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

3.5. GAME CHARACTERIZATION OF BISIMILARITY 73
- if s1 α → s ′ 1 , then there is a sequence of transitions s2 τ ⇒ s ′′
2
τ
⇒ s ′ 2 such that s′ 1 ≈ s′ 2 ;
s ′′′
2
- if s2 α → s ′ 2 , then there is a sequence of transitions s1 τ ⇒ s ′′
1
τ
⇒ s ′ 1 such that s′ 1 ≈ s′ 2 .
s ′′′
1
Prove the following claims.
1. The relation ≈ c is an equivalence relation.
2. The relation ≈ c is preserved by the operators of CCS—that is, show that if
P ≈ c Q, then
• α.P ≈ c α.Q, for each action α;
• P + R ≈ c Q + R and R + P ≈ c R + Q, for each process R;
• P | R ≈ c Q | R and R | P ≈ c R | Q, for each process R;
• P [f] ≈ c Q[f], for each relabelling f; and
• P \ L ≈ c Q \ L, for each set of labels L.
3. Argue that ≈ c is included in weak bisimilarity.
4. Find an example of two weakly bisimilar processes that are not related with
respect to ≈ c .
Which of the τ-laws from Exercise 3.26 holds with respect to ≈ c ?
3.5 Game characterization of bisimilarity
We can naturally ask ourselves the following question:
What techniques do we have to show that two states are not bisimilar?
In order to prove that for two given states s and t it is the case that s ∼ t, we should
by Definition 3.2 enumerate all binary relations over the set of states and for each
of them show that if it contains the pair (s, t) then it is not a strong bisimulation.
For the transition system from Example 3.1 on page 44 this translates into investigating
225 different candidates and in general for a transition system with n states
one would have to go through 2n2 different binary relations. (Can you see why?)
In what follows, we will introduce a game characterization of strong bisimilarity,
which will enable us to determine in a much more perspicuous way whether two
states are strongly bisimilar or not.
The idea is that there are two players in the bisimulation game, called ‘attacker’
and ‘defender’. The attacker is trying to show that two given states are not bisimilar
while the defender aims to show the opposite. The formal definition follows.
α
→
α
→