Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 239
We proceed with the construction of a probability space for sets of paths and
traces of P. We define the set of outcomes Ω as the set Path(P) containing all
finite and infinite paths in P. Furthermore,letC (α) ⊆ Ω denote the set of all
paths starting with the sequence α (also called a cylinder set) andletC be the
set of all C (α), where α is an alternating sequence of states and actions. We
define a probability measure PrP by induction on C as follows.
PrP(C (s0)) = 1 if s0 = sP and 0 otherwise,
PrP(C (s0 a0 s1 a1 ...an−1 sn a ′ s ′ )) =
PrP(C (s0 a0 s1 a1 ...an−1 sn)) · Pr a′
P (sn, s ′ ).
PrP can be extended to a unique probability measure Pr path
P
on σ(C). We briefly
write Pr path
P (α) forPr path
P (C (α)).
We have a similar construction for traces of a fully probabilistic process P. The
set of outcomes is the set of all traces and the σ-algebra σ(C) is built with
cylinder sets of traces. We have for the probability of the cylinder set of all
traces starting with the sequence β ∈ Act ∗ :
Pr ′ P
path
(C (β)) = PrP ({α ∈ Path(P) | β is a prefix of trace(α)}).
to a unique probability
on σ(C). We briefly write Pr trace
P (β) forPr trace
P (C (β)). We call
the trace distribution of P, although �
trace
ω
β∈Act PrP (β) mightbe
(β) =1instead.
Again it is possible to extend the probability measure Pr ′ P
measure Pr trace
P
Pr trace
P
greater as one. We have � β=trace(α),
α∈Path(P)
Pr trace
P
Example. For the paths α1,α2 in Example 9.3.1 (see also Figure 9.1) we have
that
Pr path
P (sP av1au1) =0.5 · 0.2 =0.1,
Pr path
P (sP bv2τ u3) =0.5 · 0.3 =0.15,
Pr trace
P (a a)=0.5 · 0.2+0.5 · 0.8 =0.5,
Pr trace
P (b) =0.5.
9.4 Probabilistic Processes
Probabilistic processes are another basic model for systems with probabilistic
phenomena. Probabilistic processes are the action-labeled extension of Markov
decision processes [Put94] and are also known (sometimes in a more general
form) as probabilistic automata [SL94]. Probabilistic processes are more abstract
than fully probabilistic processes because they can represent nondeterministic
behavior. They combine modeling probabilistic behavior and interaction with
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