ComputerAided_Design_Engineering_amp_Manufactur.pdf

Theorem 10: If pi → pk in a synthesized net, then � M1 and M2 such that M1( pi) � 0,
M1( pk) � 0,
M2( pk) � 0,
M2( pi) � 0,
and
The arc-ratio rules are summarized as follows.
Arc-Ratio Rules:
Given an NP connecting from node g to node j,
M1( pi) m
---------------- � [ Rik]�. M2( pk) (ARR.1) If � paths in N1 between g and j, then
(ARR.1.a) The weight in NP must be such that � q .
(ARR.1.b) If g is a place, and some output transitions of g do not have a common output place, then
all (except one with arbitrary input ratio) paths between g and j must have input ratio of one.
(ARR.1.c) Same as Rule TT.2, except that the statement in Rule TT.2 should be replaced with “If,
f
without firing tj , there does not exist a � to enable tg( Rjg) times, insert enough tokens
in to enable times.”
l
� w
f
tj( Rjg) u
Possible sets of {g, j, q} are {transition, transition,
place, R }, and {place, transition, }.
}, {transition, place, }, {place,
(ARR.2) Otherwise, there must be another accompanying NP (Rule TT.3) from g�to j� such that
the NP from g, via j and g�, to j� satisfies ARR.1.a.
m
R mf
Example: Figure 8.18 shows a PN with two originally disconnected subnets, entity 1 and entity 2. A
TT-path connects from t1 to t�2 . To synchronize entity 1 and entity 2, another TT-path (Rule ARR.2) is
f
f 2
generated from t3� to t4. Note that R14 � [ R( 14)�]
� -- where subscript (14)� indicates the path from
1
t1 to t4 ( t1p 6 t� 2 p�3 t�3 p7 t4 ) (Rule ARR.1.a).
Figure 8.19 shows that a � [ ] is generated using the TT rule from of to of
and their synchronic distance . We generate another ( ) such that d ,
) � 1. � 2/8 for the path from to , indicating that
gets 4 tokens for every firing of . Thus we connect an arc, from to , with weight 1/
(�4) by Rules TT.4.1 and ARR.1.a. � 4/4 along the only path from to : (
). Thus we connect an arc with weight (� 1), from to , by Rule ARR.1.a.
In Figure 8.19, is generated using the PP rule from of to of . ,
� 4/4 along the only path from to , . � 2/1. We connect a VP
with weight 2 from to (Rules PP.2.2 and ARR.1.a). Based on Rule PP.2.1, we connect a TP-Path
from to with weight 2, since � 1�2.
For GPNs, the synchronic distance between any two transitions is no longer 1 or � as the synthesized
OPNs. Since synchronic distance depends on the marking, the determination of synchronic distance
between all pairs of transitions may take exponential time complexity. Therefore, it is better that we
compute them during the synthesis process.
w t2p 6t3� t2 �1 t� 3 �2.t 2 � t3� d23� � �
� w t�4 p7t 4 ( { �1, �3} fm
{ �2, �4} R56 t5 p6 ( t5 p5 t6 p4 t4 p1 t1 p2 t2 p6), fm
p6 t5 t5 p6 [ R56 ]�
mf
R66� p6 t� 6 p6t� 3 p�4 t� 4 p� 1
mf
t� 1 p� 2 t� 5 p�5 t� 6 [ R66�] p6 t�6 � w ( p2 tg p6 tj p2� ) p2 �1 p� 2 �2 p2 p2� m
mf
R22� p2 p� 2 ( p2 t2 p3 t3 p4 p1 t1� p2� ) R2g p5 tg fm
tj p� 5 Rj5� Updating of Structure Synchronic Distance
The steps for updating the structure synchronic distance, d , is as follows:
1. For each basic process generation, the between it and any other PSP is �.
2. For each new generation, the between any two is one.
s
d s
d s
q NP
� w
lit gj
R f
R fm