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

If dual is thought of as an operation instead of a sub-structure, the above three theorems can be
summarized as follows: the dual operation preserves the unmarked, dual and subnet properties.
4.3 Other important Petri net structures
Below mentioned are some important definitions on Petri net structures. The idea behind
grouping them separately is that they are not sub-structures of a Petri net. This section introduces
the concepts of composition of nets which is a super-structure of nets, projection of nets which is
a transformed structure of nets, connectivity and strong connectivity of nets which is a special
structure of nets and synchronic distance which is a special measurement structure for transitions
of nets.
4.3.1 Composition of Petri nets and Composed PN
The composition operation can be thought of set theoretic analogue of union. So the problem is
how to unite two or more Petri nets to get a single PN, called Composed PN.
Given two nets N
1
= ( P1
, T1
, I1,
O1
) and N
2
= ( P2
, T2
, I
2
, O2
) with initial marking m0
and m
1 0
.
2
Let, Θ = { θ1,
θ
2
,..., θ
k
} is a set of simple paths present in both the nets.
Let’s assume, P1 ∩ P2
\ { p | ( ∃θ
∈ Θ)[
p ∈θ
]} = ∅ and T
1
∩ T2
\ { t | ( ∃θ
∈ Θ)[
t ∈θ
]} = ∅.
Let’s also assume that ( ∀ p ∈ P1 ∩ P2
)[ m ( p)
= m0
( p)]
.
01
2
The Concurrent Composition of N1
and N
2
is the net N = ( P,
T,
I,
O)
with initial marking m 0
.
Where, P = P ∪ P T = T ∪ ;
1 2; 1
T2
, ) ( ∃ i ∈{1,2})[
p ∈ Pi
∧ t ∈Ti
I( p,
t)
= I ( p t If ]
i
= 0 otherwise.
O( p,
t)
= O ( p,
t)
If ∃ i ∈{1,2})[
p ∈ P ∧ t ∈T
]
i
(
i
i
= 0 otherwise.
m
0
( p)
= m0
( p)
If p ∈ P
1
1
= m ( 0
p)
otherwise.
2
The composed net N is denoted as N = N || N 1 2
.
4.3.2 Projection of Petri nets and Projected PN
Let N be a composed net: N = N1
|| N
2
|| N
3
|| ... || N
n
and let m be a marking. Let ρ σ be a firing
count vector (will be defined in Chapter 5) and let σ be a firing sequence (will be defined in
Chapter 5) defined onσ ρ .
The projection of m over N , denoted as π (m)
, is the vector obtained from m by removing all
i
the components associated to the places not present in N .
The projection of ρ σ over N , denoted as π (σ
ρ )
i
i
i
, is the vector obtained byσ ρ , removing all the
components associated to transitions not present in N
i
.
The projection of σ over N
i
, denoted as π i
(σ ) , is the firing sequence obtained byσ ,
removing all the transitions not present in N .
It follows that,
(1) N generates the string σ sequencing from m0
to m .
N generates π i
(σ ) sequencing from π m ) to (m)
(2)
i
i
i
i
(
0
π .
i
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