## On specification logics for algebra-coalgebra structures: Reconciling reachability and observability (2002)

Venue: | Proceedings of FoSSaCS 2002, volume 2303 of LNCS |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Cîrstea02onspecification,

author = {Corina Cîrstea},

title = {On specification logics for algebra-coalgebra structures: Reconciling reachability and observability},

booktitle = {Proceedings of FoSSaCS 2002, volume 2303 of LNCS},

year = {2002},

pages = {82--97},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. The paper builds on recent results regarding the expressiveness of modal logics for coalgebras in order to introduce a specification framework for coalgebraic structures which offers support for modular specification. An equational specification framework for algebraic structures is obtained in a similar way. The two frameworks are then integrated in order to account for structures comprising both a coalgebraic (observational) component and an algebraic (computational) component. The integration results in logics whose sentences are either coalgebraic (modal) or algebraic (equational) in nature, but whose associated notions of satisfaction take into account both the coalgebraic and the algebraic features of the structures being specified. Each of the logics thus obtained also supports modular specification. 1

### Citations

507 |
Institutions: Abstract model theory for specication and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...mbined structures, with both algebraic and coalgebraic features playing a rôle in defining the associated notions of satisfaction. Both notions of satisfaction were shown to give rise to institutions =-=[6]-=-. The equational sentences used in [5], although similar in their expressiveness to the equational formulae of [2] and the modal formulae of [9], have a strong semantic flavour, being indexed by class... |

319 | Universal coalgebra, a theory of systems
- Rutten
- 2000
(Show Context)
Citation Context ...subalgebras, homomorphic images and products 1 ). In recent years, coalgebras (the categorical duals of algebras) have been used to study structures that involve observation (e.g. systems with state) =-=[14, 16]-=-, and various modal logics have been used to specify and reason about such structures [13, 15, 10, 7]. Moreover, the results in [2] have been dualised in [9], where it was shown that modally-definable... |

254 |
Abstract and Concrete Categories
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ...ing. Such use is supported by the existence of characterisability results for classes of algebras, both in the concrete setting of many-sorted algebras [3] and in a more abstract, categorical setting =-=[2, 1]-=-, whereby equationally-specifiable classes of algebras coincide with varieties (that is, classes of algebras closed under subalgebras, homomorphic images and products 1 ). In recent years, coalgebras ... |

187 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...a cosignature. T-algebras are taken as models for a signature (C, F) 10 . The functor taking T-algebras to their carrier is denoted UT, while its left adjoint (whose existence is guaranteed by Def. 6 =-=(3)-=-) is denoted LT. 7 The notion of (co)monad morphism considered here generalises the standard one, as defined e.g. in [1, Def. 20.55], being given by a pair (U,ρ)withU:C ′ →Cand ρ : U ◦ D ′ ⇒ D ◦ U, ra... |

151 |
Handbook of categorical algebra
- Borceux
- 1994
(Show Context)
Citation Context ...G)))-diagonalisation property of Coalg(G) and the injectivity of Z are used for the only if direction. 14 The preservation of monomorphisms by G ensures the uniqueness of 〈R, γR〉. 15 Hence, by Rem. 1 =-=(4)-=-, 〈R, π1,π2〉 gives precisely the bisimilarity relation on 〈C, γ〉.s96 C. Cîrstea A similar result holds for the satisfaction of equations up to observability: a Tσ-algebra satisfies an equation up to o... |

142 | Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...mputational features are considered. As in [5], the choice of models incorporating both coalgebraic and algebraic structure, and of the syntax used to specify these models is based on the approach in =-=[17]-=-, and results in a compatibility between the two categories of features, in that computations preserve observational indistinguishability whereas observations preserve reachability. The sentences used... |

96 |
An approach to object semantics based on terminal co-algebras
- Reichel
- 1995
(Show Context)
Citation Context ...subalgebras, homomorphic images and products 1 ). In recent years, coalgebras (the categorical duals of algebras) have been used to study structures that involve observation (e.g. systems with state) =-=[14, 16]-=-, and various modal logics have been used to specify and reason about such structures [13, 15, 10, 7]. Moreover, the results in [2] have been dualised in [9], where it was shown that modally-definable... |

92 | Coalgebraic logic
- Moss
- 1999
(Show Context)
Citation Context ... duals of algebras) have been used to study structures that involve observation (e.g. systems with state) [14, 16], and various modal logics have been used to specify and reason about such structures =-=[13, 15, 10, 7]-=-. Moreover, the results in [2] have been dualised in [9], where it was shown that modally-definable classes of coalgebras coincide with covarieties (the duals of varieties). A framework which integrat... |

61 |
Specifying coalgebras with modal logic. Theoret
- Kurz
- 2001
(Show Context)
Citation Context ...δ〉 �� ι RGZ f �� Modal formulae ι : 〈D, δ〉 →RGZwith Z final in C (and hence RGZ final in Coalg(G)) specify properties up to observational equivalence. Existing specification frameworks for coalgebras =-=[13, 15, 10, 7]-=- only employ such modal formulae. The satisfaction of modal formulae by coalgebras is preserved along homomorphic images and reflected along subcoalgebras. These are consequences of [9, Coroll. 2.4.6]... |

54 | Many-sorted coalgebraic modal logic: a model-theoretic study
- Jacobs
(Show Context)
Citation Context ... duals of algebras) have been used to study structures that involve observation (e.g. systems with state) [14, 16], and various modal logics have been used to specify and reason about such structures =-=[13, 15, 10, 7]-=-. Moreover, the results in [2] have been dualised in [9], where it was shown that modally-definable classes of coalgebras coincide with covarieties (the duals of varieties). A framework which integrat... |

54 |
Logics for Coalgebras and Applications to Computer Science
- Kurz
- 2000
(Show Context)
Citation Context ...observation (e.g. systems with state) [14, 16], and various modal logics have been used to specify and reason about such structures [13, 15, 10, 7]. Moreover, the results in [2] have been dualised in =-=[9]-=-, where it was shown that modally-definable classes of coalgebras coincide with covarieties (the duals of varieties). A framework which integrates algebraic and coalgebraic specification methods in or... |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ... duals of algebras) have been used to study structures that involve observation (e.g. systems with state) [14, 16], and various modal logics have been used to specify and reason about such structures =-=[13, 15, 10, 7]-=-. Moreover, the results in [2] have been dualised in [9], where it was shown that modally-definable classes of coalgebras coincide with covarieties (the duals of varieties). A framework which integrat... |

22 | An axiomatics for categories of coalgebras - Power, Watanabe - 1998 |

15 |
Subcategories defined by implications
- Banaschewski, Herrlich
- 1976
(Show Context)
Citation Context ...ing. Such use is supported by the existence of characterisability results for classes of algebras, both in the concrete setting of many-sorted algebras [3] and in a more abstract, categorical setting =-=[2, 1]-=-, whereby equationally-specifiable classes of algebras coincide with varieties (that is, classes of algebras closed under subalgebras, homomorphic images and products 1 ). In recent years, coalgebras ... |

10 |
On Institutions for Modular Coalgebraic Specifications
- Kurz, Hennicker
- 2000
(Show Context)
Citation Context ...e morphisms (U,τ,ξ) inducing reduct functors U (τ,ν) : Alg(T ′ σ ′) → Alg(Tσ), with U (τ,ν) taking a T ′ σ ′-algebra 〈〈C ′ ,γ ′ 〉,α ′ 〉 to the Tσ-algebra 〈〈UC ′ ,τC ′ ◦Uγ′ 〉,Uα ′ ◦νC ′〉. Remark 6. In =-=[12, 11]-=-, combined structures are captured using pairs consisting of an algebra and a coalgebra structure on the same carrier, with the additional requirement that bisimulation on the coalgebraic structure is... |

10 | Behavioural equivalence, bisimulation, and minimal realisation
- Malcolm
- 1996
(Show Context)
Citation Context ...e morphisms (U,τ,ξ) inducing reduct functors U (τ,ν) : Alg(T ′ σ ′) → Alg(Tσ), with U (τ,ν) taking a T ′ σ ′-algebra 〈〈C ′ ,γ ′ 〉,α ′ 〉 to the Tσ-algebra 〈〈UC ′ ,τC ′ ◦Uγ′ 〉,Uα ′ ◦νC ′〉. Remark 6. In =-=[12, 11]-=-, combined structures are captured using pairs consisting of an algebra and a coalgebra structure on the same carrier, with the additional requirement that bisimulation on the coalgebraic structure is... |

1 |
Integrating observational and computational features in the specification of state-based, dynamical systems
- Cîrstea
(Show Context)
Citation Context ...ls of varieties). A framework which integrates algebraic and coalgebraic specification methods in order to specify structures comprising both computational and observational features was described in =-=[5]-=-. The approach taken there was to clearly ⋆ Research supported by St. John’s College, Oxford. 1 In an abstract setting, the notions of subalgebra and homomorphic image are defined relatively to a fact... |