SEKE 2012 Proceedings - Knowledge Systems Institute

see the 5 th column of C is deleted in D. So j=5 and C is
5
shown above.
In Step 4, the elements in Y are as follows:
S 2
1 1 0 0 0 1 0 0
0 1 1 1 0 1 1 1
y det( D) 1; y det( D ( C )) 1* 1 5
2
S
0 0 1 1 0 0 1 1
S
:5 :1 2 2
0 0 0 1 1 0 0 1
Similarly, y det( D ( C )) 2 , y det( D ( C )) 1 ,
S2 : 2 2 5
S2 : 3 3 5
T
y det( D ( C )) 1 Y 2 2 1 1 1
.
. Thus
S2 : 4 4 5
In step 5, Y 1Y
2 2 1 1 1
S 2
T
.
S2
S2
In step 6, the elements of Y is as follows:
y( s ) 0,
1
y( s ) 0,
2
y( s ) y 3
( s ) 2, y( s ) y
3 4
( s ) 2,
4
S S 1 1
y( s ) y ( s ) 1, y( s ) y ( s ) 1, y( s ) y
( s ) 1
5 S 5 6 S 6 7 S 7
1 1 1
Thus we get an S-invariant Y
supported by S s , s , s , s , s .
2 3 4
5 6 7
0 0 2 2 1 1 1 T
In Algorithm 2, th e time complexity of Step 1 is
2
O(| S1
| | T |)
[10] 3
. For Step 2, it is O(| S1
| )
[10] ; For Step 3, it is
4
O(| S 1 |) ; For Step 4, it is O(| S1
| )
[10] ; For Step 5-7, the time
complexity is all O(| S |) . So the time complexity of Algorithm
2 4
2 is O(| S | | T | | S | | S | ) .
1 1
VI. CASE STUDY
Taking Petri net and place su bset S
1
2 s , s , s
1 4 5
in
Fig.1 as a case.
2 0 1 2 0 1
r
0 0 0
4
r
1
0 0 0 2 0 1 2 0
r4
r
3
A C D
S 2
0 1 1 0 1 1
0 1 1 0 1
2 1 0 0 0 0
With Algorithm 1, AS 2
can be transformed to a ladder-type
matrix shown above. Obviously, R( A ) 2 S 1
holds.
2
With Gaussian elimination method we get Y 1/2 1 1
T
S 2
which is a solu tion of A Y 0
S2 S2
. The solution is positive, so
S is a minimal support of S-invariants.
2
With Algorithm 2, the parameters C,D are shown above too.
Based on this, we get that C
3 1 1
T
and the elements in
Y
are as follows:
S 2
2 0 1 0
y det( D) 2; y det( D C
) 1;
S2:3 S2:1 1 3
0 1 1 1
S 2
Similarly y det( D C
) 2 .Thus we get
Y 1 2 2
S 2
S 2 :2 2 3
T
and Y
1Y
S2
S
1 2
T
2
2
Eventually we get a S-invariant Y
supported by S s , s , s .
2 1 4 5
1 0 0 2 2 0 T
VII. CONCLUSIONS
In this paper, after a deep research to th e properties of
supports of S-invariants, a sufficient and necessary condition
for a place s ubset to be a m inimal support of S-invariants is
obtained. Based on the condition, two polynomial algorithms
for the decidability of minimal supports of S-invariants and for
the computation of an S-invariant supported by a given
minimal support are presented.
Compared to those existed methods for the computation of
S-invariants, the algorithms presented in this paper have
polynomial time complexity. And based on these algorithms,
we can further give some effective algorithms for the S-
coverability verification of workflow nets, as well as th e
decidability of the existence of S-invariants.
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