ComputerAided_Design_Engineering_amp_Manufactur.pdf

The correctness of these rules is established in the sequel. We first show the correctness for a synthesized
net where each home place holds eactly one token. Let N denote such a net, the net after adding
tokens to , and and also . We then show that the synthesized net is marking monotonic; that
is, it remains well-behaved by adding tokens to places in the synthesized net.
Rules TT.1, TT.2, and PP.1 do not require further generations. They are simple and have been dealt
with in much of the literature using the concept of reductions and expansions. The rest rules are
established to satisfy the following guidelines for adding NPs.
a
N b
N a
N 1a
N a
N 1
1. No disturbance to : inhibiting any intrusion into normal transition firings of prior to the
generations. This guarantees that neither unbounded places nor dead transitions would occur in
N1 since it was live and bounded prior to the generations.
2. Well-behaved NP: inhibiting any dead transitions and unbounded places in NP.
There are two kinds of intrusions. One is to chnage the marking of ; the other is to eliminate some
reachable markings. In order for NP to be live, it must be able to get enough tokens from pg or by firing
tg . When tokens in NP disappear, the resultant marking of the subnet N1 in N2 must be reachable in
N1 prior to the generation of NP. Furthermore, the marking of NP must be reversible in N2. The second intrusion may occur when the joint is a transition that may not be enabled in case of
backward TT generation, hence, causing some markings in N1 to be unreachable. Thus if the joint is a
transition, NP must be able to get tokens from a generation point within each iteration.
Based on the concept of no intrusion, the rules are constructed according to the following guidelines:
1. From M0, each tg must be potentially able to be fired (always fires or has fired �� in N1 enabling
the joint that is a transition).
2. These tokens must disappear from NP before it gets unbounded tokens, and return to an N1 to
reach a marking M2. The marking of the subnet N1 in N2, M2( N1) of all possible sets of such M2 is identical to those in without the new paths.
N 1
Theorem 1: Any PN resulting from a singular application of Rules TT.1 and PP.1 to an synthesized
from a basic process is well-behaved.
Proof: This theorem can be proved by repetitively applying reduction and expansion. 30,34,36
Theorem 2: Any PN resulting from a singular application of Rules TT.2 to a synthesized from a
basic process is well-behaved.
Proof: This theorem can be proved by repetitively applying reduction and expansion. 30,34,36
N 2
Lemma 1: For an , any pair of PSPs cannot be both concurrent and exclusive to each other.
Proof: If one assumes to the contrary, then there are two , one a t � T and another a p � P. This
can only happen when they are PT or PP generations between concurrent PSPs or TP or TT generations
between exclusive PSPs. These four types of generations are forbidden.
This lemma implies that any time start node of a pair of PSPs must be either transition or place and
cannot be both. Thus, similar to an SC, the two prime nodes of a pure pair must both be transitions or
places. This lemma leads to the equivalence of structural and temporal relationship, which is markingor
time-related, rather than structurally related. Two transitions (places) are temporally concurrent to
each other if they can fire (have tokens) simultaneously in a safe PN. Without this lemma, p1 and p2 may not (may) be able to hold tokens simultaneously, even if p1�(�) p2 since there may be two DEPs
from a pps ( tps) to p1 and p2, respectively.
Lemma 2: For an , let M� � (the R for ), where all �s in C12 are marked, then M� �
2
M� � � ( C12) � � ( C21) is a reachable marking, where � ( C12) (� ( C21) ) denotes that each � in C12 ( )
holds exactly one token.
C 21
N 2
2
R 2
N 2
n ps
N 1
N 1a
N 1
N 1a
2