<s>
For	O
the	O
computer	O
p-System	B-Language
,	O
see	O
UCSD	B-Language
p-System	I-Language
.	O
</s>
<s>
A	O
P	B-Application
system	I-Application
is	O
a	O
computational	O
model	O
in	O
the	O
field	O
of	O
computer	B-General_Concept
science	I-General_Concept
that	O
performs	O
calculations	O
using	O
a	O
biologically	O
inspired	O
process	O
.	O
</s>
<s>
The	O
concept	O
was	O
first	O
introduced	O
in	O
a	O
1998	O
report	O
by	O
the	O
computer	O
scientist	O
Gheorghe	O
Păun	O
,	O
whose	O
last	O
name	O
is	O
the	O
origin	O
of	O
the	O
letter	O
P	B-Language
in	O
'	O
P	B-Application
Systems	I-Application
 '	O
.	O
</s>
<s>
Variations	O
on	O
the	O
P	B-Application
system	I-Application
model	O
led	O
to	O
the	O
formation	O
of	O
a	O
branch	O
of	O
research	O
known	O
as	O
'	O
membrane	O
computing.	O
'	O
</s>
<s>
Although	O
inspired	O
by	O
biology	O
,	O
the	O
primary	O
research	O
interest	O
in	O
P	B-Application
systems	I-Application
is	O
concerned	O
with	O
their	O
use	O
as	O
a	O
computational	O
model	O
,	O
rather	O
than	O
for	O
biological	O
modeling	O
,	O
although	O
this	O
is	O
also	O
being	O
investigated	O
.	O
</s>
<s>
A	O
P	B-Application
system	I-Application
is	O
defined	O
as	O
a	O
series	O
of	O
membranes	O
containing	O
chemicals	O
(	O
in	O
finite	O
quantities	O
)	O
,	O
catalysts	B-Device
and	O
rules	O
which	O
determine	O
possible	O
ways	O
in	O
which	O
chemicals	O
may	O
react	O
with	O
one	O
another	O
to	O
form	O
products	O
.	O
</s>
<s>
Just	O
as	O
in	O
a	O
biological	O
cell	O
,	O
where	O
a	O
chemical	O
reaction	O
may	O
only	O
take	O
place	O
upon	O
the	O
chance	O
event	O
that	O
the	O
required	O
chemical	O
molecules	O
collide	O
and	O
interact	O
(	O
possibly	O
also	O
with	O
a	O
catalyst	B-Device
)	O
,	O
the	O
rules	O
in	O
a	O
P	B-Application
system	I-Application
are	O
applied	O
at	O
random	O
.	O
</s>
<s>
A	O
P	B-Application
system	I-Application
continues	O
until	O
it	O
reaches	O
a	O
state	O
where	O
no	O
further	O
reactions	O
are	O
possible	O
.	O
</s>
<s>
Although	O
many	O
varieties	O
of	O
P	B-Application
system	I-Application
exist	O
,	O
most	O
share	O
the	O
same	O
basic	O
components	O
.	O
</s>
<s>
Each	O
element	O
has	O
a	O
specific	O
role	O
to	O
play	O
,	O
and	O
each	O
has	O
a	O
founding	O
in	O
the	O
biological	O
cell	O
architecture	O
upon	O
which	O
P	B-Application
systems	I-Application
are	O
based	O
.	O
</s>
<s>
The	O
environment	O
is	O
the	O
surroundings	O
of	O
the	O
P	B-Application
system	I-Application
.	O
</s>
<s>
In	O
the	O
initial	O
state	O
of	O
a	O
P	B-Application
system	I-Application
it	O
contains	O
only	O
the	O
container-membrane	O
,	O
and	O
while	O
the	O
environment	O
can	O
never	O
hold	O
rules	O
,	O
it	O
may	O
have	O
objects	O
passed	O
into	O
it	O
during	O
the	O
computation	O
.	O
</s>
<s>
Membranes	O
are	O
the	O
main	O
“	O
structures	O
”	O
within	O
a	O
P	B-Application
system	I-Application
.	O
</s>
<s>
A	O
membrane	O
is	O
a	O
discrete	O
unit	O
which	O
can	O
contain	O
a	O
set	O
of	O
objects	O
(	O
symbols/catalysts	O
)	O
,	O
a	O
set	O
of	O
rules	O
,	O
and	O
a	O
set	O
of	O
other	O
membranes	O
contained	O
within	O
.	O
</s>
<s>
Some	O
P	B-Application
system	I-Application
variants	O
allow	O
for	O
a	O
membrane	O
to	O
divide	O
,	O
possess	O
a	O
charge	O
or	O
have	O
varying	O
permeability	O
by	O
changing	O
membrane	O
thickness	O
.	O
</s>
<s>
In	O
a	O
P	B-Application
system	I-Application
,	O
each	O
type	O
of	O
symbol	O
is	O
typically	O
represented	O
by	O
a	O
different	O
letter	O
.	O
</s>
<s>
Because	O
the	O
multiplicity	O
of	O
symbols	O
in	O
a	O
region	O
matters	O
,	O
multisets	B-Language
are	O
commonly	O
used	O
to	O
represent	O
the	O
symbol	O
content	O
of	O
a	O
region	O
.	O
</s>
<s>
Catalysts	B-Device
are	O
similar	O
to	O
their	O
namesakes	O
in	O
chemistry	O
.	O
</s>
<s>
A	O
rule	O
has	O
a	O
required	O
set	O
of	O
input	O
objects	O
(	O
symbols	O
or	O
catalysts	B-Device
)	O
that	O
must	O
be	O
present	O
in	O
order	O
for	O
it	O
to	O
be	O
applied	O
.	O
</s>
<s>
There	O
are	O
three	O
(	O
in	O
the	O
basic	O
P	B-Application
system	I-Application
model	O
)	O
distinct	O
ways	O
in	O
which	O
a	O
rule	O
may	O
handle	O
its	O
output	O
objects	O
.	O
</s>
<s>
The	O
in	O
modifier	O
causes	O
the	O
object	O
to	O
be	O
passed	O
to	O
one	O
of	O
the	O
current	O
membrane	O
's	O
children	O
(	O
travelling	O
inwards	O
relative	O
to	O
the	O
structure	O
of	O
the	O
P	B-Application
system	I-Application
)	O
,	O
chosen	O
at	O
random	O
during	O
the	O
computation	O
.	O
</s>
<s>
The	O
out	O
modifier	O
causes	O
the	O
object	O
to	O
be	O
passed	O
out	O
of	O
the	O
current	O
membrane	O
and	O
into	O
either	O
its	O
parent	O
membrane	O
or	O
to	O
a	O
sibling	O
membrane	O
,	O
specified	O
during	O
specification	O
of	O
the	O
P	B-Application
system	I-Application
.	O
</s>
<s>
Each	O
step	O
involves	O
iterating	O
through	O
all	O
membranes	O
in	O
the	O
P	B-Application
system	I-Application
and	O
the	O
application	O
of	O
rules	O
,	O
which	O
occurs	O
in	O
both	O
a	O
maximally	O
parallel	O
and	O
non-deterministic	O
manner	O
.	O
</s>
<s>
Most	O
P	B-Application
systems	I-Application
variants	O
are	O
computationally	O
universal	O
.	O
</s>
<s>
This	O
extends	O
even	O
to	O
include	O
variants	O
that	O
do	O
not	O
use	O
rule	O
priorities	O
,	O
usually	O
a	O
fundamental	O
aspect	O
of	O
P	B-Application
systems	I-Application
.	O
</s>
<s>
As	O
a	O
model	O
for	O
computation	O
,	O
P	B-Application
systems	I-Application
offer	O
the	O
attractive	O
possibility	O
of	O
solving	O
NP-complete	O
problems	O
in	O
less-than	O
exponential	O
time	O
.	O
</s>
<s>
Some	O
P	B-Application
system	I-Application
variants	O
are	O
known	O
to	O
be	O
capable	O
of	O
solving	O
the	O
SAT	B-Algorithm
(	O
boolean	B-Algorithm
satisfiability	I-Algorithm
)	O
problem	O
in	O
linear	O
time	O
and	O
,	O
owing	O
to	O
all	O
NP-complete	O
problems	O
being	O
equivalent	O
,	O
this	O
capability	O
then	O
applies	O
to	O
all	O
such	O
problems	O
.	O
</s>
<s>
As	O
there	O
is	O
no	O
current	O
method	O
of	O
directly	O
implementing	O
a	O
P	B-Application
system	I-Application
in	O
its	O
own	O
right	O
,	O
their	O
functionality	O
is	O
instead	O
emulated	O
and	O
therefore	O
solving	O
NP-complete	O
problems	O
in	O
linear	O
time	O
remains	O
theoretical	O
.	O
</s>
<s>
However	O
,	O
it	O
has	O
also	O
been	O
proven	O
that	O
any	O
deterministic	O
P	B-Application
system	I-Application
may	O
be	O
simulated	O
on	O
a	O
Turing	B-Architecture
Machine	I-Architecture
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
image	O
shown	O
depicts	O
the	O
initial	O
state	O
of	O
a	O
P	B-Application
system	I-Application
with	O
three	O
membranes	O
.	O
</s>
<s>
Because	O
of	O
their	O
hierarchical	O
nature	O
,	O
P	B-Application
systems	I-Application
are	O
often	O
depicted	O
graphically	O
with	O
drawings	O
that	O
resemble	O
Venn	B-Application
diagrams	I-Application
or	O
David	O
Harel	O
's	O
Higraph	O
(	O
see	O
Statechart	B-Application
)	O
.	O
</s>
<s>
The	O
outermost	O
membrane	O
,	O
1	O
,	O
is	O
the	O
container	O
membrane	O
for	O
this	O
P	B-Application
system	I-Application
and	O
contains	O
a	O
single	O
out	O
rule	O
.	O
</s>
<s>
Because	O
of	O
the	O
non-deterministic	O
nature	O
of	O
P	B-Application
systems	I-Application
,	O
there	O
are	O
many	O
different	O
paths	O
of	O
computation	O
a	O
single	O
P	B-Application
system	I-Application
is	O
capable	O
of	O
,	O
leading	O
to	O
different	O
results	O
.	O
</s>
<s>
The	O
following	O
is	O
one	O
possible	O
path	O
of	O
computation	O
for	O
the	O
P	B-Application
system	I-Application
depicted	O
.	O
</s>
