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

80 CHAPTER 3.5. GAME CHARACTERIZATION OF BISIMILARITY
We remind the reader of the fact that, in the weak bisimulation game from the
current configuration (s, t), if the attacker chooses a move under the silent action
τ (let us say s τ → s ′ ) then the defender can (as one possibility) simply answer by
doing ‘nothing’, i.e., by idling in the state t (as we always have t
τ
⇒ t). In that
case, the current configuration becomes (s ′ , t).
Again, the notions of play and universal winning strategy in the weak bisimulation
game are best explained by means of an example.
Example 3.8 Consider the following transition system.
s
a
τ
s1
t1
b
a
a
a
s2 s3 t2
τ
t3
We will show that s ≈ t by defining a universal winning strategy for the attacker
in the weak bisimulation game from (s, t).
In the first round, the attacker selects the left-hand side and action a, and plays
the move s a → s1. The defender has three possible moves to answer: (i) t a ⇒ t2 via
t1, (ii) t a ⇒ t2 via t1 and t3, and (iii) t a ⇒ t3 via t1. In case (i) and (ii) the current
configuration becomes (s1, t2) and in case (iii) it becomes (s1, t3).
From the configuration (s1, t2) the attacker wins by playing s1 b → s3, and the
defender loses because t2 b .
From the configuration (s1, t3) the attacker plays the τ-move from the right-
τ
τ
hand side: t3 → t2. Defender’s only answer from s1 is s1 ⇒ s1 because no τ
actions are enabled from s1. The current configuration becomes (s1, t2) and, as
argued above, the attacker has a winning strategy from this pair.
This concludes the proof and shows that s ≈ t because we found a universal
winning strategy for the attacker.
Exercise 3.41 In the weak bisimulation game the attacker is allowed to use a →
moves for the attacks, and the defender can use a ⇒ moves in response. Argue that
if we modify the rules of the game so that the attacker can also use moves of the
form a ⇒ then this does not provide any additional power for the attacker. Conclude
that both versions of the game provide the same answer about bisimilarity/nonbisimilarity
of two processes.
a
t4
t
b
t5