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(iii)
A transition in
T
has one and only one input place and at least
two output places.
(iv)
Its initial marking
M
0
satisfies
M
0
ð
p
0
Þ¼
1 and
M
0
ð
p
Þ¼
0,
8
p
2
P
f
p
0
g
.
(v)
When no transition is enabled, the final marking
M
f
satisfies
M
f
ð
P
0
and
M
f
ð
P
0
.
p
Þ¼
1,
8
p
2
p
Þ¼
0,
8
p
2
P
Definition 5.2.
Given that a disassembly Petri net is an ordinary Petri
net (
P
,
T
,
F
,
W
) with
M
0
and
M
f
as their initial and final markings,
respectively, an assembly Petri net is (
P
,
T
,
F
0
,
W
0
) with initial and final
markings
M
0
0
and
M
0
f
satisfying the following conditions:
(i)
F
0
¼
{(
y
,
x
)
j
(
x
,
y
)
2
F
},
(ii)
W
0
(
x
,
y
)
¼
W
(
y
,
x
), and
(iii)
M
0
0
M
f
and
M
0
f
¼
¼
M
0
.
Postset of
t
is the set of output places of
t
, denoted
t
. Preset of
t
is
the set of input places of
t
, denoted
t
. Post (Pre) set of
p
is the set of
output (input) transitions of
p
, denoted
p
and
p
, respectively. In
Figure 5.2,
t
1
¼f
p
5
Þ
¼
p
2
4
g
,
t
1
¼f
p
1
g
,
p
5
¼f
t
2
[
t
2
3
g
, and
ð
t
3
¼f
p
6
7
g
. A place is called a leaf place if
p
¼
1
. In Figure 5.2,
p
2
4
and
p
6
7
are leaf places. The net's execution rules, as discussed in
Chapter 2, are applicable to assembly and disassembly Petri nets.
A circuit in a Petri net is a sequence of nodes
x
1
,
x
2
,...,and
x
n
if
x
i
{
x
1
}. An acyclic Petri
net is a net such that it contains no circuit in it.
¼
{
x
iþ
1
},
i
¼
1, 2, . . . , and
n
1, and
x
n
¼
Definition 5.3.
is a
finite set of real Web services, and
s
dummy
is dummy service that denotes
a composition or selection of real Web services. We call a Web service
an atomic Web service if it does not reference other services.
Service set
S
¼
WS
[f
s
dummy
g
where
WS
¼f
s
1
z
g
Definition5.4.
Service configuration net (SC-net): An acyclic Petri net is
an SC-net if
there is a place denoted
p
0
with no input arc, which corre-
sponds to the service requested by a user.
M
0
(i)
p
0
Þ¼
ð
1 and
p
0
;
M
0
ð
p
Þ¼
0,
8
p
6¼
