Lecture Notes in Computer Science 3472

262 Verena Wolf
Now let Q min be the fully probabilistic process with the smallest success
probability, i.e. W Q min =minQ ′{WQ ′ | Q ′ ∈ fully {0,1} ( ˆ Q�T )}. LetP ′′ be
the process such that the equation above holds with Q ′ = Q min .Thenwe
have
minP ′ WP ′ � WP ′′ � WQmin .
So ˆ P ⊑ must
JY
andomittedhere.
ˆQ. The proof for deriving ˆ P ⊑ must
JY
ˆQ from ˆ P � must
SE
ˆQ is similar
(3) (⊑JY , ⊑DH ) : Jonsson and Yi defined a testing relation which boils down to
ordinary simulation but not to the ordinary testing relation, i.e.
ˆP ⊑JY ˆ Q �=⇒ φnp( ˆ P) ⊑DH φnp( ˆ Q).
It is easy to construct a counterexample where ˆ P ⊑JY ˆ Q and φnp( ˆ P) �⊑ may
DH
φnp( ˆ Q). But ⊑ must
JY
have relationship (1) and (2).
(4) (⊑CH , ⊑DH ):Weshowthat
boils down to ⊑ must
DH
P ⊑CH Q =⇒ φnp(P) ⊑DH φnp(Q).
which is not surprising since we
Recall that for ⊑CH test processes T ∈T np,re are applied (compare Definition
9.14) and with Theorem 9.23 also for ⊑DH . Assume P ⊑CH Q and
φnp(P) may T , i.e. there exists a path α in φnp(P)�T ending up in a success
state w. SinceT ∈T np,re ,wehavethattrace(α) =β corresponds to a single
path in T , i.e. there is only a single path αT in T with trace(αT )=β. Fur-
thermore, all paths α ′ in Q�T with trace(α ′ )=β lead to w. Together with
P ⊑CH Q we derive 0 < Pr trace
P�T
(β) � Pr trace
Q�T
(β). So there is at least one
successful path in Q�T with a nonzero probability and φnp(P) ⊑ may
DH φnp(Q)
follows. With a similar argument we can derive that φnp(P) ⊑ must
DH φnp(Q).
(5) (⊑CL, ⊑DH ):Itholds[CSZ92]
P ⊑CL Q =⇒ φnp(P) ⊑DH φnp(Q).
First recall that φnp(P) isnotequaltoφnp(φpp(P)) in general. So we cannot
derive this statement from (1) and (6). We show that this statement is true
as follows. First observe that P ⊑CL Q iff
WP�T � p =⇒ WQ�T � p, ∀ T ∈Tfp τ , ∀ p ∈ [0, 1].
Furthermore, we have that φnp(P�T ) is isomorphic to φnp(P) � φnp(T )and
T np,re ⊂{T ′ ∈T np
τ
|∃T ∈T fp
τ : φnp(T )=T ′ }.
Moreover, we have that for each T ′ ∈T np
τ there exists at least one test
process T ∈Tfp τ with φnp(T )=T ′ . Hence, if P ⊑CL Q and φnp(P) may T
then there exists a successful path in φnp(P�T )andWP�T (x )=p with p ∈