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

46 CHAPTER 3. BEHAVIOURAL EQUIVALENCES
This completes the proof that R is a strong bisimulation and, since (s, t) ∈ R, we
get that s ∼ t.
In order to prove that, e.g., s1 ∼ s2 we can use the following relation
R = {(s1, s2), (s2, s2)} .
The reader is invited to verify that R is indeed a strong bisimulation.
Example 3.2 In this example we shall demonstrate that it is possible for the initial
state of a labelled transition system with infinitely many reachable states to be
strongly bisimilar to a state from which only finitely many states are reachable.
Consider the labelled transition system (Proc, Act, { α →| α ∈ Act}) where
• Proc = {si | i ≥ 1} ∪ {t},
• Act = {a}, and
• a →= {(si, si+1) | i ≥ 1} ∪ {(t, t)}.
Here is a graphical representation of this labelled transition system.
s1
t
a
a
s2
a
s3
a
s4
We can now observe that s1 ∼ t because the relation
R = {(si, t) | i ≥ 1}
a
. . .
is a strong bisimulation and it contains the pair (s1, t). The reader is invited to
verify this simple fact.
Consider now the two coffee and tea machines in our running example. We can
argue that CTM and CTM ′ are not strongly bisimilar thus. Assume, towards a
contradiction, that CTM and CTM ′ are strongly bisimilar. This means that there is
a strong bisimulation R such that
Recall that
CTM R CTM ′
CTM ′ coin
→ tea.CTM ′
.
.