SEKE 2012 Proceedings - Knowledge Systems Institute

p 1
5
t 5
t 1
p 6
2
p 9
p 2
p 5
t 2
4
t 6
r
p 2
3
1 p 7
t 3
p 10 t 7
r 3 p 8
p 11 2
t 4
p 4
r 1
t 10
4
p 5 p 4
p 16
t 4 p 8 t 5 t 6
p 6
4
p 1
p 3 p 2
t 1
t 2 t 3
p 15
p 14 p 13
p 7
p
t
9 p 7
10
p 11
t 8 p 12
t 9
t 8
Fig. 2 Example weakly 2-compound siphon. 0
= 1 + 2 - 3.
Fig. 1. Example S 3 PR with stronglydependent siphon. η 3 = η 1 + η 2 .
Fig. 2. Example weakly 2-compoundsiphon. η 0 = η 1 + η 2 − η 3 .
TABLE I
TYPES OF SIPHONS FOR THE NET IN FIG. 1
SMS [S] η Set of places c
S 1 {p 2 ,p 7 } [−t 1 + t 2 + t 6 − t 7 ] {p 9 ,p 10 ,p 3 ,p 6 } c 1 =[p 9 t 6 p 10 t 2 p 9 ]
S 2 {p 3 , p 8 } [-t 2 + t 3 + t 7 − t 8 ] {p 10 ,p 11 ,p 4 ,p 7 } c 2 =[p 10 t 7 p 11 t 3 p 10 ]
S 3 {p 2 ,p 3 ,p 7 ,p 8 } [−t 1 + t 3 + t 6 − t 8 ] {p 9 ,p 10 ,p 6 ,p 11 ,p 4 } c 3 = c 1 oc 2
TABLE II
FOUR SMS IN FIG. 2 AND THEIR η. η 4 = η 1 + η 2 − η 3 .
SMS [S] η Set of places c
S 1 {p 2 ,p 3 ,p 8 ,p 9 ,p 10 ,p 11 } [+t 2 − t 4 + t 8 − t 9 ] {p 4 ,p 12 ,p 13 ,p 14 ,p 15 } c 1 =[p 15 t 2 p 14 t 3 p 13 t 8 p 15 ]
S 2 {p 3 ,p 4 ,p 7 ,p 8 ,p 9 ,p 10 } [+t 1 − t 3 + t 7 − t 10 ] {p 5 ,p 11 ,p 14 ,p 15 ,p 16 } c 2 =[p 14 t 4 p 16 t 1 p 15 t 2 p 14 ]
S 3 {p 3 ,p 8 ,p 9 ,p 10 } [+t 2 − t 3 − t 4 + t 7 ] {p 4 ,p 11 ,p 14 ,p 15 } c 3 =[p 15 t 2 p 14 t 6 p 15 ]
S 4 {p 2 , p 3 , p 4 , p 7 , p 8 , p 9 ,
p 10 , p 11 }
[+t 1 + t 8 − t 9 − t 10 ] {p 5 ,p 12 ,p 13 ,p 14 ,p 15 ,p 16 } c 4 = c 1 ⊕ c 2
Thus, we prove the uniform formula for weakly n-
compound siphons. It also holds for strongly dependent
siphons as shown earlier. Thus, the uniform formula holds
irrespective to whether the compound siphon is strongly or
weakly. This further enhances the uniformity of the formula.
A more complicated example in the next section illustrates
this for a weakly 3-compound siphon.
V. EXAMPLE
This section employs Fig.3 to demonstrate Theorem 4. As
shown in Tables III-VII, S, [S] and η for weakly compound
siphons all share the same formula verifying Theorem 4.
For strongly dependent siphons, the same formula also holds
except S ij =[S ij ]=∅ , η ij =0,∀i ≠ j.
VI. CONCLUSION
In summary, we propose a uniform formula to compute
SMS, their complementary set and characteristic T-vectors for
both strongly and weakly n-compound siphons based on the
same underlying physics. We further propose to generalize it
to a compound siphon consisting of n basic siphons. This helps
to retain the formula in brain without consulting the references
and simplify the computation plus its implementation to reduce
the lines of codes. Future work can be directed to large S 3 PR
and more complicated systems.
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