Information Technology Reference
In-Depth Information
By
S
MS
we denote the set of all multisets over
S
. The nonnegative
integers {
m
(
s
)
j
s
2
S} are the coefficients of the multiset. s
2
m
if
m
(
s
)
0.
Suppose
S
is the set of fruits sold by a supermarket, that is,
6¼
S
¼
{
apple
,
banana
,
orange
}
If there is a shopping bag that contains two apples, one banana,
and three oranges, then this bag can be represented as a multiset over
S
,
that is,
{2
0
apple
,1
0
banana
,3
0
orange
}
S
MS
In
m
1
, the coefficients of apple, banana, and orange are two, one,
and three, respectively. We often write
m
1
m
1
¼
2
¼
2
apple
þ
1
banana
þ
3
orange
Let us assume that there is another shopping bag that contains one
apple and two oranges. Then this bag can be represented as another
multiset over
S
, that is,
m
2
S
MS
In
m
2
the coefficients of
apple
,
banana
, and
orange
are 1, 0, and 2,
respectively.
banana 2
¼
{1
0
apple
,2
0
orange
}
2
m
2
because
m
2
(
banana
)
¼
0.
Here we introduce the formal definition of colored Petri nets
following Reference [82]. We deliberately omit the lengthy introduction
of concepts such as
expression
and
type
. Instead, we briefly explain
them in the remarks right after the definition. For more details on them,
and the firing rule and analysis of CPN, please refer to Reference [82].
Definition 2.6. (Colored Petri Nets)
A colored Petri net is a nine-
tuple C
PN
¼
(
P
,
T
,
F
,
S
,
V
,
C
,
G
,
E
,
I
), where
1.
P
is a finite set of places;
2.
T
is a finite set of transitions,
P
\
T
¼
1
and
P
[
T
6¼
1
;
3.
F
is a finite set of arcs and
F
P
);
4.
S
is a finite set of nonempty types, or color sets;
5.
C
:
P
(
P
T
)
[
(
T
!
S
is a color set function that assigns a color set to each
place;
