<s>
In	O
theoretical	O
computer	O
science	O
a	O
simulation	O
is	O
a	O
relation	B-Algorithm
between	O
state	B-Application
transition	I-Application
systems	I-Application
associating	O
systems	O
that	O
behave	O
in	O
the	O
same	O
way	O
in	O
the	O
sense	O
that	O
one	O
system	O
simulates	O
the	O
other	O
.	O
</s>
<s>
The	O
basic	O
definition	O
relates	O
states	O
within	O
one	O
transition	B-Application
system	I-Application
,	O
but	O
this	O
is	O
easily	O
adapted	O
to	O
relate	O
two	O
separate	O
transition	B-Application
systems	I-Application
by	O
building	O
a	O
system	O
consisting	O
of	O
the	O
disjoint	O
union	O
of	O
the	O
corresponding	O
components	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
</s>
<s>
a	O
relation	B-Algorithm
is	O
a	O
simulation	O
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
in	O
:	O
</s>
<s>
The	O
relation	B-Algorithm
is	O
called	O
the	O
simulation	B-Application
preorder	I-Application
,	O
and	O
it	O
is	O
the	O
union	O
of	O
all	O
simulations	O
:	O
precisely	O
when	O
for	O
some	O
simulation	O
.	O
</s>
<s>
The	O
set	O
of	O
simulations	O
is	O
closed	O
under	O
union	O
;	O
therefore	O
,	O
the	O
simulation	B-Application
preorder	I-Application
is	O
itself	O
a	O
simulation	O
.	O
</s>
<s>
From	O
this	O
follows	O
that	O
the	O
largest	O
simulation	O
—	O
the	O
simulation	B-Application
preorder	I-Application
—	O
is	O
indeed	O
a	O
preorder	O
relation	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
there	O
can	O
be	O
more	O
than	O
one	O
relation	B-Algorithm
which	O
is	O
both	O
a	O
simulation	O
and	O
a	O
preorder	O
;	O
the	O
term	O
simulation	B-Application
preorder	I-Application
refers	O
to	O
the	O
largest	O
one	O
of	O
them	O
(	O
which	O
is	O
a	O
superset	O
of	O
all	O
the	O
others	O
)	O
.	O
</s>
<s>
Similarity	O
is	O
thus	O
the	O
maximal	O
symmetric	O
subset	O
of	O
the	O
simulation	B-Application
preorder	I-Application
,	O
which	O
means	O
it	O
is	O
reflexive	O
,	O
symmetric	O
,	O
and	O
transitive	O
;	O
hence	O
an	O
equivalence	O
relation	B-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
not	O
necessarily	O
a	O
simulation	O
,	O
and	O
precisely	O
in	O
those	O
cases	O
when	O
it	O
is	O
not	O
a	O
simulation	O
,	O
it	O
is	O
strictly	O
coarser	O
than	O
bisimilarity	B-Application
(	O
meaning	O
it	O
is	O
a	O
superset	O
of	O
bisimilarity	B-Application
)	O
.	O
</s>
<s>
Since	O
it	O
is	O
symmetric	O
,	O
it	O
is	O
a	O
bisimulation	B-Application
.	O
</s>
<s>
It	O
must	O
then	O
be	O
a	O
subset	O
of	O
bisimilarity	B-Application
,	O
which	O
is	O
the	O
union	O
of	O
all	O
bisimulations	B-Application
.	O
</s>
<s>
Yet	O
it	O
is	O
easy	O
to	O
see	O
that	O
similarity	O
is	O
always	O
a	O
superset	O
of	O
bisimilarity	B-Application
.	O
</s>
<s>
From	O
this	O
follows	O
that	O
if	O
similarity	O
is	O
a	O
simulation	O
,	O
it	O
equals	O
bisimilarity	B-Application
.	O
</s>
<s>
And	O
if	O
it	O
equals	O
bisimilarity	B-Application
,	O
it	O
is	O
naturally	O
a	O
simulation	O
(	O
since	O
bisimilarity	B-Application
is	O
a	O
simulation	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
similarity	O
is	O
a	O
simulation	O
if	O
and	O
only	O
if	O
it	O
equals	O
bisimilarity	B-Application
.	O
</s>
<s>
If	O
it	O
does	O
not	O
,	O
it	O
must	O
be	O
its	O
strict	O
superset	O
;	O
hence	O
a	O
strictly	O
coarser	O
equivalence	O
relation	B-Algorithm
.	O
</s>
<s>
When	O
comparing	O
two	O
different	O
transition	B-Application
systems	I-Application
(	O
S	O
 '	O
,	O
Λ	O
 '	O
,	O
→	O
'	O
)	O
and	O
(	O
S	O
"	O
,	O
Λ	O
"	O
,	O
→	O
"	O
)	O
,	O
the	O
basic	O
notions	O
of	O
simulation	O
and	O
similarity	O
can	O
be	O
used	O
by	O
forming	O
the	O
disjoint	O
composition	O
of	O
the	O
two	O
machines	O
,	O
(	O
S	O
,	O
Λ	O
,	O
→	O
)	O
with	O
S	O
=	O
S	O
 '	O
∐	O
S	O
"	O
,	O
Λ	O
=	O
Λ	O
 '	O
∪	O
Λ	O
"	O
and	O
→	O
=	O
→	O
'	O
∪	O
→	O
"	O
,	O
where	O
∐	O
is	O
the	O
disjoint	O
union	O
operator	O
between	O
sets	O
.	O
</s>
