SEKE 2012 Proceedings - Knowledge Systems Institute

(r 1 r 2 ...r k ) 1 denotes Path [r 1 t 1 r 2 ...r i t i ...r k−1 t k−1 r k ] 1 for
simple presentation.
t 2 in [r 2 tr 1 ] in Fig. 1 is an internal one of c 3 since η 3 (t 2 )=
0.
III. MOTIVATION
First we observe that (Table I) for a strongly dependent 2-
compound siphon,
ς 3 = ς 1 + ς 2 − ς 12 (1)
where ς 12 is the ς value for the siphon with R(S) =R(S 1 ∩
S 2 ), where ς = S, [S], and η, R(S) is the set of resource
places in S. Next extend this equation to a weakly dependent
2-compound siphon (Table II).
In Fig. 2, there are 3 elementary siphons S 1 -S 3 and 1 weakly
dependent siphon S 4 ; their characteristic T-vectors η are shown
in Table I. Since the number of SMS grows exponentially with
the size of a net, the time complexity of computing η for
elementary siphons is exponential and quite time consuming.
IV. THEORY
In the sequel, we assume that all core circuits extend
between two processes is an S 3 PR. Eq. (1) for 2-compound
siphon will be proved first based on Theorem 2 followed by
the theory for n-compound (n>2) siphons.
We first deal with strongly 2-compound siphons based on
the following lemma.
Lemma 1: Let r ∈ P R , the minimal siphon containing r
is S = ρ(r) ={r} ∪H(r) (also the support of a minimal
P-invariant) with [S] =∅ and η S =0.
Theorem 1: For every compound circuit made of c b1 and
c b2 in an S 3 PR corresponding to an SMS S 0 such that c b1 ∩
c b2 = {r},
1) η 0 = η 1 + η 2 , where r ∈ P R and η 0 is the η value for
S 0
2) η 0 = η 1 + η 2 − η 12 , where η 12 is the characteristic T-
vector of the minimal siphon containing r.
3) [S 0 ]=[S 1 ]+[S 2 ] − [S 12 ].
4) S 0 = S 1 + S 2 − S 12 .
5) ς = ς 1 + ς 2 - ς 12 , where ς = S, [S], η.
This theorem proves Eq. (1) for a strongly 2-compound
siphon. We now deal with weakly 2-compound siphons. We
shows that if S 0 weakly depends on S 1 and S 2 , then there
exists a third siphon S 3 — synthesized from core circuits
formed by c 1 ∩ c 2 , respectively, such that η 0 = η 1 + η 2 − η 3 .
We [5] show that in an S 3 PR, an SMS can be synthesized
from a strongly connected resource subnet and any strongly
dependent siphon corresponds to a compound circuit where the
intersection between any two elementary circuits is at most a
resource place.
Let S 0 be a strongly dependent siphon, S 1 , S 2 ..., and S n
be elementary siphons, with η S0 = η S1 + η S2 + ... + η Sn .We
show in [5] that c 0 (the core circuit from which to synthesize
S 0 ) is a compound resource circuit containing c 1 , c 2 ...,c n
and the intersection between any two c i and c j , i =j-1>0,
is exactly a resource place, where c i (i =0,1,2, ..., n)isthe
core circuit from which to synthesize S i . Thus, if S 0 is a WDS
(weakly dependent siphon), the intersection between any two
c i and c j ,i = j − 1>0 must contain more than one resource
place.
The following theorem from [7] shows that if S 0 weakly
depends on S 1 and S 2 , then η 0 = η 1 + η 2 − η 3 .
Theorem 2: (Theorem 2 in [7]) For every compound circuit
made of c b1 and c b2 in an S 3 PR corresponding to an SMS S∗,
if c 1 ∩c 2 contains a resource path that contains transitions, and
there is a minimal siphon S 12 with R(S 12 )=R(S 1 ∩S 2 ). Then
η 0 = η 1 + η 2 − η 3 , where η 3 is the characteristic T-vector of
S 12 . (Let c i be the core circuit for SMS S i , i =1, 2 and c 1 ∩
c 2 ≠ ∅. Then there is a third core circuit formed by parts of
c 1 and c 2 (c a 1 and c b 2, respectively; i.e., c 3 = c a 1 ∪ c b 2).
The theorem definitely holds for the net in Fig. 2 as shown
in Table II where Γ=c 1 ∩ c 2 =[p 15 t 2 p 14 ] is in Process 1.
Γ 1 =[p 15 t 2 p 14 t 3 p 13 ] 1 plus [p 13 t 8 p 15 ] 2 forms basic circuit c 1 ,
Γ 2 =[p 16 t 1 p 15 t 2 p 14 ] 1 plus [p 14 t 4 p 16 ] 2 form basic circuit c 2 ,
Γ 3 =[p 15 t 2 p 14 ] 1 plus [p 14 t 6 p 15 ] 2 forms basic circuit c 3 .
In general, the resource path of c 1 ∪ c 2 in Process 1
can be expressed as Γ α ΓΓ λ (serial concatenation of Γ α ,
Γ, and Γ λ ) where Γ α Γ belongs to c 2 , ΓΓ λ belongs to
c 1 , Γ α = [r 1 t 1 r 2 ...r i t i r j ...t k−1 r k ] 1 (In Fig. 2, Γ α =
[p 14 t 3 p 13 ] 1 ), Γ λ =[r i t i r j ...t k−1 r k t k r k+1 ...t m−1 r m ] 1 (In
Fig. 2, Γ λ = [p 16 t 1 p 15 ] 1 ), and Γ = [r i t i r j ...t k−1 r k ] 1
(In Fig. 2, Γ=c 1 ∩ c 2 =[p 15 t 2 p 14 ] 1 ). It remains true that
η 0 (t) =η 1 (t)+η 2 (t) − η 3 (t) for every transition in Processes
1 and 2 since each of Γ α , Γ, Γ λ is expanded by adding internal
transitions t with η(t) =0.
Define S 1,2 = S 3 and S 0 = S 1 ⊕ S 2 since R(S 1 ∩ S 2 )=
R(S 3 ). S 1 ⊕ S 2 is similar to S 1 oS 2 in terms of controllability
to be shown later. S 1 ⊕ S 2 is different than S 1 oS 2 in that
R(S 1 ∩ S 2 ) for the former contains more than one resource
place while the latter contains only one resource place.
Definition 2: : Let (N 0 ,M 0 ) be a net system and S 0 =
S 1 ⊕ S 2 denotes the fact that S 0 is a weakly dependent SMS
w.r.t. elementary siphons S 1 ,S 2 , and S 1,2 = S 3 such that
η 0 = η 1 + η 2 − η 3 .
We now propose the following:
Theorem 3: Let S 0 = S 1 ⊕ S 2 as defined in Definition 1,
then
1) [S 0 ]=[S 1 ] ∪ [S 2 ],
2) [S 0 ]=[S 1 ]+[S 2 ] − [S 3 ],
3) [S 1 ] ∩ [S 2 ]=S 1,2 ],
4) S 0 = S 1 + S 2 − [S 12 ,
5) ς = ς 1 + ς 2 - ς 12 , where ς = S, [S], η, and
6) M([S 0 ]) = M([S 1 ]) + M([S 2 ]) − M([S 1,2 ]).
Consider the S 3 PR in Fig. 3(a), [S 0 ]=
{p 2 ,p 3 ,p 7 ,p 8, p 9, p 10 }. Based on Table I, one can verify
that 1) [S 0 ]= [S 1 ] ∪[S 2 ] and 2) [S 1 ] ∩ [S 2 ]= [S 1,2 ]= S 3 .
This theorem confirms the uniform computation for a
weakly 2-compound siphon. The following theorem extends
the uniform computation to a weakly n-compound siphon.
Theorem 4: Let S 0 = S 1 ⊕ S 2 ⊕ ...⊕S n Then
1) η 0 = η 1 + η 2 + ... + η n - η 1,2 - η 2,3 - ...- η n−1,n .
2) [S 0 ]=[S 1 ]+[S 2 ]+...+[S n ]-[S 1,2 ]-[S 2,3 ] - ...-
[S n−1,n ].
3) S 0 = S 1 + S 2 + ... + S n - S 1,2 -S 2,3 - ...- S n−1,n .
4) ς = ς 1 + ς 2 + ... + ς n - ς 1,2 - ς 2,3 - ...- ς n−1,n .
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