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The reduction process consists of the following three
steps.
Step 1: ( N , M 1 1)
(Fig. 5.2) is obtained by p-subnet
reduction method. By Theorem 3.1-3.3, ( N , M
0
) is
bounded and live, iff N , ) is bounded and live.
p 11
( M 1 1
p 21
t 31
t 21
t 11
p 23
p 12 p
p 22
13
p 33
t 12
t
t 22
32
p
p 15
p 24
r1 p 31 p 34 p 32
p 14
t 23
t 33
p r2
p 25
t 13
p 26
p t 24
35
p
p 17
p 27
16
t t 25
34 p 29
t 14 p 28
Fig. 5.2 The reduced net
( N , M 1 1)
Step 2: ( N
2
, M 2
) (Fig. 5.3) is obtained by t-subnet
reduction method. By Theorem 4.1-4.3, ( N 1,
M 1)
is
bounded and live, iff N , ) is bounded and live.
t 1 '
p 11
( M 2 2
p r1
p 21
t 3 ' p 24
p r2
t 5 '
p 14
t 2 '
Fig. 5.3
( N , M )
The reduced net 2 2
Step 3: ( N
3,
M
3)
(Fig. 5.4) is obtained by deleted
•
•
places p
r1
and p
r 2
. Since p
r1 = pr1
,
( )
•
•
M
2
p r 1
> 0 and p
r 2
= pr
2
, M ( ) 2
p r 2
> 0 , then
( N , M 2 2
) is bounded and live, iff ( N , 3
M
3)
is
bounded and live.
t 6 '
p
p 21
11
t 4 '
t 1 '
t 3 ' p 24
p 14
t 2 '
p 27
t 4 '
p 27
( N , )
Fig. 5.4 The reduced net 3
M
3
t 26
t 5 '
t 6 '
It is easy to see that ( N
3,
M
3)
is bounded, live ([5]).
By Theorem 3.1-3.3 and Theorem 4.1-4.3, the original
net system N , M ) is bounded and live.
(
0
Ⅵ. APPLICATIONS
In this paper we investigate property preservations of
Petri reduction net. Two Petri net reduction methods are
proposed, which are the key methods to ensure the
reduced net preserving well behaved properties.
ACKNOWLEDGMENT
This work was financially supported by the National
Natural Science Foundation of China under Grant No.
60573012, 60672180 and 60721061, the National Grand
Fundamental Research 973 Program of China under
Grant No. 2002cb312200, and CAS Key Laboratory of
Computer Sciences (SYSKF0903).
REFERENCES
[1] I. Suzuki, T. Murata, A method for stepwise refinement
and abstraction of Petri nets. J. Comput. System Sci. 27
(1983) 51-76.
[2] J. Desel, Reduction and design of well-behaved concurrent
systems, Lecture Notes in Comput. Sci. 458 (1990) 166-
181.
[3] L. Jiao, T. Y. Cheung, W. M. Lu, On liveness and
boundedness of asymmetric choice nets, Theoretical
Computer Science 311 (2004) 165-197.
[4] C. Xia, Reduction rules for Petri net based representation
for embedded systems, 2008 ATPN’s workshop: Protocols
Engineering Based on Petri Nets and Modeling of
Concurrent Systems, (2008) 16-33.
[5] T. Murata. Petri nets: properties, analysis and applications,
Proceedings of the IEEE 77(4):541-580, 1989.
[6] R. H. Sloan, U. Bug, Reduction rules for time Petri nets,
Acta Informatica 33 (1996), 687-706.
[7] M. D. Jeng, F. DeCesare, Synthesis using resource control
nets for modeling shared-resource systems, IEEE Trans,
Robotics Automat. 11(3) (1995) 317-327.
[8] W. M. Mak. Verifying property preservation for
component-based software systems (a Petri-net based
methodology), Ph. D. Thesis, Department of Computer
Science, City University of Hong Kong, June 2001.
[9] J. Esparza. Reduction and synthesis of live and bounded
free choice Petri nets, Inform. Comput. 114(1) (1994) 50-
87.
[10] J. Esparza, C. Schröter, Net reductions for LTL modelchecking,
CHARME 2001, LNCS 2144, pp. 310-324.
[11] Y. Huang, H. Wang, P. Yu, Y. Xia, Property-transitionnet-based
workflow process modeling and verification,
Electronic Notes in Theoretical Computer Science 159
(2006) 155-170.
[12] J. Jiang, X. Zhou, Y. Sun, Component-level reduction rules
for time Petri nets based on DTPN, Journal of Information
and Computing Science, Vol.1 No.1, 2006, pp.37-46.
[13] Victor R. L. Shen, A PN-based approach to the high-level
synthesis of digital systems, INTEGRATION, the VLSI
journal 39 (2006) 182-204.
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