Lecture Notes in Computer Science 3472

246 Verena Wolf
We present two approaches of constructing testing relations for probabilistic
processes in Section 9.8.
A fully probabilistic process P is analyzed by running in parallel with a nonprobabilistic
test process or a fully probabilistic test process. Fully probabilistic
test processes have more distinguishing power than non-probabilistic ones as can
be seen by Example 6, page 263. We will discuss both approaches here and start
with the description of two different classes of non-probabilistic test processes
[Chr90].
Definition 9.9. Let T np,re ⊂T np be the set of all T =(ST , →T , sT ) ∈T np
with
(t a −→T t ′ ∧ t a −→T t ′′ )=⇒ t ′′ = t ′ .
Let T np,re
seq
⊂Tnp,re be the set of all T =(ST , →T , sT ) ∈T np,re with
(t a −→T t ′ ∧ t a′
−→T t ′′ )=⇒ (a = a ′ ∧ t ′′ = t ′ ).
T np,re is the set of ”reactive” test processes, reactive in the sense that T ∈T np,re
is the set of all sequential test processes
because in T ∈Tnp,re seq there is no choice at all between several transitions.
We apply non-probabilistic test processes here to fully probabilistic processes.
The parallel composition is defined as follows.
Definition 9.10. The parallel composition of P = (SP, →P, sP) ∈ FPP
and T =(ST , →T , sT ) ∈T np is the fully probabilistic process P�T =(SP ×
ST , →, (sP, sT )) with
has no internal nondeterminism. T np,re
seq
(s, t) (a,
(s, t) (τ,
p
v(s,t) )
−−−−−−→ (s ′ , t ′ ) iff s (a,p)
−−−→P s ′ and t a −→T t ′ ,
p
v(s,t) )
−−−−−→ (s ′ , t ′ ) iff s (τ,p)
−−−→P s ′ and t = t ′ ,
where a �= τ and v(s, t) = �
a∈Act |{t a −→T t ′ }|·Pr a P (s)+Pr τ P (s).
⊓⊔
Note that the parallel composition of a fully probabilistic process and a nonprobabilistic
test process that is able to perform τ-transitions is not sensible
here since there is no appropriate probability for an internal move of T in P�T
and we do not want to abstract from the probabilistic information here at all.
Following the approach of Cleaveland et al. the environment of a fully probabilistic
process is simulated by a test process which is also a fully probabilistic process
[CSZ92]. The parallel composition of P ∈ FPP and T ∈T fp
τ is a fully probabilistic
process P�T that executes external actions of P and T synchronously
and internal actions in isolation. Here internal actions are treated as ”invisible”
for the environment and synchronizing over internal actions is not allowed.
Definition 9.11. The parallel composition of two fully probabilistic processes
P = (SP, →P, sP) and T = (ST , →T , sT ) is P�T = (SP × ST ,
→, (sP, sT )) ∈ FPP where for a �= τ
⊓⊔