Biology Reference
In-Depth Information
Fig. 3. The Petri net from Fig. 2 after transition
t
2
was fired. The new marking is
M
=
(3, 1, 0, 1, 0).
Cases
Some transitions are independent of each other and instead of being fired sequentially, they could be
fired concurrently. Moreover, even if they affect the same places, there are cases when a marking allows
for them being fired concurrently. We will introduce the notion of a case, to formally define this intuitive
notion.
For a given network
N
and its marking
M
, a set of transitions
Definition 7.
(Non-conflicting transitions):
A
=
{
t
1
,
...
,
t
m
}⊂
T
is said to be
non-conflicting
in marking
M
if for each permutation
σ
of
{
1,
...
,
m
}
,
∀
(
M
+
t
σ
(1)
+
...
+
t
σ
(
i
)
)[
t
σ
(
i
+1)
.
1
i<m
Intuitively, non-conflicting transitions can be fired in any order. This is of importance when modeling
chemical reactions, because in real world situations many chemical reactions take place concurrently.
Fact 1.
(Uniqueness):
Let
N
be a net with a marking
M
and
A
be a non-conflicting set, then the
marking
M
reached by sequential firing of transitions from
A
is independent of the order in which these
transitions were fired.
