<s>
In	O
automata	B-Application
theory	I-Application
,	O
McNaughton	B-Application
's	I-Application
theorem	I-Application
refers	O
to	O
a	O
theorem	O
that	O
asserts	O
that	O
the	O
set	O
of	O
ω-regular	O
languages	O
is	O
identical	O
to	O
the	O
set	O
of	O
languages	O
recognizable	O
by	O
deterministic	O
Muller	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
This	O
theorem	O
is	O
proven	O
by	O
supplying	O
an	O
algorithm	O
to	O
construct	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
for	O
any	O
ω-regular	O
language	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Since	O
(	O
non-deterministic	O
)	O
Büchi	B-General_Concept
automata	I-General_Concept
and	O
ω-regular	O
languages	O
are	O
equally	O
expressive	O
,	O
the	O
theorem	O
implies	O
that	O
Büchi	B-General_Concept
automata	I-General_Concept
and	O
deterministic	O
Muller	B-General_Concept
automata	I-General_Concept
are	O
equally	O
expressive	O
.	O
</s>
<s>
Since	O
complementation	O
of	O
deterministic	O
Muller	B-General_Concept
automata	I-General_Concept
is	O
trivial	O
,	O
the	O
theorem	O
implies	O
that	O
Büchi	O
automata/ω	O
-regular	O
languages	O
are	O
closed	O
under	O
complementation	O
.	O
</s>
<s>
Following	O
McNaughton	O
's	O
definition	O
,	O
an	O
ω-event	O
is	O
a	O
finite-state	O
event	O
if	O
there	O
exists	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
it	O
.	O
</s>
<s>
One	O
direction	O
of	O
the	O
theorem	O
can	O
be	O
proven	O
by	O
showing	O
that	O
any	O
given	O
Muller	B-General_Concept
automaton	I-General_Concept
recognizes	O
an	O
ω-regular	O
language	O
.	O
</s>
<s>
Suppose	O
A	O
=(	O
Q	O
,	O
Σ	O
,	O
δ	O
,	O
q0	O
,	O
F	O
)	O
is	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
Let	O
α	O
be	O
the	O
regular	B-General_Concept
language	I-General_Concept
whose	O
elements	O
will	O
take	O
A	O
from	O
q0	O
to	O
q1	O
.	O
</s>
<s>
For	O
1≤i≤n	O
,	O
let	O
βi	O
be	O
a	O
regular	B-General_Concept
language	I-General_Concept
whose	O
elements	O
take	O
A	O
from	O
qi	O
to	O
q(i mod n )	O
+1	O
without	O
passing	O
through	O
any	O
state	O
outside	O
of	O
 { q1 , ... , qn } 	O
.	O
</s>
<s>
It	O
is	O
claimed	O
that	O
α( β1	O
...	O
βn	O
)	O
ω	O
is	O
the	O
ω-regular	O
language	O
recognized	O
by	O
the	O
Muller	B-General_Concept
automaton	I-General_Concept
A	O
.	O
</s>
<s>
The	O
other	O
direction	O
of	O
the	O
theorem	O
can	O
be	O
proven	O
by	O
showing	O
that	O
there	O
exists	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
a	O
given	O
ω-regular	O
language	O
.	O
</s>
<s>
The	O
union	O
of	O
finitely	O
many	O
deterministic	O
Muller	B-General_Concept
automata	I-General_Concept
can	O
be	O
easily	O
constructed	O
;	O
therefore	O
without	O
loss	O
of	O
generality	O
we	O
assume	O
that	O
the	O
given	O
ω-regular	O
language	O
is	O
of	O
the	O
form	O
αβω	O
.	O
</s>
<s>
Let	O
w(i,j )	O
be	O
the	O
finite	O
segment	O
ai+1	O
,...,	O
aj-1aj	O
of	O
w	O
.	O
For	O
building	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
for	O
αβω	O
,	O
we	O
introduce	O
the	O
following	O
two	O
concepts	O
with	O
respect	O
to	O
w	O
.	O
</s>
<s>
Let	O
p	O
be	O
the	O
number	O
of	O
states	O
in	O
the	O
minimum	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
Aβ*	O
to	O
recognize	O
language	O
β*	O
.	O
</s>
<s>
Proof	O
:	O
The	O
finite	O
automaton	O
Aβ*	O
is	O
minimum	O
;	O
therefore	O
it	O
does	O
not	O
contain	O
equivalent	B-General_Concept
states	I-General_Concept
.	O
</s>
<s>
Now	O
,	O
we	O
are	O
going	O
to	O
use	O
the	O
lemma	O
to	O
construct	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
for	O
language	O
αβω	O
.	O
</s>
<s>
Note	O
that	O
this	O
machine	O
will	O
be	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
machine	O
contains	O
p+2	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
and	O
a	O
master	O
controller	O
,	O
where	O
p	O
is	O
the	O
size	O
of	O
Aβ*	O
.	O
</s>
<s>
This	O
finishes	O
the	O
construction	O
of	O
the	O
desired	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
ω-regular	O
languages	O
can	O
be	O
shown	O
equiv-expressive	O
to	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Büchi	B-General_Concept
automata	I-General_Concept
can	O
be	O
shown	O
to	O
equiv-expressive	O
to	O
semi-deterministic	B-Application
Büchi	I-Application
automata	I-Application
.	O
</s>
<s>
Semi-deterministic	B-Application
Büchi	I-Application
automata	I-Application
can	O
be	O
shown	O
to	O
be	O
equiv-expressive	O
to	O
deterministic	O
Muller	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Safra	O
's	O
construction	O
transforms	O
a	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
to	O
a	O
deterministic	O
Rabin	O
automaton	O
.	O
</s>
<s>
There	O
is	O
a	O
purely	O
algebraic	O
proof	O
of	O
McNaughton	B-Application
's	I-Application
theorem	I-Application
.	O
</s>
