<s>
In	O
automata	B-Application
theory	I-Application
,	O
complementation	O
of	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
construction	O
of	O
another	O
Büchi	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
the	O
complement	O
of	O
the	O
ω-regular	O
language	O
recognized	O
by	O
the	O
given	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
This	O
construction	O
is	O
particularly	O
hard	O
relative	O
to	O
the	O
constructions	O
for	O
the	O
other	O
closure	O
properties	O
of	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Here	O
,	O
we	O
have	O
his	O
construction	O
in	O
the	O
modern	O
notation	O
used	O
in	O
automata	B-Application
theory	I-Application
.	O
</s>
<s>
Let	O
A	O
=(	O
Q	O
,	O
Σ	O
,	O
Δ	O
,	O
Q0	O
,	O
F	O
)	O
be	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
Theorem	O
1	O
:	O
~	O
A	O
has	O
finitely	O
many	O
equivalent	O
classes	O
and	O
each	O
class	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
Now	O
we	O
show	O
that	O
Lf	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
be	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
,	O
</s>
<s>
Since	O
the	O
regular	B-General_Concept
languages	I-General_Concept
are	O
closed	O
under	O
finite	O
intersections	O
and	O
under	O
relative	O
complements	O
,	O
every	O
αp	O
,	O
Q	O
 '	O
,	O
βp	O
,	O
Q	O
 '	O
,	O
and	O
γp	O
,	O
Q	O
 '	O
is	O
regular	O
.	O
</s>
<s>
We	O
can	O
translate	O
the	O
language	O
into	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
