## GSOS for probabilistic transition systems (Extended Abstract) (2002)

Venue: | Proc. Coalgebraic Methods in Computer Science (CMCS 2002), volume 65 of Electronic Notes in Theoretical Computer Science |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Bartels02gsosfor,

author = {Falk Bartels},

title = {GSOS for probabilistic transition systems (Extended Abstract)},

booktitle = {Proc. Coalgebraic Methods in Computer Science (CMCS 2002), volume 65 of Electronic Notes in Theoretical Computer Science},

year = {2002},

publisher = {Elsevier Science Publishers}

}

### OpenURL

### Abstract

We introduce probabilistic GSOS, an operator specification format for (reactive) probabilistic transition systems which arises as an adaptation of the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a congruence and the up-to-context proof principle is valid. Moreover, every specification has a final model which can be shown to offer unique solutions for guarded recursive equations. The format covers operator specifications from the literature, so that the well-behavedness results given for those arise as instances of our general one.

### Citations

438 | Bisimulation Through Probabilistic Testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...or nondeterministic labelled transition systems only, but it would be interesting to have variants of it for other types as well, which are also studied intensively in the literature. Larsen and Skou =-=[LS91-=-] for instance considered the case where (nite) nondeterminism is resolved probabilistically: when there are several states to which the process may make a move with the same label, then the choice is... |

339 | Universal coalgebra: a theory of systems - Rutten |

251 | A tutorial on (co)algebras and (co)induction - Jacobs, Rutten - 1997 |

200 |
Bisimulation canâ€™t be traced
- Bloom, Istrail, et al.
- 1995
(Show Context)
Citation Context ...er terms, in particular its subterms 2 . One can guarantee several desirable properties of the resulting models by restricting oneself to rules of a certain shape. A famous example is the GSOS format =-=-=-[BIM95], which allows specications given by sets of rules of the type 3 x i b ## b 2 R i ; 1 i n x i b ## = b 2 P i ; 1 i n x i y l y ## y (y 2 Y ) (x 1 ; : : : ; x n ) a ## t where is one of the... |

167 | Reactive, generative and stratified models of probabilistic processes - Glabbeek, Smolka, et al. - 1995 |

164 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ... systems are B-coalgebras for a suitable functor B which further exhibits probabilistic bisimulations according to the famous definition by Larsen and Skou as B-bisimulations in the categorical sense =-=[AM89]-=-. Moreover they show that the functor under consideration is well behaved in that it weakly preserves pullbacks and possesses asnal system, two important properties for the structure theory of coalgeb... |

147 | Towards a mathematical operational semantics
- Turi
- 1997
(Show Context)
Citation Context ... some ad hoc adaptation of the format to suit this type of system and then trying to adapt the known proofs, we reached this variant via an abstract theory of specication formats by Turi and Plotkin [=-=TP97-=-]: They have introduced a categorical framework where specications come as natural transformations of a certain shape which identify a class of bialgebras as their models. It can easily be shown that ... |

91 | On the bisimulation proof method
- Sangiorgi
- 1998
(Show Context)
Citation Context ...e greatest bisimulation with respect to its coalgebra structure is a congruence for its algebra operations. But it guarantees further nice properties: e.g. the principle of bisimulation up-to-context =-=[San98-=-] is valid for contexts built from operators dened by GSOS rules and systems of guarded recursive equations involving them have unique solutions in thesnal model. The format is developed for nondeterm... |

84 | Bisimulation for probabilistic transition systems: a coalgebraic approach
- Vink, Rutten
- 1999
(Show Context)
Citation Context ...r add the attribute deterministic to distinguish them from a more complex setting where nondeterministic and probabilistic choice are incorporated as independent concepts [JLY01]). De Vink and Rutten =-=[dVR99] argue 2 -=-In process algebra the states of a system are usually the terms. We will not make this identication here, so in the above statement one should read \term" as \state represented by a term" to... |

72 | Probabilistic extensions of process algebras
- Jonsson, Yi, et al.
- 2001
(Show Context)
Citation Context ...bilistic processes [vGSS95] or add the attribute deterministic to distinguish them from a more complex setting where nondeterministic and probabilistic choice are incorporated as independent concepts =-=[JLY01]). D-=-e Vink and Rutten [dVR99] argue 2 In process algebra the states of a system are usually the terms. We will not make this identication here, so in the above statement one should read \term" as \st... |

40 | Compositional verification of probabilistic processes - Larsen, Skou |

29 | Coalgebraic logic. Annals of Pure and Applied Logic 96 - Moss - 1999 |

27 | H.: Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads
- Lenisa, Power, et al.
- 2000
(Show Context)
Citation Context ... r to r. 3 Abstract GSOS In this section we recall the categorical specication format introduced by Turi and Plotkin [TP97] along with extensions taken from the work of Lenisa, Power, and Watanabe [L=-=PW00]-=- and a previous paper [Bar01]. For an introduction into the theory of (co)algebras we further refer the reader to the tutorial/overview articles of Jacobs and Rutten [JR96,Rut00]. Dynamical systems su... |

22 |
Reactive, generative, and strati models of probabilistic processes
- Glabbeek, Smolka, et al.
(Show Context)
Citation Context ...he choice is made according to a given simple probability distribution. They call these systems probabilistic transition systems (other authors call them the reactive model of probabilistic processes =-=[vGSS95]-=- or add the attribute deterministic to distinguish them from a more complex setting where nondeterministic and probabilistic choice are incorporated as independent concepts [JLY01]). De Vink and Rutte... |

19 | Generalised coinduction
- Bartels
- 2003
(Show Context)
Citation Context ... this section we recall the categorical specication format introduced by Turi and Plotkin [TP97] along with extensions taken from the work of Lenisa, Power, and Watanabe [LPW00] and a previous paper [=-=Bar01]-=-. For an introduction into the theory of (co)algebras we further refer the reader to the tutorial/overview articles of Jacobs and Rutten [JR96,Rut00]. Dynamical systems such as transition systems, aut... |

13 | GSOS for probabilistic transition systems
- Bartels
- 2002
(Show Context)
Citation Context ...he result tuple. It can be shown that all natural transformations m as in (8) arise as convex combinations of thesesS; (for a proof we refer the interested reader to a forthcoming technical report [Bar02]) 13 Bartels Theorem 6.3 For a set E and m 2 IN , every natural transformation m as in (8) arises as m = X hS;i2supp() (hS; i) sS; for some distribution 2 D ! fhS; i j S 2 Part m ; : S ... |

11 |
Bialgebraic modelling of timed processes
- Kick
- 2002
(Show Context)
Citation Context ...crete examples for several instances of the abstract format, but no rule format was developed out of these considerations 22 Bartels and non of the examples involved probabilistic systems. Marco Kick =-=[Kic02]-=- is at this moment also working on the derivation of novel concrete formats from the abstract framework, namely for timed systems. We used (reactive) probabilistic transition systems as studied by Lar... |

9 | Categorical modelling of structural operational rules: Case studies
- Turi
- 1997
(Show Context)
Citation Context ..., the categorical framework generalizing GSOS rules is taken from the work of Turi and Plotkin [TP97], with additions made by Lenisa, Power and Watanabe [LPW00] and in our previous work [Bar01]. Turi =-=[Tur97]-=- has himself worked out concrete examples for several instances of the abstract format, but no rule format was developed out of these considerations 22 Bartels and non of the examples involved probabi... |

8 |
Compositional veri of probabilistic processes
- Larsen, Skou
- 1992
(Show Context)
Citation Context ...ic transition systems as studied by Larsen and Skou [LS91], or van Glabbeek, Smolka, and Steen [vGSS95]. The coalgebraic denition we adopted was given by de Vink and Rutten [dVR99]. Larsen and Skou [L=-=S9-=-2] furthermore dened a set of basic operators to construct (nite) probabilistic transition systems and stated that their bisimulation is a congruence for them. A similar set of operators, but this tim... |