Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems 261
Now we will consider the relationships between the preorders depicted in Figure
9.14. The numbers of the arrows connecting the preorders are equal to the
numbers in the following enumeration. We have only picked out the most interesting
pairs of testing relations here and not all possible combinations. To the
best of our knowledge the relationship between pairs (2),(4),(6) and (8) have not
been examined before.
Let M1, M2 ∈ ACTMC , P, Q ∈ FPP and ˆ P, ˆ Q ∈ PP and recall that φem maps
an aCTMC to a fully probabilistic process.
(1) ( ⊑SE, ⊑DH ) : Segala states that ⊑ may
SE
[Seg96], i.e.
ˆP ⊑SE ˆ Q =⇒ φnp( ˆ P) ⊑DH φnp( ˆ Q).
This can be seen as follows: First observe that
T np
τ
= {T ′ ∈T np
τ
|∃T ∈T pp
τ
is a ”natural” extension of ⊑may
DH
: φnp(T )=T ′ }
and φnp( ˆ P)�T ′ is isomorphic to φnp( ˆ P�T ) for all T ∈Tpp τ with φnp(T )=
T ′ . Assume ˆ P ⊑ may ˆQ SE and φnp( ˆ P) may T ′ , i.e. there exists a successful
path in φnp( ˆ P)�T ′ . Therefore, there exists a P ′ ∈ fully( ˆ P�T )anda
success state w in T with WP ′(w) > 0. But for every P ′ ∈ fully( ˆ P �T )
there exists a Q ′ ∈ fully( ˆ Q�T )withWP ′(w) � WQ ′(w) (compare Definition
9.16, page 253). Hence, WQ ′(w) > 0 and there exists a successful path
in φnp( ˆ Q�T ).
ˆQ and φnp( ˆ P) must T ′ , i.e. all paths in φnp( ˆ P)�T ′ are
successful. Hence, for all P ′ ∈ fully( ˆ P �T )allpathsinφnp(P ′ ) are successful
Now assume ˆ P ⊑ must
SE
and �
w∈A
WP ′(w) =1whereAis the set of success actions in T . Since it
holds for all Q ′ ∈ fully( ˆ Q�T )thatWP ′(w) � WQ ′(w) forsomeP ′ ,wehave
1= �
�
w∈A
WP ′(w) � w∈A WQ ′(w) = 1. Thus all paths in Q ′ must be
successful.
We can conclude that Segala’s relations are a natural extension to the probabilistic
setting. That the converse of the statement does not hold can be easily
seen with Example 9.8. In Figure 9.9, page 254 we have φnp(P) ⊑DH φnp(Q)
but P �⊑ may
SE Q.
(2) (⊑JY , ⊑SE ) : We have to consider ⊑SE in a more restrictive way. Only τfree
probabilistic processes are now of interest and instead of fully(·) we
take fully {0,1} (·) in Definition 9.16. Furthermore, we do not compare the
success probabilities of each single success state rather than the total success
probability. Let the resulting relations be denoted by � may
SE and �must SE .Itis
easy to see that � may
SE is not equivalent to ⊑JY or ⊑ may
JY . But we have that
�must SE is equivalent to ⊑must JY .Toseethis,considertwoτ-free probabilistic
ˆQ. Thenwehave
processes ˆ P and ˆ Q and T ∈T pp with ˆ P � must
SE
∀ Q ′ ∈ fully {0,1} ( ˆ Q�T ): ∃ P ′ ∈ fully {0,1} ( ˆ P�T ): WP ′ � WQ ′.