<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
automata	B-Application
theory	I-Application
,	O
a	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
a	O
theoretical	O
machine	O
which	O
either	O
accepts	O
or	O
rejects	O
infinite	O
inputs	O
.	O
</s>
<s>
Some	O
states	O
are	O
accepting	B-Architecture
states	I-Architecture
and	O
one	O
state	O
is	O
the	O
start	B-Architecture
state	I-Architecture
.	O
</s>
<s>
The	O
machine	O
accepts	O
an	O
input	O
if	O
and	O
only	O
if	O
it	O
will	O
pass	O
through	O
an	O
accepting	B-Architecture
state	I-Architecture
infinitely	O
many	O
times	O
as	O
it	O
reads	O
the	O
input	O
.	O
</s>
<s>
A	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
,	O
later	O
referred	O
to	O
just	O
as	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
,	O
has	O
a	O
transition	O
function	O
which	O
may	O
have	O
multiple	O
outputs	O
,	O
leading	O
to	O
many	O
possible	O
paths	O
for	O
the	O
same	O
input	O
;	O
it	O
accepts	O
an	O
infinite	O
input	O
if	O
and	O
only	O
if	O
some	O
possible	O
path	O
is	O
accepting	O
.	O
</s>
<s>
Deterministic	O
and	O
non-deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
generalize	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
and	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
to	O
infinite	O
inputs	O
.	O
</s>
<s>
Each	O
are	O
types	O
of	O
ω-automata	B-General_Concept
.	O
</s>
<s>
Büchi	B-General_Concept
automata	I-General_Concept
recognize	O
the	O
ω-regular	O
languages	O
,	O
the	O
infinite	O
word	O
version	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Büchi	B-General_Concept
automata	I-General_Concept
are	O
often	O
used	O
in	O
model	B-Application
checking	I-Application
as	O
an	O
automata-theoretic	O
version	O
of	O
a	O
formula	O
in	O
linear	O
temporal	O
logic	O
.	O
</s>
<s>
Formally	O
,	O
a	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
a	O
tuple	O
A	O
=(	O
Q	O
,	O
Σ	O
,	O
δ	O
,	O
q0	O
,	O
F	O
)	O
that	O
consists	O
of	O
the	O
following	O
components	O
:	O
</s>
<s>
In	O
a	O
(	O
non-deterministic	O
)	O
Büchi	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
,	O
and	O
the	O
single	O
initial	O
state	O
q0	O
is	O
replaced	O
by	O
a	O
set	O
I	O
of	O
initial	O
states	O
.	O
</s>
<s>
Generally	O
,	O
the	O
term	O
Büchi	B-General_Concept
automaton	I-General_Concept
without	O
qualifier	O
refers	O
to	O
non-deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
For	O
more	O
comprehensive	O
formalism	O
see	O
also	O
ω-automaton	B-General_Concept
.	O
</s>
<s>
The	O
set	O
of	O
Büchi	B-General_Concept
automata	I-General_Concept
is	O
closed	O
under	O
the	O
following	O
operations	O
.	O
</s>
<s>
Let	O
and	O
be	O
Büchi	B-General_Concept
automata	I-General_Concept
and	O
be	O
a	O
finite	B-Architecture
automaton	I-Architecture
.	O
</s>
<s>
Proof	O
:	O
The	O
Büchi	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
is	O
constructed	O
in	O
two	O
stages	O
.	O
</s>
<s>
First	O
,	O
we	O
construct	O
a	O
finite	B-Architecture
automaton	I-Architecture
A	O
 '	O
such	O
that	O
A	O
 '	O
also	O
recognizes	O
but	O
there	O
are	O
no	O
incoming	O
transitions	O
to	O
initial	O
states	O
of	O
A	O
 '	O
.	O
</s>
<s>
Second	O
,	O
we	O
will	O
construct	O
the	O
Büchi	B-General_Concept
automaton	I-General_Concept
A	O
"	O
that	O
recognize	O
by	O
adding	O
a	O
loop	O
back	O
to	O
the	O
initial	O
state	O
of	O
A	O
 '	O
.	O
</s>
<s>
Proof	O
:	O
The	O
proof	O
is	O
presented	O
here	B-Application
.	O
</s>
<s>
Büchi	B-General_Concept
automata	I-General_Concept
recognize	O
the	O
ω-regular	O
languages	O
.	O
</s>
<s>
Using	O
the	O
definition	O
of	O
ω-regular	O
language	O
and	O
the	O
above	O
closure	O
properties	O
of	O
Büchi	B-General_Concept
automata	I-General_Concept
,	O
it	O
can	O
be	O
easily	O
shown	O
that	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
can	O
be	O
constructed	O
such	O
that	O
it	O
recognizes	O
any	O
given	O
ω-regular	O
language	O
.	O
</s>
<s>
For	O
converse	O
,	O
see	O
construction	O
of	O
a	O
ω-regular	O
language	O
for	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
class	O
of	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
does	O
not	O
suffice	O
to	O
encompass	O
all	O
omega-regular	O
languages	O
.	O
</s>
<s>
In	O
particular	O
,	O
there	O
is	O
no	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
the	O
language	O
,	O
which	O
contains	O
exactly	O
words	O
in	O
which	O
1	O
occurs	O
only	O
finitely	O
many	O
times	O
.	O
</s>
<s>
We	O
can	O
demonstrate	O
it	O
by	O
contradiction	O
that	O
no	O
such	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
exists	O
.	O
</s>
<s>
Let	O
us	O
suppose	O
A	O
is	O
a	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
that	O
recognizes	O
with	O
final	O
state	O
set	O
F	O
.	O
A	O
accepts	O
.	O
</s>
<s>
The	O
class	O
of	O
languages	O
recognizable	O
by	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
is	O
characterized	O
by	O
the	O
following	O
lemma	O
.	O
</s>
<s>
Lemma	O
:	O
An	O
ω-language	O
is	O
recognizable	O
by	O
a	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
if	O
it	O
is	O
the	O
limit	O
language	O
of	O
some	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
Proof	O
:	O
Any	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
can	O
be	O
viewed	O
as	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
and	O
vice	O
versa	O
,	O
since	O
both	O
types	O
of	O
automaton	O
are	O
defined	O
as	O
5-tuple	O
of	O
the	O
same	O
components	O
,	O
only	O
the	O
interpretation	O
of	O
acceptance	O
condition	O
is	O
different	O
.	O
</s>
<s>
Model	B-Application
checking	I-Application
of	O
finite	B-Architecture
state	I-Architecture
systems	O
can	O
often	O
be	O
translated	O
into	O
various	O
operations	O
on	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
the	O
following	O
are	O
some	O
useful	O
operations	O
for	O
the	O
applications	O
of	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Since	O
deterministic	O
Büchi	B-General_Concept
automata	I-General_Concept
are	O
strictly	O
less	O
expressive	O
than	O
non-deterministic	O
automata	O
,	O
there	O
can	O
not	O
be	O
an	O
algorithm	O
for	O
determinization	O
of	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
But	O
,	O
McNaughton	B-Application
's	I-Application
Theorem	I-Application
and	O
Safra	O
's	O
construction	O
provide	O
algorithms	O
that	O
can	O
translate	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
into	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
or	O
deterministic	O
Rabin	O
automaton	O
.	O
</s>
<s>
The	O
language	O
recognized	O
by	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
non-empty	O
if	O
and	O
only	O
if	O
there	O
is	O
a	O
final	O
state	O
that	O
is	O
both	O
reachable	O
from	O
the	O
initial	O
state	O
,	O
and	O
lies	O
on	O
a	O
cycle	O
.	O
</s>
<s>
An	O
effective	O
algorithm	O
that	O
can	O
check	O
emptiness	O
of	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
:	O
</s>
<s>
Run	O
a	O
search	O
(	O
e.g.	O
,	O
the	O
depth-first	B-Algorithm
search	I-Algorithm
)	O
to	O
find	O
which	O
SCCs	O
are	O
reachable	O
from	O
the	O
initial	O
state	O
.	O
</s>
<s>
The	O
algorithm	O
for	O
minimizing	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
also	O
correctly	O
minimizes	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
The	O
algorithm	O
does	O
not	O
guarantee	O
minimum	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
However	O
,	O
the	O
algorithms	O
for	O
minimizing	B-General_Concept
deterministic	I-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
does	O
not	O
work	O
for	O
deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
A	O
translation	O
from	O
a	O
Linear	O
temporal	O
logic	O
formula	O
to	O
a	O
generalized	B-Application
Büchi	I-Application
automaton	I-Application
is	O
given	O
here	B-Application
.	O
</s>
<s>
And	O
,	O
a	O
translation	O
from	O
a	O
generalized	B-Application
Büchi	I-Application
automaton	I-Application
to	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
is	O
presented	O
above	O
.	O
</s>
<s>
A	O
given	O
Muller	B-General_Concept
automaton	I-General_Concept
can	O
be	O
transformed	O
into	O
an	O
equivalent	O
Büchi	B-General_Concept
automaton	I-General_Concept
with	O
the	O
following	O
automata	O
construction	O
.	O
</s>
<s>
Let	O
's	O
suppose	O
A	O
=	O
(	O
Q	O
,	O
Σ	O
,	O
∆	O
,	O
Q0	O
,	O
{	O
F0	O
,...,	O
Fn}	O
)	O
is	O
a	O
Muller	B-General_Concept
automaton	I-General_Concept
,	O
where	O
F0	O
,...,	O
Fn	O
are	O
sets	O
of	O
accepting	B-Architecture
states	I-Architecture
.	O
</s>
<s>
The	O
Büchi	B-General_Concept
automaton	I-General_Concept
A	O
 '	O
simulates	O
the	O
Muller	B-General_Concept
automaton	I-General_Concept
A	O
as	O
follows	O
:	O
At	O
the	O
beginning	O
of	O
the	O
input	O
word	O
,	O
the	O
execution	O
of	O
A	O
follows	O
the	O
execution	O
of	O
A	O
,	O
since	O
initial	O
states	O
are	O
same	O
and	O
∆	O
 '	O
contains	O
∆	O
.	O
</s>
<s>
This	O
construction	O
closely	O
follows	O
the	O
first	O
part	O
of	O
the	O
proof	O
of	O
McNaughton	B-Application
's	I-Application
Theorem	I-Application
.	O
</s>
<s>
Let	O
the	O
given	O
Kripke	B-Application
structure	I-Application
be	O
defined	O
by	O
where	O
Q	O
is	O
the	O
set	O
of	O
states	O
,	O
is	O
the	O
set	O
of	O
initial	O
states	O
,	O
R	O
is	O
a	O
relation	O
between	O
two	O
states	O
also	O
interpreted	O
as	O
an	O
edge	O
,	O
L	O
is	O
the	O
label	O
for	O
the	O
state	O
and	O
AP	O
are	O
the	O
set	O
of	O
atomic	O
propositions	O
that	O
form	O
L	O
.	O
</s>
<s>
The	O
Büchi	B-General_Concept
automaton	I-General_Concept
will	O
have	O
the	O
following	O
characteristics	O
:	O
</s>
<s>
Note	O
however	O
that	O
there	O
is	O
a	O
difference	O
in	O
the	O
interpretation	O
between	O
Kripke	B-Application
structures	I-Application
and	O
Büchi	B-General_Concept
automata	I-General_Concept
.	O
</s>
