Determinacy \(\to\) (observation equivalence \(=\) trace equivalence).

*(English)*Zbl 0571.68018If an experiment s is conducted on a parallel process p, then, in general, different processes may result from the experiment, due to the nondeterministic behaviour of p (in the notation of Milner [R. Milner, A calculus of communicating systems (1980; Zbl 0452.68027)]:p\(\Rightarrow^{s}p'\) for different p’). Process p is called determinate if the resulting processes are all equivalent (i.e., if \(p\Rightarrow^{s}p'\) and \(p\Rightarrow^{s}p''\), then p’ and p” are equivalent). This means that, although p behaves nondeterministically, this cannot be detected by an observer of p. Let \(\simeq\) denote observation equivalence, used in CCS (Milner, loc. cit.) let \(\simeq_ f\) denote (the much weaker) failure equivalence, used for CSP [S. D. Brookes, Lect. Notes Comput. Sci. 154, 83-96 (1983; Zbl 0516.68024)], and let \(\simeq_ t\) denote (the still weaker) trace equivalence. We show that the three corresponding notions of determinacy are the same, and that for determinate processes \(\simeq\), \(\simeq_ f\), and \(\simeq_ t\) are the same. Determinacy is preserved under \(\simeq\) and \(\simeq_ f\), but not under \(\simeq_ t\).

##### MSC:

68N25 | Theory of operating systems |

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DOI

##### References:

[1] | Brookes, S.D., On the relationship of CCS and CSP, (), 83-96 |

[2] | Brookes, S.D.; Rounds, W.C., Behaviourial equivalence of relations induced by programming logics, (), 97-108 |

[3] | Hoare, C.A.R.; Brookes, S.D.; Roscoe, A.W.; Hoare, C.A.R.; Brookes, S.D.; Roscoe, A.W., A theory of communicating sequential processes, (), J. ACM, 31, 560-599, (1984), also · Zbl 0628.68025 |

[4] | Milner, R., A calculus of communicating systems, () · Zbl 0452.68027 |

[5] | Milner, R., Calculi for synchrony and asynchrony, Theoret. comput. sci., 25, 3, 267-310, (1983) · Zbl 0512.68026 |

[6] | Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pacific J. math., 5, 285-309, (1955) · Zbl 0064.26004 |

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