Lecture Notes in Computer Science 3472

270 Verena Wolf
T np,re
seq
⊑ tr CH
ordinary
traces
fully probabilistic
processes
T np,re
probabilistic
processes
aCTMCs
⊑ ste
CH
⊑CL ⊑ ⊑BC
may
SE ⊑ may
JY
extended
traces
fp
Tτ prob.
traces
T pp
τ
trace distr.
precongr.
T pp
prob.
simulation
T pa
τ
extended
traces+time
Fig. 9.15. Characterizations for probabilistic testing relations: The upper vertical arrows
connect the respective class of processes with the matching testing relation and
the label corresponds to the applied set of test processes. The lower vertical arrows connect
each testing relation with the respective characterization and the lower horizontal
arrows show that extended traces are a special case of probabilistic traces and a special
case of extended traces with an additional time requirement. Moreover, ordinary traces
are a special case of extended traces.
Figure 9.15 shows the characterizations presented in this section and the applied
sets of test processes, respectively. Moreover, the underlying models are
depicted. Of course, extended traces, enriched with a time function, boil down
to ordinary extended traces and probabilistic traces are the probabilistic extension
of extended traces. It is clear that extended traces are an extension of
ordinary traces.
9.12 Connecting Testing and Probabilistic Bisimulation
In the following, we discuss the work of Larsen and Skou that connects an intuitive
testing approach for a subclass of probabilistic processes with probabilistic
bisimulation [LS91]. Larsen and Skou apply non-probabilistic test processes to
τ-free reactive probabilistic processes . A probabilistic process P =(SP, →P, sP)
is reactive if s a −→P µ and s a −→P λ implies µ = λ for all s ∈ SP. The intuitive
idea is that in each step the external environment chooses an action and there is
no ”internal” nondeterminism between two transitions with equal actions. For
s a a
−→ µ we put µ s,a = µ. Wewrites−−→/
P if there exists no µ s,a .
Larsen and Skou defined the probabilistic bisimulation equivalence such that for
two bisimulation equivalent states the probability to move with an a-transition
to an equivalence class E is equal for all a ∈ Actτ.