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

3.3. STRONG BISIMILARITY 59
B 2 0
in
B2 1
in
B2
2
out
out
B 1 1 | B1 0
in
in
out
out
B1 0 | B1
0
B 1 1 | B1 1
Figure 3.2: A bisimulation showing B 2 0 ∼ B1 0 | B1 0
in
in
out
B1 0 | B1
1
certainly met when n = 2 because, as you can readily check, the relation depicted
in Figure 3.2 is a bisimulation showing that
B 2 0 ∼ B 1 0 | B 1 0 .
That this holds regardless of the size of the buffer to be implemented is the import
of the following result.
Proposition 3.2 For each natural number n ≥ 1,
B n 0 ∼ B 1 0 | B 1 0 | · · · | B 1 0
n times
Proof: Construct the following binary relation, where i1, i2, . . . , in ∈ {0, 1}:
R = { B n i , B1 i1 | B1 i2 | · · · | B1 in
|
.
out
n
ij = i} .
Intuitively, the above relation relates a buffer of capacity n holding i items with a
parallel composition of n buffers of capacity one, provided that exactly i of them
are full.
It is not hard to see that
• Bn 0 , B1 0 | B1 0 | · · · | B1
0 ∈ R, and
• R is a strong bisimulation.
It follows that
B n 0 ∼ B1 0 | B1 0 | · · · | B1 0 ,
n times
which was to be shown. We encourage you to fill in the details in this proof. ✷
j=1