ComputerAided_Design_Engineering_amp_Manufactur.pdf

home place of the basic process (called ) from which the synthesis begins (called the time-saving
method). We will briefly compare the two after the presentation of the latter method.
The algorithm then assigns a global variable (initialized to 1) to pbh . This variable Hc indicates the
minimum tokens required in pbh for the synthesized net to be live. It also assigns two variables to each
node ni in the net: upper[i] and lower [i] representing ( Rih) and , respectively. Their ratio is
called the least home ratio. Lower[i] indicates the minimum tokens, among all reachable markings,
required in for , if a place, to gain at least one token (actually upper [i]), or, if a transition, to fire
at least one (actually upper[i]) time by firing transitions along any path from to .
Whenever we generate a new arc and a new node at its end, we calculate the upper and lower of the
new node using the following rule.
u
( Rih) l
pbh ni pbh ni Rule New-Ratio
Let [ n1n 2]
be an arc with only two nodes with weight w and the least home ratio of n1 is known, then
n 2
(1) if is a place
n 2
(2) if is a transition
p bh
upper[ 2]
---------------------lower[
2]
� upper[ 1]w
--------------------------,
lower[ 1]
i.e., upper[ 2]
� upper[ 1]�w,
lower[ 2]
� lower[ 1],
and Hc unchanged.
upper[ 2]
1 upper[ 1]
---------------------- � --------------------- ⎛---------------------- ⎞ ,
lower[ 2]
lower[ 1]
⎝ w ⎠
i.e; lower[ 2]
� lower[ 1]�w/gcd,
and
upper[ 2]
� upper[ 1]/gcd,
and Hc � Hc�w/gcd, where gcd is the greatest
common denominatorof upper[ 1]
and w.
(1) is obvious; (2) is to avoid tokens left in the place between and when all tokens in are
consumed and proceed to fire .
When we stop at an , which is in N1 , we require that its least home ratio be primarily equal, but not
necessarily literally smaller (unlike Rule Arr.1.a) than that in N1 . This is to reduce the time to find the
firing ratio between and , which cannot be deferred from the mere least home ratios of and
. As a result, extra time complexity would be incurred to trace all paths between and to find
the firing ratio between them.
By “primarily,” we mean the resulting ratio after the cancellation of the GCD (greatest common
denominator) between upper and lower. By “literally,” we mean we do not perform the above cancellation
of the GCD between upper and lower.
The two ratios must be primarily equal and there are no restrictions on the magnitudes of upper and
lower in N1 n1 n2 Hc pbh n2 nj ng nj ng nj ng nj . They may be smaller or greater than those in the NP. In the former (smaller) case, there is
no need to update all uppers and lowers. In the latter (greater), there are two cases: forward and backward
generations.
For a backward generation, there is no need to update for a TT-generation, since it creates a new home
place as will be discussed later. In the former case of a forward TT-generation plus backward PPgenerations,
the updates at tj are needed for any node n1, nj → n1 must have its upper and lower updated
if its lower is smaller than that of tj . And Hc must be updated also if any updated lower is greater than
the current Hc. Otherwise, it will not be able to get a token or fire at least once if the amount of tokens
in the home place equals the old lower (not updated) at the node. If the lower of the above is smaller
n 1