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

The Reachability Problem is stated as follows. Given a Petri net, given an initial marking m 0
,
given another marking m r
; the question is whether there exists a sequential firing of transitions
which will bring the net from m
0
to m r
. If the answer is 'yes', then mr
is said to be Reachable
from m
0
.The set of all possible markings reachable from m
0
, is called the Reachability Set,
denoted by the symbol R( m
0
) for a given PN. Note that reachability set is defined for a given PN,
for a given initial marking m 0
. This dependency on initial marking clearly reveals that reachability
is a behavioral property.
It may happen that mr
is reached from m
0
by the firing of a single transition, in that case m
r
is
said to be immediately reachable from m 0
. In the general case, m
r
is reached via the sequential
firing of r transitions, called the firing sequence, denoted by = t t ... t
σ ; where r ∈ [ 1, m]
,
m being the total number of transitions present in the PN. This means that firing sequence
σ
r
contains an ordered string of transitions, the length of the string being equal to r . Note that
m
r
j1
j 2
j r
σ
r
m
r
.
the transition string defining firing sequence may contain repetition. Symbolically,
0
The fact that m
r
is reachable from m
0
via σ
r
, is sometimes represented [14] by the notation
m
0
[ σ
r
> m r
. For a given PN, the set of all possible firing sequences from initial marking m
0
is
denoted by N,
m )
Example 5.1
L or simply m )
(
0
L .
(
0
Suppose a PN has total 7 transitions ( m = 7)
. Let there exists a firing sequence of length
4 ( r = 4)
, which brings the PN from an initial marking m
0
(given) to another marking m
4
(given).
This firing sequence which brings m
0
to m
4
is given by σ
4
= t
1t3t2t3
implying
j = , j = 3, j 2 and 3
m . In alternative notation,
1
1
2 3
=
m
0
[ σ
4
> m 4
.
j . Symbolically, 0
4 =
m σ
4 4
σ , there exists an associated ( m ×1)
With every firing sequence r
Firing Count Vector, which is
a column vector (single column, multiple rows),
ρ σ
r
whose elements correspond to number of
times that particular transition has fired in that firing sequenceσ r
. Since this vector keeps a count
of the number of times a particular transition gets fired in a particular firing sequence, hence the
term Firing Count Vector.
Example 5.2
1t
t t
In the previous example, corresponding to the firing sequenceσ 4
=
3 2 3
, there exists an
associated (7 x 1) Firing vector ρ σ
4
, which is a column vector, given by ρ T
σ
4
= ( 1120000) , which
means t 1
has fired only once, t2
once, t 3
twice and t4 , t5,
t6
, t7
never, in that firing sequenceσ 4
.
With the above understanding of firing count vector, now one can formally represent Firing
Count vector as
ρ T
σ r
= ( n
1
n2n3....
n m
) ; where nk
refers to the number of firings oft k
.
A somewhat subtle point is to be noted here. It was mentioned that with every firing sequence
there exists an associated firing count vector. But this is not a one-to-one mapping, in the sense
that, given a firing count vector, there exists more than one firing sequences. The reason of this is
that firing sequence gives very precise information about which transitions are fired, how many
t
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