## Stone coalgebras (2003)

Venue: | ILLC, University of Amsterdam |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Kupke03stonecoalgebras,

author = {Clemens Kupke and Alexander Kurz and Yde Venema},

title = {Stone coalgebras},

booktitle = {ILLC, University of Amsterdam},

year = {2003},

pages = {394--406},

publisher = {Springer}

}

### OpenURL

### Abstract

### Citations

456 | Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...to find, for a given type constructor T : X → X , solutions to the equation X ∼ = T X. The typical situation is the following. X is a category of topological spaces as, for example, domains (see e.g. =-=[4]-=-) or (ultra)metric spaces (see e.g. [11,32]), T is a functor, and the favoured solution of X ∼ = T X is the final T -coalgebra X → T X. The Vietoris functor is known in domain theory as the Plotkin po... |

443 |
Y.: Modal Logic
- Blackburn, Rijke, et al.
- 2002
(Show Context)
Citation Context ...tive general frames. This category in its turn is dual to that of modal algebras, and hence, unlike Kripke frames, descriptive general frames form a mathematically adequate semantics for modal logics =-=[8]-=-. The connection with modal logic thus forms a second reason as to why Stone coalgebras are of interest. Since coalgebras can be seen as a very general model of state-based dynamics, and modal logic a... |

298 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...ms We investigated coalgebras over Stone spaces as models for modal logic. But what is the significance of Stone-coalgebras from the point of view of systems (that is, coalgebras over Set, cf. Rutten =-=[30]-=-)? What is the relationship between Set-coalgebras and Stone-coalgebras? An interesting observation is here that their notions of behavioural equivalence coincide. Recall that two elements of two coal... |

236 |
Outline of General Topology
- Engelking
- 1968
(Show Context)
Citation Context ... V : Stone → Stone, a topological analogue of the power set functor on Set. V is a functorial extension of a well-known topological construction which associates with a topology its Vietoris topology =-=[12]-=-. This construction preserves a number of nice topological properties; in particular, it turns Stone spaces into Stone spaces [18]. As we will see further on, the category Coalg(V) of coalgebras for t... |

228 | Domain theory in logical form
- Abramsky
- 1991
(Show Context)
Citation Context ... that the duality of descriptive general frames and modal algebras is an instance of the general relationship between syntax and semantics as laid out by Abramsky in his domain theory in logical form =-=[3]-=-. He was also the first to observe the duality of Coalg(V) and modal algebras in [1]. Modal algebras and (descriptive) general frames play a crucial role in the theory of modal logic, providing an imp... |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...been considered in Abramsky [1]. The category of Stone spaces with a countable base and their connection to SFP-domains have been investigated by Alessi, Baldan, and Honsell [5]. Compared to Abramsky =-=[2]-=-, our work might be seen as a variation based on the use of Stone spaces instead of SFP-domains. Motivated by a different perspective, coalgebras over Stone spaces have been considered recently also b... |

100 |
Topology via Logic
- Vickers
- 1989
(Show Context)
Citation Context ...ids reasoning element-wise about ultrafilters. It uses that each of the type constructors ×, +, (−) D , V has a dual on Boolean algebras that can be described by generators and relations, see Vickers =-=[33]-=-. For example, V ∂ A is generated by symbols [V]a, a ∈ A, and relations expressing that the insertion of generators [V] : A → V ∂ A preserves finite meets (V ∂ is the H of Remark 3.13 where we write n... |

89 | The Logic of
- Baltag, Moss, et al.
- 1998
(Show Context)
Citation Context ...e seen as a very general model of state-based dynamics, and modal logic as a logic for dynamic systems, the relation between modal logic and coalgebras is rather tight. Starting with the work of Moss =-=[25]-=-, this has been an active research area [28,17,6,27,15,9]. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra [20,19], an important differ... |

54 |
Logics for Coalgebras and Applications to Computer Science
- Kurz
- 2000
(Show Context)
Citation Context ...g with the work of Moss [25], this has been an active research area [28,17,6,27,15,9]. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra =-=[20,19]-=-, an important difference being that the relation with Set-based coalgebras works smoothly only for modal languages that allow infinitary formulas. In the case of the Vietoris functor however, it foll... |

53 |
Metamathematics of modal logics
- Goldblatt
- 1976
(Show Context)
Citation Context ...emark that when it comes down to the technicalities, this section contains little news; most of the results in this section can be 5sobtained by exposing existing material from Esakia [13], Goldblatt =-=[16]-=-, Johnstone [18], and Sambin and Vaccaro [31] in a new coalgebraic light. Moreover, it will turn out that the duality of descriptive general frames and modal algebras is an instance of the general rel... |

53 | Many-sorted coalgebraic modal logic: a model-theoretic study
- Jacobs
(Show Context)
Citation Context ...d dynamics, and modal logic as a logic for dynamic systems, the relation between modal logic and coalgebras is rather tight. Starting with the work of Moss [25], this has been an active research area =-=[28,17,6,27,15,9]-=-. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra [20,19], an important difference being that the relation with Set-based coalgebras wo... |

46 |
Topologies on spaces of subsets
- Michael
- 1951
(Show Context)
Citation Context ...In particular, if X is a Stone space, then the set VClp X forms a subbasis for υX. The Vietoris construction preserves various nice topological properties; proofs of this can be found in for instance =-=[24]-=-. Lemma 2.8 Let X = (X, τ) be a topological space. (1) If X is compact then (K(X), υX) is compact. (2) If X is compact and Hausdorff, then (K(X), υX) is compact and Hausdorff. (3) If X is a Stone spac... |

42 |
Control Flow Semantics
- Bakker, Vink
- 1996
(Show Context)
Citation Context ... : X → X , solutions to the equation X ∼ = T X. The typical situation is the following. X is a category of topological spaces as, for example, domains (see e.g. [4]) or (ultra)metric spaces (see e.g. =-=[11,32]-=-), T is a functor, and the favoured solution of X ∼ = T X is the final T -coalgebra X → T X. The Vietoris functor is known in domain theory as the Plotkin powerdomain and its version on Stone has been... |

37 | A hierarchy of probabilistic system types. Theoretical Computer Science - Bartels, Sokolova, et al. - 2004 |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ...d dynamics, and modal logic as a logic for dynamic systems, the relation between modal logic and coalgebras is rather tight. Starting with the work of Moss [25], this has been an active research area =-=[28,17,6,27,15,9]-=-. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra [20,19], an important difference being that the relation with Set-based coalgebras wo... |

30 | Semantical principles in the modal logic of coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context ...d dynamics, and modal logic as a logic for dynamic systems, the relation between modal logic and coalgebras is rather tight. Starting with the work of Moss [25], this has been an active research area =-=[28,17,6,27,15,9]-=-. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra [20,19], an important difference being that the relation with Set-based coalgebras wo... |

30 | On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces
- Turi, Rutten
- 1998
(Show Context)
Citation Context ... : X → X , solutions to the equation X ∼ = T X. The typical situation is the following. X is a category of topological spaces as, for example, domains (see e.g. [4]) or (ultra)metric spaces (see e.g. =-=[11,32]-=-), T is a functor, and the favoured solution of X ∼ = T X is the final T -coalgebra X → T X. The Vietoris functor is known in domain theory as the Plotkin powerdomain and its version on Stone has been... |

29 |
A logic for coalgebraic simulation
- Baltag
- 2000
(Show Context)
Citation Context |

25 | A calculus of transition systems (towards universal co-algebra
- Rutten
- 1995
(Show Context)
Citation Context ...dicating possible future research directions. Stone Coalgebras and Modal Logic Research on the relation between coalgebras and modal logic started with Moss [25] although earlier work, e.g. by Rutten =-=[29]-=- already showed that Kripke frames and models are instances of coalgebras. [20,19] showed that modal logic for coalgebras dualises equational logic for algebras, the idea being that equations describe... |

24 | A co-variety-theorem for modal logic
- Kurz
- 1998
(Show Context)
Citation Context ...g with the work of Moss [25], this has been an active research area [28,17,6,27,15,9]. The relation between modal logic and coalgebras can be seen to dualise that between equational logic and algebra =-=[20,19]-=-, an important difference being that the relation with Set-based coalgebras works smoothly only for modal languages that allow infinitary formulas. In the case of the Vietoris functor however, it foll... |

21 | A cook’s tour of the finitary non-wellfounded sets
- Abramsky
(Show Context)
Citation Context ...een Coalg(V) and the category MA of modal algebras, that Coalg(V) provides an adequate semantics for finitary modal logics. Although probably not widely known, this insight is in fact due to Abramsky =-=[1]-=-. In Sections 4 and 5 we further substantiate our case for Stone spaces as a coalgebraic base category by considering so-called Vietoris polynomial functors as the Stone-based analogues of Kripke poly... |

21 |
Topology and duality in modal logic
- Sambin, Vaccaro
- 1988
(Show Context)
Citation Context ...lities, this section contains little news; most of the results in this section can be 5sobtained by exposing existing material from Esakia [13], Goldblatt [16], Johnstone [18], and Sambin and Vaccaro =-=[31]-=- in a new coalgebraic light. Moreover, it will turn out that the duality of descriptive general frames and modal algebras is an instance of the general relationship between syntax and semantics as lai... |

21 |
Terminal sequences for accessible endofunctors, Coalgebraic Methods in Computer Science
- Worrell
- 1999
(Show Context)
Citation Context ... (X2, c2), x2) are behavioural equivalent. — Proof: ‘only if’ is immediate. The converse follows from the fact that the final T -coalgebra appears as the ω-limit of the terminal sequence (see Worrell =-=[34]-=-) of ˘ T . Generalising Stone Coalgebras Coalgebras over Stone spaces can be generalised in different ways. We have seen that replacing the topologies by represented Boolean algebras leads to general ... |

18 |
Topological Kripke models
- Esakia
(Show Context)
Citation Context .... We hasten to remark that when it comes down to the technicalities, this section contains little news; most of the results in this section can be 5sobtained by exposing existing material from Esakia =-=[13]-=-, Goldblatt [16], Johnstone [18], and Sambin and Vaccaro [31] in a new coalgebraic light. Moreover, it will turn out that the duality of descriptive general frames and modal algebras is an instance of... |

14 | Coalgebraic modal logic of finite rank
- Kurz, Pattinson
- 2005
(Show Context)
Citation Context ... provide a natural and adequate semantics for finitary modal logics, but there is ample room for clarification here. Another approach to a coalgebraic semantics for finitary modal logics was given in =-=[21]-=-. There, the idea is to modify coalgebra morphisms in such a way that they capture not bisimulation but only bisimulation up to rank ω. Since finitary modal logics capture precisely bisimulation up to... |

14 |
Category theory for the working mathematician
- Lane
(Show Context)
Citation Context ...naturality of j). αΦ(I) ◦ Clp(rΦ(T ) ◦ Sp(next)) = jΦ(I) ◦ Clp(Sp(next)) ◦ Clp(rΦ(T )) = next ◦ jΦ(T ) ◦ Clp(rΦ(T )) = next ◦ αΦ(T ). qed Proof of Theorem 5.5. For the adjunction it suffices to show (=-=[23]-=-, p. 81) that for all (X, c) ∈ Stone and for all u : C(Φ) → (X, c) there is a unique v : A(X, c) → Φ such that the following diagram in Coalg(T ) commutes: CA(X, c) C(v) CΦ 22 γX u (X, c)sIndeed, defi... |

8 | Coalgebraic semantics for positive modal logic
- Palmigiano
(Show Context)
Citation Context ...sider other topological spaces as base categories. From the point of view of modal logic, it will be interesting to investigate the Vietoris functors on other base categories. For example, Palmigiano =-=[26]-=- shows that the Vietoris functor can be defined on Priestley spaces, leading to an adequate semantics for positive modal logic. From the point of view of the theory of coalgebras, the value of the mov... |

7 | A calculus of terms for coalgebras of polynomial functors
- Goldblatt
- 2001
(Show Context)
Citation Context |

5 | On specification logics for algebra-coalgebra structures: Reconciling reachability and observability
- Cirstea
- 2002
(Show Context)
Citation Context |

5 |
The order structure of Stone spaces and the TD-separation axiom, Zeitschrift für Mathematische Logik und Grundlagen der
- Gehrke
- 1991
(Show Context)
Citation Context ... of objects and the paths as morphisms between them. ✁ Remark 4.2 All of our results could have been generalised to a setting that allows infinite constants and infinite topologised sums as in Gehrke =-=[14]-=-. We confine ourselves to the functors of Definition 4.1 in order to stay as close as possible to existing work on Kripke polynomial functors. We will now define the Boolean algebras with operators as... |

4 | Partializing Stone Spaces Using SFP Domains (extended abstract
- Alessi, Baldan, et al.
- 1997
(Show Context)
Citation Context ... its version on Stone has been considered in Abramsky [1]. The category of Stone spaces with a countable base and their connection to SFP-domains have been investigated by Alessi, Baldan, and Honsell =-=[5]-=-. Compared to Abramsky [2], our work might be seen as a variation based on the use of Stone spaces instead of SFP-domains. Motivated by a different perspective, coalgebras over Stone spaces have been ... |

4 |
A coalgebraic view of Heyting duality
- Davey, Galati
(Show Context)
Citation Context ... seen as a variation based on the use of Stone spaces instead of SFP-domains. Motivated by a different perspective, coalgebras over Stone spaces have been considered recently also by Davey and Galati =-=[10]-=-. Acknowledgements We would like to thank the participants of the ACG-meetings at the Amsterdam Centrum voor Wiskunde en Informatica (CWI); we benefited in particular from discussions with Marcello Bo... |

4 | Modal predicates and coequations
- Kurz, Rosick´y
- 2002
(Show Context)
Citation Context ...c for algebras, the idea being that equations describe quotients of free algebras and modal formulae describe subsets of final (or cofree) coalgebras. Another account of the duality has been given in =-=[22]-=- where it was shown that modalities dualise algebraic operations. But whereas, usually, any quotient of a free algebra can be defined by a set of ordinary equations, one needs infinitary modal formula... |