Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 259
We call a path successful if its last state is a success state. We are applying
elements of T np
τ here but it is easy to see that the set of test processes can be
reduced to τ-free test processes without loss of expressiveness. The following
steps show how to delete τ-transitions in test processes.
• Fix T =(ST , →T , sT ) ∈T np
τ
with a set A of success states and assume
without loss of generality that T has a tree-like structure, i.e. every state
t �= sT has exactly one predecessor.
• We define prec : ST → ST inductively by
⎧
⎪⎨
sT if s = sT ,
prec(s) =
⎪⎩
s if ∃ s ′ : s ′ a −→T s, a �= τ,
prec(s ′ ) if s ′ a −→T s implies a = τ.
Note that prec(s) is constructed such that s can be reached from prec(s)
with a sequence of internal moves.
• Let RMτ(T )=({s ∈ ST |∃s ′ : s = prec(s ′ )}, →, sT ) ∈T np be a nonprobabilistic
test process where → is such that
prec(s) a −→ s ′ iff a �= τ and s a −→T s ′ .
Let A ′ = {prec(s) | s ∈A}be the set of success states of RMτ(T ).
Intuitively speaking, RMτ (T )istheτ-free copy of T .
Proposition 9.21. Let C ∈ NP and T ∈T np
τ .
C may T iff C may RMτ(T ),
C must T iff C must RMτ(T ).
Informally speaking, for each path α in C �T there is a corresponding path α ′
in the process C �RMτ (T ) such that the RMτ(T )-part of α ′ is the τ-free copy of
the T -part of α. In our setting, removing τ-transitions in non-probabilistic test
processes does not change the relation ⊑DH . So we apply the set T np instead of
for ⊑DH and the resulting relations coincide.
T np
τ
T np can be decreased to T np,re by ”splitting” a test process in several ”reactive”
ones:
For T =(ST , →T , sT ) ∈Tnp with set of success states A let RMnd (T ) ⊆Tnp,re be the set of all test processes T ′ =(S, →, sT )where
•→is a largest subset of →T with
s a −→ s ′ , s a −→ s ′′ ⇒ s ′ = s ′′ ,
• S is the set of all states in the process (ST , →, sT ) reachable from sT and
•A ′ = A∩S is the set of success states of T ′ .