Lecture Notes in Computer Science 3472

240 Verena Wolf
1
2
sP
a b
µ λ
1
2
2
3
1
3
u1 u2 u3
Fig. 9.2. The probabilistic process P.
the environment by resolving nondeterminism. So one probabilistic process P
describes a set of fully probabilistic processes (here called resolutions of P). We
show how to find a way of splitting P into several resolutions, i.e. how to remove
nondeterminism and obtain a set of fully probabilistic processes (resolutions)
from P. A more detailed comparison between several models for probabilistic
systems can be found in the work of Glabbeek, Smolka and Steffen [vGSS95].
Definition 9.3. A probabilistic process is a tuple P =(SP, →P, sP), where
• SP is a countable set of states,
•→P⊆SP × Actτ × Distr(SP) is a transition relation,
• sP ∈ SP is an initial state.
⊓⊔
We write s a −→P µ for a transition (s, a,µ) ∈→P and define PP as the set of
all probabilistic processes. A state s can have several nondeterministic alternatives
for the next transition. The destination of a transition s a −→ µ is chosen
probabilistically with regard to the distribution µ. For a transition s a −→ µ the
probability µ(s ′ )forsomes ′ ∈ supp(µ) can be seen as the conditional probability
of the target state s ′ given that transition s a −→ µ is chosen. This is motivated
by the idea that the external environment decides which action is performed
whereas the probabilistic choice determining the next target state is resolved
by the process itself. Of course, the external user cannot choose between two
transitions s a −→P µ and s a −→P µ ′ , µ �= µ ′ . We therefore have a kind of ”true”
nondeterminism in addition. Later on we will remove nondeterminism by adding
weights to every transition such that analyzing the process becomes less difficult.
Example. Figure 9.2 shows P =(SP, →P, sP) ∈ PP with SP = {sP, u1, u2, u3},
→P= {(sP, a,µ), (sP, b,λ)}, µ(u1) =µ(u2) = 1
2 , λ(u2) = 2
3 and λ(u3) = 1
3 .
States are drawn as circles and distributions as boxes. Transitions s a −→P µ are
drawn as solid arrows and probabilistic choices are drawn as dashed arrows.
We can state that the probability of reaching u1 lies in the interval [0, 1
2
a higher probability than 1
2
] because
is not possible even if the environment schedules the