Lecture Notes in Computer Science 3472

238 Verena Wolf
(a, 0.2)
v1
sP
(a, 0.5) (b, 0.5)
(a, 0.8) (τ,0.3)
v2
(b, 0.7)
u1 u2 u3 u4
Fig. 9.1. An example of a fully probabilistic process.
9.3.1 Paths and Traces of Fully Probabilistic Processes
Similar as to non-probabilistic processes we define paths and traces in a fully
probabilistic process.
Definition 9.2. Let P =(SP, →P, sP) ∈ FPP.
• A finite path α of P is a sequence
α = s0 a0 s1 a1 ...an−1 sn,
(ai ,p)
−−−→ P si+1, p > 0for0� i < n. As
where s0 = sP, sn is terminal and si
before, lstate(α) =sn denotes the last state of a finite path α.
• An infinite path α of P is an infinite sequence
α = s0 a0 s1 a1 ...,
where s0 = sP and for 0 � i : si
• A finite trace β is a sequence
β = a0 a1 ...an−1 ∈ Act ∗ .
• An infinite trace β is a sequence
β = a0 a1 ... ∈ Act ω .
(ai ,p)
−−−→ P si+1, p > 0.
Let trace(α) ∈ Act ω be the ordered sequence of all external actions occurring
in a finite/infinite path α.
⊓⊔
Example. Consider P ∈ FPP in Example 9.3. We have, for instance, the finite
paths α1 = sP av1 au1 and α2 = sP bv2 τ u3 of P. P has no infinite paths.
Furthermore, we have trace(α1) =aaand trace(α2) =b.
⊓⊔