## Bistructures, Bidomains and Linear Logic (1997)

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Venue: | in Proc. 21st ICALP |

Citations: | 9 - 3 self |

### BibTeX

@INPROCEEDINGS{Curien97bistructures,bidomains,

author = {Pierre-Louis Curien and Gordon Plotkin and Glynn Winskel},

title = {Bistructures, Bidomains and Linear Logic},

booktitle = {in Proc. 21st ICALP},

year = {1997},

pages = {352--363},

publisher = {Springer-Verlag}

}

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### Abstract

Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has an associated co-Kleisli category which is closely related to that of Berry's bidomains (both have equivalent non-trivial full sub-cartesian closed categories).

### Citations

354 |
Proofs and types
- Girard, Taylor, et al.
- 1989
(Show Context)
Citation Context ... this paper we link Winskel's bistructures [25], Girard's linear logic [10] and Berry's bidomains [25]. We show how bistructures provide a model of classical linear logic extending Girard's web model =-=[10, 11]; we show -=-too that a certain class of bistructures represent bidomains. We hope that the structures isolated here will help in the search for a direct, extensional and "mathematically natural" account... |

263 |
Event structures
- Winskel
- 1986
(Show Context)
Citation Context ... 0se 2 x ) e 0 2 x. Ordered by inclusion, the configurations form a coherent prime algebraic domain (\Gamma(E); `) [21]; such domains are precisely the infinitely distributive, coherent Scott domains =-=[27]-=-. An instance of the causal dependency ordering e 0se when e and e 0 are distinct, is understood as meaning that the event e causally depends on the event e 0 , in that the event e can only occur afte... |

239 |
The formal semantics of Programming languages
- Winskel
- 1993
(Show Context)
Citation Context ...least element and lubs of all directed sets; continuous functions between cpos are those monotonic functions preserving all the directed lubs. For other domain-theoretic terminology see, for example, =-=[28]-=-. 2 The categories of partial orders and topological spaces are finitary, but structures involving reference to the natural numbers, e.g., measure spaces, are not. The notion can be formalised; one re... |

171 |
G.: Petri nets, event structures and domains
- Nielsen, Plotkin, et al.
- 1981
(Show Context)
Citation Context ... "approximations " to the desired sequential structures. Event structures are partial orders of events equipped with a conflict relation and obeying an axiom of finite causes. They were intr=-=oduced in [21]-=- as a model of concurrency, and turned out to have close connections with concrete domains [14] and hence sequentiality [5]; they are also a natural generalisation of Girard's webs. Winskel introduced... |

102 | Linear logic, *-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ...ion of E 0 ( E 1 . As such it is a relation between the events of E 0 and E 1 . Composition in the category is that of relations. The category is a model of intuitionistic linear logic, as defined in =-=[24, 4]-=-. For instance, its tensor is given in a coordinatewise fashion. For event structures E i = (E i ;si ;si ), for i = 0; 1, define: E 0\Omega E 1 = (E 0 \Theta E 1 ; ; ) where (e 0 ; e 1 )s(e 0 0 ; e 0 ... |

99 | What is a categorical model of intuitionistic linear logic
- Bierman
- 1995
(Show Context)
Citation Context ...ion of E 0 ( E 1 . As such it is a relation between the events of E 0 and E 1 . Composition in the category is that of relations. The category is a model of intuitionistic linear logic, as defined in =-=[24, 4]-=-. For instance, its tensor is given in a coordinatewise fashion. For event structures E i = (E i ;si ;si ), for i = 0; 1, define: E 0\Omega E 1 = (E 0 \Theta E 1 ; ; ) where (e 0 ; e 1 )s(e 0 0 ; e 0 ... |

75 |
Categorical combinators, sequential algorithms and functional programming
- Curien
- 1986
(Show Context)
Citation Context ...l semantics. For instance, Berry and Curien's 1 Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 category of concrete data structures and sequential algorithms =-=[5]-=- may be obtained as the co-Kleisli category of a games model [6, 16]. The connection between games and sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstr... |

61 |
Events in computation
- Winskel
- 1980
(Show Context)
Citation Context ...sli category which is closely related to that of Berry's bidomains (both have equivalent non-trivial full sub-cartesian closed categories). 1 Introduction In this paper we link Winskel's bistructures =-=[25]-=-, Girard's linear logic [10] and Berry's bidomains [25]. We show how bistructures provide a model of classical linear logic extending Girard's web model [10, 11]; we show too that a certain class of b... |

59 | Hypercoherences: a strongly stable model of linear logic
- Ehrhard
- 1993
(Show Context)
Citation Context ...uctures (Theorem 2). In Section 8 we discuss possible variations and connections with other work; in particular we consider strengthenings of bistructures incorporating Ehrhard's hypercoherences (see =-=[8]-=-) thereby accounting for strong stability within our approach. In this paper, cpos are partial orders with a least element and lubs of all directed sets; continuous functions between cpos are those mo... |

46 |
Fully abstract models of typed -calculi
- Milner
- 1977
(Show Context)
Citation Context .... We hope that the structures isolated here will help in the search for a direct, extensional and "mathematically natural" account of sequentiality and thereby of Milner's fully abstract mod=-=el of PCF [20]-=-. Girard has given an analysis of intuitionistic logic in terms of his more primitive linear logic. When we consider models, this is reflected in the fact that cartesian closed categories (categorical... |

41 |
Modèles complètement adéquats et stables des lambda-calculs typés, Thèse de doctorat d’état, Université Paris VII
- Berry
- 1979
(Show Context)
Citation Context ...rs (x; e 1 ) such that e 1 2 f(x) and e 1 62 f(y) if y ae x. The inclusion of configurations determines an ordering on stable functions, refining the pointwise ordering and called the stable ordering =-=[2]-=-. 4 The definition of E 0 ! E 1 is asymmetric in that configurations are paired with events, rather than events with events. This led Girard to two successive decompositions, each of which turned out ... |

37 |
Event Structures. In Petri Nets: Applications and Relationships to Other Models of Concurrency, volume
- Winskel
- 1987
(Show Context)
Citation Context ...e, for all events e. The event structures satisfying this axiom yield the dI-domains [2] which are coherent, and therefore lead to a cartesian closed category of stably ordered stable functions. (See =-=[26]-=- where an alternative description of event structures 3 In [21], an axiom relating causal dependency and conflict is imposed; however it is inessential in that it does not affect the class of domains ... |

37 |
Logic of Domains
- Zhang
- 1991
(Show Context)
Citation Context ...′ 0 ≤ e0}). The inclusion ordering on configurations reflects the pointwise ordering on functions; in particular, the function events (e0, e1) correspond to the prime additive one-step functions (see =-=[31]-=-); and the order ≤ to the pointwise order between them. A morphism E0 → E1 is defined to be a configuration of E0 ⊸ E1. As such it is a relation between the events of E0 and E1. Composition in the cat... |

34 |
On the symmetry of sequentiality
- Curien
- 1993
(Show Context)
Citation Context ...n Computer Science, Centre of the Danish National Research Foundation. 1 category of concrete data structures and sequential algorithms [5] may be obtained as the co-Kleisli category of a games model =-=[6, 16]-=-. The connection between games and sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstract extensional collapse [1, 12]. After Berry isolated the mathemati... |

31 | Kripke logical relations and PCF
- O’Hearn, Riecke
- 1995
(Show Context)
Citation Context ...esult [19] there is no hope of finding a direct presentation of the fully abstract model for PCF in this way, adjoining a suitable coherence structure to the two orders of bistructures, unless, as in =-=[22], this is -=-of an infinitary nature. However, the fact that there are hypercoherence-based bistructure models refining the Scott model shows (adapting the "Definability Lemma", e.g., Proposition 4.1.10 ... |

27 |
Fully abstract models of typed λ-calculi
- Milner
- 1977
(Show Context)
Citation Context .... We hope that the structures isolated here will help in the search for a direct, extensional and “mathematically natural” account of sequentiality and thereby of Milner’s fully abstract model of PCF =-=[20]-=-. Girard has given an analysis of intuitionistic logic in terms of his more primitive linear logic. When we consider models, this is reflected in the fact that cartesian closed categories (categorical... |

25 | Finitary PCF is not decidable
- Loader
- 2001
(Show Context)
Citation Context ...table ordering by a structure for sequentiality. If one does not know the right axioms, one could instead look for suitable variants of bistructures of events. However, Loader's undecidability result =-=[19] for the f-=-initary fragment of PCF shows that there is a major obstacle to finding a category of structured sets providing a fully abstract model of PCF. Such a category cannot be "finitary," in a cert... |

20 | Full abstraction and the context lemma
- Jim, Meyer
- 1997
(Show Context)
Citation Context ...es both the stable and pointwise order, its PCF theory (those inequalities on terms which hold in the bidomain model) does not include that of the Scott model. The argument follows Jim and Meyer's in =-=[13], and is b-=-ased on the bidomain model failing to eliminate an "or-tester", the first of Curien's examples in [5], Proposition 4.4.2. The same argument applies to the variation sketched above, where coh... |

9 |
games and linear logic
- Lamarche, Sequentiality
- 1992
(Show Context)
Citation Context ...n Computer Science, Centre of the Danish National Research Foundation. 1 category of concrete data structures and sequential algorithms [5] may be obtained as the co-Kleisli category of a games model =-=[6, 16]-=-. The connection between games and sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstract extensional collapse [1, 12]. After Berry isolated the mathemati... |

9 |
Logic of Domains. Birkhauser
- Zhang
- 1991
(Show Context)
Citation Context ...e 0 g). The inclusion ordering on configurations reflects the pointwise ordering on functions; in particular, the function events (e 0 ; e 1 ) correspond to the prime additive one-step functions (see =-=[31]-=-); and the ordersto the pointwise order between them. A morphism E 0 ! E 1 is defined to be a configuration of E 0 ( E 1 . As such it is a relation between the events of E 0 and E 1 . Composition in t... |

6 |
A relative PCF-definability result for strongly stable functions and some corollaries
- Ehrhard
- 1999
(Show Context)
Citation Context ...) account of sequentiality. At first order, the strongly stable model contains exactly the sequential functions. At higher orders, it is the extensional collapse of the model of sequential algorithms =-=[8, 9]-=-. Generalisations encompassing both Girard's webs and hypercoherences have been proposed independently by Lamarche [18] (based on quantale-valued sets) and by Winskel [30] (based on a notion of indexi... |

3 |
The full abstraction problem: state of the art
- Berry, Curien, et al.
- 1985
(Show Context)
Citation Context ... sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstract extensional collapse [1, 12]. After Berry isolated the mathematically natural notion of stability =-=[3]-=-, it was soon realized that sequential functions are stable. While there is a cartesian closed category of stable functions, at higher orders the extensional ordering is not respected. It was therefor... |

2 |
Games and full abstraction for PCF, manuscript available from http://www.dcs.ed.ac.uk
- Abramsky, Jagadeesan, et al.
- 1995
(Show Context)
Citation Context ...o-Kleisli category of a games model [6, 16]. The connection between games and sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstract extensional collapse =-=[1, 12]-=-. After Berry isolated the mathematically natural notion of stability [3], it was soon realized that sequential functions are stable. While there is a cartesian closed category of stable functions, at... |

2 |
On full abstraction for PCF: I, II and III, available by ftp at ftp.comlab.ox.ac.uk, in directory /pub/Documents/techpapers/Luke.Ong/pcf.ps.gz
- Hyland, Ong
- 1994
(Show Context)
Citation Context ...o-Kleisli category of a games model [6, 16]. The connection between games and sequentiality has in turn informed recent work on intensional models of PCF and their fully-abstract extensional collapse =-=[1, 12]-=-. After Berry isolated the mathematically natural notion of stability [3], it was soon realized that sequential functions are stable. While there is a cartesian closed category of stable functions, at... |

2 |
A large cartesian closed category of domains
- Lamarche
(Show Context)
Citation Context ...oof. But the precise programming constructs associated with these extensions is a mystery, and would need a deeper understanding of the operational nature of bistructure models. Connections. Lamarche =-=[17] (followed-=- by Kegelmann [15]) has proposed a large cartesian closed category of "interpolative domains" encompassing, as full sub-cartesian closed categories, categories of continuous functions on one... |

2 |
Generalizing coherence spaces and hypercoherences
- Lamarche
- 1995
(Show Context)
Citation Context ...r orders, it is the extensional collapse of the model of sequential algorithms [8, 9]. Generalisations encompassing both Girard's webs and hypercoherences have been proposed independently by Lamarche =-=[18]-=- (based on quantale-valued sets) and by Winskel [30] (based on a notion of indexing inspired by logical relations). We believe that our biordering treatment can be applied to all these categories. We ... |

1 |
Coherence bistructures, unpublished manuscript
- Curien
- 1994
(Show Context)
Citation Context ... be enough symmetry that one has a model of classical linear logic. In Sections 2 and 3 we give two approaches to bistructures; these represent two independent developments of the ideas of this paper =-=[23, 7]-=-. Section 2 starts from the world of webs and stable domain theory; Section 3 proceeds from that of event structures and continuous domain theory. We introduce bistructures in Section 4, and bistructu... |

1 |
Concrete Domains, in A Collection of Contributions
- Kahn, Plotkin
- 1993
(Show Context)
Citation Context ... events equipped with a conflict relation and obeying an axiom of finite causes. They were introduced in [21] as a model of concurrency, and turned out to have close connections with concrete domains =-=[14]-=- and hence sequentiality [5]; they are also a natural generalisation of Girard's webs. Winskel introduced bistructures (of events) in [25], representing a full sub-cartesian closed category of bidomai... |

1 |
Factorization systems on domains
- Kegelmann
- 1996
(Show Context)
Citation Context ...ming constructs associated with these extensions is a mystery, and would need a deeper understanding of the operational nature of bistructure models. Connections. Lamarche [17] (followed by Kegelmann =-=[15]) has prop-=-osed a large cartesian closed category of "interpolative domains" encompassing, as full sub-cartesian closed categories, categories of continuous functions on one hand, and categories of sta... |

1 | Stable bistructure models of PCF
- Winskel
- 1994
(Show Context)
Citation Context ...ing to eliminate an "or-tester", the first of Curien's examples in [5], Proposition 4.4.2. The same argument applies to the variation sketched above, where coherence is replaced by hypercohe=-=rence. In [29]-=- it is shown how to adjoin an additional conflict relation to bistructures to obtain a model of PCF whose 37 theory strictly includes that of the Scott model. Several strengthenings of this idea are p... |

1 |
handwritten notes
- Winskel, Hypercoherences
- 1994
(Show Context)
Citation Context ...l of sequential algorithms [8, 9]. Generalisations encompassing both Girard's webs and hypercoherences have been proposed independently by Lamarche [18] (based on quantale-valued sets) and by Winskel =-=[30]-=- (based on a notion of indexing inspired by logical relations). We believe that our biordering treatment can be applied to all these categories. We checked this for hypercoherences. There, the binary ... |