## Dialectica and Chu Constructions: Cousins? (2006)

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Venue: | In this Volume |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Paiva06dialecticaand,

author = {Valeria De Paiva},

title = {Dialectica and Chu Constructions: Cousins?},

booktitle = {In this Volume},

year = {2006}

}

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### Abstract

This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove that the dialectica construction can be done over a symmetric monoidal closed basis. We also point out several interesting open problems concerning the Dialectica construction.

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Citation Context ....2 Lambek Calculus An unrelated application of the dialectica construction has been described in [dP91c]. In this case we use a modification of the dialectica construction to model the Lambek calculus=-=[Lam58]-=-, a syntactic formalism devised by Lambek in the late fifties as an explanation of the methematics of sentence structure. The interested reader is referred to the paper. Here we briefly describe its m... |

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Citation Context ... appropriate category. Of course theorems, which are provable from no assumptions, are a special case and get mapped to special morphisms. This style of semantics of proofs, prototypically defined in =-=[LS85]-=- is usually associated with intuitionistic logics, formalized as Natural Deduction systems. This is the celebrated extended Curry-Howard isomorphism. Linear Logic makes an interesting case for the ext... |

121 |
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Citation Context ...a symmetric monoidal closed category, the special case of a cartesian closed category, or even the rather special cartesian closed category Sets, but the original version, proposed by Barr and Chu in =-=[Barr79]-=- was 1sthe most general one. Meanwhile all the published versions of Dialectica constructions are over cartesian closed categories. It has been known for a while ([HdP91]) that the Dialectica construc... |

116 |
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Citation Context ... complication is which logical structure should the set of natural numbers N have, so that we are allowed to define Dial N(Sets). It turns out that Lawvere had done work on a similar problem 5 before =-=[Law]-=- and the only thing we need to do is consider truncated 4 These results have not been formally published, as my co-authors (responsible for the concurrency side of the work) decided to leave academia,... |

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Citation Context ...rd isomorphism. Linear Logic makes an interesting case for the extended Curry-Howard isomorphism. On the one hand, it was the first non-intuitionistic logic for which semantics of proofs was obtained =-=[See89]-=-. The point here is that Linear Logic has an involutive negation, which unlike the involutive negation of classical logic, does not collapse the category into a poset. On the other 1 The way the gener... |

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autonomous categories and linear logic
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Citation Context ... is symmetric monoidal closed and locally presentable and Ω is an internal cogenerator in C (thm 4.8, page 12). But in its generality, the paper gives us no direct construction for the operator !. I=-=n [Barr91] he -=-proved the existence of ! for the subcategory of separated objects of ChuΩC, when C is a cocomplete and complete cartesian closed category and Ω is an internal cogenerator (thm 9.2, page 19). In p... |

85 |
Quantales and (non-commutative) linear logic
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Citation Context ...that Dial N(Sets) is a sound categorical model of the Lambek calculus. More importantly, the paper goes on to show that modalities could be produced for the Lambek calculus, following ideas of Yettter=-=[Yet88]-=-. Thus if one adds additives and restricted forms of permutation, the categorical model allows us to define a Girard-style modality. All this is done only categorically, i.e. semantically. Thus this p... |

76 | A Term Calculus for Intuitionistic Linear Logic - Benton, Bierman, et al. - 1993 |

49 |
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Citation Context ...lectica construction [BGdP91] seems more appropriate. On the other hand, since one usually tries to avoid gigantic collections of morphisms, the more restrained Chu morphisms seem a much better choice=-=[Gup94]. -=-But the comparison turns trickier when exponentials (or modalities) enter the picture. 11s3.4 Modalities in Chu2(Sets) and Dial2(Sets) The Chu construction[Barr79], as a general way of building ∗-au... |

48 |
A categorical linear framework for Petri nets, in
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Citation Context ... and shows the versatility of the construction. 5.1 Petri Nets The Dialectica construction has been applied to the modelling of Petri nets in two different, but related settings. First, Brown and Gurr=-=[BG90]-=- have developed a category of safe Petri nets, using the original dialectica construction GC from my thesis [dP91a] as a blueprint. The basic idea here is that a Petri net can be described (following ... |

48 |
Games Semantics for Linear Logic
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Citation Context ...he dialectica construction [dP89a, dP89b] has, from the beginning, discussed modalities and their modeling, since the cartesian closed structure they provide was our initial goal. Lafont and Streicher=-=[LS90]-=- produced modalities for the Chu construction based on the dialectica modalities of [dP89b]. Of course Barr has done extensive work([Barr91, Barr90, Barr96, Barr98]) on Chu categories from the perspec... |

28 | A Linear Specification Language for Petri Nets
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Citation Context ...ich one to choose. For instance, when dealing with Petri nets (both categories have been used for this application), if one desires morphisms which behave like simulations the dialectica construction =-=[BGdP91]-=- seems more appropriate. On the other hand, since one usually tries to avoid gigantic collections of morphisms, the more restrained Chu morphisms seem a much better choice[Gup94]. But the comparison t... |

27 | A dialectica-like model of linear logic
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(Show Context)
Citation Context ...itionistic implication and insist that morphisms consists only of logical implication and not logical bi-implication, we have (the main variant) of the Dialectica construction. This was introduced in =-=[dP89b]. The-=- formal definition is as follows. Definition 2. The category Dial2(Sets) has as objects A triples such as (U, X, α), where U and X are sets and α: U × X → 2 is a relation. Given two objects A and... |

25 | Categorical models of linear logic revisited. Prépublication électronique - Melliès - 2003 |

23 |
The Dialectica categories, in
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Citation Context ...rd category over C, in this case Sets. 6sthe name of the dialectica categories. The connection between the interpretation and a different, but similar categorical construction, was first presented in =-=[dP89a]. T-=-here a category DSets, which is an internalized version of Gödel’s dialectica interpretation is presented and shown to be a model of intuitionistic linear logic (ILL). For the sake of completeness ... |

22 | On an intuitionistic modal logic - Bierman, Paiva - 2000 |

21 |
A category of labelled Petri nets and compositional proof system
- Winskel
- 1988
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Citation Context ...gory of safe Petri nets, using the original dialectica construction GC from my thesis [dP91a] as a blueprint. The basic idea here is that a Petri net can be described (following Winskel’s suggestion=-=s [Win88]-=-) as a set of events E, a set of conditions B and two relations, the precondition and the postcondition relations, relating events and conditions. Given that the basic objects of the dialectica constr... |

14 | Relating categorical semantics for intuitionistic linear logic
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- 2001
(Show Context)
Citation Context ...ponential comonad and a corresponding linear exponential monad. The term linear exponential comonad was coined by Hyland ([HS99]) and is a neat short-hand for all that is involved. To be precise (cf. =-=[MMPR01]-=-) one could say that a “linear exponential comonad” is a monoidal comonad, whose category of Eilenberg-Moore coalgebras is cartesian. But then one would have to explain what is a “monoidal comonad”, w... |

13 |
Accessible categories and models of linear logic
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- 1990
(Show Context)
Citation Context ...nstruction based on the dialectica modalities of [dP89b]. Of course Barr has done extensive work([Barr91, Barr90, Barr96, Barr98]) on Chu categories from the perspective of models of Linear Logic. In =-=[Barr90], fo-=-r instance he proved that ! exists for Chu Ω(C) if C is symmetric monoidal closed and locally presentable and Ω is an internal cogenerator in C (thm 4.8, page 12). But in its generality, the paper... |

10 | The Chu construction. Theory and - Barr - 1996 |

10 |
Abstract Games for Linear Logic. Extended Abstract
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- 1999
(Show Context)
Citation Context ...ucts and coproducts can be added at will. 4. To model modalities/exponentials we need a linear exponential comonad and a corresponding monad. The term linear exponential comonad was coined by Hyland (=-=[HS99]-=-) and is a neat short-hand for all that is involved. To be precise [MMPR01] one could say that a linear exponential comonad is a monoidal comonad, whose category of Eilenberg-Moore coalgebras is carte... |

9 | The separated extensional Chu category - Barr - 1998 |

7 | Information transfer across Chu spaces - Benthem |

7 | A Dialectica Model of the Lambek Calculus
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- 1991
(Show Context)
Citation Context ...oth by the logic of C and the ‘logic’ of Ω, 7swhile the logic of Chu Ω(C) only depends on C. This makes for flexibility of modelling: getting a non-commutative version of the Dialectica constr=-=uction ([dP91c]-=-) was much easier than getting a non-commutative version of the Chu construction ([Barr96, Barr95]), but also means fewer examples. Since we settled on the same representatin for dialectica objects as... |

6 | autonomous categories: once more around the track. Theory and Applications of Categories 6 - Barr - 1999 |

6 |
Fuzzy models of linear logic
- Barr
- 1996
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Citation Context ...n of morphism of linear logic models proposed by Hyland and Schalk. Barr has done extensive work([Barr91, Barr90, Barr96, Barr98]) on Chu categories from the perspective of models of Linear Logic. In =-=[Barr90]-=-, for instance he proved that ! exists for Chu Ω(C) if C is symmetric monoidal closed and locally presentable and Ω is an internal cogenerator in C (thm 4.8, page 12). But in its generality, the paper... |

6 | Autonomous categories, revisited - Barr - 1996 |

4 |
On a Hitherto Unexploited Extension of the Finitary Standpoint
- Gödel
- 1980
(Show Context)
Citation Context ...ooses the connection with Chu spaces, which we are trying to emphasize here. Before starting to compare the constructions, we pause to explain the name “dialectica”. Gödel’s Dialectica Interpre=-=tation [G58]-=- is the origin of 2 The old name GC came from Girard category over C, in this case Sets. 6sthe name of the dialectica categories. The connection between the interpretation and a different, but similar... |

4 |
Lineales: Algebras and Categories in the Semantics of Linear Logic
- Paiva
- 2002
(Show Context)
Citation Context ...ctions Dial N(Sets) (where the object 2 was generalised to N the set of natural numbers) and DialΩ(C), where C is still a cartesian closed category 3 , but instead of the object 2, we use a lineale =-=Ω [dP02]. -=-In this paper, we generalize from a category C cartesian closed to a category V symmetric monoidal closed, to give us Dial Ω(V). First of all our base category V has to be a symmetric monoidal close... |

3 | Categorical Multirelations, Linear Logic and Petri Nets
- Paiva
- 1991
(Show Context)
Citation Context ...lectica Category In this section we present the generalization of the category Dial2(Sets) to Dial Ω(V), which is the goal of this paper. This generalization has, historically, proceeded by steps. I=-=n [dP91b] w-=-e described two constructions Dial N(Sets) (where the object 2 was generalised to N the set of natural numbers) and DialΩ(C), where C is still a cartesian closed category 3 , but instead of the obje... |

3 | Whats is a categorical model of linear logic - Schalk - 2004 |

2 | Modified realizability interpretation of classical linear logic
- Oliva
- 2007
(Show Context)
Citation Context ...ad of one for intuitionistic logic) based on this work, see Shirahata’s contribution in this special volume. Also connections to other kinds of functional interpretations have been uncovered by Oliva =-=[Oli07]-=-. More work on functional interpretations and categorical models would be welcome though. 3.4. Comparing Chu2(Sets) and Dial2(Sets). The first main similarity between the two constructions is that the... |

2 |
Spaces, Course notes for the School
- Pratt, Chu
- 1999
(Show Context)
Citation Context ...cts. One first, big difference between the constructions has to do with the structure required to obtain the categories Chu2(Sets) and Dial2(Sets). As Pratt remarks on his lectures notes on Chu spaces=-=[Pra99]-=-, as far as the Chu construction is concerned, the order on 2 and the consequent logical reading of equality as logical bi-implication is in the “eye of the beholder”. For the Chu construction, whatev... |

2 | The Dialectica interpretation of firstorder classical linear logic - Shirahata |

1 | Dual Intuitionistic Linear Logic. Doctoral thesis, available as ECS-LFCS-96-347 technical report - Barber |

1 |
Full completeness of the multiplicative linear logic
- Devarajan, Hughes, et al.
- 1999
(Show Context)
Citation Context ...need working out. • Some of the connections to games are still to be worked out, it appears. In particular, it would be nice to know whether there is (or not) a modification of Devarajan et al’s p=-=roof[DHPP99] t-=-hat Chu spaces are fully complete that would work for dialectica. • Generalisations of the dialectica construction to 2-categories or bicategories, in the style of Koslowski [Kos00] might be possibl... |

1 | Essays Dedicated to Johan van Benthem on the Occasion of his Fiftieth Birthday, Amsterdam Univ - In - 1999 |

1 |
A 2-dimensional view of the Chu-construction Presented at
- Koslowski
- 2000
(Show Context)
Citation Context ...an et al’s proof[DHPP99] that Chu spaces are fully complete that would work for dialectica. • Generalisations of the dialectica construction to 2-categories or bicategories, in the style of Koslow=-=ski [Kos00] migh-=-t be possible. • Work on model theory of the chu construction, done by van Benthem [vBen00] and Feferman[F03] might be easy to adapt. If so, there would be relevant connections to Gödel’s interpr... |

1 |
Personal communication
- Schalk
- 2002
(Show Context)
Citation Context ...ike the involutive negation of classical logic, does not collapse the category into a poset. On the other 1 The way the generalization must be done can be deduced from recent work of Hyland and Schalk=-=[S02]-=- on categories of games, but reconstructing it would require a reader with considerable expertise on the subject. 2shand, a Natural Deduction formulation for Intuitionistic Linear Logic was harder to ... |

1 |
Quantales and (non-commutative) Linear Logic
- Winskel
- 1990
(Show Context)
Citation Context ...hat Dial N (Sets) is a sound categorical model of the Lambek calculus. More importantly, the paper goes on to show that modalities could be produced for the Lambek calculus, following ideas of Yettter=-=[Yet88]-=-. Thus 19sif one adds additives and restricted forms of permutation, the categorical model allows us to define a Girard-style modality. All this is done only categorically, i.e. semantically. Thus thi... |

1 |
Proof theory in the abstract Ann
- Hyland
- 2002
(Show Context)
Citation Context ...cation in Gödel’s Dialectica. The fact that the category obtained also corresponds to a model of a version of linear logic was a pleasant surprise in 1987. This work is also discussed and improved in =-=[H02]-=-. Recently Shirahata proposed a dialectica interpretation of linear logic (instead of one for intuitionistic logic) based on this work, see Shirahata’s contribution in this special volume. Also connec... |