## A General Completeness Result in Refinement (1999)

Venue: | in: Proceedings of the 14th International Workshop on Algebraic Development Techniques, no. 1827 in Lecture Notes in Computer Science |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Kinoshita99ageneral,

author = {Yoshiki Kinoshita and A. J. Power},

title = {A General Completeness Result in Refinement},

booktitle = {in: Proceedings of the 14th International Workshop on Algebraic Development Techniques, no. 1827 in Lecture Notes in Computer Science},

year = {1999},

publisher = {Springer}

}

### OpenURL

### Abstract

. In a paper in 1986, Hoare, He and Sanders proposed a formulation of refinement for a system equivalent to the #-calculus using a relation based semantics. To give a proof method to show that one program is a refinement of another, they introduced downward simulation and upward simulation, but the proof method based upon either of them is not complete with respect to their notion of refinement, so they claimed "joint" completeness based upon both notions of simulation with respect to their notion of refinement. We give a new definition of refinement in terms of structure respecting lax transformations, and show that the proof method based upon downward simulation is complete with respect to this notion of refinement. Although our theory works for the #-calculus, we present the result for the -calculus to make the presentation simpler. We use results in enriched category theory to show this, and the central notion here is that of algebraic structure on locally ordered categories, not o...

### Citations

165 |
Data Refinement: ModelOriented Proof Methods and their Comparison. Cambridge Tracts in Theoretical Computer Science
- Roever, Engelhardt
- 1998
(Show Context)
Citation Context ...ness of downward simulation alone. In the setting of the -calculus, de Roever and Engelhardt explicitly gave an example of data refinement which cannot arise as an extension of downward simulation in =-=[2]-=-. Later, Gardiner and # The authors acknowledge the support of STA through the COE budget for Global Information Processing Project and a British Council grant. The second author also acknowledges the... |

48 |
Data refinement of predicate transformers
- Gardiner, Morgan
- 1991
(Show Context)
Citation Context ...d author also acknowledges the support of EPSRC grants GR/J84205 and GR/M56333. A part of this work was done while the first author was staying at McGill University funded by an NSERC grant. 2 Morgan =-=[3, 4]-=- worked on predicate transformer semantics and obtained a singlescompleteness result for simulation and cosimulation, providing the operators preserve some properties. The aim of our paper is to give ... |

34 |
A single complete rule for data refinement
- Gardiner, Morgan
- 1992
(Show Context)
Citation Context ...d author also acknowledges the support of EPSRC grants GR/J84205 and GR/M56333. A part of this work was done while the first author was staying at McGill University funded by an NSERC grant. 2 Morgan =-=[3, 4]-=- worked on predicate transformer semantics and obtained a singlescompleteness result for simulation and cosimulation, providing the operators preserve some properties. The aim of our paper is to give ... |

29 |
Lawvere, Functorial semantics of algebraic theories
- William
(Show Context)
Citation Context ...guage in terms of Fix-algebras, which are given by algebraic structure on locally ordered categories. The technique we use here is an enriched version of functorial semantics as originated by Lawvere =-=[10]-=- (see also [11].) In Section 5, we define refinement as a structure respecting lax transformation between models and prove soundness and completeness of lax transformations with respect to refinements... |

11 |
An outline of functorial semantics
- Linton
- 1969
(Show Context)
Citation Context ...of Fix-algebras, which are given by algebraic structure on locally ordered categories. The technique we use here is an enriched version of functorial semantics as originated by Lawvere [10] (see also =-=[11]-=-.) In Section 5, we define refinement as a structure respecting lax transformation between models and prove soundness and completeness of lax transformations with respect to refinements. The fundament... |

7 | Algebraic structure for bicategory enriched categories - Gordon, Power - 1998 |

7 | Lax naturality through enrichment
- Kinoshita, Power
- 1996
(Show Context)
Citation Context ...y categories LocOrd and Fix-Alg # become insu#cient, as they do not contain lax transformations among their data, which is why we consider enrichment in Theorem 2. We refer the reader to Appendix and =-=[9, 8]-=- for the background mathematical setting and proofs. Theorem 2 is the key to prove our main theorem, which states the soundness and completeness result. Soundness says that any downward simulation bet... |

4 |
de Bakker and Erik de Vink. Control Flow Semantics
- Jaco
- 1996
(Show Context)
Citation Context ...his paper which is focused on the -calculus. The paper is organised as follows. In Section 2, we give the syntax for our example language, which is a mild generalisation of the -calculus presented in =-=[1]-=-. Our example language has the least fixpoint construct in contrast to the greatest fixpoint provided in the #-calculus studied in [7]. Our theory works both for the -calculus and the #-calculus, but ... |

2 |
Data refinement refined. volume 213
- He, Hoare, et al.
- 1986
(Show Context)
Citation Context ...ets. Our definition of refinement is neither a restriction nor a generalisation of Hoare, He and Sanders' definition, but we include all their important examples. 1 Introduction Hoare, He and Sanders =-=[6, 7]-=- introduced the notion of downward simulation and upward simulation between interpretations of base statements and showed joint completeness of downward and upward simulations with respect to data ref... |

2 | Data refinement and algebraic structure. ETL - Kinoshita, Power - 1996 |