## Behavioural Satisfaction and Equivalence in Concrete Model Categories (1996)

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Citations: | 30 - 9 self |

### BibTeX

@MISC{Bidoit96behaviouralsatisfaction,

author = {Michel Bidoit and A. Tarlecki},

title = {Behavioural Satisfaction and Equivalence in Concrete Model Categories},

year = {1996}

}

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### Abstract

. We use the well-known framework of concrete categories to show how much of standard universal algebra may be done in an abstract and still rather intuitive way. This is used to recast the unifying view of behavioural semantics of specications based on behavioural satisfaction and, respectively, on behavioural equivalence of models abstracting away from many particular features of standard algebras. We also give an explicit representation of behavioural equivalence between models in terms of behavioural correspondences. 1 Introduction Behavioural semantics for specications plays a crucial role in the formalisation of the development process, where a specication need not be implemented exactly but only so that the required system behaviour is achieved | the idea goes back to [GGM76], [Hoa72]; see e.g. [ST95] for the context in which we view it now. There have been two basic approaches to behavioural semantics of speci cations. One introduces a new behavioural satisfaction o...

### Citations

474 |
Categories for the working mathematician
- MacLane
- 1998
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Citation Context ...R at the University of Bergen, Norway (AT) and by European ESPRIT II Basic Research WG 6112 COMPASS (MB, AT). 2 S-concrete categories In this section we will recall the concept of a concrete category =-=[MacL71]-=-. The basic intuition to follow is that objects of a concrete category are sets with some extra structure, or in other words that each object comes equipped with a carrier set. Similarly, morphisms of... |

413 |
Proof of correctness of data representations
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(Show Context)
Citation Context ...le in the formalisation of the development process, where a specification need not be implemented exactly but only so that the required system behaviour is achieved --- the idea goes back to [GGM76], =-=[Hoa72]-=-; see e.g. [ST95] for the context in which we view it now. There have been two basic approaches to behavioural semantics of specifications. One introduces a new behavioural satisfaction of formulae, b... |

397 | LCF considered as a programming language - Plotkin - 1977 |

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66 | On observational equivalence and algebraic specification
- Sannella, Tarlecki
- 1987
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Citation Context .... The other is based on an external behavioural equivalence of algebras, where two algebras are considered equivalent if they cannot be distinguished by any computation the user can perform, see e.g. =-=[ST87]-=-. There have also been attempts to unify the two views [Rei85], recently concluded in [BHW95]. One goal of this paper is to show how the above ideas work in the framework of an arbitrary category of m... |

42 |
Data Abstraction and the Correctness of Modular Programming
- Schoett
- 1987
(Show Context)
Citation Context ...y a particularly simple concept of observational correspondence. Our abstract observational correspondences retain the most crucial properties of correspondences from the standard algebraic framework =-=[Sch87]-=-, and seem to yield equally elegant and useful concepts in other frameworks we study. By providing a more abstract view of behavioural semantics we indirectly propose in this paper a way of introducin... |

36 |
Behavioural validity of conditional equations in abstract data types
- Reichel
- 1985
(Show Context)
Citation Context ...f algebras, where two algebras are considered equivalent if they cannot be distinguished by any computation the user can perform, see e.g. [ST87]. There have also been attempts to unify the two views =-=[Rei85]-=-, recently concluded in [BHW95]. One goal of this paper is to show how the above ideas work in the framework of an arbitrary category of models, rather than just in that of standard algebras. The stru... |

32 | Behavioural theories and the proof of behavioural properties - Bidoit, Hennicker - 1996 |

30 | Modular correctness proofs of behavioural implementations - Bidoit, Hennicker - 1998 |

29 |
Observability concepts in abstract data type specification
- GIARRATANA, F, et al.
- 1976
(Show Context)
Citation Context ...rucial role in the formalisation of the development process, where a specification need not be implemented exactly but only so that the required system behaviour is achieved --- the idea goes back to =-=[GGM76]-=-, [Hoa72]; see e.g. [ST95] for the context in which we view it now. There have been two basic approaches to behavioural semantics of specifications. One introduces a new behavioural satisfaction of fo... |

28 | Initial behaviour semantics for algebraic specifications - Nivela, Orejas - 1988 |

26 |
Institutions: Abstract model theory for computer science
- Goguen, Burstall
- 1992
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Citation Context ...nt practical tool for proving behavioural equivalence in all the specific frameworks we study. This work is very much in the spirit of the theory of institutions, as introduced by Goguen and Burstall =-=[GB92]-=- and used in a number of similar attempts to generalise and clarify some fundamental ideas and concepts of algebraic specification and program development. We follow a similar pattern in [BT96], assum... |

25 | On behavioural abstraction and behavioural satisfaction in higher-order logic
- Hofmann, Sannella
- 1996
(Show Context)
Citation Context ...deas should work as well for order-sorted and for error algebras. Another framework with behavioural semantics likely to fit into the mould presented here is that of higher-order logic, as studied in =-=[HS95]-=-. Of course, there also exist frameworks where some of our basic assumptions are not satisfied, and some expected facts and constructions break down. As shown in [BT95b], this is the case for instance... |

22 | Quasi-Varieties in Abstract Algebraic Institutions - Tarlecki - 1986 |

13 |
Behavioural and abstractor specifications. Science of Computer Programming 25:149–186
- Bidoit, Hennicker, et al.
- 1995
(Show Context)
Citation Context ...are considered equivalent if they cannot be distinguished by any computation the user can perform, see e.g. [ST87]. There have also been attempts to unify the two views [Rei85], recently concluded in =-=[BHW95]-=-. One goal of this paper is to show how the above ideas work in the framework of an arbitrary category of models, rather than just in that of standard algebras. The structure of an arbitrary category ... |

5 | Regular algebras: a framework for observational speci with recursive de Report LIENS-95-12, Ecole Normale Superieure - Bidoit, Tarlecki - 1995 |

5 |
Model--theoretic foundations for formal program development: basic concepts and motivation
- Sannella, Tarlecki
- 1995
(Show Context)
Citation Context ...ation of the development process, where a specification need not be implemented exactly but only so that the required system behaviour is achieved --- the idea goes back to [GGM76], [Hoa72]; see e.g. =-=[ST95]-=- for the context in which we view it now. There have been two basic approaches to behavioural semantics of specifications. One introduces a new behavioural satisfaction of formulae, based on the inter... |

4 | Behavioural and abstractor speci cations - Bidoit, Hennicker, et al. - 1995 |

4 | On Observational Equivalence and Algebraic Speci cations - Sanella, Tarlecki - 1987 |

1 |
Hiding and behaviour: an institutional approach. A Classical Mind
- Burstall, Diaconescu
- 1994
(Show Context)
Citation Context ...nce modelling relations (and hence kernels, etc) between objects as spans (as in Sect. 4) and then defining quotients and subobjects w.r.t. some given factorization system for the category of models. =-=[BD94] may -=-provide a good starting point in this direction, although technicalities there are quite different. For instance, in [BD94] an extra assumption about the existence of a terminal "abstract behavio... |