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(2) In a process (tokens from outside flow into p
i
, pass
N0
and then flow out from p o
), the number of tokens
flowing into pi
is equal to the number of tokens flowing
out from p
o
.
Definition 3.2 p-subnet reduction operation: a reduced
net N ' = ( P',
T '; F',
W ')
is obtained from original Petri
net N = ( P,
T;
F,
W ) by using ~ p to replace a p-
subnet N
p
= ( Pp
, Tp;
Fp,
Wp)
, where
(1) P' = P ∪{ ~ p}
− Pp
,
(2) T ' = T − Tp
,
(3) ~ ) |
•
} {( ~ •
F ' F ∪{(
t,
p t ∈ P ∪ p,
t) | t ∈P
}
=
i
o
•
o
} − Fp
( , M
0
•
−{( t,
pi
) | t∈
pi}
−{(
po,
t)
| t ∈ p .
Definition 3.3 ( N ', M
0'
) obtained from N ) by
p-subnet reduction operation comprises net N'
and
marking M where
'
0
⎪⎧
[ M
M
0'
= ⎨
⎪⎩
[ M
M
(
( P \ Pp
)
( P \ Pp
)0
( P \ Pp
)0
,0]
, n]
M ( p ) = 0
0
M ( p ) = n
0
i
i
( n ≥ 1)
is obtained from M by deleted the vector
corresponding to P
p
).
Definition 3.4 A net ( N , M p po)
is said to be a closed
p-subnet if adding a transition set
~•
T = { t | t corresponding to t ∈ p } and arc set
tp
p
p
{( po,
t
p),(
t
p,
pi
) | t ∈Ttp}
to ( N
p,
M p 0)
, and
preserving the marking of ( N
p,
M p 0)
.
Note that in this section, let
( N ', M
0
') : the original net; N
p
= ( Pp
, Tp;
Fp,
Wp)
:
the p-subnet ; N , ) : the p-subnet system;
(
p
M p 0
( N
P
, M P 0
) : the closed p-subnet system; ( N , M
0
) :
the reduced net.
Theorem 3.1 Suppose that N , M ) is obtained from
(
0
(
0
N ', M ')
by p-subnet reduction operation. Then
( N ', M
0'
) is bounded iff ( N,
M
0)
and ( N , M p p0)
are bounded.
Proof. (1) Since N,
M ) is bound, then ∀ p ∈ P ,
(
0
∃k > 0 1 , such that M ( p)
≤ k1
, ∀ M ∈ R( M
0)
.
Obviously, ∀ p ∈ P −{p
~ } , M
( P \ ~ p
( p)
≤ k
) 1
。 Since
( N , M p p0)
is bound, then ∀ p ∈ P
, ∃k > 2
0 , such
that M P
( p)
≤ k2
, ∀ M
P
∈ R( M P 0)
. Let k = k 1
+ k2
,
by Supposition 3.1, ∀ p ∈ P'
,
M' ( p)
= [ MP
\ ~ p,
MP](
p)
≤ k , ∀ M ' ∈ R(
M
0'
) , so
( N',
M
0')
is bound.
(2) Suppose that ( N,
M
0)
is unbound,
then ∃ p ∈ P , ∀k > 0 , ∃ M ∈ R( M
0)
and M ( p)
> k .
By Supposition 3.1 and Definition 3.1-3.4,
∀k > 0 , ∃M ' ∈ R(
M
0'
) and M '(
p)
> k . This
contradicts with the fact that ( N ', M
0'
) is bounded.
Theorem 3.2 Suppose that N , M ) is obtained from
(
0
(
0
N ', M ')
by p-subnet reduction operation. If
( N ', M
0'
) is live and p i
∈{ p|(
p∈P')
∧(
M0 '( p)
> 0)}
,
then ( N , M
0)
and ( N , M p p0)
are live.
Theorem 3.3 Suppose that N , M ) is obtained from
(
0
(
0
N ', M ')
by p-subnet reduction operation. If
( N , M
0
) and ( N , M p p0)
are live, then ( N ', M ' 0
) is
live.
Ⅳ. THE SECOND REDCTION METHOD
In this section we present the second reduction
operation. This operation preserves boundedness and
liveness.
Definition 4.1 A net N
0
= ( P0
, T0
; F0
, W0
) is said to
be a t-subnet of N = ( P,
T;
F,
W ) iff,
(1) • P0 ∪ P
•
0
⊆ T0
,
(2) N
0
is connected, { ti,
to}
⊆ T0
and t
i
is the only
input transition of N
0
, t o
is the only output transition of
N
0
.
Supposition 4.1 A t-subnet system ( N
0,
M
t 0)
contains
t-subnet N
0
and initial marking M
t 0
, satisfy
(1) In a process (tokens from outside flow into t i
, pass
N
0
and then flow out from t o
), t
o
is fired, iff t i
is
fired.
(2)Before t
i
is fired and after t
o
is fired,
∀ t ∈T0 −{
t i
, to}
, t can not be enabled.
(3) If P
0
dose not contain token in initial state, P
0
dose
not contain token after a process; If P
0
contains token(s)
in initial state, the token(s) will come back to the initial
state after a process.
Definition 4.2 t-subnet reduction operation: a reduced
net N ' = ( P',
T';
F',
W ')
is obtained from original Petri
net N = ( P,
T;
F,
W ) by using ~ t to replace a t-subnet
N
t
= ( Pt
, Tt
; Ft
, Wt
) , where
(1) P' = P − Pt
;
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