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Determinacy $$\to$$ (observation equivalence $$=$$ trace equivalence). (English) Zbl 0571.68018
If 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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##### 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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