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

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

Citations: | 25 - 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.

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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... |

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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... |

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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... |

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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... |

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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... |

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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 ... |

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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 [... |

7 |
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(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 ... |

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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 |
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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... |

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1 |
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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... |