<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
mathematical	O
logic	O
,	O
an	O
infinite-tree	B-Application
automaton	I-Application
is	O
a	O
state	B-Architecture
machine	I-Architecture
that	O
deals	O
with	O
infinite	O
tree	O
structures	O
.	O
</s>
<s>
It	O
can	O
be	O
seen	O
as	O
an	O
extension	O
of	O
top-down	O
finite-tree	B-Application
automata	I-Application
to	O
infinite	O
trees	O
or	O
as	O
an	O
extension	O
of	O
infinite-word	B-General_Concept
automata	I-General_Concept
to	O
infinite	O
trees	O
.	O
</s>
<s>
A	O
finite	B-Architecture
automaton	I-Architecture
which	O
runs	O
on	O
an	O
infinite	O
tree	O
was	O
first	O
used	O
by	O
Michael	O
Rabin	O
for	O
proving	O
decidability	O
of	O
S2S	O
,	O
the	O
monadic	O
second-order	O
theory	O
with	O
two	O
successors	O
.	O
</s>
<s>
It	O
has	O
been	O
further	O
observed	O
that	O
tree	B-Application
automata	I-Application
and	O
logical	O
theories	O
are	O
closely	O
connected	O
and	O
it	O
allows	O
decision	O
problems	O
in	O
logic	O
to	O
be	O
reduced	O
into	O
decision	O
problems	O
for	O
automata	O
.	O
</s>
<s>
A	O
(	O
nondeterministic	O
)	O
infinite-tree	B-Application
automaton	I-Application
is	O
a	O
tuple	O
with	O
the	O
following	O
components	O
.	O
</s>
<s>
An	O
infinite-tree	B-Application
automaton	I-Application
is	O
deterministic	O
if	O
for	O
every	O
,	O
,	O
and	O
,	O
the	O
transition	O
relation	O
has	O
exactly	O
one	O
-tuple	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
run	O
of	O
a	O
tree	B-Application
automaton	I-Application
on	O
an	O
input	O
tree	O
assigns	O
automaton	O
states	O
to	O
the	O
tree	O
nodes	O
in	O
a	O
way	O
that	O
satisfies	O
the	O
automaton	O
transition	O
relation	O
.	O
</s>
<s>
A	O
bit	O
more	O
formally	O
,	O
a	O
run	O
of	O
a	O
tree	B-Application
automaton	I-Application
over	O
a	O
-labeled	O
tree	O
is	O
a	O
-labeled	O
tree	O
as	O
follows	O
.	O
</s>
<s>
If	O
the	O
automaton	O
is	O
nondeterministic	O
,	O
there	O
may	O
be	O
several	O
different	O
runs	O
on	O
the	O
same	O
input	O
tree	O
;	O
for	O
deterministic	B-Architecture
automata	I-Architecture
,	O
the	O
run	O
is	O
unique	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
accepted	O
-labeled	O
trees	O
is	O
called	O
tree	B-Application
language	I-Application
which	O
is	O
recognized	O
by	O
the	O
tree	B-Application
automaton	I-Application
.	O
</s>
<s>
Nondeterministic	O
Muller	O
,	O
Rabin	O
,	O
Streett	O
,	O
and	O
parity	O
tree	B-Application
automata	I-Application
recognize	O
the	O
same	O
set	O
of	O
tree	B-Application
languages	I-Application
,	O
and	O
thus	O
have	O
the	O
same	O
expressive	O
power	O
.	O
</s>
<s>
But	O
nondeterministic	O
Büchi	O
tree	B-Application
automata	I-Application
are	O
strictly	O
weaker	O
,	O
i.e.	O
,	O
there	O
exists	O
a	O
tree	B-Application
language	I-Application
that	O
can	O
be	O
recognized	O
by	O
a	O
Rabin	O
tree	B-Application
automaton	I-Application
but	O
cannot	O
be	O
recognized	O
by	O
any	O
Büchi	O
tree	B-Application
automaton	I-Application
.	O
</s>
<s>
(	O
For	O
example	O
,	O
there	O
is	O
no	O
Büchi	O
tree	B-Application
automaton	I-Application
that	O
recognizes	O
the	O
set	O
of	O
-labeled	O
trees	O
whose	O
every	O
path	O
has	O
only	O
finitely	O
many	O
s	O
,	O
see	O
e.g.	O
</s>
<s>
Furthermore	O
,	O
deterministic	O
tree	B-Application
automata	I-Application
(	O
Muller	O
,	O
Rabin	O
,	O
Streett	O
,	O
parity	O
,	O
Büchi	O
,	O
looping	O
)	O
are	O
strictly	O
less	O
expressive	O
than	O
their	O
nondeterministic	O
variants	O
.	O
</s>
<s>
For	O
example	O
,	O
there	O
is	O
no	O
deterministic	B-Application
tree	I-Application
automaton	I-Application
that	O
recognizes	O
the	O
language	O
of	O
binary	O
trees	O
whose	O
root	O
has	O
its	O
left	O
or	O
right	O
child	O
marked	O
with	O
.	O
</s>
<s>
This	O
is	O
in	O
sharp	O
contrast	O
to	O
automata	O
on	O
infinite	O
words	O
,	O
where	O
nondeterministic	O
Büchi	O
ω-automata	B-General_Concept
have	O
the	O
same	O
expressive	O
power	O
as	O
the	O
others	O
.	O
</s>
<s>
The	O
languages	O
of	O
nondeterministic	O
Muller/Rabin/Streett/parity	O
tree	B-Application
automata	I-Application
are	O
closed	O
under	O
union	O
,	O
intersection	O
,	O
projection	O
,	O
and	O
complementation	O
.	O
</s>
