A Study of Petri Nets: Modeling, Analysis and Simulation - Tamu.edu

Example 5.6
Suppose it is desired to solve the Equality problem of PN, which says that, whether reachability
sets of given Petri nets are equal or not. Formally, given two Petri nets N1
and N
2
given
by: N
1
= ( P1
, T1
, I1,
O1
, m0
) and N ( , , , , )
1
2
= P2
T2
I
2
O2
m0
; it is required to find
2
whether R ( N1,
m ) = R(
N
2,
m0
) . Another important problem is the Subset Problem, which
01
2
seeks to determine whether R( N1,
m ) ⊆ R(
N
2,
m0
) .Now the equality problem can be
01
2
reduced to two subset problems since to show R ( N1,
m ) = R(
N
2,
m0
) , it is sufficient to show
01
2
that R( N1,
m ) ⊆ R(
N
2
, m0
) and R( N , ) ( , )
01
2
2
m0 ⊆ R N
2
1
m0
. Thus, the Equality Problem is
1
reducible to Subset Problem. This illustrates the concept of reducibility.
In the previous discussion, many types of reachability problems were introduced. Instead of
solving each of them separately, isn't it a nice idea if it can be shown that, at least some of them
are reducible to other problems? The following figure shows which reachability problems are
reducible to which problems.
Reachability Problem
Zero-reachability Problem
Sub-marking reachability Problem
Single-place zero reachability Problem
Figure 5.3: Reducibility among reachability problems [7]
From now on, this report will use the term reachability tree for non-existence of omega case
and coverability tree for the trees where (ω) will appear. But, in a coverability tree, information is
lost through the use of the symbol (omega). Hence, if someone enquires whether a particular
state (marking) is reachable or not, then one may not be able to give any conclusive answer
using coverability tree, since information was lost during introduction of the symbol (ω). Thus the
reachability test from coverability tree is inconclusive.
5.2.1.2 Boundedness
A PN is called k-bounded with respect to an initial marking m 0
, if each place in the net gets at
most k tokens for all markings belonging to the reachability set R ( m0
) , where k is a finite positive
integer.
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