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L-type language as example in the following discussion and the finite final marking
set is defined as the subset of its reachable marking set
R
(
M
0
).
Q
T
ↆ
R
(
M
0
)
∧
(
∀
M
e
∈
Q
T
,
∀
p
∈
P
−
P
f
:
M
e
(
p
)=0), where
M
0
is the initial marking,
P
is
the place set of a given Petri net
P
is an appointed place set.
The formal definition of Petri net language is presented in Definition 5.1. Let
T
*
stands for all possible strings of transition
T
including the empty one.
Definition 5.1
[12]
.
Let
ʣ
=(
P,
T
;
F, M
0
), and
P
f
∈
ʣ
=(
P,
T
;
F, M
0
) be a Petri net, and
P
f
∈
P
.
L
(
ʣ
) is the language
of
ʣ
if
L
(
ʣ
)={
ʴ
|
ʴ∈
T
*
∧
M
0
[
ʴ
>
M
e
∧
(
∀
p
∈
P
−
P
f
:
M
e
(
p
)=0)}.
P
f
is named as the end place
set of
ʣ
.
During the following discussions about Petri net language, a Petri net is denoted di-
rectly as
=(
P,
T
;
F, M
0
,
P
f
) where
P
f
is the end place set. The end place set can be
decided by users with different purposes based on the physical system.
The operation of synchronized shuffle of Petri net language is defined and applied
in [13]-[14]. To present the definition of synchronized shuffle operation, several nota-
tions are introduced first.
Definition 5.2
[13]
.
Let
X
be an alphabet. For any
a
ʣ
∈
X
, we denote
#
(
a
,
w
) as the num-
ber of
a
occurring in the word
w
.
Definition 5.3
[13]
.
Let
X
be an alphabet. For any language
L
X
*
, we denote the al-
ↆ
phabet of
L
by
ʴ
(
L
)={
x
∈
X
|
∃
w
∈
L
:
#
(
x
,
w
)
>
0}.
For example,
ʴ
(
a
+
b
)
=
ʴ
(
aa
+
aab
)={
a
,
b
}.
Definition 5.4
[13]
.
Let
X
be an alphabet. We define the projection of
X
over the sub-
alphabet
Y
such that
Y
ↆ
X
, for each
x
∈
X
, if
x
∈
Y
then
ʠ
Y
(
x
)=
x
, otherwise
X
ₒ
ʠ
X
ₒ
Y
(
x
)=
ʵ
.
The definition of synchronized shuffle of two Petri net languages is defined in
Definition 5.5.
Definition 5.5
[13]
.
Let
X
be an alphabet and
L
(
X
*
be two different lan-
ʣ
1
),
L
(
ʣ
2
)
ↆ
guages on
X
. The synchronized shuffle of
L
(
ʣ
1
) and
L
(
ʣ
2
)
,
denoted as
L
(
ʣ
1
)
ʘ
L
(
ʣ
2
), is
defined as:
.
*
L
(
ʣʘ ʣ =
)
L
(
)
{
wXXL
∈
|
=
ʴ
(
(
ʣ
))
∪
ʴ
(
L
(
ʣ
)),
ʠ
(
wL
)
∈ ʣ
(
),
i
∈
{1, 2}}
1
2
1
2
XL
ₒʣ
ʴ
(
(
))
i
i
k
=
ʘʣ
L
()
Similarly, the sharing synthesis of
L
(
ʣ
i
) (
i
∈
{1, 2, 3…
k
}) denoted by
, is
i
i
1
k
k
−
1
defined as:
ʘ ʣ=
L
(
)
((
L
(
ʣʘʣ ʘʣ
)
L
(
))
L
(
))
ʘʣ =ʘ ʣʘʣ
L
(
)
L
(
)
L
(
).
i
1
2
3
k
i
k
i
=
1
i
=
1
According to Definition 5.5, it is easy to obtain that: (1)
bab
ʘ
cac
={
bcabc
,
bcacb,
cbabc, cbacb
}, (2)
abc
ʘ
cba
=
∅
, and (3)
ab
ʘ
a
={
ab
}.
