Lecture Notes in Computer Science 3472

258 Verena Wolf
u1
s M ′ 1
(a, r1) (b, r2)
u2
v1
Fig. 9.12. M1 �∼BC M2.
s M ′ 2
(b, 2r2) (a, 2r1)
The example in Figure 9.12 shows that even processes that differ only by expected
duration can be different under ⊑BC . The probability of an a-transition
equals
. The probability of a b-transition is
r1
r1+r2
= 2r1
2(r1+r2) in M ′ 1 and in M ′ 2
also equal for both. Thus, a preorder solely based on these transition probabilities
can never distinguish M ′ 1 and M ′ 2 whereas ∼BC can because the expected
residence time is 1
r1+r2 for sM ′ 1 but 1
2(r1+r2) for sM ′ 2 . Hence M ′ 1 �∼BC M ′ 2.
9.10 Relationships Between Different Testing Relations
In this section we look more closely at the relationship between the preorders
previously presented and also the relationship between the classical (non-probabilistic)
testing relations of De Nicola and Hennessy [dNH84] and the probabilistic
extensions in this chapter. Figure 9.14 on page 265 gives an overview of
the relationships and is discussed at the end of this section.
We start with the introduction of the classical testing relations where nonprobabilistic
test processes are applied to non-probabilistic processes (see also
Chapter 5).
Definition 9.19. The parallel composition of C =(SC , →C , sC ) ∈ NP and
is a non-probabilistic process C �T =((SC × ST ), →, (sC , sT )) with
T ∈Tnp τ
(s, t) a −→ (s ′ , t ′ ) iff (s a −→C s ′ ∧ t a −→T t ′ for a �= τ),
(s, t) τ −→ (s ′ , t) iff s τ −→C s ′ ,
(s, t) τ −→ (s, t ′ ) iff t τ −→T t ′ .
Definition 9.20. [dNH84] Let C1, C2 ∈ NP and T ∈T np
τ
v2
⊓⊔
⊓⊔
with set A of success
states.
• C1 may T iff ∃ α ∈ Path(C1�T ):lstate(α) =(s, t) witht∈A, s ∈ SC1.
• C1 must T iff ∀ α ∈ Path(C1�T ):lstate(α) =(s, t) witht∈A, s ∈ SC1.
• C1 ⊑ may
DH C2 iff ∀ T ∈Tnp τ : C1 may T =⇒ C2 may T .
• C1 ⊑must DH C2 iff ∀ T ∈Tnp τ : C1 must T =⇒ C2 mustT.
• C1 ⊑DH C2 iff C1 ⊑ may
DH C2 and C1 ⊑must DH C2.
• C1 ∼DH C2 iff C1 ⊑DH C2 and C2 ⊑DH C1.
⊓⊔