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

98 CHAPTER 4. THEORY OF FIXED POINTS
We calculate the sequence F i (Proc × Proc) for i ≥ 1 thus:
F 1 (Proc × Proc) = {(Q1, Q4), (Q4, Q1), (Q2, Q3), (Q3, Q2)} ∪ I
F 2 (Proc × Proc) = {(Q2, Q3), (Q3, Q2)} ∪ I and finally
F 3 (Proc × Proc) = F 2 (Proc × Proc) .
Therefore, the only distinct processes that are related by the largest strong bisimulation
over this labelled transition system are Q2 and Q3, and indeed Q2 ∼ Q3.
Exercise 4.11 Using the iterative algorithm described above, compute the largest
strong bisimulation over the labelled transition system described by the following
defining equations in CCS:
P1 = a.P2
P2 = a.P1
P3 = a.P2 + a.P4
P4 = a.P3 + a.P5
P5 = 0 .
You may find it useful to draw the labelled transition system associated with the
above CCS definition first.
Exercise 4.12 Use the iterative algorithm described above to compute the largest
bisimulation over the labelled transition system in Example 3.7.
Exercise 4.13 What is the worst case complexity of the algorithm outlined above
when run on a labelled transition system consisting of n states and m transitions?
Express your answer using O-notation, and compare it with the complexity of the
algorithm due to Page and Tarjan mentioned in Section 3.6.
Exercise 4.14 Let (Proc, Act, { α → | α ∈ Act}) be a labelled transition system.
For each i ≥ 0, define the relation ∼i over Proc as follows:
• s1 ∼0 s2 holds always;
• s1 ∼i+1 s2 holds iff for each action α:
- if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 ∼i s ′ 2 ;
- if s2 α → s ′ 2 , then there is a transition s1 α → s ′ 1 such that s′ 1 ∼i s ′ 2 .