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

3.4. WEAK BISIMILARITY 65
Our order of business will now be to use the new transition relations presented
above to define a notion of bisimulation that can be used to equate processes that
offer the same observable behaviour despite possibly having very different amounts
of internal computations. The idea underlying the definition of the new notion of
bisimulation is that a transition of a process can now be matched by a sequence of
transitions from the other that has the same ‘observational content’ and leads to a
state that is bisimilar to that reached by the first process.
Definition 3.4 [Weak bisimulation and observational equivalence] A binary relation
R over the set of states of an LTS is a weak bisimulation iff whenever s1 R s2
and α is an action (including τ):
- if s1 α → s ′ 1 , then there is a transition s2 α ⇒ s ′ 2 such that s′ 1 R s′ 2 ;
- if s2 α → s ′ 2 , then there is a transition s1 α ⇒ s ′ 1 such that s′ 1 R s′ 2 .
Two states s and s ′ are observationally equivalent (or weakly bisimilar), written
s ≈ s ′ , iff there is a weak bisimulation that relates them. Henceforth the relation
≈ will be referred to as observational equivalence or weak bisimilarity.
Example 3.4 Let us consider the following labelled transition system.
s
τ
s1
a
s2 t
Obviously s ∼ t. On the other hand s ≈ t because the relation
R = {(s, t), (s1, t), (s2, t1)}
a
t1
is a weak bisimulation such that (s, t) ∈ R. It remains to verify that R is indeed a
weak bisimulation.
• Let us examine all possible transitions from the components of the pair (s, t).
If s τ τ
a
→ s1 then t ⇒ t and (s1, t) ∈ R. If t → t1 then s a ⇒ s2 and
(s2, t1) ∈ R.
• Let us examine all possible transitions from (s1, t). If s1 a → s2 then t a ⇒ t1
and (s2, t1) ∈ R. Similarly if t a → t1 then s1 a ⇒ s2 and again (s2, t1) ∈ R.
• Consider now the pair (s2, t1). Since neither s2 nor t1 can perform any
transition, it is safe to have this pair in R.
Hence we have shown that each pair from R satisfies the conditions given in Definition
3.4, which means that R is a weak bisimulation, as claimed.