<s>
Automata	B-Application
theory	I-Application
is	O
the	O
study	O
of	O
abstract	B-Application
machines	I-Application
and	O
automata	O
,	O
as	O
well	O
as	O
the	O
computational	O
problems	O
that	O
can	O
be	O
solved	O
using	O
them	O
.	O
</s>
<s>
An	O
automaton	O
with	O
a	O
finite	O
number	O
of	O
states	B-Application
is	O
called	O
a	O
Finite	B-Architecture
Automaton	I-Architecture
(	O
FA	O
)	O
or	O
Finite-State	B-Architecture
Machine	I-Architecture
(	O
FSM	O
)	O
.	O
</s>
<s>
The	O
figure	O
on	O
the	O
right	O
illustrates	O
a	O
finite-state	B-Architecture
machine	I-Architecture
,	O
which	O
is	O
a	O
well-known	O
type	O
of	O
automaton	O
.	O
</s>
<s>
This	O
automaton	O
consists	O
of	O
states	B-Application
(	O
represented	O
in	O
the	O
figure	O
by	O
circles	O
)	O
and	O
transitions	O
(	O
represented	O
by	O
arrows	O
)	O
.	O
</s>
<s>
As	O
the	O
automaton	O
sees	O
a	O
symbol	O
of	O
input	O
,	O
it	O
makes	O
a	O
transition	O
(	O
or	O
jump	O
)	O
to	O
another	O
state	O
,	O
according	O
to	O
its	O
transition	B-Application
function	I-Application
,	O
which	O
takes	O
the	O
previous	O
state	O
and	O
current	O
input	O
symbol	O
as	O
its	O
arguments	O
.	O
</s>
<s>
Automata	B-Application
theory	I-Application
is	O
closely	O
related	O
to	O
formal	O
language	O
theory	O
.	O
</s>
<s>
Automata	O
play	O
a	O
major	O
role	O
in	O
the	O
theory	O
of	O
computation	O
,	O
compiler	O
construction	O
,	O
artificial	B-Application
intelligence	I-Application
,	O
parsing	B-Language
and	O
formal	O
verification	O
.	O
</s>
<s>
The	O
theory	O
of	O
abstract	O
automata	O
was	O
developed	O
in	O
the	O
mid-20th	O
century	O
in	O
connection	O
with	O
finite	B-Architecture
automata	I-Architecture
.	O
</s>
<s>
Automata	B-Application
theory	I-Application
was	O
initially	O
considered	O
a	O
branch	O
of	O
mathematical	O
systems	O
theory	O
,	O
studying	O
the	O
behavior	O
of	O
discrete-parameter	O
systems	O
.	O
</s>
<s>
Early	O
work	O
in	O
automata	B-Application
theory	I-Application
differed	O
from	O
previous	O
work	O
on	O
systems	O
by	O
using	O
abstract	O
algebra	O
to	O
describe	O
information	O
systems	O
rather	O
than	O
differential	O
calculus	O
to	O
describe	O
material	O
systems	O
.	O
</s>
<s>
The	O
theory	O
of	O
the	O
finite-state	B-Architecture
transducer	I-Architecture
was	O
developed	O
under	O
different	O
names	O
by	O
different	O
research	O
communities	O
.	O
</s>
<s>
The	O
earlier	O
concept	O
of	O
Turing	B-Architecture
machine	I-Architecture
was	O
also	O
included	O
in	O
the	O
discipline	O
along	O
with	O
new	O
forms	O
of	O
infinite-state	O
automata	O
,	O
such	O
as	O
pushdown	B-Application
automata	I-Application
.	O
</s>
<s>
With	O
the	O
publication	O
of	O
this	O
volume	O
,	O
"	O
automata	B-Application
theory	I-Application
emerged	O
as	O
a	O
relatively	O
autonomous	O
discipline	O
"	O
.	O
</s>
<s>
The	O
book	O
included	O
Kleene	O
's	O
description	O
of	O
the	O
set	O
of	O
regular	O
events	O
,	O
or	O
regular	B-General_Concept
languages	I-General_Concept
,	O
and	O
a	O
relatively	O
stable	O
measure	O
of	O
complexity	O
in	O
Turing	B-Architecture
machine	I-Architecture
programs	O
by	O
Shannon	O
.	O
</s>
<s>
The	O
study	O
of	O
linear	B-Application
bounded	I-Application
automata	I-Application
led	O
to	O
the	O
Myhill	O
–	O
Nerode	O
theorem	O
,	O
which	O
gives	O
a	O
necessary	O
and	O
sufficient	O
condition	O
for	O
a	O
formal	O
language	O
to	O
be	O
regular	O
,	O
and	O
an	O
exact	O
count	O
of	O
the	O
number	O
of	O
states	B-Application
in	O
a	O
minimal	O
machine	O
for	O
the	O
language	O
.	O
</s>
<s>
The	O
pumping	O
lemma	O
for	O
regular	B-General_Concept
languages	I-General_Concept
,	O
also	O
useful	O
in	O
regularity	O
proofs	O
,	O
was	O
proven	O
in	O
this	O
period	O
by	O
Michael	O
O	O
.	O
Rabin	O
and	O
Dana	O
Scott	O
,	O
along	O
with	O
the	O
computational	O
equivalence	O
of	O
deterministic	O
and	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
While	O
any	O
finite	B-Architecture
automaton	I-Architecture
can	O
be	O
simulated	O
using	O
a	O
universal	O
gate	O
set	O
,	O
this	O
requires	O
that	O
the	O
simulating	O
circuit	O
contain	O
loops	O
of	O
arbitrary	O
complexity	O
.	O
</s>
<s>
By	O
the	O
end	O
of	O
the	O
decade	O
,	O
automata	B-Application
theory	I-Application
came	O
to	O
be	O
seen	O
as	O
"	O
the	O
pure	O
mathematics	O
of	O
computer	O
science	O
"	O
.	O
</s>
<s>
An	O
automaton	O
has	O
a	O
set	O
of	O
states	B-Application
.	O
</s>
<s>
At	O
each	O
moment	O
during	O
a	O
run	O
of	O
the	O
automaton	O
,	O
the	O
automaton	O
is	O
in	O
one	O
of	O
its	O
states	B-Application
.	O
</s>
<s>
When	O
the	O
automaton	O
receives	O
new	O
input	O
it	O
moves	O
to	O
another	O
state	O
(	O
or	O
transitions	O
)	O
based	O
on	O
a	O
transition	B-Application
function	I-Application
that	O
takes	O
the	O
previous	O
state	O
and	O
current	O
input	O
symbol	O
as	O
parameters	O
.	O
</s>
<s>
The	O
automaton	O
reads	O
the	O
symbols	O
of	O
the	O
input	O
word	O
and	O
transitions	O
between	O
states	B-Application
until	O
the	O
word	O
is	O
read	O
completely	O
,	O
if	O
it	O
is	O
finite	O
in	O
length	O
,	O
at	O
which	O
point	O
the	O
automaton	O
halts	O
.	O
</s>
<s>
To	O
investigate	O
the	O
possible	O
state/input/output	O
sequences	O
in	O
an	O
automaton	O
using	O
formal	O
language	O
theory	O
,	O
a	O
machine	O
can	O
be	O
assigned	O
a	O
starting	O
state	O
and	O
a	O
set	O
of	O
accepting	B-Architecture
states	I-Architecture
.	O
</s>
<s>
Then	O
,	O
depending	O
on	O
whether	O
a	O
run	O
starting	O
from	O
the	O
starting	O
state	O
ends	O
in	O
an	O
accepting	B-Architecture
state	I-Architecture
,	O
the	O
automaton	O
can	O
be	O
said	O
to	O
accept	O
or	O
reject	O
an	O
input	O
sequence	O
.	O
</s>
<s>
is	O
a	O
set	O
of	O
states	B-Application
,	O
</s>
<s>
is	O
the	O
next-state	O
function	O
or	O
transition	B-Application
function	I-Application
mapping	O
state-input	O
pairs	O
to	O
successor	O
states	B-Application
,	O
</s>
<s>
If	O
is	O
finite	O
,	O
then	O
is	O
a	O
finite	B-Architecture
automaton	I-Architecture
.	O
</s>
<s>
A	O
sequence	O
of	O
states	B-Application
,	O
where	O
such	O
that	O
for	O
,	O
is	O
a	O
run	O
of	O
the	O
automaton	O
on	O
an	O
input	O
starting	O
from	O
state	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
at	O
first	O
the	O
automaton	O
is	O
at	O
the	O
start	B-Architecture
state	I-Architecture
,	O
and	O
receives	O
input	O
.	O
</s>
<s>
For	O
and	O
every	O
following	O
in	O
the	O
input	O
string	O
,	O
the	O
automaton	O
picks	O
the	O
next	O
state	O
according	O
to	O
the	O
transition	B-Application
function	I-Application
,	O
until	O
the	O
last	O
symbol	O
has	O
been	O
read	O
,	O
leaving	O
the	O
machine	O
in	O
the	O
final	O
state	O
of	O
the	O
run	O
,	O
.	O
</s>
<s>
The	O
transition	B-Application
function	I-Application
is	O
extended	O
inductively	O
into	O
to	O
describe	O
the	O
machine	O
's	O
behavior	O
when	O
fed	O
whole	O
input	O
words	O
.	O
</s>
<s>
For	O
the	O
empty	O
string	O
,	O
for	O
all	O
states	B-Application
,	O
and	O
for	O
strings	O
where	O
is	O
the	O
last	O
symbol	O
and	O
is	O
the	O
(	O
possibly	O
empty	O
)	O
rest	O
of	O
the	O
string	O
,	O
.	O
</s>
<s>
called	O
accept	B-Architecture
states	I-Architecture
.	O
</s>
<s>
A	O
word	O
is	O
an	O
accepting	O
word	O
for	O
the	O
automaton	O
if	O
,	O
that	O
is	O
,	O
if	O
after	O
consuming	O
the	O
whole	O
string	O
the	O
machine	O
is	O
in	O
an	O
accept	B-Architecture
state	I-Architecture
.	O
</s>
<s>
For	O
finite	B-Architecture
automata	I-Architecture
the	O
recognizable	O
languages	O
are	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Such	O
automata	O
are	O
called	O
ω-automata	B-General_Concept
.	O
</s>
<s>
Tree	O
input	O
:	O
The	O
input	O
may	O
be	O
a	O
tree	B-Application
of	I-Application
symbols	I-Application
instead	O
of	O
sequence	O
of	O
symbols	O
.	O
</s>
<s>
Such	O
an	O
automaton	O
is	O
called	O
a	O
tree	B-Application
automaton	I-Application
.	O
</s>
<s>
Such	O
an	O
automaton	O
is	O
called	O
an	O
infinite	B-Application
tree	I-Application
automaton	I-Application
.	O
</s>
<s>
Finite	B-Architecture
states	I-Architecture
:	O
An	O
automaton	O
that	O
contains	O
only	O
a	O
finite	O
number	O
of	O
states	B-Application
.	O
</s>
<s>
Infinite	O
states	B-Application
:	O
An	O
automaton	O
that	O
may	O
not	O
have	O
a	O
finite	O
number	O
of	O
states	B-Application
,	O
or	O
even	O
a	O
countable	O
number	O
of	O
states	B-Application
.	O
</s>
<s>
Stack	B-Application
memory	O
:	O
An	O
automaton	O
may	O
also	O
contain	O
some	O
extra	O
memory	O
in	O
the	O
form	O
of	O
a	O
stack	B-Application
in	O
which	O
symbols	O
can	O
be	O
pushed	O
and	O
popped	O
.	O
</s>
<s>
This	O
kind	O
of	O
automaton	O
is	O
called	O
a	O
pushdown	B-Application
automaton	I-Application
.	O
</s>
<s>
Queue	B-Application
memory	O
:	O
An	O
automaton	O
may	O
have	O
memory	O
in	O
the	O
form	O
of	O
a	O
queue	B-Application
.	O
</s>
<s>
Such	O
a	O
machine	O
is	O
called	O
queue	B-Application
machine	I-Application
and	O
is	O
Turing-complete	O
.	O
</s>
<s>
Some	O
machines	O
have	O
additional	O
working	O
tapes	O
,	O
including	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
,	O
linear	B-Application
bounded	I-Application
automaton	I-Application
,	O
and	O
log-space	O
transducer	O
.	O
</s>
<s>
Deterministic	O
:	O
For	O
a	O
given	O
current	O
state	O
and	O
an	O
input	O
symbol	O
,	O
if	O
an	O
automaton	O
can	O
only	O
jump	O
to	O
one	O
and	O
only	O
one	O
state	O
then	O
it	O
is	O
a	O
deterministic	B-Application
automaton	I-Application
.	O
</s>
<s>
Nondeterministic	O
:	O
An	O
automaton	O
that	O
,	O
after	O
reading	O
an	O
input	O
symbol	O
,	O
may	O
jump	O
into	O
any	O
of	O
a	O
number	O
of	O
states	B-Application
,	O
as	O
licensed	O
by	O
its	O
transition	O
relation	O
.	O
</s>
<s>
Notice	O
that	O
the	O
term	O
transition	B-Application
function	I-Application
is	O
replaced	O
by	O
transition	O
relation	O
:	O
The	O
automaton	O
non-deterministically	O
decides	O
to	O
jump	O
into	O
one	O
of	O
the	O
allowed	O
choices	O
.	O
</s>
<s>
Alternation	O
:	O
This	O
idea	O
is	O
quite	O
similar	O
to	O
tree	B-Application
automata	I-Application
but	O
orthogonal	O
.	O
</s>
<s>
Such	O
automata	O
are	O
called	O
alternating	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
Acceptance	O
of	O
infinite	O
words	O
:	O
an	O
ω-automaton	B-General_Concept
cannot	O
have	O
final	O
states	B-Application
,	O
as	O
infinite	O
words	O
never	O
terminate	O
.	O
</s>
<s>
Rather	O
,	O
acceptance	O
of	O
the	O
word	O
is	O
decided	O
by	O
looking	O
at	O
the	O
infinite	O
sequence	O
of	O
visited	O
states	B-Application
during	O
the	O
run	O
.	O
</s>
<s>
For	O
example	O
,	O
quantum	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
,	O
geometric	O
automata	O
and	O
metric	B-General_Concept
automata	I-General_Concept
have	O
probabilistic	O
acceptance	O
.	O
</s>
<s>
Automata	B-Application
theory	I-Application
is	O
a	O
subject	O
matter	O
that	O
studies	O
properties	O
of	O
various	O
types	O
of	O
automata	O
.	O
</s>
<s>
Automata	B-Application
theory	I-Application
also	O
studies	O
the	O
existence	O
or	O
nonexistence	O
of	O
any	O
effective	O
algorithms	O
to	O
solve	O
problems	O
similar	O
to	O
the	O
following	O
list	O
:	O
</s>
<s>
Is	O
it	O
possible	O
to	O
transform	O
a	O
given	O
non-deterministic	O
automaton	O
into	O
a	O
deterministic	B-Application
automaton	I-Application
without	O
changing	O
the	O
language	O
recognized	O
?	O
</s>
<s>
Normally	O
automata	B-Application
theory	I-Application
describes	O
the	O
states	B-Application
of	O
abstract	B-Application
machines	I-Application
but	O
there	O
are	O
discrete	O
automata	O
,	O
analog	B-Application
automata	I-Application
or	O
continuous	O
automata	O
,	O
or	O
hybrid	B-Application
discrete-continuous	I-Application
automata	I-Application
,	O
which	O
use	O
digital	O
data	O
,	O
analog	O
data	O
or	O
continuous	O
time	O
,	O
or	O
digital	O
and	O
analog	O
data	O
,	O
respectively	O
.	O
</s>
<s>
Each	O
model	O
in	O
automata	B-Application
theory	I-Application
plays	O
important	O
roles	O
in	O
several	O
applied	O
areas	O
.	O
</s>
<s>
Finite	B-Architecture
automata	I-Architecture
are	O
used	O
in	O
text	B-Device
processing	I-Device
,	O
compilers	O
,	O
and	O
hardware	B-General_Concept
design	I-General_Concept
.	O
</s>
<s>
Context-free	O
grammar	O
(	O
CFGs	O
)	O
are	O
used	O
in	O
programming	O
languages	O
and	O
artificial	B-Application
intelligence	I-Application
.	I-Application
</s>
<s>
Some	O
other	O
examples	O
which	O
could	O
be	O
explained	O
using	O
automata	B-Application
theory	I-Application
in	O
biology	O
include	O
mollusk	O
and	O
pine	O
cone	O
growth	O
and	O
pigmentation	O
patterns	O
.	O
</s>
<s>
Automata	O
also	O
appear	O
in	O
the	O
theory	O
of	O
finite	O
fields	O
:	O
the	O
set	O
of	O
irreducible	O
polynomials	O
that	O
can	O
be	O
written	O
as	O
composition	O
of	O
degree	O
two	O
polynomials	O
is	O
in	O
fact	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
Another	O
problem	O
for	O
which	O
automata	O
can	O
be	O
used	O
is	O
the	O
induction	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Automata	O
simulators	O
are	O
pedagogical	O
tools	O
used	O
to	O
teach	O
,	O
learn	O
and	O
research	O
automata	B-Application
theory	I-Application
.	O
</s>
<s>
The	O
mathematical	O
category	O
of	O
deterministic	B-Architecture
automata	I-Architecture
,	O
sequential	O
machines	O
or	O
sequential	O
automata	O
,	O
and	O
Turing	B-Architecture
machines	I-Architecture
with	O
automata	O
homomorphisms	O
defining	O
the	O
arrows	O
between	O
automata	O
is	O
a	O
Cartesian	B-Application
closed	I-Application
category	I-Application
,	O
it	O
has	O
both	O
categorical	O
limits	O
and	O
colimits	O
.	O
</s>
