<s>
In	O
theoretical	O
computer	O
science	O
a	O
bisimulation	B-Application
is	O
a	O
binary	O
relation	B-Algorithm
between	O
state	B-Application
transition	I-Application
systems	I-Application
,	O
associating	O
systems	O
that	O
behave	O
in	O
the	O
same	O
way	O
in	O
that	O
one	O
system	O
simulates	O
the	O
other	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Intuitively	O
two	O
systems	O
are	O
bisimilar	B-Application
if	O
they	O
,	O
assuming	O
we	O
view	O
them	O
as	O
playing	O
a	O
game	O
according	O
to	O
some	O
rules	O
,	O
match	O
each	O
other	O
's	O
moves	O
.	O
</s>
<s>
Given	O
a	O
labeled	B-Application
state	I-Application
transition	I-Application
system	I-Application
(	O
,	O
,	O
)	O
,	O
</s>
<s>
a	O
bisimulation	B-Application
is	O
a	O
binary	O
relation	B-Algorithm
,	O
</s>
<s>
From	O
this	O
follows	O
that	O
the	O
symmetric	O
closure	O
of	O
a	O
bisimulation	B-Application
is	O
a	O
bisimulation	B-Application
,	O
and	O
that	O
each	O
symmetric	O
simulation	B-Application
is	O
a	O
bisimulation	B-Application
.	O
</s>
<s>
Thus	O
some	O
authors	O
define	O
bisimulation	B-Application
as	O
a	O
symmetric	O
simulation	B-Application
.	O
</s>
<s>
Equivalently	O
,	O
is	O
a	O
bisimulation	B-Application
if	O
and	O
only	O
if	O
for	O
every	O
pair	O
of	O
states	O
in	O
and	O
all	O
labels	O
α	O
in	O
:	O
</s>
<s>
Given	O
two	O
states	O
and	O
in	O
,	O
is	O
bisimilar	B-Application
to	O
,	O
written	O
,	O
if	O
and	O
only	O
if	O
there	O
is	O
a	O
bisimulation	B-Application
such	O
that	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
bisimilarity	B-Application
relation	B-Algorithm
is	O
the	O
union	O
of	O
all	O
bisimulations	B-Application
:	O
precisely	O
when	O
for	O
some	O
bisimulation	B-Application
.	O
</s>
<s>
The	O
set	O
of	O
bisimulations	B-Application
is	O
closed	O
under	O
union	O
;	O
therefore	O
,	O
the	O
bisimilarity	B-Application
relation	B-Algorithm
is	O
itself	O
a	O
bisimulation	B-Application
.	O
</s>
<s>
Since	O
it	O
is	O
the	O
union	O
of	O
all	O
bisimulations	B-Application
,	O
it	O
is	O
the	O
unique	O
largest	O
bisimulation	B-Application
.	O
</s>
<s>
Bisimulations	B-Application
are	O
also	O
closed	O
under	O
reflexive	O
,	O
symmetric	O
,	O
and	O
transitive	O
closure	O
;	O
therefore	O
,	O
the	O
largest	O
bisimulation	B-Application
must	O
be	O
reflexive	O
,	O
symmetric	O
,	O
and	O
transitive	O
.	O
</s>
<s>
From	O
this	O
follows	O
that	O
the	O
largest	O
bisimulation	B-Application
—	O
bisimilarity	B-Application
—	O
is	O
an	O
equivalence	O
relation	B-Algorithm
.	O
</s>
<s>
Bisimulation	B-Application
can	O
be	O
defined	O
in	O
terms	O
of	O
composition	O
of	O
relations	O
as	O
follows	O
.	O
</s>
<s>
From	O
the	O
monotonicity	O
and	O
continuity	O
of	O
relation	B-Algorithm
composition	O
,	O
it	O
follows	O
immediately	O
that	O
the	O
set	O
of	O
bisimulations	B-Application
is	O
closed	O
under	O
unions	O
(	O
joins	O
in	O
the	O
poset	O
of	O
relations	O
)	O
,	O
and	O
a	O
simple	O
algebraic	O
calculation	O
shows	O
that	O
the	O
relation	B-Algorithm
of	O
bisimilarity	B-Application
—	O
the	O
join	O
of	O
all	O
bisimulations	B-Application
—	O
is	O
an	O
equivalence	O
relation	B-Algorithm
.	O
</s>
<s>
This	O
definition	O
,	O
and	O
the	O
associated	O
treatment	O
of	O
bisimilarity	B-Application
,	O
can	O
be	O
interpreted	O
in	O
any	O
involutive	O
quantale	O
.	O
</s>
<s>
Bisimilarity	B-Application
can	O
also	O
be	O
defined	O
in	O
order-theoretical	O
fashion	O
,	O
in	O
terms	O
of	O
fixpoint	O
theory	O
,	O
more	O
precisely	O
as	O
the	O
greatest	O
fixed	O
point	O
of	O
a	O
certain	O
function	O
defined	O
below	O
.	O
</s>
<s>
Given	O
a	O
labelled	B-Application
state	I-Application
transition	I-Application
system	I-Application
(	O
,	O
,	O
)	O
,	O
define	O
to	O
be	O
a	O
function	O
from	O
binary	O
relations	O
over	O
to	O
binary	O
relations	O
over	O
,	O
as	O
follows	O
:	O
</s>
<s>
Let	O
be	O
any	O
binary	O
relation	B-Algorithm
over	O
.	O
</s>
<s>
Bisimilarity	B-Application
is	O
then	O
defined	O
to	O
be	O
the	O
greatest	O
fixed	O
point	O
of	O
.	O
</s>
<s>
Bisimulation	B-Application
can	O
also	O
be	O
thought	O
of	O
in	O
terms	O
of	O
a	O
game	O
between	O
two	O
players	O
:	O
attacker	O
and	O
defender	O
.	O
</s>
<s>
By	O
the	O
above	O
definition	O
the	O
system	O
is	O
a	O
bisimulation	B-Application
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
winning	O
strategy	O
for	O
the	O
defender	O
.	O
</s>
<s>
A	O
bisimulation	B-Application
for	O
state	B-Application
transition	I-Application
systems	I-Application
is	O
a	O
special	O
case	O
of	O
coalgebraic	O
bisimulation	B-Application
for	O
the	O
type	O
of	O
covariant	O
powerset	O
functor	B-Language
.	O
</s>
<s>
In	O
special	O
contexts	O
the	O
notion	O
of	O
bisimulation	B-Application
is	O
sometimes	O
refined	O
by	O
adding	O
additional	O
requirements	O
or	O
constraints	O
.	O
</s>
<s>
An	O
example	O
is	O
that	O
of	O
stutter	B-Application
bisimulation	I-Application
,	O
in	O
which	O
one	O
transition	O
of	O
one	O
system	O
may	O
be	O
matched	O
with	O
multiple	O
transitions	O
of	O
the	O
other	O
,	O
provided	O
that	O
the	O
intermediate	O
states	O
are	O
equivalent	O
to	O
the	O
starting	O
state	O
(	O
"	O
stutters	O
"	O
)	O
.	O
</s>
<s>
A	O
different	O
variant	O
applies	O
if	O
the	O
state	B-Application
transition	I-Application
system	I-Application
includes	O
a	O
notion	O
of	O
silent	O
(	O
or	O
internal	O
)	O
action	O
,	O
often	O
denoted	O
with	O
,	O
i.e.	O
</s>
<s>
actions	O
that	O
are	O
not	O
visible	O
by	O
external	O
observers	O
,	O
then	O
bisimulation	B-Application
can	O
be	O
relaxed	O
to	O
be	O
weak	O
bisimulation	B-Application
,	O
in	O
which	O
if	O
two	O
states	O
and	O
are	O
bisimilar	B-Application
and	O
there	O
is	O
some	O
number	O
of	O
internal	O
actions	O
leading	O
from	O
to	O
some	O
state	O
then	O
there	O
must	O
exist	O
state	O
such	O
that	O
there	O
is	O
some	O
number	O
(	O
possibly	O
zero	O
)	O
of	O
internal	O
actions	O
leading	O
from	O
to	O
.	O
</s>
<s>
A	O
relation	B-Algorithm
on	O
processes	O
is	O
a	O
weak	O
bisimulation	B-Application
if	O
the	O
following	O
holds	O
(	O
with	O
,	O
and	O
being	O
an	O
observable	O
and	O
mute	O
transition	O
respectively	O
)	O
:	O
</s>
<s>
This	O
is	O
closely	O
related	O
to	O
bisimulation	B-Application
up	O
to	O
a	O
relation	B-Algorithm
.	O
</s>
<s>
Typically	O
,	O
if	O
the	O
state	B-Application
transition	I-Application
system	I-Application
gives	O
the	O
operational	O
semantics	O
of	O
a	O
programming	O
language	O
,	O
then	O
the	O
precise	O
definition	O
of	O
bisimulation	B-Application
will	O
be	O
specific	O
to	O
the	O
restrictions	O
of	O
the	O
programming	O
language	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
general	O
,	O
there	O
may	O
be	O
more	O
than	O
one	O
kind	O
of	O
bisimulation	B-Application
,	O
(	O
bisimilarity	B-Application
resp	O
.	O
)	O
</s>
<s>
Since	O
Kripke	O
models	O
are	O
a	O
special	O
case	O
of	O
(	O
labelled	O
)	O
state	B-Application
transition	I-Application
systems	I-Application
,	O
bisimulation	B-Application
is	O
also	O
a	O
topic	O
in	O
modal	O
logic	O
.	O
</s>
<s>
In	O
fact	O
,	O
modal	O
logic	O
is	O
the	O
fragment	O
of	O
first-order	O
logic	O
invariant	O
under	O
bisimulation	B-Application
(	O
van	O
Benthem	O
's	O
theorem	O
)	O
.	O
</s>
<s>
Checking	O
that	O
two	O
finite	O
transition	B-Application
systems	I-Application
are	O
bisimilar	B-Application
can	O
be	O
done	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
fastest	O
algorithms	O
are	O
quasilinear	O
time	O
using	O
partition	B-Data_Structure
refinement	I-Data_Structure
through	O
a	O
reduction	O
to	O
the	O
coarsest	O
partition	O
problem	O
.	O
</s>
