ComputerAided_Design_Engineering_amp_Manufactur.pdf

FIGURE 8.5(a) An example of the application of completeness rule 1. (From Reference 2a. With permission.)
1 2 3 4 5 6
1 EX EX CN CN CN
2 EX EX CN CN CN
3 EX EX CN CN CN
4 CN CN CN EX EX
5 CN CN CN EX EX
6 CN CN CN EX EX
FIGURE 8.5(b) Original T-Matrix (excluding the new IG). LEX (� 2, � 5) � {� 1, � 2, � 3}, LEX (� 5, � 2) � {� 4,
� 5, � 6}. (From Reference 2a. With permission.)
to �k, then Aik � Aki � CN, where CN stands for concurrent. Such a matrix is termed a Temporal-
Matrix (T-Matrix) because the structural relationship between two PSPs resembles their temporal relationship
as explained below; see also Lemma 1.
It is easy to see that if we execute a PN starting from the initial marking involving �1 , then �1 → �2 implies that �1 is executed earlier than �2; �1��2 implies that there is no need to execute �2 to proceed;
�1��2 implies that both of them need to be executed to proceed. Intuitively, �1 ↔ �2 , if they are
subject to an intra-iteration precedence relationship; �1��2, if they can proceed in parallel; and �1��2 if it can return to its initial marking with only one of them executed.
Complementing the prime start node is the prime end node defined as follows:
Definition: Let DEP1 � [ �1n k…n 2npe] and DEP2� [ �2n k�…n 2�n pe]
, where DEP1 � DEP2� { npe} ,
�1 and �2 are two PSPs, and there are no DEPs (called bridges) from a node ( �npe) in DEP1 to a
node ( �npe) of DEP2 and vice versa. npe is called a prime end node.
The definitions of prime (start and end) nodes can be extended to a pair of nodes rather than PSPs.
Note in an SC, the nps and npe of any pair of nodes may not be both transition or both places. In such
cases, however, the corresponding two handles must have bridges across them. An example is shown in
Figure 8.5(a) where nodes t4 and t5 are mutually exclusive since their prime start node is p1 , a place,
but their prime end node is t6 , a transition. The corresponding two PT handles [ p1t 1p3t 4p2t 6]
and
[ p1t 3p4t 5pgt� 1 p� 2 t6] have TP bridges [ t3p 2]
and [ t1p g]
across them. Also note that on the two handles
from the prime start node p1 to the prime end node t6 , there are other prime end nodes p2 and pg ,
respectively. There are no bridges between the two PP handles [ p1t 1p3t 4p2] and [ p1t 3p2] . Such a pair
of nodes and is defined as a pure pair of prime nodes.
p 1
p 2