## Axioms for Probability and Nondeterminism (2003)

Venue: | ENTCS |

Citations: | 24 - 1 self |

### BibTeX

@INPROCEEDINGS{Mislove03axiomsfor,

author = {Michael Mislove and Joël Ouaknine and James Worrell},

title = {Axioms for Probability and Nondeterminism},

booktitle = {ENTCS},

year = {2003},

pages = {7--28}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a domain model for a process algebra featuring both probabilistic and nondeterministic choice. The former is modelled using the probabilistic powerdomain of Jones and Plotkin, while the latter is modelled by a geometrically convex variant of the Plotkin powerdomain. The main result is to show that the expected laws for probability and nondeterminism are sound and complete with respect to the model. We also present an operational semantics for the process algebra, and we show that the domain model is fully abstract with respect to probabilistic bisimilarity.

### Citations

418 | Bisimulation through probabilistic testing - Larsen, Skou - 1991 |

287 | Probabilistic simulations for probabilistic processes
- Lynch
- 1994
(Show Context)
Citation Context ...there is only one type of process, and the operators + and p can be applied without restriction. To accommodate both types of choice, following the probabilistic automaton model of Segala and Lynch [=-=17]-=-, we model the initial capabilities of a process as a set of probability distributions. Nondeterministic choice is modelled by union, and probabilistic choice by pointwise lifting the natural probabil... |

141 |
Probabilistic Non-determinism
- Jones
- 1989
(Show Context)
Citation Context ...den by importing standard constructions and results from general domain theory. This paper gives a domain-theoretic semantics for a probabilistic process algebra. We use the probabilistic powerdomain =-=[11]-=- to model probabilistic choice, and we use a geometrically convex variant of the Plotkin powerdomain, 1 Support by the US National Science Foundation and the US Oce of Naval Research is gratefully ack... |

126 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ...isimilarity for calculi featuring probabilistic choice but not nondeterministic choice. The development in [18] closely follows the treatment of equations for bisimilarity insnite-state CCS by Milner =-=[14]-=-. Our completeness proof follows the same pattern, except that arguments based on operational semantics get replaced by domain theory. Following Milner [14] and Stark and Smolka [18], we include a uni... |

123 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...ain VD of a domain D. The elements of VD are valuations on D. These are like probability measures, but can be dened using purely topological data. Technically, a valuation on D is a mapping : D ! [0; 1] from the lattice (D; ) of Scott open subsets of D to the unit interval satisfying the following laws. strictness ; = 0 and D = 1 modularity (U [ V ) + (U \ V ) = U + V for all U; V . Sc... |

67 | W.: Probabilistic extensions of process algebras
- Jonsson, Larsen, et al.
- 2001
(Show Context)
Citation Context ... distributive law tends to arise in those models where the guiding philosophy is that probabilistic choices should be resolved before nondeterministic choices. Examples of such models can be found in =-=[12,16]-=-. 1.1 Related Work Stark and Smolka [18] and Baeten et al. [4] give complete axiomatizations of probabilistic bisimilarity for calculi featuring probabilistic choice but not nondeterministic choice. T... |

66 |
Axiomatizing probabilistic processes: ACP with generative probabilities
- Baeten, Bergstra, et al.
- 1995
(Show Context)
Citation Context ... philosophy is that probabilistic choices should be resolved before nondeterministic choices. Examples of such models can be found in [12,16]. 1.1 Related Work Stark and Smolka [18] and Baeten et al. =-=[4]-=- give complete axiomatizations of probabilistic bisimilarity for calculi featuring probabilistic choice but not nondeterministic choice. The development in [18] closely follows the treatment of equati... |

42 | The troublesome probabilistic powerdomain
- Jung, Tix
- 1998
(Show Context)
Citation Context ... directed family fU i g i2I . The set of all valuations VD becomes a domain when equipped with the pointwise order: v i U 6 U for all U 2 D. Furthermore, VD is coherent whenever D is coherent [8]. The Scott topology on VD can be described directly in terms of the Scott topology on D using a probabilistic modality. The sub-basic opens are the sets } p U = f 2 VD : U > pg where 0 6 p 6 1 and ... |

41 |
Axiomatizations for probabilistic bisimulation
- Bandini, Segala
- 2001
(Show Context)
Citation Context ...een obtained by Aceto, Esik and Ingolfsdottir [2]. Their approach involved adding axioms for probabilistic choice to an underlying equational formulation of iteration theories. Bandini and Segala [6] give a complete axiomatization of strong and weak probabilistic bisimilarity for an algebra including nondeterministic and probabilistic choice, but not recursion. One signicant dierence between th... |

38 | Refinementoriented probability for CSP
- Morgan, McIver, et al.
- 1996
(Show Context)
Citation Context ... distributive law tends to arise in those models where the guiding philosophy is that probabilistic choices should be resolved before nondeterministic choices. Examples of such models can be found in =-=[12,16]-=-. 1.1 Related Work Stark and Smolka [18] and Baeten et al. [4] give complete axiomatizations of probabilistic bisimilarity for calculi featuring probabilistic choice but not nondeterministic choice. T... |

33 | Process algebra with probabilistic choice
- Andova
- 1999
(Show Context)
Citation Context ...erdomain. Using this distributive law he denes a combined monad of probabilistic and nondeterministic choice. The corresponding equational theory however does not include the law E p E = E. Andova [3=-=]-=- considers the addition of probabilistic choice to the algebra ACP , studied extensively by de Bakker and his students. The pivotal issue in Andova's work is how to extend the parallel composition ope... |

25 | Nondeterminism and probabilistic choice: obeying the laws
- Mislove
(Show Context)
Citation Context ...arch Agency and the Army Research Oce under contract DAAD19-01-1-0485 is gratefully acknowledged. cs2003 Published by Elsevier Science B. V. Mislove, Ouaknine and Worrell independently due to Mislove =-=[15]-=- and Tix [19], to model nondeterministic choice. The main result of the paper is to present a set of laws for probabilistic and nondeterministic choice which is sound and complete with respect to our ... |

21 | B.: Reactive, generative and strati¯ed models of probabilistic processes - Glabbeek, Smolka, et al. - 1995 |

17 | Equational axioms for probabilistic bisimilarity (preliminary report - Aceto, Esik, et al. - 2002 |

17 | Domain equations for probabilistic processes
- Baier, Kwiatkowska
- 1997
(Show Context)
Citation Context ...roduce syntactic restrictions on the application of probabilistic choice and nondeterministic choice. For instance, the distributive law (1) would be ill-typed in their setting. Baier and Kwiatkowska =-=[5]-=- consider the combination of probability and nondeterminism from a domain-theoretic perspective. Like us they produce a model for a process algebra by solving a domain equation involving the probabili... |

16 |
Continuous D-Cones: Convexity and Powerdomain Constructions
- Tix
- 1999
(Show Context)
Citation Context ...ve their inspiration in two preceding works: the development of a domain equation for bisimulation from [1], and the presentation of a powerdomain combining nondeterminism and probabilistic choice in =-=[15,19]-=-. Analogous results have been devised for other models for probabilistic choice, a number of which have been mentioned in Subsection 1.1. While some of these related works consider probabilistic choic... |

10 | The powerdomain of indexed valuations
- Varacca
- 2002
(Show Context)
Citation Context ...istic choice operator that we model. Finally they do not axiomatize their semantics. A radically dierent way to model probability and nondeterminism in domain theory has been investigated by Varacca [=-=20]-=-. He models probabilistic choice using a monad of indexed valuations. This structure is more rigid than the usual probabilistic powerdomain (which is a quotient), and allows a categorical distributive... |

8 | Probabilistic extensions of semantical models
- Hartog
- 2002
(Show Context)
Citation Context ...tend the parallel composition operator of ACP to the algebra extended to include probabilistic choice, an issue that requires considerable delicate reasoning to resolve. Finally, the paper den Hartog =-=[10-=-] presents an extensive consideration to the interactions between diering forms of nondeterministic and probabilistic choice, called local and global, and which correspond to internal and external cho... |

2 |
A complete axiom system for probabilistic processes
- Stark, Smolka
- 2000
(Show Context)
Citation Context ...ondeterministic choice. In giving an operational semantics to such languages one must not only specify which transitions are possible, but also the probabilities with which they are taken (see, e.g., =-=[18]-=-). In the presence of unguarded recursion the resultant bookkeeping becomes quite complicated, and arguably undermines the usual advantages of an operational approach. On the other hand, in a domain m... |