<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
branch	O
of	O
theoretical	O
computer	O
science	O
,	O
an	O
ω-automaton	B-General_Concept
(	O
or	O
stream	B-General_Concept
automaton	I-General_Concept
)	O
is	O
a	O
variation	O
of	O
finite	B-Architecture
automata	I-Architecture
that	O
runs	O
on	O
infinite	O
,	O
rather	O
than	O
finite	O
,	O
strings	O
as	O
input	O
.	O
</s>
<s>
Since	O
ω-automata	B-General_Concept
do	O
not	O
stop	O
,	O
they	O
have	O
a	O
variety	O
of	O
acceptance	O
conditions	O
rather	O
than	O
simply	O
a	O
set	O
of	O
accepting	B-Architecture
states	I-Architecture
.	O
</s>
<s>
ω-automata	B-General_Concept
are	O
useful	O
for	O
specifying	O
behavior	O
of	O
systems	O
that	O
are	O
not	O
expected	O
to	O
terminate	O
,	O
such	O
as	O
hardware	O
,	O
operating	B-General_Concept
systems	I-General_Concept
and	O
control	O
systems	O
.	O
</s>
<s>
Classes	O
of	O
ω-automata	B-General_Concept
include	O
the	O
Büchi	B-General_Concept
automata	I-General_Concept
,	O
Rabin	O
automata	O
,	O
Streett	O
automata	O
,	O
parity	O
automata	O
and	O
Muller	B-General_Concept
automata	I-General_Concept
,	O
each	O
deterministic	O
or	O
non-deterministic	O
.	O
</s>
<s>
These	O
classes	O
of	O
ω-automata	B-General_Concept
differ	O
only	O
in	O
terms	O
of	O
acceptance	O
condition	O
.	O
</s>
<s>
They	O
all	O
recognize	O
precisely	O
the	O
regular	O
ω-languages	O
except	O
for	O
the	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
,	O
which	O
is	O
strictly	O
weaker	O
than	O
all	O
the	O
others	O
.	O
</s>
<s>
Although	O
all	O
these	O
types	O
of	O
automata	O
recognize	O
the	O
same	O
set	O
of	O
ω-languages	O
,	O
they	O
nonetheless	O
differ	O
in	O
succinctness	O
of	O
representation	O
for	O
a	O
given	O
ω-language	O
.	O
</s>
<s>
Formally	O
,	O
a	O
deterministic	O
ω-automaton	B-General_Concept
is	O
a	O
tuple	O
A	O
=(	O
Q	O
,	O
Σ	O
,	O
δ	O
,	O
Q0	O
,	O
Acc	O
)	O
that	O
consists	O
of	O
the	O
following	O
components	O
:	O
</s>
<s>
The	O
main	O
purpose	O
of	O
an	O
ω-automaton	B-General_Concept
is	O
to	O
define	O
a	O
subset	O
of	O
the	O
set	O
of	O
all	O
inputs	O
:	O
The	O
set	O
of	O
accepted	O
inputs	O
.	O
</s>
<s>
Whereas	O
in	O
the	O
case	O
of	O
an	O
ordinary	O
finite	B-Architecture
automaton	I-Architecture
every	O
run	O
ends	O
with	O
a	O
state	O
rn	O
and	O
the	O
input	O
is	O
accepted	O
if	O
and	O
only	O
if	O
rn	O
is	O
an	O
accepting	B-Architecture
state	I-Architecture
,	O
the	O
definition	O
of	O
the	O
set	O
of	O
accepted	O
inputs	O
is	O
more	O
complicated	O
for	O
ω-automata	B-General_Concept
.	O
</s>
<s>
The	O
set	O
of	O
accepted	O
input	O
ω-words	O
is	O
called	O
the	O
recognized	O
ω-language	O
by	O
the	O
automaton	O
,	O
which	O
is	O
denoted	O
as	O
L(A )	O
.	O
</s>
<s>
The	O
difference	O
between	O
various	O
types	O
of	O
ω-automata	B-General_Concept
(	O
Büchi	O
,	O
Rabin	O
etc	O
.	O
)	O
</s>
<s>
Formally	O
,	O
a	O
nondeterministic	O
ω-automaton	B-General_Concept
is	O
a	O
tuple	O
A	O
=(	O
Q	O
,	O
Σ	O
,	O
Δ	O
,	O
Q0	O
,	O
Acc	O
)	O
that	O
consists	O
of	O
the	O
following	O
components	O
:	O
</s>
<s>
Unlike	O
a	O
deterministic	O
ω-automaton	B-General_Concept
,	O
which	O
has	O
a	O
transition	O
function	O
δ	O
,	O
the	O
non-deterministic	O
version	O
has	O
a	O
transition	O
relation	O
Δ	O
.	O
</s>
<s>
A	O
nondeterministic	O
ω-automaton	B-General_Concept
may	O
admit	O
many	O
different	O
runs	O
on	O
any	O
given	O
input	O
,	O
or	O
none	O
at	O
all	O
.	O
</s>
<s>
Whether	O
a	O
run	O
is	O
accepting	O
depends	O
only	O
on	O
Acc	O
,	O
as	O
for	O
deterministic	O
ω-automata	B-General_Concept
.	O
</s>
<s>
Every	O
deterministic	O
ω-automaton	B-General_Concept
can	O
be	O
regarded	O
as	O
a	O
nondeterministic	O
ω-automaton	B-General_Concept
by	O
taking	O
Δ	O
to	O
be	O
the	O
graph	O
of	O
δ	O
.	O
</s>
<s>
The	O
definitions	O
of	O
runs	O
and	O
acceptance	O
for	O
deterministic	O
ω-automata	B-General_Concept
are	O
then	O
special	O
cases	O
of	O
the	O
nondeterministic	O
cases	O
.	O
</s>
<s>
Acceptance	O
conditions	O
may	O
be	O
infinite	O
sets	O
of	O
ω-words	O
.	O
</s>
<s>
A	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
an	O
ω-automaton	B-General_Concept
A	O
that	O
uses	O
the	O
following	O
acceptance	O
condition	O
,	O
for	O
some	O
subset	O
F	O
of	O
Q	O
:	O
</s>
<s>
there	O
is	O
an	O
accepting	B-Architecture
state	I-Architecture
that	O
occurs	O
infinitely	O
often	O
inρ	O
.	O
</s>
<s>
A	O
is	O
an	O
ω-automaton	B-General_Concept
A	O
that	O
uses	O
the	O
following	O
acceptance	O
condition	O
,	O
for	O
some	O
set	O
Ω	B-Language
of	O
pairs	O
(	O
Bi	O
,	O
Gi	O
)	O
of	O
sets	O
of	O
states	O
:	O
</s>
<s>
Rabin	O
condition	O
A	O
accepts	O
exactly	O
those	O
runs	O
ρ	O
for	O
which	O
there	O
exists	O
a	O
pair	O
(	O
Bi	O
,	O
Gi	O
)	O
in	O
Ω	B-Language
such	O
that	O
Bi∩	O
Inf(ρ )	O
is	O
empty	O
and	O
Gi∩	O
Inf(ρ )	O
is	O
not	O
empty	O
.	O
</s>
<s>
A	O
Streett	O
automaton	O
is	O
an	O
ω-automaton	B-General_Concept
A	O
that	O
uses	O
the	O
following	O
acceptance	O
condition	O
,	O
for	O
some	O
set	O
Ω	B-Language
of	O
pairs	O
(	O
Bi	O
,	O
Gi	O
)	O
of	O
sets	O
of	O
states	O
:	O
</s>
<s>
Streett	O
condition	O
A	O
accepts	O
exactly	O
those	O
runs	O
ρ	O
such	O
that	O
for	O
all	O
pairs	O
(	O
Bi	O
,	O
Gi	O
)	O
in	O
Ω	B-Language
,	O
Bi∩	O
Inf(ρ )	O
is	O
empty	O
or	O
Gi∩	O
Inf(ρ )	O
is	O
not	O
empty	O
.	O
</s>
<s>
A	O
parity	B-General_Concept
automaton	I-General_Concept
is	O
an	O
automaton	O
A	O
whose	O
set	O
of	O
states	O
is	O
Q={0,1,2,...,k}	O
for	O
some	O
natural	O
numberk	O
,	O
and	O
that	O
has	O
the	O
following	O
acceptance	O
condition	O
:	O
</s>
<s>
A	O
Muller	B-General_Concept
automaton	I-General_Concept
is	O
an	O
ω-automaton	B-General_Concept
A	O
that	O
uses	O
the	O
following	O
acceptance	O
condition	O
,	O
for	O
a	O
subset	O
F	O
of	O
P(Q )	O
(	O
the	O
power	O
set	O
of	O
Q	O
)	O
:	O
</s>
<s>
Every	O
Büchi	B-General_Concept
automaton	I-General_Concept
can	O
be	O
regarded	O
as	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
Similarly	O
every	O
Rabin	O
,	O
Streett	O
or	O
parity	B-General_Concept
automaton	I-General_Concept
can	O
also	O
be	O
regarded	O
as	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
following	O
ω-language	O
L	O
over	O
the	O
alphabet	O
Σ={0,1},	O
which	O
can	O
be	O
recognized	O
by	O
a	O
nondeterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
:	O
</s>
<s>
L	O
consists	O
of	O
all	O
ω-words	O
in	O
Σω	O
in	O
which	O
1	O
occurs	O
only	O
finitely	O
many	O
times	O
.	O
</s>
<s>
A	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
recognizing	O
L	O
needs	O
only	O
two	O
states	O
q0	O
(	O
the	O
initial	O
state	O
)	O
and	O
q1	O
.	O
</s>
<s>
Notice	O
that	O
above	O
language	O
cannot	O
be	O
recognized	O
by	O
a	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
,	O
which	O
is	O
strictly	O
less	O
expressive	O
than	O
its	O
non-deterministic	O
counterpart	O
.	O
</s>
<s>
An	O
ω-language	O
over	O
a	O
finite	O
alphabet	O
Σ	O
is	O
a	O
set	O
of	O
ω-words	O
over	O
Σ	O
,	O
i.e.	O
</s>
<s>
it	O
is	O
a	O
subset	O
of	O
Σω	O
.	O
</s>
<s>
An	O
ω-language	O
over	O
Σ	O
is	O
said	O
to	O
be	O
recognized	O
by	O
an	O
ω-automaton	B-General_Concept
A	O
(	O
with	O
the	O
same	O
alphabet	O
)	O
if	O
it	O
is	O
the	O
set	O
of	O
all	O
ω-words	O
accepted	O
by	O
A	O
.	O
</s>
<s>
The	O
expressive	O
power	O
of	O
a	O
class	O
of	O
ω-automata	B-General_Concept
is	O
measured	O
by	O
the	O
class	O
of	O
all	O
ω-languages	O
that	O
can	O
be	O
recognized	O
by	O
some	O
automaton	O
in	O
the	O
class	O
.	O
</s>
<s>
The	O
nondeterministic	O
Büchi	O
,	O
parity	O
,	O
Rabin	O
,	O
Streett	O
,	O
and	O
Muller	B-General_Concept
automata	I-General_Concept
,	O
respectively	O
,	O
all	O
recognize	O
exactly	O
the	O
same	O
class	O
of	O
ω-languages	O
.	O
</s>
<s>
These	O
are	O
known	O
as	O
the	O
ω-Kleene	O
closure	O
of	O
the	O
regular	O
languages	O
or	O
as	O
the	O
regular	O
ω-languages	O
.	O
</s>
<s>
Using	O
different	O
proofs	O
it	O
can	O
also	O
be	O
shown	O
that	O
the	O
deterministic	O
parity	O
,	O
Rabin	O
,	O
Streett	O
,	O
and	O
Muller	B-General_Concept
automata	I-General_Concept
all	O
recognize	O
the	O
regular	O
ω-languages	O
.	O
</s>
<s>
It	O
follows	O
from	O
this	O
that	O
the	O
class	O
of	O
regular	O
ω-languages	O
is	O
closed	O
under	O
complementation	O
.	O
</s>
<s>
However	O
,	O
the	O
example	O
above	O
shows	O
that	O
the	O
class	O
of	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
is	O
strictly	O
weaker	O
.	O
</s>
<s>
Because	O
nondeterministic	O
Muller	O
,	O
Rabin	O
,	O
Streett	O
,	O
parity	O
,	O
and	O
Büchi	B-General_Concept
automata	I-General_Concept
are	O
equally	O
expressive	O
,	O
they	O
can	O
be	O
translated	O
to	O
each	O
other	O
.	O
</s>
<s>
Let	O
us	O
use	O
the	O
following	O
abbreviation	O
:	O
for	O
example	O
,	O
NB	O
stands	O
for	O
nondeterministic	O
Büchi	O
ω-automaton	B-General_Concept
,	O
while	O
DP	O
stands	O
for	O
deterministic	O
parity	O
ω-automaton	B-General_Concept
.	O
</s>
<s>
ω-automata	B-General_Concept
can	O
be	O
used	O
to	O
prove	O
decidability	O
of	O
S1S	O
,	O
the	O
monadic	O
second-order	O
(	O
MSO	O
)	O
theory	O
of	O
natural	O
numbers	O
under	O
successor	O
.	O
</s>
<s>
Infinite-tree	O
automata	O
extend	O
ω-automata	B-General_Concept
to	O
infinite	O
trees	O
and	O
can	O
be	O
used	O
to	O
prove	O
decidability	O
of	O
S2S	O
,	O
the	O
MSO	O
theory	O
with	O
two	O
successors	O
,	O
and	O
this	O
can	O
be	O
extended	O
to	O
the	O
MSO	O
theory	O
of	O
graphs	O
with	O
bounded	O
(	O
given	O
a	O
fixed	O
bound	O
)	O
treewidth	O
.	O
</s>
