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mutual exclusion (only one process at a time may access the database for
writing) and choice (an access can either be a read or a write). The detailed
explanation of the net is delayed after the definition of the system dynamics.
According to our PN definition, the model in Fig.
2.1
can be described by
the 8-tuple
M
, where P and T are as given earlier, PAR =
{
K
}
, PRED =
{
K
≥
1
}
, MP associates the parameter K with p
1
, the value 1 with p
5
and the value 0 with all other places. Examples of the input, output, and
inhibition functions are the following: I(t
1
) =
{
p
1
}
, O(t
1
) =
{
p
2
}
, H(t
1
) =
∅
and, for example, I(t
5
) =
{
p
4
,p
5
}
, I(t
5
,p
4
) = 1, O(t
5
) =
{
p
7
}
and H(t
5
) =
{
p
6
}
.
2.2
Models, Systems, and Nets
PN models are characterized by a basic structure and by a parametric ini-
tial marking. Therefore, a PN model describes a set of real systems, whose
precise model can be obtained by assigning a value to each parameter com-
prised in the initial marking. A PN model in which the initial marking is
completely specified is called a PN system, and can be formally defined as
follows:
Definition 2.2.1 A PN system is the 6-tuple
S
=
{
P,T,I,O,H,M
0
}
(2.2)
where
P is the set of places;
T is the set of transitions, T
∩
P =
∅
;
I,O,H : T
→
Bag(P), are the input, output and inhibition functions, re-
spectively, where Bag(P) is set of all possible multisets on P,
M
0
: P
→
IN is the initial marking:, a function that associates with each
place a natural number.
A PN system can be obtained from a PN model by appropriately assign-
ing a value to the parameters of the model. Given a PN model
M
=
{
P,T,I,O,H,PAR,PRED,MP
}
we can define the PN system derived from
model
M
under the substitution function sub : PAR
→
IN as
S
=
{
P
0
,T
0
,I
0
,O
0
,H
0
,M
0
}
(2.3)
where P
0
= P, T
0
= T, I
0
= I, O
0
= O, H
0
= H and M
0
0
is defined as
(
if MP(p)
∈
IN
MP(p)
M
0
0
(p) =
if MP(p)
∈
PAR
sub(MP(p))
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