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Theorem 5.1. Let W i =( P i , T i ; F i , M 0 i , P fi ) ( i
{1, 2, 3… k }) be the WF-net of a
workflow process and W CW =( P , T ; F , M 0 , P f ) is their corresponding cross-organization
workflow net with task synchronization pattern. We have L ( W CW )=
ʘ 1≤ i k L ( W i ).
Proof. Let M f
R ( M 0 ), and
M
M f such that
p
P
P f , M ( p )=0, M fi = {
ʓ
Pi | M
M f },
P
L ( W CW )={
M f )}.
Next, we prove this theorem by induction on |
ʴ
|
ʴ∈
T *
M 0 [
ʴ
> M
( M
ʴ
|.
(1) If |
ʴ
|=1,
ʴ i =
ʠ T Ti (
ʴ
)=
ʴ
if
ʴ∈
T i and
ʴ i =
ʠ T Ti (
ʴ
)=
ʵ
otherwise, ( i
{1, 2, 3… k }).
With
ʴ∈
L ( W CW ) iff M 0 [
ʴ
> M 1
M 1
M f , and iff (
ʓ
Pi ( M 0 )[
ʴ
i >
ʓ
Pi ( M 1 ))
P
P
(
ʓ P Pi ( M 1 )
{1, 2, 3… k }).
(2) Suppose the conclusion is correct when|
M fi ),
ʴ i
L ( W i ) ( i
ʴ
|= n . In the following, we prove that
the conclusion is also correct when|
ʴ
|= n +1.
Let
ʴ
=
ʴ′•
t
, where t
is the ( n +1) th element of
ʴ
and |
ʴ′
|= n .
Based on
ʴ∈
L ( W CW ) iff ( M 0 [
ʴ′•
t
> M n +1 )
( M n +1
M f ),
M 0 [
ʴ′
> M n [ t
> M n +1 iff
M n
R ( M 0 ); and
M f
={ M n |( M 0 [
ʴ′
> M n [ t
> M n +1 )
( M n +1
M f )}, M fi
={
ʔ P Pi M n | M n
M f
} ( i
{1, 2, 3… k }).
According to the supposition,
∃ʴ
=
ʠ
Ti (
ʴ′
) and
ʴ
′∈
L ( W i ) such that
i
T
i
ʴ i =
ʠ T Ti (
ʴ
)=
ʴ′•
t
if t
′∈
T i and
ʴ i =
ʠ T Ti (
ʴ
)=
ʴ′
otherwise; and
ʴ i
L ( W i ) iff M 0 i [
ʴ i
> M i
M i ′∈
M fi
; iff M 0 i [
ʴ i > M ( n +1) i
M ( n +1) i
M fi , such that
ʴ i
L ( W i ) and
ʴ i =
ʠ T Ti (
ʴ
).
Therefore, the theorem is proved.
According to Theorem 5.1, we know that L ( W CW )=
k L ( W i ) is the language ex-
pression of the cross-organization workflow with task synchronization pattern if the
language expression L ( W i ) ( i
ʘ
1
i
{1, 2, 3… k }) of each WF-net can be obtained.
Take the CWF-net of the medical diagnosis cross-organization workflow with syn-
chronous collaboration patterns in Fig. 3 as an example, its language behaviors can be
obtained with the above-mentioned approach. Assume that its end place set is defined
as P f ={ O 1 , O 2 }. Then the behavior description of the W CW in Fig. 3 can be obtained
with the following steps:
(1) The language expression of W S and W C as shown in Figs. 1-2 are easy to
present, which are formally expressed as L ( W S )= t 1 t 2 t 3 t 4 ( t 6 // t 5 t 9 t 10 ) t 12 and
L ( W C )= t 7 t 8 t 9 t 10 t 11 ; and
(2) Based on the Theorem 5.1, the behaviors for the cross-organization medical di-
agnosis workflow with task synchronization pattern in Fig. 3 can be expressed as
L ( W CW )= L ( W S )
ʘ
L ( W C ), i.e. L ( W CW )= ( t 1 t 2 t 3 t 4 ( t 6 // t 5 t 9 t 10 ) t 12 )
ʘ
( t 7 t 8 t 9 t 10 t 11 ).
6
Conclusion and Future Work
This paper presents the behavior description approach for a kind cross-organizational
workflow with task synchronization pattern using Petri net language, which mainly
includes the following three steps: (1) modeling the cross-organizational workflow