Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 241
left transition. Furthermore, this probability can be 0 if the right transition is
chosen. In a similar manner we derive that the probability of reaching u2 lies in
the interval [ 1 2
2 , 3 ]. In general we can assume that the left transition is taken with
probability p1, the right transition with probability p2 and that with probability
1 − (p1 + p2) no transition is chosen (the external user decides to do nothing).
Accordingly, in the following section we add weights to every transition.
⊓⊔
9.4.1 Removing Nondeterminism
The idea of adding weights to nondeterministic alternatives is also known as
randomized policies, schedulers or adversaries [Put94]. Sometimes the possibility
of scheduling no transition at all is omitted or all weights take only values in the
set {0, 1} (also called deterministic scheduler).
Adding weights means reducing P to a fully probabilistic resolution of P where
the (unique) probability of reaching certain states or performing certain actions
can be computed.
Definition 9.4.
• Let P =(SP, →P, sp) ∈ PP and ⊥�∈ SP, stop �∈ Actτ .WeextendAct to the
set Act ∪{stop} and P to Stop(P) =(SP ∪{⊥}, →, sP) ∈ PP with
→=→P ∪{(s, stop,χ⊥) | s ∈ SP}.
• Let Q =(SQ, →Q, sQ) ∈ PP and let δ be a weight function on (SQ × Actτ ×
Distr(SQ)) such that for all non-terminal states s ∈ SQ :
�
a,µ:s a δ(s, a,µ)=1.
−→µ
In this case δ is called a weight function for P and δ(Q) =(SQ, →, sQ) ∈
FPP is given by
s (a,p)
−−−→ s ′ iff δ(s, a,µ) · �
µ:s a
−→µ µ(s′ )=p.
• Let fully(P) be the set of fully probabilistic processes of P ∈ PP constructed
in the way previously described, i.e.
fully(P) ={δ(Stop(P)) | δ is a weight function for Stop(P)}.
⊓⊔
The stop-action models that no transition of P is chosen. This action always
ends in the terminal ⊥-state and no further executions are possible. Sometimes
the Stop(P) extension is called the halting extension of P.
Example. Figure 9.3 shows δ(Stop(P)) ∈ fully(P) wherePis the probabilistic
process of Figure 9.2 and δ(sP , a,µ)= 1
2 , δ(sP, b,λ)= 1
3 , δ(sP, stop,χ⊥) = 1
6
and δ(ui, stop,χ⊥) =1fori =1, 2, 3.
⊓⊔
We have already defined paths and traces in fully probabilistic processes. With
the help of the previous construction we analyze paths and traces in a probabilistic
process P by considering fully(P).