## Event Spaces and Their Linear Logic (1991)

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Venue: | In AMAST’91: Algebraic Methodology and Software Technology, Workshops in Computing |

Citations: | 22 - 9 self |

### BibTeX

@INPROCEEDINGS{Pratt91eventspaces,

author = {Vaughan Pratt},

title = {Event Spaces and Their Linear Logic},

booktitle = {In AMAST’91: Algebraic Methodology and Software Technology, Workshops in Computing},

year = {1991},

pages = {1--23},

publisher = {Springer-Verlag}

}

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### Abstract

Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not symmetrically. Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language. Event spaces resemble finite dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them i...

### Citations

263 |
Event structures
- Winskel
- 1986
(Show Context)
Citation Context ...etri nets whose operations correspond to respectively nondeterministic choice and concurrence of nets while preserving that symmetry. Conversely Winskel's event structures and prime algebraic domains =-=[Win86]-=- have an attractive algebraic theory but are not at all symmetric. In this paper we give an algebraic model of computation that treats choice and concurrency symmetrically. It subsumes event structure... |

236 | Modeling concurrency with partial orders
- PRATT
- 1986
(Show Context)
Citation Context ... b is the notion of interacting particle systems such as colliding galaxies and a sequence of trains passing through a sequence of stations. Interaction is defined, and example applications given, in =-=[Pra86]-=- (where it is notated a \Theta b because the coincidence a \Theta b = a\Omega b in Pos led us to believe interaction was ordinary product) and in [CCMP91] by which time we had understood the distincti... |

189 | Toposes, Triples, and Theories - Barr, Wells - 1985 |

171 |
G.: Petri nets, event structures and domains
- Nielsen, Plotkin, et al.
- 1981
(Show Context)
Citation Context ...ation this duality has a natural interpretation in terms of schedules and automata, whose elements denote respectively events and states. This interpretation was found by Nielsen, Plotkin and Winskel =-=[NPW81]-=-, who then extended the duality to incorporate a symmetric irreflexive binary conflict relation #, with x#y forbidding the occurrence of x together with y. They required conflict to be persistent: if ... |

155 |
The theory of representations for boolean algebras
- Stone
- 1936
(Show Context)
Citation Context ... the join only of sets containing x; equivalently, when the join of the set of elements strictly less than x is itself strictly less than x. Stone's original duality put the Stone topology on the set =-=[Sto36]-=- or poset [Sto37] side. In the latter case this view of a poset topologized with a Stone topology was not apparent from Stone's paper and was found much later by Priestley [Pri70]. In the case of the ... |

124 | Modeling concurrency with geometry
- Pratt
- 1991
(Show Context)
Citation Context ...tice ideals of A, in such a way that A is isomorphic to 2 S op if and only if S is isomorphic to 2 A op . We offer a brief tutorial on this duality at the end of this paper and a longer one elsewhere =-=[Pra91]-=-. In the context of computation this duality has a natural interpretation in terms of schedules and automata, whose elements denote respectively events and states. This interpretation was found by Nie... |

101 |
Relevant Logic and Entailment
- Dunn
- 1986
(Show Context)
Citation Context ...s a\Omega b = b\Omega a. This phenomenon of having two parallel lists of arithmetical operations is not unusual, occurring for example with relation algebras [JT48], vector spaces, and relevant logic =-=[Dun86]-=-. Concentrating on both rows of column 2, products, we find respectively a structured and unstructured product as follows for each of these examples. For relations we have composition R; S and interse... |

100 | Towards a geometry of interaction - Girard - 1989 |

87 |
Autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...phenomena in Ev in terms of unreachable events and initial states took time. That !a even made sense in Ev was completely unobvious to me until I had digested the relevant parts of the papers of Barr =-=[Bar91a]-=- and Seely [See89], after which I wondered why I did not see it right away. Thanks also to Michael Barr for much helpful email correspondence concerning monads and comonads. Guo Qiang Zhang, Stefano K... |

54 |
Topological representation of distributive lattices and Brouwerian logic, Čas. mat. fys
- Stone
- 1937
(Show Context)
Citation Context ... sets containing x; equivalently, when the join of the set of elements strictly less than x is itself strictly less than x. Stone's original duality put the Stone topology on the set [Sto36] or poset =-=[Sto37]-=- side. In the latter case this view of a poset topologized with a Stone topology was not apparent from Stone's paper and was found much later by Priestley [Pri70]. In the case of the distributive latt... |

43 |
On partial languages
- Grabowski
- 1981
(Show Context)
Citation Context ... interaction or orthocurrence [Pra86, CCMP91]. For schedules defined as posets, concurrence PkQ can be defined simply as juxtaposition or poset coproduct P + Q, a point of view advocated by Grabowski =-=[Gra81] and the a-=-uthor [Pra85, Pra86]. For event spaces that "mimic" a poset P , namely the free event space F (P ), coproduct remains the appropriate definition because F is a left adjoint and hence preserv... |

35 |
Process logic
- Pratt
- 1979
(Show Context)
Citation Context ...minating computations. This uses the poset structure to code with an inf what Hoare has represented with p , and the absence of which we represented as the limbo statesin an early process logic paper =-=[Pra79]-=-, coded here with an infinite chain lacking an inf. 6 Programming with Arithmetic We give here the sense in which sum is concurrence, product is choice, and tensor product is interaction or orthocurre... |

29 | Temporal structures
- Casley, Crew, et al.
- 1991
(Show Context)
Citation Context ...uth of xsy yields an inherently flexible triangle inequality, namely reflexivity. (We develop this relationship between posets and metric spaces in concurrency modeling considerably further elsewhere =-=[CCMP91]-=-.) Third, sum and product of vector spaces degenerate to the same operation, and similarly for tensor sum and tensor product, whereas for event spaces these are all distinct operations, respectively c... |

28 |
On the combination of subalgebras
- Birkhoff
- 1933
(Show Context)
Citation Context ...cs are most succinctly described as being those of full linear logic. Our starting point is Winskel's notion of event structure, based on Birkhoff's duality of finite posets and distributive lattices =-=[Bir33]-=-, extended to the infinite case by Stone [Sto36, Sto37]. This duality associates to each poset S = (X; ) the distributive lattice 2 S op consisting of the order ideals of S, and to each distributive l... |

24 |
Representation problems for relation algebras
- Jónsson, Tarski
- 1192
(Show Context)
Citation Context ...fi a ? = a \Gammaffi b ? immediately yields a\Omega b = b\Omega a. This phenomenon of having two parallel lists of arithmetical operations is not unusual, occurring for example with relation algebras =-=[JT48]-=-, vector spaces, and relevant logic [Dun86]. Concentrating on both rows of column 2, products, we find respectively a structured and unstructured product as follows for each of these examples. For rel... |

13 | Accessible categories and models of linear logic - Barr - 1990 |

8 | Some constructions for order-theoretic models of concurrency - Pratt - 1985 |

7 |
logic, *-autonomous categories and cofree algebras
- Linear
- 1989
(Show Context)
Citation Context ... thus excludes demonic choice and concatenation. (Though we have been faithful to the basic language of linear logic our choice of notation for its operations and constants is closer to that of Seely =-=[See89]-=- and Barr [Bar91a, Bar91b] than of Girard.) In assessing the significance of linear logic it is tempting to focus on the closed but not cartesian closed aspect. But that aspect is already present in r... |

6 |
Representation of distributive lattices
- Priestley
- 1995
(Show Context)
Citation Context ... topology on the set [Sto36] or poset [Sto37] side. In the latter case this view of a poset topologized with a Stone topology was not apparent from Stone's paper and was found much later by Priestley =-=[Pri70]-=-. In the case of the distributive lattice 2 X op the compact elements are exactly those subsets Y of X for which there exists an element y 2 Y such that no proper subset of Y containing y belongs to 2... |

1 |
A note on complete semilattices
- Johnstone
- 1978
(Show Context)
Citation Context ... semilattice, a poset with all joins. The notion of event space is obtained from that of complete semilattice by replacing the empty join or bottom by the empty meet or top. As P. Johnstone has noted =-=[Joh78], th-=-e category CSLat of complete semilattices is self-dual, CSLat �� = CSLat op . 2 Johnstone obtains this very neatly as a direct consequence of Freyd's adjoint functor theorem. The category Ev of ev... |

1 | Arbitrary products are coproducts in complete (-)semilattices - Latch - 1976 |