ComputerAided_Design_Engineering_amp_Manufactur.pdf

Note if the pair of nodes are both output (input) nodes of a prime start (end) node, then the corresponding
prime end (start) node must be the same type as the prime start (end) node; i.e., they must both be
transitions or both be places. For instance, t1 and t3 (p2 and p�2 ) are both output (input) nodes of
p1( t6); their prime end is also a place (transition) p2 or pg ( t1 or t2 or t3) . Thus, in an SC, the two
prime nodes of a pure pair must both be transitions or places.
Definition: If �� , n1�� and n2��, then n1 ↔ n2 . If n1 � �1 and n2 � �2 , �1 � �2 ; the
structural relationship between n1 and n2 is A12 .
Independent, uncoordinated action may cause an N to be unbounded and nonlive. A single generation
may thus necessitate that additional generations maintain coordination and the associated searching of
additional ng and nj as in Rules TT.4 and PP.2 below. This searching motivates the following two
definitions.
Definition: A local exclusive set (LEX) of �i with respect to �k, Xik , is the maximal set of all PSPs that
are exclusive to each other and are equal or exclusive to �i, but not to �k. That is, Xik � LEX ( �i,� k)
� {�z��z � �i or �z��i, ¬ ( �z��k) , ��z1,� z2 � Xik,� z1 ��z2} .
Definition: A local concurrent set (LCN) of �i with respect to �k, Cik, is the maximal set of all PSPs
which are concurrent to each other and are equal or concurrent to �i, but not to �k, i.e., Cik � LCN
( �i,� k)
�{�z��z � �i or �z���i, ¬ ( �z||�k) , ��z1,� z2 � Cik,� z1 || �z2} .
Examples of LEX and LCN appear in Figures 8.5 and 8.6, and Rules TT.4 and PP.2, respectively, employ
them.
The definition of LEX (LCN) between two PSPs can be extended to the LEX (LCN) between two
transitions (places). Thus LEX(ti, tk) [LCN ( pi,p k)
] is a set of transitions (places), instead of PSPs. Let
G(J)denote the set of all �g( �j) involved in a single application of the TT or PP rule. To avoid the
unboundedness problem, it is necessary to have a new directed path from each PSP in Xgj to each PSP
in Xjg . Both Xgj and Xjg can be determined from the T-Matrix even though dgj � �,
which is,
therefore, of no use to the synthesis.
Since synchronic distance depends on the marking, the determination of synchronic distance between
all pairs of transitions may take exponential time complexity. It is easier, however, to determine the
synchronic distance due to the structure of the net; such a synchronic distance is called the structure
synchronic distance d as defined below. Since we construct most new generations based on the net
structure, rather than on the marking, of interest is the structure of synchronic distance, rather than the
synchronic distance itself. We may find the structure synchronic distance of a synthesized net by examining
the synchronic distance of the same net with each home place holding only one token; i.e., by
making the net safe. However, not all such synchronic distances are useful to the synthesis. For instance,
we will observe that when and d � 1, the information of is of no use and we
define that . On the contrary, when d , the TT-generation from to may
cause unbounded places in the TT-path. This problem cannot be detected by checking the T-Matrix only.
The information of is indeed useful here, and the value of is assigned to be . Hence we
have following definition:
s
( )
tg � tj ( Xgj, Xjg) dgj � �
s
dgj � 1
( Xgj, Xjg) � �
tg tj s
dgj � �
dgj �
Definition (Structure Synchronic Distance): The structure synchronic distance between transitions
s
s
and tj is dgj � 1 if d ( Xgj, Xjg) is finite and dgj � � if d ( Xgj, Xjg) � �.
8.3 The Synthesis Rules
We first present possible types of generations followed by the definitions of the rules.
In order for the rules to be complete, all possible generations must be considered. The types of
generations depend on ng and nj in two factors: (1) whether they are transitions or places and (2) their
structural relationship. For factor 1, there are four types, TT, PP, TP, and PT generations, defined as
follows:
t g