ComputerAided_Design_Engineering_amp_Manufactur.pdf

C 12
Proof: After each � in fires once, there exists a � for one of to gain a token. By the equivalence
of temporal and structural relationship, each � in C21 will gain a token. Hence the lemma.
This lemma ensures that the application of Rule PP.2 will not alter the reachable markings of N . The
following lemma also comes from the equivalence of temporal and structural relationship.
Lemma 3: Using Rule TT.4, if one in gets a token in each of its input places except the one from
, then any subsequent firing sequence, if long enough, will inject a token to and fires the above .
Proof: This lemma comes from the fact that, prior to the generation, d( ) � 1, since ↔ or � .
This lemma, along with Lemma 4, ensures that no tokens get accumulated in the NP indefinitely. Note:
if → , then the NP is marked by Rule TT.2. can fire without the token injected by firing a . But
subsequent firing will always lead to the firing of a to restore .
The above firing sequence, denotd as , including both and and � t � , t � (the set
of transitions in the NP), is called a complete firing sequence of the NP. Lemma 4 shows that after a ,
the marking of in , will be a reachable one in . The correctness of this lemma is based
on the following:
Observation 1: �t � , if , , and
1.
2. � pair of and � , ¬ using Rule TT.4, and ¬ using Rule PP.2.
3. Any TP (PT) generation occurs between exclusive (concurrent) PSPs.
4. Any pair of PSPs ad , joining at a node in the NP, then , if � .
1
tj Xjg n˙j n˙ j tj Xgj, Xjg Xgj Xjg tj tg tj tg w
tg M0 � w
tg tj � w
T w
� w
N 1a N 2a M 1/2
N 1a
T w
t�ngt�n˙j t�n˙g |•t � 1.
t1 t2 T w
( t1 || t2) ( t1 | t2) �1 �2 nk �1� ()� | 2 nk T w P w
( )
Proof: (1) |•t| � 1 only for transition joints within the NP. But this occurs only under Rule PP.2.1
where the joint is an n˙g . (2) Consider Rule TT.4 first. Any pair of new PSPs corresponding to the TPpath
using Rule TT.4.1 are mutually exclusive since they have the same pps as the two corresponding
�g . The same result applies to Rule TT.4.2 since the nps is a place pj . The case for Rule PP.2 can be
proved in a dual fashion. Cases 3 and 4 can be similarly proved.
The observation derives from the fact that there are only TP and PT generations beyond the first
generation using Rules TT.4 and PP.2. From (2) of Observation I, any pair of PSPs �1 and �2 using
Rule TT.4 (PP.2) is eithter �1 ↔ �2 or �1 ()� � 2 . Thus, a complete firing path through the NP is a
single DEP using Rule TT.4 and includes all transitions in the NP using Rule PP.2.
Lemma 4: For the N after applying Rule TT.4 (PP.2), after injecting a token (tokens) into the NP by
firing only one (the ), the subsequent firing sequence will lead to ,
where both and .
Proof: We first show that the injected tokens will eventually disappear from the NP due to the following
facts: 1. There are no internal TP-paths whose and 2. Injected tokens can flow freely inside the
NP since every t inside the NP has only one input place by Observation 1. We then show that upon the
disappearing of the injected tokens from the NP, the resultant contains a � . This is true for
Rule TT.4 since the subfiring sequence in leading to the firings of and is exactly the same as
that in prior to the generation. For Rule PP.2, the token disappearing is equivalent to the switching
of tokens from to , which leads to a submarking by Lemma 2.
Lemma 5: ��, such that .
2
1a w
tg n˙g ( M� , M0 )[� w 1a w
→ ( M� , M0 )
1a
1a 1a
M� M� �R
pj � nj M 2
M 1/2 R 1a
N 1/2
tg tj N 1a
Cgj Cjg M 1/2 �R 1a
2 2 w
[�M0 , either � ( � or � t�� , ¬ ) t�T w
�
( )
M 0
Proof: This lemma comes directly from Lemmas 3 and 4.
Theorem 3: Any PNs resulting from the singular applications of Rules TT.4 and PP.2 to a N synthesized
from a basic process are well behaved.
1a
Proof: Prove by induction. It is easy to see that a basic process is live, bounded, and reversible. Then,
assuming the N i.e., N from the (N-1)th synthesis steps is bounded, live, and reversible, the proof
1a
( )
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