Lecture Notes in Computer Science 3472

256 Verena Wolf
• P ⊑ must
JY
Q iff for all T ∈Tpp :
min P ′ ∈fully {0,1} (P�T ) WP ′ � min Q ′ ∈fully {0,1} (Q�T ) WQ ′.
Q and P ⊑must JY Q.
⊓⊔
Two processes P and Q are related in ⊑ may
JY if the ”best” fully probabilistic
resolution of P�T has a total success probability that is at least the total success
We write P ⊑JY Q iff P ⊑ may
JY
probability of the best fully probabilistic resolution of Q�T .Notethat⊑ may
JY boils
down to ordinary simulation [Jon91] (when probabilities are omitted) which is
another implementation relation for non-probabilistic processes [Jon91].
A non-probabilistic process Q simulates a non-probabilistic process P if Q can
”simulate” every step of P. The converse must not necessarily hold.
Jonsson and Yi also showed that ⊑ may
JY coincides with their probabilistic simulation
[JY02].
Note that if P ⊑JY Q the process Q is bounded in some sense by P (so again
we assume that Q has the role of the implementation and P theroleofthe
specification). The total success probabilities of Q have to lie in the interval
[min P ′ ∈fully {0,1} (P�T ) WP ′, max P ′ ∈fully {0,1} (P�T ) WP ′].
Thus P is a more abstract resolution of Q and the requirements in P are fulfilled
in this case because the ”worst” resolution of Q�T is at least as good as the
worst resolution of P�T .
Example. Consider P and Q in Figure 9.9, page 254. It can be shown that
Q ⊑JY P. The composition with T , for example, yields
{0.5} = {WQ ′ | Q ′ ∈ fully {0,1} (Q�T )}
⊂{WP ′ | P ′ ∈ fully {0,1} (P�T )} = {0, 0.5, 1}.
9.9 Compositional Testing of Markovian Processes
In this section, we briefly sketch a testing theory for action-labeled Markov chains
as proposed by Bernardo and Cleaveland [BC00]. The idea of defining a notion
of testing for aCTMCs is very similar to the approaches for fully probabilistic
processes in Section 9.7 but here two quantities are taken into account when
testing M1, M2 ∈ ACTMC with a Markovian test process T ∈T pa
τ :
(1) For i ∈ {1, 2} consider all paths α in the fully probabilistic embeddings
M ′
i = φem(Mi�T ) ∈ FPP with success state lstate(α) and calculate their
probabilities where Pr path path
Mi �T := PrM ′
5 .
i
5 HerethepathdistributionofanaCTMCM is equal to the path distribution of the
embedded fully probabilistic process φem(M )
⊓⊔