ComputerAided_Design_Engineering_amp_Manufactur.pdf

FIGURE 8.9(a) After the interaction-generation (IG), there are 18 PSPs in Petri nets. Each number stands for a
PSP. (From Reference 2a. With permission.)
For case (a), prior to the generation, no path passes through both �r and �q (otherwise �r → �q );
after the generation, there is a path passing through �r , �l , and �a. Hence, Arq � SE; this completes
the proof for case (1). For case (b), note the two paths �g2… �h and �l� j2 …�q intersect at the
generation point of �l ; this might change the relationship between �h and �q . However, it is easy to
see that Ahq � Agj and from the structure definitions of “�”and “�”, we have A�hq � Ahq . Hence, except
for mz � rq, A�mz �A mz where m, z �{r, q, u, v}. Hence case (2) proved.
Case (3) can be proved similarly to cases (1) and (2).
The corresponding time complexity is O n mainly incurred by the operation �SE. To complete
the step of the single application of a TT or PP rule, additional TP- or PT-paths may have to be generated
from/to the NPs. The matrix update after each such generation can be performed in a similar fashion to
that in Lemma 6 with the same time complexity.
Figure 8.9 shows the PN after the IG and the updated T-Matrix (refer to Figure 8.6). Any alphabetic
number in Figure 8.9(a) stands for a PSP. in Figure 8.6 has been separated into two PSPs, and
, respectively. Note this PN does not belong to the class of asymmetric-choice nets; note also that all
pairs of exclusive (concurrent) PSPs have places (transitions) as their and ; i.e., the PN belongs
to the class of synchronized-choice nets. The synthesized PN in Figure 8.5(a) also belogns to the class of
synchronized-choice nets.
2
( ) A�rq �3 �3 �4 nps npe Complexity of the Algorithm
The complexity for the algorithm consists of the follwing components:
1. Determining which rule is applicable and checking the rule violation.
2. Updating matrix entries.