Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 271
Definition 9.32. [LS91] Let P be a reactive probabilistic process. An equivalence
relation R⊆SP ×SP is a probabilistic bisimulation iff for all (s, s ′ ) ∈R
we have that
�
v∈E µ s,a (v) =�v∈E
µ s ′ ,a (v) for all E ∈ SP/R, a ∈ Actτ,
where SP/R denotes the quotient space of R.
Two reactive probabilistic processes P, Q are probabilistically bisimilar if
there exists a probabilistic bisimulation R on (SP ∪SQ) such that (sP, sQ)areina
probabilistic bisimulation in the probabilistic process (SP ∪SQ, →P ∪→Q, sP) 7 .
We write P ∼ bs Q in this case.
⊓⊔
The probabilistic bisimulation extends the standard bisimulation [HM85] for
non-probabilistic processes. It was motivated by a probabilistic modal logic PML
[LS91] that is a probabilistic extension of the Hennessy-Milner logic HML, also
introduced by Hennessy and Milner [HM85]. Two non-probabilistic processes
(probabilistic processes) are bisimilar (probabilistic bisimilar, respectively) if
and only if they satisfy exactly the same HML (PML, respectively) formulas.
For a more detailed discussion about probabilistic bisimulation see the work of
Baier et al. [BHKW03] where also a weak 8 notion of ∼ bs is defined and where
the relationship between probabilistic (bi-)simulation and probabilistic logics is
examined in the discrete-time and also in the continuous-time case.
In the following, we present a testing approach that is, opposed to the previous
sections, not based on the parallel composition of a test process and a
tested process and it turns out that this approach yields a relation equivalent
to probabilistic bisimulation. Without loss of generality we can assume that
T =(ST , →T , sT ) ∈T np has a tree-like structure, i.e. each t ∈ ST , t �= sT has
exactly one predecessor. We define a set of observations OT that are produced
if T is applied:
Definition 9.33. Let OT (s) denotethesetofobservations obtained from the
state t ∈ ST inductively given by OT (t) ={1ω} if t is terminal and
OT (t)=({0a1}∪{1a1 : o | o ∈ OT(t1)}) ×...× ({0an }∪{1an : o | o ∈ OT (tn)})
if t ai
−→T ti, 1� i � n. LetOT = OT(sT ).
⊓⊔
Note that OT is well-defined since test processes are finite-state, finitely branching
and acyclic. Intuitively, 1a denotes that action a is observed and 0a that a is
not observed. The observed actions are concatenated with ”:”. The observation
1a : o, for instance, means that a is observed and followed by the observation o.
If the test process branches, the observation is a tuple. For example, 1a :(0a, 1b)
means that first action a is observed and then for the a-branch (in T )noaaction
is performed (0a) and for the b-branch a b-action is executed (1b). Of
course, a = b is possible.
7 Without loss of generality we can assume that SP ∩ SQ = ∅.
8 ’’Weak” in the sense that τ-actions are treated in a special way.