## Classical linear logic of implications (2002)

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Venue: | In Proc. Computer Science Logic (CSL'02), Springer Lecture Notes in Comp. Sci. 2471 |

Citations: | 10 - 4 self |

### BibTeX

@INPROCEEDINGS{Hasegawa02classicallinear,

author = {Masahito Hasegawa},

title = {Classical linear logic of implications},

booktitle = {In Proc. Computer Science Logic (CSL'02), Springer Lecture Notes in Comp. Sci. 2471},

year = {2002},

pages = {458--472},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for category-theoretic models given by ∗-autonomous categories with linear exponential comonads. 1

### Citations

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λµ–calculus: an algorithmic interpretation of classical natural deduction
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- 1992
(Show Context)
Citation Context ...x:k (L x))): ut 2.2 Alternative formulations of DCLL Formulation based on the -calculus. Instead of the combinator C for the double-negation elimination, we could use the syntax of the -calculus [21] for expressing the duality, as done in [17] for the multiplicative fragment (MLL). We do not take this approach here as our presentation using C seems suciently simple, while the -calculus style fo... |

210 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ...s in fact amounts to the infamous (in)equality known as \triple unit problem" (which asks if two canonical endomorphisms on ((A ( I) ( I) ( I are the same in a symmetric monoidal closed category,=-=-=- see [19, 16]) if one replaces ? by I. 3 DILL in DCLL The primitive constructs of DILL (summarized in Appendix A) can be dened in DCLL as follows: I ? ( ? 1 2 ( 1 ( 2 ( ?) ( ? ! ( ! ?) ( ? x ... |

114 | autonomous categories - Barr - 1979 |

96 | What is a categorical model of intuitionistic linear logic
- Bierman
- 1995
(Show Context)
Citation Context ...ruction), is that the term model of DCLL forms a model of DILL, i.e., a symmetric monoidal closed category equipped with a symmetric monoidal comonad satisfying certain coherence conditions (see e.g. =-=[7]) which w-=-e shall call a \linear exponential comonad" (following [15]). 3 Denition 1 (linear exponential comonad). A symmetric monoidal comonad ! = (!; "; ; mA;B ; m I ) on a symmetric monoidal catego... |

93 | A mixed linear non-linear logic: proofs, terms and models
- Benton
- 1995
(Show Context)
Citation Context ...lds for every such models. ut 3 In [2] a model of DILL is described as a symmetric monoidal adjunction between a cartesian closed category and a symmetric monoidal closed category (Benton's LNL model =-=[5]). It-=- is known that such an \adjunction model" gives rise to a linear exponential comonad on the symmetric monoidal closed category part. Conversely, a symmetric monoidal closed category with a linear... |

86 |
autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...m (!A;sA ) to (!B;sB ) is also a comonoid morphism from (!A; e A ; dA ) to (!B; e B ; dB ). Moreover, the symmetric monoidal closed category given by the term model of DCLL is a -autonomous category [=-=3,-=- 4] if we take ? as the dualizing object. Recall that a -autonomous category can be characterized as a symmetric monoidal closed category with an object ? such that the canonical morphism from to ( (... |

37 |
Linear Type Theories, Semantics and Action Calculi
- Barber
- 1997
(Show Context)
Citation Context ...tive exponential fragment of Classical Linear Logic [10] (often called MELL in the literature). It can be regarded as an extension of the Dual Intuitionistic Linear Logic (DILL) of Barber and Plotkin =-=[1, 2]-=-. The main feature of DCLL is its simplicity: just three logical connectives (intuitionistic implication →, linear implication ⊸ and the bottom type ⊥) andsix axioms for the equational theory on terms... |

37 |
Linear logic.Theoretical
- Girard
- 1987
(Show Context)
Citation Context ...es with linear exponential comonads. 1 Introduction We propose a linear lambda calculus called Dual Classical Linear Logic (DCLL) for the multiplicative exponential fragment of Classical Linear Logic =-=[10]-=- (often called MELL in the literature). It can be regarded as an extension of the Dual Intuitionistic Linear Logic (DILL) of Barber and Plotkin [1, 2]. The main feature of DCLL is its simplicity: just... |

30 | Linear continuations
- Filinski
- 1992
(Show Context)
Citation Context ...tion monad by letting be ?: ! ' (! ( ?) ( ? ' ( ! ?) ( ?. It is also interesting to re-examine the previous work on applying Classical Linear Logic to programming languages with control features [9, =-=20] using DCL-=-L; in particular Filinski's work [9] seems to share several ideas with the design of DCLL. 6.2 Is \!" better than \!"? A possible criticism on DCLL is on its indirect treatment of the expone... |

30 | Glueing and orthogonality for models of linear logic
- Hyland, Schalk
- 2003
(Show Context)
Citation Context ...., a symmetric monoidal closed category equipped with a symmetric monoidal comonad satisfying certain coherence conditions (see e.g. [7]) which we shall call a \linear exponential comonad" (follo=-=wing [15]). 3-=- Denition 1 (linear exponential comonad). A symmetric monoidal comonad ! = (!; "; ; mA;B ; m I ) on a symmetric monoidal category C is called a linear exponential comonad when the category of its... |

11 | Logical predicates for intuitionistic linear type theories, Typed Lambda Calculi and Applications
- Hasegawa
- 1999
(Show Context)
Citation Context ...r instance, it is much easier to describe and analyze the translations between type systems if we use term calculi like DCLL instead of graph-based systems. Also techniques of logical relations (e.g. =-=[11, 23]-=-) seem to work more smoothly on term-based systems. As future work, we plan to study the compilations of call-by-value programming languages into linearly typed intermediate languages [6, 13] using DC... |

8 | Linearly used continuations
- Berdine, O’Hearn, et al.
- 2001
(Show Context)
Citation Context ...s (e.g. [11, 23]) seem to work more smoothly on term-based systems. As future work, we plan to study the compilations of call-by-value programming languages into linearly typed intermediate languages =-=[6, 13]-=- using DCLL as a target calculus. In fact, our choice of the logical connectives has been motivated by this research direction { see the discussion in Sec. 6. Despite its simplicity, it is shown that ... |

8 | A classical linear lambda calculus
- Bierman
- 1999
(Show Context)
Citation Context ...that it may give a con uent and normalizing reduction system (which cannot be expected for DCLL); also it allows natural treatment of the connective & (by introducing the binary -bindings). See also [8] for relevant results. Axiomatization without C. In DCLL, the following equations are provable: Lemma 2. 1. C? = m (?(?)(? :m (x ? :x) 2. C ! = m ((!)(?)(? :x :C (k (? :m (f ! :k (f ... |

7 | Explicit substitution internal languages for autonomous and ∗-autonomous categories
- Koh, Ong
- 1999
(Show Context)
Citation Context ...s of DCLL Formulation based on the -calculus. Instead of the combinator C for the double-negation elimination, we could use the syntax of the -calculus [21] for expressing the duality, as done in [17] for the multiplicative fragment (MLL). We do not take this approach here as our presentation using C seems suciently simple, while the -calculus style formulation requires to introduce yet another ... |

5 |
Dual intuitionistic linear logic, submitted
- Barber, Plotkin
- 1997
(Show Context)
Citation Context ...tive exponential fragment of Classical Linear Logic [10] (often called MELL in the literature). It can be regarded as an extension of the Dual Intuitionistic Linear Logic (DILL) of Barber and Plotkin =-=[1, 2]-=-. The main feature of DCLL is its simplicity: just three logical connectives (intuitionistic implication →, linear implication ⊸ and the bottom type ⊥) andsix axioms for the equational theory on terms... |

4 | Girard translation and logical predicates
- Hasegawa
- 2000
(Show Context)
Citation Context ...hnically simpler presentation. 4 This result is shown by mildly extending the proof of full completeness of Girard's translation from the simply typed lambda calculus into the f!; (g-fragment of DILL =-=[12]-=-. 5 This is not a monad on the term model of DILL; it is a monad on a suitable subcategory of the category of !-coalgebras. 6 In particular Plotkin's system [22] is the second-order f!;(g-calculus in ... |

3 |
Linearly Used Eects: Monadic and CPS Transformations into the Linear Lambda Calculus
- Hasegawa
(Show Context)
Citation Context ...s (e.g. [11, 23]) seem to work more smoothly on term-based systems. As future work, we plan to study the compilations of call-by-value programming languages into linearly typed intermediate languages =-=[6, 13]-=- using DCLL as a target calculus. In fact, our choice of the logical connectives has been motivated by this research direction { see the discussion in Sec. 6. Despite its simplicity, it is shown that ... |

3 | Categorical models for intuitionistic and linear type theory
- Maietti, Paiva, et al.
- 2000
(Show Context)
Citation Context ...? A possible criticism on DCLL is on its indirect treatment of the exponentials, which have been regarded as the central feature of Linear Logic by many people (though there are some exceptions, e.g. =-=[24,-=- 22, 18] 6 ). We used to consider ! as a primitive and ! as a derived connective as ! ! ( , but not in the other way (i.e. ! ( ! ?) ( ? as we do in DCLL). However, even in the Intuitionistic Linea... |

2 |
Exhausting strategies, joker games and imll with units
- Murawski, Ong
- 1999
(Show Context)
Citation Context ...s in fact amounts to the infamous (in)equality known as \triple unit problem" (which asks if two canonical endomorphisms on ((A ( I) ( I) ( I are the same in a symmetric monoidal closed category,=-=-=- see [19, 16]) if one replaces ? by I. 3 DILL in DCLL The primitive constructs of DILL (summarized in Appendix A) can be dened in DCLL as follows: I ? ( ? 1 2 ( 1 ( 2 ( ?) ( ? ! ( ! ?) ( ? x ... |

2 |
Programs with continuations and linear logic
- Nishizaki
- 1993
(Show Context)
Citation Context ...tion monad by letting be ?: ! ' (! ( ?) ( ? ' ( ! ?) ( ?. It is also interesting to re-examine the previous work on applying Classical Linear Logic to programming languages with control features [9, =-=20] using DCL-=-L; in particular Filinski's work [9] seems to share several ideas with the design of DCLL. 6.2 Is \!" better than \!"? A possible criticism on DCLL is on its indirect treatment of the expone... |

1 |
autonomous categories andlinear logic
- Barr
- 1991
(Show Context)
Citation Context ...rphism from (!A, δA) to (!B,δB) is also a comonoid morphism from (!A, eA,dA) to (!B,eB,dB). Moreover, the symmetric monoidal closed category given by the term model of DCLL is a ∗-autonomous category =-=[3, 4]-=- ifwetake⊥asthe dualizing object. Recall that a ∗-autonomous category can be characterized as a symmetric monoidal closed category with an object ⊥ such that the canonical morphism from σ to (σ ⊸ ⊥) ⊸... |

1 |
A mixedlinear andnon-linear logic: proofs, terms andmodels (extended abstract
- Benton
- 1995
(Show Context)
Citation Context ... holds for every such models. ⊓⊔ 3 In [2] a model of DILL is described as a symmetric monoidal adjunction between a cartesian closedcategory anda symmetric monoidal closedcategory (Benton’s LNL model =-=[5]-=-). It is known that such an “adjunction model” gives rise to a linear exponential comonadon the symmetric monoidal closedcategory part. Conversely, a symmetric monoidal closedcategory with a linear ex... |