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

3.3. STRONG BISIMILARITY 57
Recall that we defined the specification of a counter thus:
Counter0
Countern
def
= up.Counter1
def
= up.Countern+1 + down.Countern−1 (n > 0) .
Moreover, we stated that we expect that process to be ‘behaviourally equivalent’ to
the process C defined by
C def
= up.(C | down.0) .
We can now show that, in fact, C and Counter0 are strongly bisimilar. To this end,
note that this follows if we can show that the relation R below
{(C | Π k i=1 Pi, Countern) | (1) k ≥ 0 ,
(2) Pi = 0 or Pi = down.0, for each i ,
(3) the number of is with Pi = down.0 is n}
is a strong bisimulation. (Can you see why?) The following result states that this
does hold true.
Proposition 3.1 The relation R defined above is a strong bisimulation.
Proof: Assume that
(C | Π k i=1Pi) R Countern .
By the definition of the relation R, each Pi is either 0 or down.0, and the number
of Pi = down.0 is n. We shall now show that
1. if C | Π k i=1 Pi α → P for some action α and process P , then there is some
process Q such that Countern α → Q and P R Q, and
2. if Countern α → Q for some some action α and process Q, then there is some
process P such that C | Π k i=1 Pi α → P and P R Q.
We establish these two claims separately.
1. Assume that C | Π k i=1 Pi α → P for some some action α and process P . Then
• either α = up and P = C | down.0 | Π k i=1 Pi
• or n > 0, α = down and P = C | Πk i=1P ′
i
processes (P1, . . . , Pk) and (P ′ 1 , . . . , P ′ k
, where the vectors of
) differ in exactly one position
ℓ, and at that position Pℓ = down.0 and P ′ ℓ = 0.