## Stochastic transition systems for continuous state spaces and non-determinism (2005)

Venue: | In FoSSaCS’05, LNCS 3441 |

Citations: | 12 - 3 self |

### BibTeX

@INPROCEEDINGS{Cattani05stochastictransition,

author = {Stefano Cattani and Roberto Segala and Marta Kwiatkowska and Gethin Norman},

title = {Stochastic transition systems for continuous state spaces and non-determinism},

booktitle = {In FoSSaCS’05, LNCS 3441},

year = {2005},

pages = {125--139},

publisher = {SpringerVerlag}

}

### OpenURL

### Abstract

Abstract. We study the interaction between non-deterministic and probabilistic behaviour in systems with continuous state spaces, arbitrary probability distributions and uncountable branching. Models of such systems have been proposed previously. Here, we introduce a model that extends probabilistic automata to the continuous setting. We identify the class of schedulers that ensures measurability properties on executions, and show that such measurability properties are preserved by parallel composition. Finally, we demonstrate how these results allow us to define an alternative notion of weak bisimulation in our model. 1

### Citations

3204 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context .... 6 Weak Transitions and Weak Bisimulation In this section, we show how the results of the previous sections enable us to define weak transitions and weak bisimulation in our model. A weak transition =-=[17]-=- abstracts from internal computation and considers sequences of actions of the form τ ∗ aτ ∗ , where τ denotes a generic internal action. In the case of probabilistic automata, this is achieved by con... |

596 | A Compositional Approach to Performance Modeling
- Hillston
- 1996
(Show Context)
Citation Context ... creates a requirement to model non-determinism, in addition to such stochastic features. Several models capable of representing probabilistic behaviour have been proposed in the literature, see e.g. =-=[1, 10, 13, 16, 24]-=-. Particular attention has been paid to the nature of interaction between probabilistic and non-deterministic behaviour; though these can be seen as orthogonal, the way they interact in the model has ... |

403 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ... creates a requirement to model non-determinism, in addition to such stochastic features. Several models capable of representing probabilistic behaviour have been proposed in the literature, see e.g. =-=[1, 10, 13, 16, 24]-=-. Particular attention has been paid to the nature of interaction between probabilistic and non-deterministic behaviour; though these can be seen as orthogonal, the way they interact in the model has ... |

157 |
Real Analysis and Probability
- Ash
- 1972
(Show Context)
Citation Context ... is easy to show that η1 defines a legal scheduler for S1 and its measurability follows from Theorem 2. In order to prove that δη1,q 1 = π1(δη, q), we need to show that δη1,q 1 (CΛ) = δη,q(π −1 (CΛ)) =-=(2)-=- for all basic sets CΛ. Equation (2) is proved by algebraic arguments and by exploiting the properties of regular conditional probabilities. Since the two measures agree on the basic sets, which form ... |

152 | Reactive, generative, and stratified models of probabilistic processes
- Glabbeek, Smolka, et al.
- 1995
(Show Context)
Citation Context ... creates a requirement to model non-determinism, in addition to such stochastic features. Several models capable of representing probabilistic behaviour have been proposed in the literature, see e.g. =-=[1, 10, 13, 16, 24]-=-. Particular attention has been paid to the nature of interaction between probabilistic and non-deterministic behaviour; though these can be seen as orthogonal, the way they interact in the model has ... |

147 |
Time and probability in formal design of distributed systems
- Hansson
- 1994
(Show Context)
Citation Context ...odel has led to fundamental distinctions. In the discrete state, discrete time model different variants have been proposed. In some models randomisation replaces non-determinism [10], while elsewhere =-=[11]-=- states are either probabilistic or non-deterministic, such that probabilistic and deterministic choices alternate. Furthermore, one can replace conventional transitions with probabilistic transitions... |

135 | Labelled Markov processes
- Desharnais
- 1999
(Show Context)
Citation Context ...roduced, both in the case in which non-determinism is replaced with race conditions (e.g. generalised semi-Markov processes), and in the case in which it is retained [7, 4]. Labelled Markov processes =-=[8]-=- are extensions of transition systems to continuous state spaces and general distributions, but have no non-determinism, in the sense that the choice of action determines the next transition. Operatio... |

71 |
Interactive Markov Chains and the Quest for Quantified Quality. Volume 2428 of LNCS
- Hermanns
- 2002
(Show Context)
Citation Context ... memoryless property. The usual convention is to replace non-determinism with race condition, leading to continuous time Markov chains, but non-determinism can be kept (e.g. interactive Markov chains =-=[12]-=-). Generally distributed delays have also been introduced, both in the case in which non-determinism is replaced with race conditions (e.g. generalised semi-Markov processes), and in the case in which... |

56 | Weak bisimulation for fully probabilistic processes
- Baier, Hermanns
- 1997
(Show Context)
Citation Context ...s, first introduced in the context of CCS [17], are fundamental relations for concurrent systems, and have been extended to the probabilistic setting, both for discrete (strong and weak bisimulation) =-=[16, 11, 22, 3, 19]-=- and continuous state spaces (strong bisimulation) [7, 8, 5]. A notion of weak bisimulation for the continuous setting was introduced in [4], where the problem of defining a measure on paths for weak ... |

40 | A categorical approach to probability theory - Giry - 1982 |

38 | Weak bisimulation for probabilistic systems
- Philippou, Lee, et al.
- 2000
(Show Context)
Citation Context ...s, first introduced in the context of CCS [17], are fundamental relations for concurrent systems, and have been extended to the probabilistic setting, both for discrete (strong and weak bisimulation) =-=[16, 11, 22, 3, 19]-=- and continuous state spaces (strong bisimulation) [7, 8, 5]. A notion of weak bisimulation for the continuous setting was introduced in [4], where the problem of defining a measure on paths for weak ... |

36 |
Algebras and Automata for Timed and Stochastic Systems
- D’Argenio
(Show Context)
Citation Context ...tributed delays have also been introduced, both in the case in which non-determinism is replaced with race conditions (e.g. generalised semi-Markov processes), and in the case in which it is retained =-=[7, 4]-=-. Labelled Markov processes [8] are extensions of transition systems to continuous state spaces and general distributions, but have no non-determinism, in the sense that the choice of action determine... |

33 | Verifying quantitative properties of continuous probabilistic timed automata
- Kwiatkowska, Norman, et al.
- 2000
(Show Context)
Citation Context ...f stochastic transition systems with non-determinism and continuous state spaces. We identify the class of measurable schedulers that generate tractable runs, confirming the choice originally made in =-=[15]-=-; this restriction enables the definition of a measure on executions. We also show that such measurability properties are preserved through parallel composition. Structure of the paper. In Section 2 w... |

25 |
Specification and Analysis of Stochastic Real-Time Systems
- Bravetti
- 2002
(Show Context)
Citation Context ...tributed delays have also been introduced, both in the case in which non-determinism is replaced with race conditions (e.g. generalised semi-Markov processes), and in the case in which it is retained =-=[7, 4]-=-. Labelled Markov processes [8] are extensions of transition systems to continuous state spaces and general distributions, but have no non-determinism, in the sense that the choice of action determine... |

11 |
Tutte le algebre insieme: Concepts, discussions and relations of stochastic process algebras with general distributions
- Bravetti, D’Argenio
- 2004
(Show Context)
Citation Context ...and general distributions, but have no non-determinism, in the sense that the choice of action determines the next transition. Operational models with non-determinism have already been proposed, e.g. =-=[7, 4, 5]-=-, and the notions of bisimulation and parallel composition have been studied for such systems; [4] also defines weak bisimulation. When considering continuous distributions and state spaces, the notio... |

10 |
Measure and probability for concurrency theorists
- Panangaden
(Show Context)
Citation Context ...and results of measure theory that are necessary for the remainder of the paper. A basic knowledge of topology and metric spaces is assumed. Most results can be found in standard textbooks, e.g. [2]; =-=[18]-=- serves as a good introduction to measure theory. Basic definitions. Given a set X, an algebra over X is a family FX of subsets of X that includes X and is closed under complementation and finite unio... |

1 |
Formal Verification of Probabilistic Systems
- Alfaro
- 1997
(Show Context)
Citation Context |

1 |
Trace-based Process Algebras for Real-time Probabilistic Systems
- Cattani
- 2005
(Show Context)
Citation Context ...lationship between our notion of bisimulation and trace distributions is currently being investigated. Stochastic transition systems are also used as a semantic model for a stochastic process algebra =-=[6]-=-. Further work would include a logical characterisation of our equivalence relations, approximation and metrics. Acknowledgements. We would like to thank Prakash Panangaden for the helpful discussions... |