## Nondeterminism and Probabilistic Choice: Obeying the Laws (2000)

Venue: | In Proc. 11th CONCUR, volume 1877 of LNCS |

Citations: | 26 - 2 self |

### BibTeX

@INPROCEEDINGS{Mislove00nondeterminismand,

author = {Michael Mislove},

title = {Nondeterminism and Probabilistic Choice: Obeying the Laws},

booktitle = {In Proc. 11th CONCUR, volume 1877 of LNCS},

year = {2000},

pages = {350--364},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domain-theoretic techniques, we show how models can be devised using the "standard model" for probabilistic choice, and then applying modified domain-theoretic models for nondeterministic choice. These models are distinguished by the fact that the expected laws for nondeterministic choice and probabilistic choice remain valid. We also describe some potential applications of our model to aspects of security.

### Citations

421 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...ic models using coalgebraic techniques (cf., e.g., [23] for an introduction). There are several approaches that have been put forward for modeling probabilistic choice, including { approaches such as =-=[16, 18]-=- that focus on state-based models and use probabilistic transition systems to re ect the operational behavior of the system under study. In these approaches, discrete probabilistic models are consider... |

292 | N.: Probabilistic simulations for probabilistic processes
- Segala, Lynch
- 1995
(Show Context)
Citation Context ...ic models using coalgebraic techniques (cf., e.g., [23] for an introduction). There are several approaches that have been put forward for modeling probabilistic choice, including { approaches such as =-=[16, 18]-=- that focus on state-based models and use probabilistic transition systems to re ect the operational behavior of the system under study. In these approaches, discrete probabilistic models are consider... |

142 |
Probabilistic Non-determinism
- Jones
(Show Context)
Citation Context ...he work of Saheb-Dharjomi [24]. While this work was thesrst to consider modeling probabilistic choice as a domain, the most in uential work along this line is without question the PhD thesis of Jones =-=[11], whe-=-re it was shown that Saheb-Dharjommi's construction could be extended to \measures" having total variation less than 1, and that the probabilistic power domain of ? Partial support provided by th... |

129 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ...k to the seminal work of Saheb-Djarhomi [24] in which a now standard construction of a cpo supporting probabilistic choice was given, beginning with an underlying cpo. This work led to the results in =-=[11, 12-=-] that claried and expanded the nature of Saheb-Djarhomi's construction, and also showed that this construction, when applied to a continuous domain, yields a continuous domain. This is the constructi... |

123 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ... domain of a continuous domain is again continuous. The denition of the more general construction goes as follows. Denition 1. If P is a dcpo, then a continuous valuation on P is a mapping : D ! [0; 1] dened on the Scott open subsets of P that satises: 1. (;) = 0. 2. (U [ V ) = (U) + (V ) + (U \ V ), 3. is monotone, and 4. ([ i U i ) = sup i (U i ), if fU i j i 2 Ig is an increasing fami... |

96 |
A calculus for communicating systems with time and probabilities
- Hansson, Jonsson
- 1990
(Show Context)
Citation Context ...e will use the notation p + q to denote a probabilistic choice in which the process has probability of acting like p, and probability 1 of acting like q, where 0 1. 2 { approaches such as [7] and [17] that use a process algebra in which probabilistic choice is substituted for nondeterministic choice. Here, and in the next case, the focus is on the probability that the process in question ... |

94 | Full abstraction for a simple parallel programming language
- Hennessy, Plotkin
- 1979
(Show Context)
Citation Context ...rministic choice to generate an interleaving semantics for parallel composition. This approach is well-supported by the models of computation, including both the standard domain-theoretic models (cf. =-=[10]-=-), and the metric space approach (cf. [3]). These and similar approaches to modeling nondeterminism satisfy the basic assumption that nondeterministic choice is a commutative, associative and idempote... |

90 | CSP and determinism in security modelling
- Roscoe
- 1995
(Show Context)
Citation Context ...or example (a ! p) 2 (a ! q) = (a ! p) u (a ! q) holds in CSP, but it is unclear whether it holds in our model. There is another area that our model also should be of interest. In recent work, Roscoe =-=[22]-=- has shown that the security of a multilevel system can be analyzed using CSP. In particular, he shows that such a system (by which we mean a process representing the functioning of the system that us... |

78 | Bisimulation for probabilistic transition systems: a coalgebraic approach - Vink, Rutten - 1999 |

67 | An improved failures model for communicating processes - Brookes, Roscoe - 1985 |

42 | The troublesome probabilistic powerdomain
- Jung, Tix
- 1998
(Show Context)
Citation Context ...ently assert about the probabilistic power domain of a continuous domain is that it is again continuous, and that the probabilistic power domain of a coherent continuous domain is again coherent (cf. =-=[13]-=-). In particular, the probabilistic power domain is not bounded complete, and it remains an open question whether this operator leaves any cartesian closed category of continuous domains invariant. In... |

38 | Refinementoriented probability for CSP
- Morgan, McIver, et al.
- 1996
(Show Context)
Citation Context ...uthors have attempted to incorporate both nondeterministic choice and probabilistic choice within one model. None that we know of has accomplished that goal completely satisfactorily. For example, in =-=[20]-=- a model incorporating probabilistic choice is built by simply applying Jones' probabilistic power domain to the standard failures-divergences model for CSP. But, the natural extension of nondetermini... |

24 | Probabilistic domains
- Heckmann
- 1994
(Show Context)
Citation Context ...ows that (PUA (P ); ) is a continuous coherent domain. The probabilistic choice operators can be dened on PUA (P ) by X + Y = fx + y j x 2 X; y 2 Y g; and it follows from the rectangle law (cf. [9]) for P that X + Y 2 PUA (P ) for X;Y 2 PUA (P ). The continuity of these operations also is easily seen. One can argue that PUA (P ) with these operations satises the laws of Mean Values (cf. [9])... |

23 |
A Compendium of Continuous Lattices
- Lawson, Scott
- 1980
(Show Context)
Citation Context ... i 2 Ig is an increasing family of Scott open sets. We order this family pointwise: , (U) (U) (8U 2 P ), and we denote the family of continuous valuations on P by PPr (P ). 5 It was Lawson [15] whosrst showed the connection between continuous valuations and measures on the cpo P { indeed, he showed that, in the case P has a countable basis, there is a one-to-one correspondence between regul... |

17 | Domain equations for probabilistic processes
- Baier, Kwiatkowska
- 1997
(Show Context)
Citation Context ...d for nondeterministic choice. Here, and in the next case, the focus is on the probability that the process in question acts like one branch or the other from the choices listed. { approaches such as =-=[20, 4, 8]-=- that extend a process algebra by adding probabilistic choice operators. If the branches have distinct initial actions, then the focus is on the probability of the process executing a given action, ra... |

15 |
Probabilities and priorities in timed csp
- Lowe
- 1999
(Show Context)
Citation Context ...se the notation p + q to denote a probabilistic choice in which the process has probability of acting like p, and probability 1 of acting like q, where 0 1. 2 { approaches such as [7] and [17] that use a process algebra in which probabilistic choice is substituted for nondeterministic choice. Here, and in the next case, the focus is on the probability that the process in question acts like... |

10 | Topology, domain theory and theoretical computer science. Topology and Its Applications
- Mislove
- 1998
(Show Context)
Citation Context ...supremum of simple measures, for general we can form the \limit" of the family F i , where = t i x2F i r xsx . (This limit can be thought of as being taken in the convex power domain of P { cf=-=. [19].) -=-The projection mapping then send to ^ supp , for which it is routine to verify the required equations for an e-p pair. For the second claim, we note that x 7! "x : P ! PUA (P ) is a morphism of ... |

8 |
Solving re domain equations in a category of complete metric spaces
- America, Rutten
- 1989
(Show Context)
Citation Context ...ng semantics for parallel composition. This approach is well-supported by the models of computation, including both the standard domain-theoretic models (cf. [10]), and the metric space approach (cf. =-=[3]-=-). These and similar approaches to modeling nondeterminism satisfy the basic assumption that nondeterministic choice is a commutative, associative and idempotent operation. In fact, the results from [... |

8 |
CPOs of measures for nondeterminism, Theoretical Computer Science 12
- Saheb-Djahromi
- 1980
(Show Context)
Citation Context ...e recently, probabilistic choice has been added as a family of operators in the syntax of the language under study. One can trace this research in denotational semantics to the work of Saheb-Dharjomi =-=[24]-=-. While this work was thesrst to consider modeling probabilistic choice as a domain, the most in uential work along this line is without question the PhD thesis of Jones [11], where it was shown that ... |

5 | Argument duplication in probabilistic CSP
- Morgan, McIver, et al.
- 1995
(Show Context)
Citation Context ...ecise probability to this process acting like p, since we have no way to assign a probability to how u resolves its choices, precisely since it is not a probabilistic choice operator. Further work in =-=[21]-=- addresses the question of duplication arising where it is not desired. Two possible solutions are presented there. Our interest is in studying how to overcome duplication at the nondeterministic choi... |

5 |
On the foundations of semantics: Nonstandard sets, metric spaces, partial orders
- Turi, Rutten
- 1998
(Show Context)
Citation Context ...tion relies heavily on domain theory and some of the constructs it provides. The work here is closely related to the emerging area of devising semantic models using coalgebraic techniques (cf., e.g., =-=[23]-=- for an introduction). There are several approaches that have been put forward for modeling probabilistic choice, including { approaches such as [16, 18] that focus on state-based models and use proba... |

1 | CSP models, a process p re a process q if every behavior of p also is a behavior of q. That is, q is more nondeterministic than p. This notion is the basis of for reasoning about speci in CSP - Abramsky, Jung, et al. - 1994 |

1 |
de Vink, Mixing up nondeterminism and probability: A preliminary report
- Hartog, P
- 1997
(Show Context)
Citation Context ...d for nondeterministic choice. Here, and in the next case, the focus is on the probability that the process in question acts like one branch or the other from the choices listed. { approaches such as =-=[20, 4, 8]-=- that extend a process algebra by adding probabilistic choice operators. If the branches have distinct initial actions, then the focus is on the probability of the process executing a given action, ra... |