## Refinement-oriented probability for CSP (1995)

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Citations: | 40 - 7 self |

### BibTeX

@MISC{Morgan95refinement-orientedprobability,

author = {Carroll Morgan and Annabelle McIver and Karen Seidel and J.W. Sanders},

title = {Refinement-oriented probability for CSP},

year = {1995}

}

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### Abstract

Jones and Plotkin give a general construction for forming a probabilistic powerdomain over any directed-complete partial order [Jon90, JP89]. We apply their technique to the failures/divergences semantic model for Communicating Sequential Processes [Hoa85]. The resulting probabilistic model supports a new binary operator, probabilistic choice, and retains all operators of CSP including its two existing forms of choice. An advantage of using the general construction is that it is easy to see which CSP identities remain true in the probabilistic model. A surprising consequence however is that probabilistic choice distributes through all other operators; such algebraic mobility means that the syntactic position of the choice operator gives little information about when the choice actually must occur. That in turn leads to some interesting interaction between probability and nondeterminism. A simple communications protocol is used to illustrate the probabilistic algebra, and several sugg...

### Citations

3638 | Communicating Sequential Processes
- Hoare
- 1985
(Show Context)
Citation Context ....uk. include other aspects: parallelism and real-time behaviour are just two examples. In this work we attempt to include probability as an integral part of a rigorous development method. Hoare's CSP =-=[5]-=- is used for the experiment, but we believe the results to be more general. The generality is largely because we exploit Claire Jones' work [6], in which she presents a method for constructing probabi... |

3426 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...he set from which the choice will be made. Our work belongs to the second group. In the first group we find the "generative" model of [16] and the "extended failures" model of [10]=-=: the first is CCS- [11]-=- and the second CSPbased. In the generative model (but using our syntax), a probabilistic external choice a ! A p [] b ! B selects the left branch if the environment offers only a, and similarly the r... |

1533 |
A Discipline of Programming
- Dijkstra
- 1976
(Show Context)
Citation Context ...nguish the two in theory: an implementor may be asked to choose between several alternatives each of which is deterministic, which is underspecification rather than nondeterminism. Many other methods =-=[2]-=- do not make that distinction, reasoning that it is relevant only when repeated testing (or equivalently `copying') is allowed. Yet testing for probability must involve repetition, as one is looking f... |

1484 | An axiomatic basis for computer programming
- Hoare
- 1969
(Show Context)
Citation Context ...ch is characterised by the behavioural properties which specifications may describe; and the resulting implementations are intended to preserve those properties. The early formal methods (for example =-=[4, 2]-=- concentrated only on functional sequential behaviour. Over the last two decades, however, many methods have been developed subsequently that All authors are members of the Programming Research Group,... |

1143 |
Computer Networks
- Tanenbaum
- 2000
(Show Context)
Citation Context ...e deterministic fixed points (hence Flips , at (6) to come, is deterministic). 12 Example: a simple protocol Our example is taken from [17]: it is a probabilistic version of the Stop-andWait protocol =-=[15]-=-, which achieves flow control and overcomes the unreliability of a lossy communication link by the simple expedient of (re)transmitting until an acknowledgement is received. The probabilistic version ... |

438 | Testing equivalences for processes
- Nicola, Hennessy
- 1984
(Show Context)
Citation Context ...failure. The mathematics will then tell us what proportion of those im2 plementations we can expect to satisfy their specification: the refinement relation becomes a function into the closed interval =-=[0; 1]-=- of probabilities. In other situations, however, probabilistic behaviour may be desirable, because it improves expected efficiency. Then probabilistic refinement can quantify the likelihood that the s... |

427 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1989
(Show Context)
Citation Context ...results of trace-like tests, as in [12, 10, 13], and the "reactive" and generative models of [16]; yet it is coarser than the "stratified bisimulation" of [16] and the "probab=-=ilistic bisimulation" of [8]-=-. If an equivalence is very coarse (identifying `many' processes), then some operators will not be definable (or, alternatively expressed, in the presence of those operators the equivalence will not b... |

163 | Reactive, generative and stratified models of probabilistic processes
- Glabbeek, Smolka, et al.
- 1995
(Show Context)
Citation Context ...c choice (it is always the process); rather it is in who decides the set from which the choice will be made. Our work belongs to the second group. In the first group we find the "generative"=-= model of [16] and the &-=-quot;extended failures" model of [10]: the first is CCS- [11] and the second CSPbased. In the generative model (but using our syntax), a probabilistic external choice a ! A p [] b ! B selects the... |

143 |
Probabilistic nondeterminism
- Jones
- 1990
(Show Context)
Citation Context ...egral part of a rigorous development method. Hoare's CSP [5] is used for the experiment, but we believe the results to be more general. The generality is largely because we exploit Claire Jones' work =-=[6]-=-, in which she presents a method for constructing probabilistic spaces from arbitrary directed-complete partial orders. The failures-divergences model of CSP is one such, and so we are able to carry o... |

129 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ... through p \Phi whence the desired refinement to (13) follows simply by dropping the middle two terms. Finally, it is clear that the mathematical framework provided by Claire Jones and Gordon Plotkin =-=[6, 7] is -=-of great generality. We have specialised it here by exploiting CSP's algebraic character, allowing us to concentrate on probabilistic refinement (F ! �� ). A similar approach is possible for seque... |

96 | Process Algebra, volume 18 of Cambridge Tracts - Baeten, Weijland - 1990 |

68 | Axiomatizing probabilistic processes: ACP with generative probabilities - Baeten, Bergstra, et al. - 1995 |

62 |
Testing probabilistic and nondeterministic processes
- Yi, Larsen
- 1992
(Show Context)
Citation Context .... determinism-preserving and constructive (Sec. 13) functions have deterministic fixed points (hence Flips , at (6) to come, is deterministic). 12 Example: a simple protocol Our example is taken from =-=[17]-=-: it is a probabilistic version of the Stop-andWait protocol [15], which achieves flow control and overcomes the unreliability of a lossy communication link by the simple expedient of (re)transmitting... |

39 |
and branching structures in the semantics and logics of reactive systems
- Linear
(Show Context)
Citation Context ... are equal when the test (F ! �� ) yields identical results for all finite F . The (F ! �� )-equivalence is finer than those based on assigning probabilities to the results of trace-like tests=-=, as in [12, 10, 13], and the "reactive"-=- and generative models of [16]; yet it is coarser than the "stratified bisimulation" of [16] and the "probabilistic bisimulation" of [8]. If an equivalence is very coarse (identify... |

38 |
Probabilistic communicating processes
- Seidel
- 1992
(Show Context)
Citation Context ... are equal when the test (F ! �� ) yields identical results for all finite F . The (F ! �� )-equivalence is finer than those based on assigning probabilities to the results of trace-like tests=-=, as in [12, 10, 13], and the "reactive"-=- and generative models of [16]; yet it is coarser than the "stratified bisimulation" of [16] and the "probabilistic bisimulation" of [8]. If an equivalence is very coarse (identify... |

31 | Linear and Branching Structures in the Semantics and Logics of Reactive Systems - PNUELI - 1985 |

24 |
domains and predicate transformers: a topological view , Techincal monograph 126
- Power
- 1983
(Show Context)
Citation Context ...ulated operator indifferent nondeterminism (as opposed to "demonic"), we introduce it by a Smyth-like construction similar to that which can impose nondeterminism on an otherwise determinist=-=ic system [14]-=-. The resulting space NDCSP comprises certain subsets of PCSP which are up-closed (for the same reasons as in the Smyth construction): for Smyth-set S and probabilistic A; B , we have B 2 S whenever A... |

20 | Representing nondeterministic and probabilistic behaviour in reactive processes
- Lowe
- 1993
(Show Context)
Citation Context ...Phi through u is enough to establish both its lack of idempotence (example following Lem. 9.5) and its failure to distribute through other operators in the presence of probability (example drawn from =-=[9]-=-): (a ! STOP u b ! STOP) k (a ! STOP p \Phi b ! STOP) (9) =? distribute u illegally through k a ! STOP k (a ! STOP p \Phi b ! STOP) u b ! STOP k (a ! STOP p \Phi b ! STOP) = (a ! STOP p \Phi STOP) u (... |

15 |
Probabilities and priorities in timed csp
- Lowe
- 1999
(Show Context)
Citation Context ...r it is in who decides the set from which the choice will be made. Our work belongs to the second group. In the first group we find the "generative" model of [16] and the "extended fail=-=ures" model of [10]-=-: the first is CCS- [11] and the second CSPbased. In the generative model (but using our syntax), a probabilistic external choice a ! A p [] b ! B selects the left branch if the environment offers onl... |

15 | Probabilistic nondeterminism. Monograph ECS-LFCS-90-105 - Jones - 1990 |

13 | Bergstra, Process algebra with partial choice - Baeten, A - 1994 |

5 | Argument duplication in probabilistic CSP - Morgan, McIver, et al. - 1995 |

2 | A probabilistic powerdomain of evaluations - Morgan, McIver, et al. - 1989 |

1 | Probabilistic communicatingprocesses - Seidel - 1992 |