<s>
A	O
Petri	B-Operating_System
net	I-Operating_System
,	O
also	O
known	O
as	O
a	O
place/transition	O
(	O
PT	O
)	O
net	O
,	O
is	O
one	O
of	O
several	O
mathematical	O
modeling	B-Language
languages	I-Language
for	O
the	O
description	O
of	O
distributed	B-Architecture
systems	I-Architecture
.	O
</s>
<s>
A	O
Petri	B-Operating_System
net	I-Operating_System
is	O
a	O
directed	O
bipartite	O
graph	O
that	O
has	O
two	O
types	O
of	O
elements	O
:	O
places	O
and	O
transitions	O
.	O
</s>
<s>
Some	O
sources	O
state	O
that	O
Petri	B-Operating_System
nets	I-Operating_System
were	O
invented	O
in	O
August	O
1939	O
by	O
Carl	O
Adam	O
Petri	O
—	O
at	O
the	O
age	O
of	O
13	O
—	O
for	O
the	O
purpose	O
of	O
describing	O
chemical	O
processes	O
.	O
</s>
<s>
Like	O
industry	O
standards	O
such	O
as	O
UML	B-Language
activity	O
diagrams	O
,	O
Business	B-Architecture
Process	I-Architecture
Model	I-Architecture
and	I-Architecture
Notation	I-Architecture
,	O
and	O
event-driven	O
process	O
chains	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
offer	O
a	O
graphical	O
notation	O
for	O
stepwise	O
processes	O
that	O
include	O
choice	O
,	O
iteration	B-Algorithm
,	O
and	O
concurrent	B-Architecture
execution	I-Architecture
.	O
</s>
<s>
Unlike	O
these	O
standards	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
have	O
an	O
exact	O
mathematical	O
definition	O
of	O
their	O
execution	O
semantics	O
,	O
with	O
a	O
well-developed	O
mathematical	O
theory	O
for	O
process	O
analysis	O
.	O
</s>
<s>
The	O
German	O
computer	O
scientist	O
Carl	O
Adam	O
Petri	O
,	O
after	O
whom	O
such	O
structures	O
are	O
named	O
,	O
analyzed	O
Petri	B-Operating_System
nets	I-Operating_System
extensively	O
in	O
his	O
1962	O
dissertation	O
.	O
</s>
<s>
A	O
Petri	B-Operating_System
net	I-Operating_System
consists	O
of	O
places	O
,	O
transitions	O
,	O
and	O
arcs	O
.	O
</s>
<s>
Graphically	O
,	O
places	O
in	O
a	O
Petri	B-Operating_System
net	I-Operating_System
may	O
contain	O
a	O
discrete	O
number	O
of	O
marks	O
called	O
tokens	O
.	O
</s>
<s>
In	O
an	O
abstract	O
sense	O
relating	O
to	O
a	O
Petri	B-Operating_System
net	I-Operating_System
diagram	O
,	O
a	O
transition	O
of	O
a	O
Petri	B-Operating_System
net	I-Operating_System
may	O
fire	O
if	O
it	O
is	O
enabled	O
,	O
i.e.	O
</s>
<s>
a	O
strict	O
ordering	O
of	O
transitions	O
,	O
describing	O
precedence	O
)	O
is	O
defined	O
,	O
the	O
execution	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
is	O
nondeterministic	O
:	O
when	O
multiple	O
transitions	O
are	O
enabled	O
at	O
the	O
same	O
time	O
,	O
they	O
will	O
fire	O
in	O
any	O
order	O
.	O
</s>
<s>
Since	O
firing	O
is	O
nondeterministic	O
,	O
and	O
multiple	O
tokens	O
may	O
be	O
present	O
anywhere	O
in	O
the	O
net	O
(	O
even	O
in	O
the	O
same	O
place	O
)	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
well	O
suited	O
for	O
modeling	O
the	O
concurrent	B-Operating_System
behavior	O
of	O
distributed	B-Architecture
systems	I-Architecture
.	O
</s>
<s>
Petri	B-Operating_System
nets	I-Operating_System
are	O
state-transition	B-Application
systems	I-Application
that	O
extend	O
a	O
class	O
of	O
nets	O
called	O
elementary	B-Operating_System
nets	O
.	O
</s>
<s>
and	O
are	O
disjoint	B-Algorithm
finite	O
sets	O
of	O
places	O
and	O
transitions	O
,	O
respectively	O
.	O
</s>
<s>
M	O
:	O
P	O
→	O
Z	O
is	O
a	O
place	O
multiset	B-Language
,	O
where	O
Z	O
is	O
a	O
countable	O
set	O
.	O
</s>
<s>
M	O
extends	O
the	O
concept	O
of	O
configuration	O
and	O
is	O
commonly	O
described	O
with	O
reference	O
to	O
Petri	B-Operating_System
net	I-Operating_System
diagrams	O
as	O
a	O
marking	O
.	O
</s>
<s>
W	O
:	O
F	O
→	O
Z	O
is	O
an	O
arc	O
multiset	B-Language
,	O
so	O
that	O
the	O
count	O
(	O
or	O
weight	O
)	O
for	O
each	O
arc	O
is	O
a	O
measure	O
of	O
the	O
arc	O
multiplicity	O
.	O
</s>
<s>
If	O
a	O
Petri	B-Operating_System
net	I-Operating_System
is	O
equivalent	O
to	O
an	O
elementary	B-Operating_System
net	O
,	O
then	O
Z	O
can	O
be	O
the	O
countable	O
set	O
 { 0 , 1 } 	O
and	O
those	O
elements	O
in	O
P	O
that	O
map	O
to	O
1	O
under	O
M	O
form	O
a	O
configuration	O
.	O
</s>
<s>
Similarly	O
,	O
if	O
a	O
Petri	B-Operating_System
net	I-Operating_System
is	O
not	O
an	O
elementary	B-Operating_System
net	O
,	O
then	O
the	O
multiset	B-Language
M	O
can	O
be	O
interpreted	O
as	O
representing	O
a	O
non-singleton	O
set	O
of	O
configurations	O
.	O
</s>
<s>
In	O
this	O
respect	O
,	O
M	O
extends	O
the	O
concept	O
of	O
configuration	O
for	O
elementary	B-Operating_System
nets	O
to	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
In	O
the	O
diagram	O
of	O
a	O
Petri	B-Operating_System
net	I-Operating_System
(	O
see	O
top	O
figure	O
right	O
)	O
,	O
places	O
are	O
conventionally	O
depicted	O
with	O
circles	O
,	O
transitions	O
with	O
long	O
narrow	O
rectangles	O
and	O
arcs	O
as	O
one-way	O
arrows	O
that	O
show	O
connections	O
of	O
places	O
to	O
transitions	O
or	O
transitions	O
to	O
places	O
.	O
</s>
<s>
If	O
the	O
diagram	O
were	O
of	O
an	O
elementary	B-Operating_System
net	O
,	O
then	O
those	O
places	O
in	O
a	O
configuration	O
would	O
be	O
conventionally	O
depicted	O
as	O
circles	O
,	O
where	O
each	O
circle	O
encompasses	O
a	O
single	O
dot	O
called	O
a	O
token	O
.	O
</s>
<s>
In	O
the	O
given	O
diagram	O
of	O
a	O
Petri	B-Operating_System
net	I-Operating_System
(	O
see	O
right	O
)	O
,	O
the	O
place	O
circles	O
may	O
encompass	O
more	O
than	O
one	O
token	O
to	O
show	O
the	O
number	O
of	O
times	O
a	O
place	O
appears	O
in	O
a	O
configuration	O
.	O
</s>
<s>
The	O
configuration	O
of	O
tokens	O
distributed	O
over	O
an	O
entire	O
Petri	B-Operating_System
net	I-Operating_System
diagram	O
is	O
called	O
a	O
marking	O
.	O
</s>
<s>
Let	O
PN0	O
(	O
top	O
figure	O
)	O
be	O
a	O
Petri	B-Operating_System
net	I-Operating_System
with	O
a	O
marking	O
configured	O
M0	O
,	O
and	O
PN1	O
(	O
bottom	O
figure	O
)	O
be	O
a	O
Petri	B-Operating_System
net	I-Operating_System
with	O
a	O
marking	O
configured	O
M1	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
firing	O
of	O
transition	O
t	O
generates	O
a	O
map	O
that	O
has	O
the	O
marking	O
configured	O
M1	O
in	O
the	O
image	O
of	O
M0	O
and	O
results	O
in	O
Petri	B-Operating_System
net	I-Operating_System
PN1	O
,	O
seen	O
in	O
the	O
bottom	O
figure	O
.	O
</s>
<s>
The	O
precise	O
meaning	O
of	O
"	O
equal	O
to	O
or	O
greater	O
"	O
will	O
depend	O
on	O
the	O
precise	O
algebraic	O
properties	O
of	O
addition	O
being	O
applied	O
on	O
Z	O
in	O
the	O
firing	O
rule	O
,	O
where	O
subtle	O
variations	O
on	O
the	O
algebraic	O
properties	O
can	O
lead	O
to	O
other	O
classes	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
;	O
for	O
example	O
,	O
algebraic	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
S	O
and	O
T	O
are	O
disjoint	B-Algorithm
,	O
i.e.	O
</s>
<s>
is	O
a	O
multiset	B-Language
of	O
arcs	O
,	O
i.e.	O
</s>
<s>
These	O
texts	O
often	O
define	O
Petri	B-Operating_System
nets	I-Operating_System
using	O
F	O
instead	O
of	O
W	O
.	O
When	O
using	O
this	O
convention	O
,	O
a	O
Petri	B-Operating_System
net	I-Operating_System
graph	O
is	O
a	O
bipartite	O
multigraph	B-Language
with	O
node	O
partitions	O
S	O
and	O
T	O
.	O
</s>
<s>
A	O
marking	O
of	O
a	O
Petri	B-Operating_System
net	I-Operating_System
(	O
graph	O
)	O
is	O
a	O
multiset	B-Language
of	O
its	O
places	O
,	O
i.e.	O
,	O
a	O
mapping	O
.	O
</s>
<s>
is	O
a	O
Petri	B-Operating_System
net	I-Operating_System
graph	O
;	O
</s>
<s>
is	O
the	O
initial	O
marking	O
,	O
a	O
marking	O
of	O
the	O
Petri	B-Operating_System
net	I-Operating_System
graph	O
.	O
</s>
<s>
For	O
a	O
(	O
marked	O
)	O
Petri	B-Operating_System
net	I-Operating_System
,	O
we	O
are	O
interested	O
in	O
the	O
firings	O
that	O
can	O
be	O
performed	O
starting	O
with	O
the	O
initial	O
marking	O
.	O
</s>
<s>
A	O
firing	O
sequence	O
for	O
a	O
Petri	B-Operating_System
net	I-Operating_System
with	O
graph	O
and	O
initial	O
marking	O
is	O
a	O
sequence	O
of	O
transitions	O
such	O
that	O
.	O
</s>
<s>
A	O
common	O
variation	O
is	O
to	O
disallow	O
arc	O
multiplicities	O
and	O
replace	O
the	O
bag	B-Language
of	O
arcs	O
W	O
with	O
a	O
simple	O
set	O
,	O
called	O
the	O
flow	O
relation	O
,	O
.	O
</s>
<s>
This	O
does	O
not	O
limit	O
expressive	B-Language
power	I-Language
as	O
both	O
can	O
represent	O
each	O
other	O
.	O
</s>
<s>
The	O
markings	O
of	O
a	O
Petri	B-Operating_System
net	I-Operating_System
can	O
be	O
regarded	O
as	O
vectors	O
of	O
non-negative	O
integers	O
of	O
length	O
.	O
</s>
<s>
Its	O
transition	O
relation	O
can	O
be	O
described	O
as	O
a	O
pair	O
of	O
by	O
matrices	B-Architecture
:	O
</s>
<s>
One	O
thing	O
that	O
makes	O
Petri	B-Operating_System
nets	I-Operating_System
interesting	O
is	O
that	O
they	O
provide	O
a	O
balance	O
between	O
modeling	O
power	O
and	O
analyzability	O
:	O
many	O
things	O
one	O
would	O
like	O
to	O
know	O
about	O
concurrent	B-Operating_System
systems	I-Operating_System
can	O
be	O
automatically	O
determined	O
for	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
although	O
some	O
of	O
those	O
things	O
are	O
very	O
expensive	O
to	O
determine	O
in	O
the	O
general	O
case	O
.	O
</s>
<s>
Several	O
subclasses	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
have	O
been	O
studied	O
that	O
can	O
still	O
model	O
interesting	O
classes	O
of	O
concurrent	B-Operating_System
systems	I-Operating_System
,	O
while	O
these	O
problems	O
become	O
easier	O
.	O
</s>
<s>
An	O
overview	O
of	O
such	O
decision	O
problems	O
,	O
with	O
decidability	O
and	O
complexity	O
results	O
for	O
Petri	B-Operating_System
nets	I-Operating_System
and	O
some	O
subclasses	O
,	O
can	O
be	O
found	O
in	O
Esparza	O
and	O
Nielsen	O
(	O
1995	O
)	O
.	O
</s>
<s>
The	O
reachability	O
problem	O
for	O
Petri	B-Operating_System
nets	I-Operating_System
is	O
to	O
decide	O
,	O
given	O
a	O
Petri	B-Operating_System
net	I-Operating_System
N	O
and	O
a	O
marking	O
M	O
,	O
whether	O
.	O
</s>
<s>
improved	O
the	O
lower	O
bound	O
and	O
showed	O
that	O
the	O
problem	O
is	O
not	O
ELEMENTARY	B-Operating_System
.	O
</s>
<s>
Petri	B-Operating_System
nets	I-Operating_System
can	O
be	O
described	O
as	O
having	O
different	O
degrees	O
of	O
liveness	O
.	O
</s>
<s>
A	O
place	O
in	O
a	O
Petri	B-Operating_System
net	I-Operating_System
is	O
called	O
k-bound	O
if	O
it	O
does	O
not	O
contain	O
more	O
than	O
k	O
tokens	O
in	O
all	O
reachable	O
markings	O
,	O
including	O
the	O
initial	O
marking	O
;	O
it	O
is	O
said	O
to	O
be	O
safe	O
if	O
it	O
is	O
1-bounded	O
;	O
it	O
is	O
bounded	B-Algorithm
if	O
it	O
is	O
k-bounded	O
for	O
some	O
k	O
.	O
</s>
<s>
A	O
(	O
marked	O
)	O
Petri	B-Operating_System
net	I-Operating_System
is	O
called	O
k-bounded	O
,	O
safe	O
,	O
or	O
bounded	B-Algorithm
when	O
all	O
of	O
its	O
places	O
are	O
.	O
</s>
<s>
A	O
Petri	B-Operating_System
net	I-Operating_System
(	O
graph	O
)	O
is	O
called	O
(	O
structurally	O
)	O
bounded	B-Algorithm
if	O
it	O
is	O
bounded	B-Algorithm
for	O
every	O
possible	O
initial	O
marking	O
.	O
</s>
<s>
A	O
Petri	B-Operating_System
net	I-Operating_System
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
its	O
reachability	O
graph	O
is	O
finite	O
.	O
</s>
<s>
Boundedness	B-Algorithm
is	O
decidable	O
by	O
looking	O
at	O
covering	B-Algorithm
,	O
by	O
constructing	O
the	O
Karp	O
–	O
Miller	O
Tree	O
.	O
</s>
<s>
Some	O
definitions	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
explicitly	O
allow	O
this	O
as	O
a	O
syntactic	O
feature	O
.	O
</s>
<s>
Formally	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
with	O
place	O
capacities	O
can	O
be	O
defined	O
as	O
tuples	B-Application
,	O
where	O
is	O
a	O
Petri	B-Operating_System
net	I-Operating_System
,	O
an	O
assignment	O
of	O
capacities	O
to	O
(	O
some	O
or	O
all	O
)	O
places	O
,	O
and	O
the	O
transition	O
relation	O
is	O
the	O
usual	O
one	O
restricted	O
to	O
the	O
markings	O
in	O
which	O
each	O
place	O
with	O
a	O
capacity	O
has	O
at	O
most	O
that	O
many	O
tokens	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
in	O
the	O
net	O
N	O
,	O
both	O
places	O
are	O
assigned	O
capacity	O
2	O
,	O
we	O
obtain	O
a	O
Petri	B-Operating_System
net	I-Operating_System
with	O
place	O
capacities	O
,	O
say	O
N2	O
;	O
its	O
reachability	O
graph	O
is	O
displayed	O
on	O
the	O
right	O
.	O
</s>
<s>
Alternatively	O
,	O
places	O
can	O
be	O
made	O
bounded	B-Algorithm
by	O
extending	O
the	O
net	O
.	O
</s>
<s>
a	O
place	O
can	O
be	O
made	O
k-bounded	O
by	O
adding	O
a	O
"	O
counter-place	O
"	O
with	O
flow	O
opposite	O
to	O
that	O
of	O
the	O
place	O
,	O
and	O
adding	O
tokens	O
to	O
make	O
the	O
total	O
in	O
both	O
places	O
k	O
.	O
</s>
<s>
As	O
well	O
as	O
for	O
discrete	O
events	O
,	O
there	O
are	O
Petri	B-Operating_System
nets	I-Operating_System
for	O
continuous	O
and	O
hybrid	O
discrete-continuous	O
processes	O
that	O
are	O
useful	O
in	O
discrete	O
,	O
continuous	O
and	O
hybrid	O
control	O
theory	O
,	O
and	O
related	O
to	O
discrete	O
,	O
continuous	O
and	O
hybrid	O
automata	B-Application
.	O
</s>
<s>
There	O
are	O
many	O
extensions	O
to	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
coloured	O
Petri	B-Operating_System
nets	I-Operating_System
)	O
with	O
the	O
original	O
Petri	B-Operating_System
net	I-Operating_System
,	O
some	O
add	O
properties	O
that	O
cannot	O
be	O
modelled	O
in	O
the	O
original	O
Petri	B-Operating_System
net	I-Operating_System
formalism	O
(	O
e.g.	O
</s>
<s>
timed	O
Petri	B-Operating_System
nets	I-Operating_System
)	O
.	O
</s>
<s>
Although	O
backwards-compatible	O
models	O
do	O
not	O
extend	O
the	O
computational	O
power	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
they	O
may	O
have	O
more	O
succinct	O
representations	O
and	O
may	O
be	O
more	O
convenient	O
for	O
modeling	O
.	O
</s>
<s>
Extensions	O
that	O
cannot	O
be	O
transformed	O
into	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
sometimes	O
very	O
powerful	O
,	O
but	O
usually	O
lack	O
the	O
full	O
range	O
of	O
mathematical	O
tools	O
available	O
to	O
analyse	O
ordinary	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
The	O
term	O
high-level	O
Petri	B-Operating_System
net	I-Operating_System
is	O
used	O
for	O
many	O
Petri	B-Operating_System
net	I-Operating_System
formalisms	O
that	O
extend	O
the	O
basic	O
P/T	O
net	O
formalism	O
;	O
this	O
includes	O
coloured	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
hierarchical	O
Petri	B-Operating_System
nets	I-Operating_System
such	O
as	O
Nets	B-Application
within	I-Application
Nets	I-Application
,	O
and	O
all	O
other	O
extensions	O
sketched	O
in	O
this	O
section	O
.	O
</s>
<s>
The	O
term	O
is	O
also	O
used	O
specifically	O
for	O
the	O
type	O
of	O
coloured	O
nets	O
supported	O
by	O
CPN	B-Application
Tools	I-Application
.	O
</s>
<s>
an	O
inhibitor	O
arc	O
imposes	O
the	O
precondition	O
that	O
the	O
transition	O
may	O
only	O
fire	O
when	O
the	O
place	O
is	O
empty	O
;	O
this	O
allows	O
arbitrary	O
computations	O
on	O
numbers	O
of	O
tokens	O
to	O
be	O
expressed	O
,	O
which	O
makes	O
the	O
formalism	O
Turing	B-Algorithm
complete	I-Algorithm
and	O
implies	O
existence	O
of	O
a	O
universal	O
net	O
.	O
</s>
<s>
In	O
a	O
standard	O
Petri	B-Operating_System
net	I-Operating_System
,	O
tokens	O
are	O
indistinguishable	O
.	O
</s>
<s>
In	O
a	O
coloured	O
Petri	B-Operating_System
net	I-Operating_System
,	O
every	O
token	O
has	O
a	O
value	O
.	O
</s>
<s>
In	O
popular	O
tools	O
for	O
coloured	O
Petri	B-Operating_System
nets	I-Operating_System
such	O
as	O
CPN	B-Application
Tools	I-Application
,	O
the	O
values	O
of	O
tokens	O
are	O
typed	O
,	O
and	O
can	O
be	O
tested	O
(	O
using	O
guard	O
expressions	O
)	O
and	O
manipulated	O
with	O
a	O
functional	B-Language
programming	I-Language
language	I-Language
.	O
</s>
<s>
A	O
subsidiary	O
of	O
coloured	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
the	O
well-formed	B-General_Concept
Petri	I-General_Concept
nets	I-General_Concept
,	O
where	O
the	O
arc	O
and	O
guard	O
expressions	O
are	O
restricted	O
to	O
make	O
it	O
easier	O
to	O
analyse	O
the	O
net	O
.	O
</s>
<s>
Another	O
popular	O
extension	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
is	O
hierarchy	O
;	O
this	O
in	O
the	O
form	O
of	O
different	O
views	O
supporting	O
levels	O
of	O
refinement	O
and	O
abstraction	O
was	O
studied	O
by	O
Fehling	O
.	O
</s>
<s>
Another	O
form	O
of	O
hierarchy	O
is	O
found	O
in	O
so-called	O
object	O
Petri	B-Operating_System
nets	I-Operating_System
or	O
object	O
systems	O
where	O
a	O
Petri	B-Operating_System
net	I-Operating_System
can	O
contain	O
Petri	B-Operating_System
nets	I-Operating_System
as	O
its	O
tokens	O
inducing	O
a	O
hierarchy	O
of	O
nested	O
Petri	B-Operating_System
nets	I-Operating_System
that	O
communicate	O
by	O
synchronisation	O
of	O
transitions	O
on	O
different	O
levels	O
.	O
</s>
<s>
See	O
for	O
an	O
informal	O
introduction	O
to	O
object	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
A	O
vector	B-Application
addition	I-Application
system	I-Application
with	I-Application
states	I-Application
(	O
VASS	O
)	O
is	O
an	O
equivalent	O
formalism	O
to	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
However	O
,	O
it	O
can	O
be	O
superficially	O
viewed	O
as	O
a	O
generalisation	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
Consider	O
a	O
finite	B-Architecture
state	I-Architecture
automaton	I-Architecture
where	O
each	O
transition	O
is	O
labelled	O
by	O
a	O
transition	O
from	O
the	O
Petri	B-Operating_System
net	I-Operating_System
.	O
</s>
<s>
The	O
Petri	B-Operating_System
net	I-Operating_System
is	O
then	O
synchronised	O
with	O
the	O
finite	B-Architecture
state	I-Architecture
automaton	I-Architecture
,	O
i.e.	O
,	O
a	O
transition	O
in	O
the	O
automaton	O
is	O
taken	O
at	O
the	O
same	O
time	O
as	O
the	O
corresponding	O
transition	O
in	O
the	O
Petri	B-Operating_System
net	I-Operating_System
.	O
</s>
<s>
It	O
is	O
only	O
possible	O
to	O
take	O
a	O
transition	O
in	O
the	O
automaton	O
if	O
the	O
corresponding	O
transition	O
in	O
the	O
Petri	B-Operating_System
net	I-Operating_System
is	O
enabled	O
,	O
and	O
it	O
is	O
only	O
possible	O
to	O
fire	O
a	O
transition	O
in	O
the	O
Petri	B-Operating_System
net	I-Operating_System
if	O
there	O
is	O
a	O
transition	O
from	O
the	O
current	O
state	O
in	O
the	O
automaton	O
labelled	O
by	O
it	O
.	O
</s>
<s>
Prioritised	B-General_Concept
Petri	I-General_Concept
nets	I-General_Concept
add	O
priorities	O
to	O
transitions	O
,	O
whereby	O
a	O
transition	O
cannot	O
fire	O
,	O
if	O
a	O
higher-priority	O
transition	O
is	O
enabled	O
(	O
i.e.	O
</s>
<s>
For	O
these	O
cases	O
,	O
timed	O
Petri	B-Operating_System
nets	I-Operating_System
have	O
evolved	O
,	O
where	O
there	O
are	O
transitions	O
that	O
are	O
timed	O
,	O
and	O
possibly	O
transitions	O
which	O
are	O
not	O
timed	O
(	O
if	O
there	O
are	O
,	O
transitions	O
that	O
are	O
not	O
timed	O
have	O
a	O
higher	O
priority	O
than	O
timed	O
ones	O
)	O
.	O
</s>
<s>
A	O
subsidiary	O
of	O
timed	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
the	O
stochastic	B-Application
Petri	I-Application
nets	I-Application
that	O
add	O
nondeterministic	O
time	O
through	O
adjustable	O
randomness	O
of	O
the	O
transitions	O
.	O
</s>
<s>
Dualistic	O
Petri	B-Operating_System
Nets	I-Operating_System
(	O
dP-Nets	O
)	O
is	O
a	O
Petri	B-Operating_System
Net	I-Operating_System
extension	O
developed	O
by	O
E	O
.	O
Dawis	O
,	O
et	O
al	O
.	O
</s>
<s>
dP-Nets	O
balance	O
the	O
duality	O
of	O
change/no	O
-change	O
,	O
action/passivity	O
,	O
(	O
transformation	O
)	O
time/space	O
,	O
etc.	O
,	O
between	O
the	O
bipartite	O
Petri	B-Operating_System
Net	I-Operating_System
constructs	O
of	O
transformation	O
and	O
place	O
resulting	O
in	O
the	O
unique	O
characteristic	O
of	O
transformation	O
marking	O
,	O
i.e.	O
,	O
when	O
the	O
transformation	O
is	O
"	O
working	O
"	O
it	O
is	O
marked	O
.	O
</s>
<s>
A	O
zero	O
transformation	O
time	O
used	O
in	O
many	O
typical	O
Petri	B-Operating_System
Nets	I-Operating_System
may	O
be	O
mathematically	O
appealing	O
but	O
impractical	O
in	O
representing	O
real-world	O
processes	O
.	O
</s>
<s>
dP-Nets	O
also	O
exploit	O
the	O
power	O
of	O
Petri	B-Operating_System
Nets	I-Operating_System
 '	O
hierarchical	O
abstraction	O
to	O
depict	O
process	O
architecture	O
.	O
</s>
<s>
There	O
are	O
many	O
more	O
extensions	O
to	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
however	O
,	O
it	O
is	O
important	O
to	O
keep	O
in	O
mind	O
,	O
that	O
as	O
the	O
complexity	O
of	O
the	O
net	O
increases	O
in	O
terms	O
of	O
extended	O
properties	O
,	O
the	O
harder	O
it	O
is	O
to	O
use	O
standard	O
tools	O
to	O
evaluate	O
certain	O
properties	O
of	O
the	O
net	O
.	O
</s>
<s>
Instead	O
of	O
extending	O
the	O
Petri	B-Operating_System
net	I-Operating_System
formalism	O
,	O
we	O
can	O
also	O
look	O
at	O
restricting	O
it	O
,	O
and	O
look	O
at	O
particular	O
types	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
obtained	O
by	O
restricting	O
the	O
syntax	O
in	O
a	O
particular	O
way	O
.	O
</s>
<s>
Ordinary	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
the	O
nets	O
where	O
all	O
arc	O
weights	O
are	O
1	O
.	O
</s>
<s>
Restricting	O
further	O
,	O
the	O
following	O
types	O
of	O
ordinary	O
Petri	B-Operating_System
nets	I-Operating_System
are	O
commonly	O
used	O
and	O
studied	O
:	O
</s>
<s>
In	O
a	O
state	B-Architecture
machine	I-Architecture
(	O
SM	O
)	O
,	O
every	O
transition	O
has	O
one	O
incoming	O
arc	O
,	O
and	O
one	O
outgoing	O
arc	O
,	O
and	O
all	O
markings	O
have	O
exactly	O
one	O
token	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
there	O
can	O
not	O
be	O
concurrency	B-Architecture
,	O
but	O
there	O
can	O
be	O
conflict	O
(	O
i.e.	O
</s>
<s>
nondeterminism	B-Application
)	O
:	O
mathematically	O
,	O
</s>
<s>
This	O
means	O
,	O
that	O
there	O
can	O
not	O
be	O
conflict	O
,	O
but	O
there	O
can	O
be	O
concurrency	B-Architecture
:	O
mathematically	O
,	O
</s>
<s>
there	O
can	O
be	O
both	O
concurrency	B-Architecture
and	O
conflict	O
,	O
but	O
not	O
at	O
the	O
same	O
time	O
:	O
mathematically	O
,	O
</s>
<s>
Extended	O
free	O
choice	O
(	O
EFC	O
)	O
–	O
a	O
Petri	B-Operating_System
net	I-Operating_System
that	O
can	O
be	O
transformed	O
into	O
an	O
FC	O
.	O
</s>
<s>
In	O
an	O
asymmetric	O
choice	O
net	O
(	O
AC	O
)	O
,	O
concurrency	B-Architecture
and	O
conflict	O
(	O
in	O
sum	O
,	O
confusion	O
)	O
may	O
occur	O
,	O
but	O
not	O
symmetrically	O
:	O
mathematically	O
,	O
</s>
<s>
Workflow	B-Operating_System
nets	O
(	O
WF-nets	O
)	O
are	O
a	O
subclass	O
of	O
Petri	B-Operating_System
nets	I-Operating_System
intending	O
to	O
model	O
the	O
workflow	B-Operating_System
of	O
process	O
activities	O
.	O
</s>
<s>
A	O
directed	O
path	O
in	O
the	O
Petri	B-Operating_System
net	I-Operating_System
is	O
defined	O
as	O
the	O
sequence	O
of	O
nodes	O
(	O
places	O
and	O
transitions	O
)	O
linked	O
by	O
the	O
directed	O
arcs	O
.	O
</s>
<s>
An	O
elementary	B-Operating_System
path	O
includes	O
every	O
node	O
in	O
the	O
sequence	O
only	O
once	O
.	O
</s>
<s>
A	O
well-handled	O
Petri	B-Operating_System
net	I-Operating_System
is	O
a	O
net	O
in	O
which	O
there	O
are	O
no	O
fully	O
distinct	O
elementary	B-Operating_System
paths	O
between	O
a	O
place	O
and	O
a	O
transition	O
(	O
or	O
transition	O
and	O
a	O
place	O
)	O
,	O
i.e.	O
,	O
if	O
there	O
are	O
two	O
paths	O
between	O
the	O
pair	O
of	O
nodes	O
then	O
these	O
paths	O
share	O
a	O
node	O
.	O
</s>
<s>
Extended	O
WF-net	O
is	O
a	O
Petri	B-Operating_System
net	I-Operating_System
that	O
is	O
composed	O
of	O
a	O
WF-net	O
with	O
additional	O
transition	O
t	O
(	O
feedback	O
transition	O
)	O
.	O
</s>
<s>
Firing	O
of	O
the	O
transition	O
causes	O
iteration	B-Algorithm
of	O
the	O
process	O
(	O
Note	O
,	O
the	O
extended	O
WF-net	O
is	O
not	O
a	O
WF-net	O
)	O
.	O
</s>
<s>
WRI	O
(	O
Well-handled	O
with	O
Regular	O
Iteration	B-Algorithm
)	O
WF-net	O
,	O
is	O
an	O
extended	O
acyclic	O
well-handled	O
WF-net	O
.	O
</s>
<s>
The	O
DSM-nets	O
are	O
realization	O
of	O
DSM-based	O
plans	O
into	O
workflow	B-Operating_System
processes	O
by	O
Petri	B-Operating_System
nets	I-Operating_System
,	O
and	O
are	O
equivalent	O
to	O
WRI-WF-nets	O
.	O
</s>
<s>
Other	O
ways	O
of	O
modelling	O
concurrent	B-Architecture
computation	I-Architecture
have	O
been	O
proposed	O
,	O
including	O
vector	B-Application
addition	I-Application
systems	I-Application
,	O
communicating	B-Application
finite-state	I-Application
machines	I-Application
,	O
Kahn	B-Application
process	I-Application
networks	I-Application
,	O
process	O
algebra	O
,	O
the	O
actor	B-Application
model	I-Application
,	O
and	O
trace	O
theory	O
.	O
</s>
<s>
Different	O
models	O
provide	O
tradeoffs	O
of	O
concepts	O
such	O
as	O
compositionality	O
,	O
modularity	B-Architecture
,	O
and	O
locality	O
.	O
</s>
<s>
An	O
approach	O
to	O
relating	O
some	O
of	O
these	O
models	O
of	O
concurrency	B-Architecture
is	O
proposed	O
in	O
the	O
chapter	O
by	O
Winskel	O
and	O
Nielsen	O
.	O
</s>
