<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
in	O
particular	O
in	O
formal	O
language	O
theory	O
,	O
a	O
quotient	B-General_Concept
automaton	I-General_Concept
can	O
be	O
obtained	O
from	O
a	O
given	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
by	O
joining	O
some	O
of	O
its	O
states	O
.	O
</s>
<s>
A	O
(	O
nondeterministic	O
)	O
finite	O
automaton	O
is	O
a	O
quintuple	B-Application
A	O
=	O
⟨Σ	O
,	O
S	O
,	O
s0	O
,	O
δ	O
,	O
Sf⟩	O
,	O
where	O
:	O
</s>
<s>
It	O
recognizes	O
the	O
finite	O
set	O
of	O
strings	O
{	O
1	O
,	O
10	O
,	O
100	O
}	O
;	O
this	O
set	O
can	O
also	O
be	O
denoted	O
by	O
the	O
regular	B-Language
expression	I-Language
"	O
1+10+100	O
"	O
.	O
</s>
<s>
The	O
relation	B-Algorithm
( ≈	O
)	O
=	O
{	O
⟨a	O
,	O
a⟩	O
,	O
⟨a	O
,	O
b⟩	O
,	O
⟨b	O
,	O
a⟩	O
,	O
⟨b	O
,	O
b⟩	O
,	O
⟨c	O
,	O
c⟩	O
,	O
⟨c	O
,	O
d⟩	O
,	O
⟨d	O
,	O
c⟩	O
,	O
⟨d	O
,	O
d⟩	O
}	O
,	O
more	O
briefly	O
denoted	O
as	O
a≈b	O
,	O
c≈d	O
,	O
is	O
an	O
equivalence	O
relation	B-Algorithm
on	O
the	O
set	O
 { a , b , c , d } 	O
of	O
automaton	O
A	O
’s	O
states	O
.	O
</s>
<s>
{	O
ε	O
,	O
1	O
,	O
10	O
,	O
100	O
,	O
1000	O
,	O
...	O
,	O
11	O
,	O
110	O
,	O
1100	O
,	O
11000	O
,	O
...	O
,	O
111	O
,	O
...	O
}	O
;	O
this	O
set	O
can	O
also	O
be	O
denoted	O
by	O
the	O
regular	B-Language
expression	I-Language
"	O
1*0*	O
"	O
.	O
</s>
<s>
For	O
every	O
automaton	O
A	O
and	O
every	O
equivalence	O
relation	B-Algorithm
≈	O
on	O
its	O
states	O
set	O
,	O
L( 	O
A/	O
≈	O
)	O
is	O
a	O
superset	O
of	O
(	O
or	O
equal	O
to	O
)	O
L(A )	O
.	O
</s>
<s>
Given	O
a	O
finite	O
automaton	O
A	O
over	O
some	O
alphabet	O
Σ	O
,	O
an	O
equivalence	O
relation	B-Algorithm
≈	O
can	O
be	O
defined	O
on	O
Σ*	O
by	O
x	O
≈	O
y	O
if	O
∀	O
z	O
∈	O
Σ*	O
:	O
xz	O
∈	O
L(A )	O
↔	O
yz	O
∈	O
L(A )	O
.	O
</s>
