Lecture Notes in Computer Science 3472

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9 Testing Theory for Probabilistic Systems 273
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Fig. 9.17. Q ⊑LS P and P �⊑LS Q.
Example. Consider P, Q ∈ PP shown in Figure 9.17 (see also Example 9.8,
page 253). We have Q ⊑LS P and P �⊑LS Q. If we apply the test process T of
Figure 9.16, we can derive for o =1a :(1b :1c :1ω, 1b :0d)
Pr Obs 1
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P,T (o) = 2 · [(1 · 1) · (1 · 1)] + 2 · [(1 · 0) · (1 · 0)] = 2 ,
Pr Obs
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Q,T (o) =[1 2 · 1+ 2 · 0] · [ 2 · 1+ 2 · 0] = 4 .
So we obtain P �⊑LS Q, but recall we have P ∼JY Q, Q ⊑SE P and P �⊑SE Q
(see Example 9.7.2).
⊓⊔
Larsen and Skou proved that the following relationship between their notion of
testing and the probabilistic bisimulation holds:
Theorem 9.36. [LS91] Let P, Q ∈ PP be τ-free and reactive. Then
P ∼ bs QiffP∼LS Q.
⊓⊔
Note that the previous theorem also holds for probabilistic processes that are not
finitely branching but fulfill the minimal probability assumption, i.e. there exists
ɛ>0 such that whenever s a −→ µ either µ(s ′ )=0orµ(s ′ ) >ɛfor all s ′ .The
difference between ∼LS and the testing preorders defined in the previous sections
is that ∼LS considers success or failure ”after each step” whereas testing relations
for probabilistic processes like ⊑SE or ⊑JY take only the success probability after
a (maximal) trace into account. So it is clear that ∼LS distinguishes processes
that can not be distinguished by ⊑SE or ⊑JY . To the best of our knowledge this
relationship has not been further considered yet.
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