction among which: for any pair of objects A and B we have ?(A B) _ ' ?A P ?B. Any P-monoid N is a ?-algebra and any P-morphi... |