<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
is	O
a	O
type	O
of	O
an	O
ω-automaton	B-General_Concept
.	O
</s>
<s>
The	O
acceptance	O
condition	O
separates	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
from	O
other	O
ω-automata	B-General_Concept
.	O
</s>
<s>
The	O
Muller	B-General_Concept
automaton	I-General_Concept
is	O
defined	O
using	O
a	O
Muller	O
acceptance	O
condition	O
,	O
i.e.	O
</s>
<s>
Formally	O
,	O
F	O
⊆	O
P(Q )	O
where	O
P(Q )	O
is	O
powerset	B-General_Concept
of	O
Q	O
.	O
F	O
defines	O
the	O
acceptance	O
condition	O
.	O
</s>
<s>
In	O
a	O
non-deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
,	O
the	O
transition	O
function	O
δ	O
is	O
replaced	O
with	O
a	O
transition	O
relation	O
Δ	O
that	O
returns	O
a	O
set	O
of	O
states	O
and	O
the	O
initial	O
state	O
q0	O
is	O
replaced	O
by	O
a	O
set	O
of	O
initial	O
states	O
Q0	O
.	O
</s>
<s>
Generally	O
,	O
'	O
Muller	B-General_Concept
automaton	I-General_Concept
 '	O
refers	O
to	O
a	O
non-deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
For	O
more	O
comprehensive	O
formalisation	O
look	O
at	O
ω-automaton	B-General_Concept
.	O
</s>
<s>
The	O
Muller	O
automata	O
are	O
equally	O
expressive	O
as	O
parity	B-General_Concept
automata	I-General_Concept
,	O
Rabin	O
automata	O
,	O
Streett	O
automata	O
,	O
and	O
non-deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
,	O
to	O
mention	O
some	O
,	O
and	O
strictly	O
more	O
expressive	O
than	O
the	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
McNaughton	B-Application
's	I-Application
theorem	I-Application
demonstrates	O
the	O
equivalence	O
of	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
and	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
Following	O
is	O
a	O
list	O
of	O
automata	O
constructions	O
that	O
each	O
transforms	O
a	O
type	O
of	O
ω-automata	B-General_Concept
to	O
a	O
non-deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
If	O
is	O
the	O
set	O
of	O
final	O
states	O
in	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
with	O
the	O
set	O
of	O
states	O
,	O
we	O
can	O
construct	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
with	O
same	O
set	O
of	O
states	O
,	O
transition	O
function	O
and	O
initial	O
state	O
with	O
the	O
Muller	O
accepting	O
condition	O
as	O
F	O
=	O
{	O
X	O
|	O
X	O
∈	O
P(Q )	O
∧	O
X	O
∩	O
B	O
≠	O
}	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
Rabin	O
conditions	O
can	O
be	O
emulated	O
by	O
constructing	O
the	O
acceptance	O
set	O
in	O
the	O
Muller	B-General_Concept
automaton	I-General_Concept
as	O
all	O
sets	O
that	O
satisfy	O
and	O
,	O
for	O
some	O
j	O
.	O
</s>
<s>
Note	O
that	O
this	O
covers	O
the	O
case	O
of	O
parity	B-General_Concept
automata	I-General_Concept
too	O
,	O
as	O
the	O
parity	O
acceptance	O
condition	O
can	O
be	O
expressed	O
as	O
a	O
Rabin	O
acceptance	O
condition	O
easily	O
.	O
</s>
<s>
The	O
Streett	O
conditions	O
can	O
be	O
emulated	O
by	O
constructing	O
the	O
acceptance	O
set	O
in	O
the	O
Muller	B-General_Concept
automaton	I-General_Concept
as	O
all	O
sets	O
that	O
satisfy	O
,	O
for	O
all	O
j	O
.	O
</s>
<s>
McNaughton	B-Application
's	I-Application
theorem	I-Application
provides	O
a	O
procedure	O
to	O
transform	O
any	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
into	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
