Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

78 CHAPTER 3.5. GAME CHARACTERIZATION OF BISIMILARITY
We will define attacker’s universal winning strategy from (s, t) and hence show
that s ∼ t.
In the first round the attacker plays on the left-hand side the move s a → s1
and the defender can only answer by t a → t1. The current configuration becomes
(s1, t1). In the second round the attacker plays on the right-hand side according
b
b
to the transition t1 → t1 and the defender can only answer by s1 → s3. The
current configuration becomes (s3, t1). Now the attacker wins by playing again
the transition t1 b → t1 (or t1 b → t2) and the defender loses because s3 .
Exercise 3.37 Consider the following labelled transition system.
a
s
s1
b
s2
b
a
a
t
b
t1
b
t2
a
a
u
a
a
u1 u2
b
b
b
u3
a
v
a
b
v1
v2
b
b
v3
Decide whether s ∼ t, s ∼ u, and s ∼ v. Support your claims by giving a universal
winning strategy either for the attacker (in the negative case) or the defender (in
the positive case). In the positive case, you should also define a strong bisimulation
relating the pair of processes in question.
Exercise 3.38 (For the theoretically minded) Prove Proposition 3.3 on page 75.
Hint: Argue that, using the universal winning strategy for the defender, you can
find a strong bisimulation, and conversely that, given a strong bisimulation, you
can define a universal winning strategy for the defender.
Exercise 3.39 (For the theoretically minded) Recall from Exercise 3.17 that a
binary relation R over the set of states of an LTS is a simulation iff whenever
s1 R s2 and a is an action then
• if s1 a → s ′ 1 , then there is a transition s2 a → s ′ 2 such that s′ 1 R s′ 2 .
A binary relation R over the set of states of an LTS is a 2-nested simulation iff R
is a simulation and moreover R −1 ⊆ R.
b