<s>
In	O
the	O
theory	O
of	O
computation	O
,	O
a	O
branch	O
of	O
theoretical	O
computer	O
science	O
,	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
—	O
also	O
known	O
as	O
deterministic	O
finite	O
acceptor	O
(	O
DFA	B-General_Concept
)	O
,	O
deterministic	B-General_Concept
finite-state	I-General_Concept
machine	I-General_Concept
(	O
DFSM	O
)	O
,	O
or	O
deterministic	O
finite-state	B-Architecture
automaton	I-Architecture
(	O
DFSA	B-General_Concept
)	O
—	O
is	O
a	O
finite-state	B-Architecture
machine	I-Architecture
that	O
accepts	O
or	O
rejects	O
a	O
given	O
string	O
of	O
symbols	O
,	O
by	O
running	O
through	O
a	O
state	O
sequence	O
uniquely	O
determined	O
by	O
the	O
string	O
.	O
</s>
<s>
In	O
search	O
of	O
the	O
simplest	O
models	O
to	O
capture	O
finite-state	B-Architecture
machines	I-Architecture
,	O
Warren	O
McCulloch	O
and	O
Walter	O
Pitts	O
were	O
among	O
the	O
first	O
researchers	O
to	O
introduce	O
a	O
concept	O
similar	O
to	O
finite	B-Architecture
automata	I-Architecture
in	O
1943	O
.	O
</s>
<s>
The	O
figure	O
illustrates	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
using	O
a	O
state	B-Application
diagram	I-Application
.	O
</s>
<s>
In	O
this	O
example	O
automaton	O
,	O
there	O
are	O
three	O
states	B-Application
:	O
S0	O
,	O
S1	O
,	O
and	O
S2	O
(	O
denoted	O
graphically	O
by	O
circles	O
)	O
.	O
</s>
<s>
Upon	O
reading	O
a	O
symbol	O
,	O
a	O
DFA	B-General_Concept
jumps	O
deterministically	O
from	O
one	O
state	O
to	O
another	O
by	O
following	O
the	O
transition	O
arrow	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
has	O
a	O
start	B-Architecture
state	I-Architecture
(	O
denoted	O
graphically	O
by	O
an	O
arrow	O
coming	O
in	O
from	O
nowhere	O
)	O
where	O
computations	O
begin	O
,	O
and	O
a	O
set	O
of	O
accept	B-Architecture
states	I-Architecture
(	O
denoted	O
graphically	O
by	O
a	O
double	O
circle	O
)	O
which	O
help	O
define	O
when	O
a	O
computation	O
is	O
successful	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
is	O
defined	O
as	O
an	O
abstract	O
mathematical	O
concept	O
,	O
but	O
is	O
often	O
implemented	O
in	O
hardware	O
and	O
software	O
for	O
solving	O
various	O
specific	O
problems	O
such	O
as	O
lexical	B-Application
analysis	I-Application
and	O
pattern	B-Language
matching	I-Language
.	O
</s>
<s>
For	O
example	O
,	O
a	O
DFA	B-General_Concept
can	O
model	O
software	O
that	O
decides	O
whether	O
or	O
not	O
online	O
user	O
input	O
such	O
as	O
email	O
addresses	O
are	O
syntactically	O
valid	O
.	O
</s>
<s>
DFAs	B-General_Concept
have	O
been	O
generalized	O
to	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
NFA	O
)	O
which	O
may	O
have	O
several	O
arrows	O
of	O
the	O
same	O
label	O
starting	O
from	O
a	O
state	O
.	O
</s>
<s>
Using	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
method	O
,	O
every	O
NFA	O
can	O
be	O
translated	O
to	O
a	O
DFA	B-General_Concept
that	O
recognizes	O
the	O
same	O
language	O
.	O
</s>
<s>
DFAs	B-General_Concept
,	O
and	O
NFAs	O
as	O
well	O
,	O
recognize	O
exactly	O
the	O
set	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
The	O
automaton	O
accepts	O
the	O
string	O
if	O
a	O
sequence	O
of	O
states	B-Application
,	O
,	O
exists	O
in	O
with	O
the	O
following	O
conditions	O
:	O
</s>
<s>
In	O
words	O
,	O
the	O
first	O
condition	O
says	O
that	O
the	O
machine	O
starts	O
in	O
the	O
start	B-Architecture
state	I-Architecture
.	O
</s>
<s>
The	O
second	O
condition	O
says	O
that	O
given	O
each	O
character	O
of	O
string	O
,	O
the	O
machine	O
will	O
transition	O
from	O
state	O
to	O
state	O
according	O
to	O
the	O
transition	B-Application
function	I-Application
.	O
</s>
<s>
The	O
last	O
condition	O
says	O
that	O
the	O
machine	O
accepts	O
if	O
the	O
last	O
input	O
of	O
causes	O
the	O
machine	O
to	O
halt	O
in	O
one	O
of	O
the	O
accepting	B-Architecture
states	I-Architecture
.	O
</s>
<s>
A	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
without	O
accept	B-Architecture
states	I-Architecture
and	O
without	O
a	O
starting	O
state	O
is	O
known	O
as	O
a	O
transition	B-Application
system	I-Application
or	O
semiautomaton	O
.	O
</s>
<s>
For	O
more	O
comprehensive	O
introduction	O
of	O
the	O
formal	O
definition	O
see	O
automata	B-Application
theory	I-Application
.	O
</s>
<s>
The	O
following	O
example	O
is	O
of	O
a	O
DFA	B-General_Concept
,	O
with	O
a	O
binary	O
alphabet	O
,	O
which	O
requires	O
that	O
the	O
input	O
contains	O
an	O
even	O
number	O
of	O
0s	O
.	O
</s>
<s>
is	O
defined	O
by	O
the	O
following	O
state	B-Application
transition	I-Application
table	I-Application
:	O
</s>
<s>
If	O
the	O
input	O
did	O
contain	O
an	O
even	O
number	O
of	O
0s	O
,	O
will	O
finish	O
in	O
state	O
,	O
an	O
accepting	B-Architecture
state	I-Architecture
,	O
so	O
the	O
input	O
string	O
will	O
be	O
accepted	O
.	O
</s>
<s>
The	O
language	O
recognized	O
by	O
is	O
the	O
regular	B-General_Concept
language	I-General_Concept
given	O
by	O
the	O
regular	B-Language
expression	I-Language
(	O
1*	O
)	O
(	O
0	O
(	O
1*	O
)	O
0	O
(	O
1*	O
)	O
)	O
*	O
,	O
where	O
*	O
is	O
the	O
Kleene	O
star	O
,	O
e.g.	O
,	O
1*	O
denotes	O
any	O
number	O
(	O
possibly	O
zero	O
)	O
of	O
consecutive	O
ones	O
.	O
</s>
<s>
According	O
to	O
the	O
above	O
definition	O
,	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
are	O
always	O
complete	O
:	O
they	O
define	O
from	O
each	O
state	O
a	O
transition	O
for	O
each	O
input	O
symbol	O
.	O
</s>
<s>
While	O
this	O
is	O
the	O
most	O
common	O
definition	O
,	O
some	O
authors	O
use	O
the	O
term	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
for	O
a	O
slightly	O
different	O
notion	O
:	O
an	O
automaton	O
that	O
defines	O
at	O
most	O
one	O
transition	O
for	O
each	O
state	O
and	O
each	O
input	O
symbol	O
;	O
the	O
transition	B-Application
function	I-Application
is	O
allowed	O
to	O
be	O
partial	B-Algorithm
.	O
</s>
<s>
A	O
local	O
automaton	O
is	O
a	O
DFA	B-General_Concept
,	O
not	O
necessarily	O
complete	O
,	O
for	O
which	O
all	O
edges	O
with	O
the	O
same	O
label	O
lead	O
to	O
a	O
single	O
vertex	O
.	O
</s>
<s>
When	O
the	O
start	B-Architecture
state	I-Architecture
and	O
accept	B-Architecture
states	I-Architecture
are	O
ignored	O
,	O
a	O
DFA	B-General_Concept
of	O
states	B-Application
and	O
an	O
alphabet	O
of	O
size	O
can	O
be	O
seen	O
as	O
a	O
digraph	O
of	O
vertices	O
in	O
which	O
all	O
vertices	O
have	O
out-arcs	O
labeled	O
(	O
a	O
-out	O
digraph	O
)	O
.	O
</s>
<s>
In	O
a	O
random	O
DFA	B-General_Concept
,	O
the	O
maximum	O
number	O
of	O
vertices	O
reachable	O
from	O
one	O
vertex	O
is	O
very	O
close	O
to	O
the	O
number	O
of	O
vertices	O
in	O
the	O
largest	O
SCC	O
with	O
high	O
probability	O
.	O
</s>
<s>
If	O
DFAs	B-General_Concept
recognize	O
the	O
languages	O
that	O
are	O
obtained	O
by	O
applying	O
an	O
operation	O
on	O
the	O
DFA	B-General_Concept
recognizable	O
languages	O
then	O
DFAs	B-General_Concept
are	O
said	O
to	O
be	O
closed	O
under	O
the	O
operation	O
.	O
</s>
<s>
The	O
DFAs	B-General_Concept
are	O
closed	O
under	O
the	O
following	O
operations	O
.	O
</s>
<s>
For	O
each	O
operation	O
,	O
an	O
optimal	O
construction	O
with	O
respect	O
to	O
the	O
number	O
of	O
states	B-Application
has	O
been	O
determined	O
in	O
state	B-General_Concept
complexity	I-General_Concept
research	O
.	O
</s>
<s>
Since	O
DFAs	B-General_Concept
are	O
equivalent	B-General_Concept
to	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
NFA	O
)	O
,	O
these	O
closures	O
may	O
also	O
be	O
proved	O
using	O
closure	O
properties	O
of	O
NFA	O
.	O
</s>
<s>
A	O
run	O
of	O
a	O
given	O
DFA	B-General_Concept
can	O
be	O
seen	O
as	O
a	O
sequence	O
of	O
compositions	O
of	O
a	O
very	O
general	O
formulation	O
of	O
the	O
transition	B-Application
function	I-Application
with	O
itself	O
.	O
</s>
<s>
For	O
a	O
given	O
input	O
symbol	O
,	O
one	O
may	O
construct	O
a	O
transition	B-Application
function	I-Application
by	O
defining	O
for	O
all	O
.	O
</s>
<s>
(	O
This	O
trick	O
is	O
called	O
currying	B-Application
.	O
)	O
</s>
<s>
One	O
may	O
then	O
consider	O
the	O
result	O
of	O
function	B-Application
composition	I-Application
repeatedly	O
applied	O
to	O
the	O
various	O
functions	O
,	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Given	O
a	O
pair	O
of	O
letters	O
,	O
one	O
may	O
define	O
a	O
new	O
function	O
,	O
where	O
denotes	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
A	O
run	O
of	O
the	O
DFA	B-General_Concept
is	O
a	O
sequence	O
of	O
compositions	O
of	O
with	O
itself	O
.	O
</s>
<s>
Repeated	O
function	B-Application
composition	I-Application
forms	O
a	O
monoid	O
.	O
</s>
<s>
The	O
construction	O
can	O
also	O
be	O
reversed	O
:	O
given	O
a	O
,	O
one	O
can	O
reconstruct	O
a	O
,	O
and	O
so	O
the	O
two	O
descriptions	O
are	O
equivalent	B-General_Concept
.	O
</s>
<s>
DFAs	B-General_Concept
are	O
one	O
of	O
the	O
most	O
practical	O
models	O
of	O
computation	O
,	O
since	O
there	O
is	O
a	O
trivial	O
linear	O
time	O
,	O
constant-space	O
,	O
online	B-Algorithm
algorithm	I-Algorithm
to	O
simulate	O
a	O
DFA	B-General_Concept
on	O
a	O
stream	O
of	O
input	O
.	O
</s>
<s>
Also	O
,	O
there	O
are	O
efficient	O
algorithms	O
to	O
find	O
a	O
DFA	B-General_Concept
recognizing	O
:	O
</s>
<s>
the	O
complement	O
of	O
the	O
language	O
recognized	O
by	O
a	O
given	O
DFA	B-General_Concept
.	O
</s>
<s>
the	O
union/intersection	O
of	O
the	O
languages	O
recognized	O
by	O
two	O
given	O
DFAs	B-General_Concept
.	O
</s>
<s>
Because	O
DFAs	B-General_Concept
can	O
be	O
reduced	O
to	O
a	O
canonical	O
form	O
(	O
minimal	B-General_Concept
DFAs	I-General_Concept
)	O
,	O
there	O
are	O
also	O
efficient	O
algorithms	O
to	O
determine	O
:	O
</s>
<s>
DFAs	B-General_Concept
are	O
equivalent	B-General_Concept
in	O
computing	O
power	O
to	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
NFAs	O
)	O
.	O
</s>
<s>
This	O
is	O
because	O
,	O
firstly	O
any	O
DFA	B-General_Concept
is	O
also	O
an	O
NFA	O
,	O
so	O
an	O
NFA	O
can	O
do	O
what	O
a	O
DFA	B-General_Concept
can	O
do	O
.	O
</s>
<s>
Also	O
,	O
given	O
an	O
NFA	O
,	O
using	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
one	O
can	O
build	O
a	O
DFA	B-General_Concept
that	O
recognizes	O
the	O
same	O
language	O
as	O
the	O
NFA	O
,	O
although	O
the	O
DFA	B-General_Concept
could	O
have	O
exponentially	O
larger	O
number	O
of	O
states	B-Application
than	O
the	O
NFA.Sakarovitch	O
(	O
2009	O
)	O
p.105Lawson	O
(	O
2004	O
)	O
p.63	O
However	O
,	O
even	O
though	O
NFAs	O
are	O
computationally	O
equivalent	B-General_Concept
to	O
DFAs	B-General_Concept
,	O
the	O
above	O
mentioned	O
problems	O
are	O
not	O
necessarily	O
solved	O
efficiently	O
also	O
for	O
NFAs	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
is	O
universal	O
if	O
and	O
only	O
if	O
all	O
states	B-Application
are	O
final	O
states	B-Application
,	O
but	O
this	O
does	O
not	O
hold	O
for	O
NFAs	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
finite-state	B-Architecture
automata	I-Architecture
are	O
of	O
strictly	O
limited	O
power	O
in	O
the	O
languages	O
they	O
can	O
recognize	O
;	O
many	O
simple	O
languages	O
,	O
including	O
any	O
problem	O
that	O
requires	O
more	O
than	O
constant	O
space	O
to	O
solve	O
,	O
cannot	O
be	O
recognized	O
by	O
a	O
DFA	B-General_Concept
.	O
</s>
<s>
The	O
classic	O
example	O
of	O
a	O
simply	O
described	O
language	O
that	O
no	O
DFA	B-General_Concept
can	O
recognize	O
is	O
bracket	O
or	O
Dyck	O
language	O
,	O
i.e.	O
,	O
the	O
language	O
that	O
consists	O
of	O
properly	O
paired	O
brackets	O
such	O
as	O
word	O
"	O
(( )	O
(	O
)	O
)	O
"	O
.	O
</s>
<s>
Intuitively	O
,	O
no	O
DFA	B-General_Concept
can	O
recognize	O
the	O
Dyck	O
language	O
because	O
DFAs	B-General_Concept
are	O
not	O
capable	O
of	O
counting	O
:	O
a	O
DFA-like	O
automaton	O
needs	O
to	O
have	O
a	O
state	O
to	O
represent	O
any	O
possible	O
number	O
of	O
"	O
currently	O
open	O
"	O
parentheses	O
,	O
meaning	O
it	O
would	O
need	O
an	O
unbounded	O
number	O
of	O
states	B-Application
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
positive	O
words	O
and	O
a	O
set	O
of	O
negative	O
words	O
one	O
can	O
construct	O
a	O
DFA	B-General_Concept
that	O
accepts	O
all	O
words	O
from	O
and	O
rejects	O
all	O
words	O
from	O
:	O
this	O
problem	O
is	O
called	O
DFA	B-General_Concept
identification	O
(	O
synthesis	O
,	O
learning	O
)	O
.	O
</s>
<s>
While	O
some	O
DFA	B-General_Concept
can	O
be	O
constructed	O
in	O
linear	O
time	O
,	O
the	O
problem	O
of	O
identifying	O
a	O
DFA	B-General_Concept
with	O
the	O
minimal	O
number	O
of	O
states	B-Application
is	O
NP-complete	O
.	O
</s>
<s>
The	O
first	O
algorithm	O
for	O
minimal	B-General_Concept
DFA	I-General_Concept
identification	O
has	O
been	O
proposed	O
by	O
Trakhtenbrot	O
and	O
Barzdin	O
and	O
is	O
called	O
the	O
TB-algorithm	O
.	O
</s>
<s>
However	O
,	O
Traxbar	O
does	O
not	O
guarantee	O
the	O
minimality	O
of	O
the	O
constructed	O
DFA	B-General_Concept
.	O
</s>
<s>
In	O
his	O
work	O
E.M.	O
Gold	O
also	O
proposed	O
a	O
heuristic	O
algorithm	O
for	O
minimal	B-General_Concept
DFA	I-General_Concept
identification	O
.	O
</s>
<s>
Gold	O
's	O
algorithm	O
assumes	O
that	O
and	O
contain	O
a	O
characteristic	O
set	O
of	O
the	O
regular	B-General_Concept
language	I-General_Concept
;	O
otherwise	O
,	O
the	O
constructed	O
DFA	B-General_Concept
will	O
be	O
inconsistent	O
either	O
with	O
or	O
.	O
</s>
<s>
Other	O
notable	O
DFA	B-General_Concept
identification	O
algorithms	O
include	O
the	O
RPNI	O
algorithm	O
,	O
the	O
Blue-Fringe	O
evidence-driven	O
state-merging	O
algorithm	O
,	O
</s>
<s>
Another	O
research	O
direction	O
is	O
the	O
application	O
of	O
evolutionary	B-Algorithm
algorithms	I-Algorithm
:	O
the	O
smart	O
state	O
labeling	O
evolutionary	B-Algorithm
algorithm	I-Algorithm
allowed	O
to	O
solve	O
a	O
modified	O
DFA	B-General_Concept
identification	O
problem	O
in	O
which	O
the	O
training	O
data	O
(	O
sets	O
and	O
)	O
is	O
noisy	O
in	O
the	O
sense	O
that	O
some	O
words	O
are	O
attributed	O
to	O
wrong	O
classes	O
.	O
</s>
<s>
Yet	O
another	O
step	O
forward	O
is	O
due	O
to	O
application	O
of	O
SAT	B-Algorithm
solvers	O
by	O
Marjin	O
J	O
.	O
H	O
.	O
Heule	O
and	O
S	O
.	O
Verwer	O
:	O
the	O
minimal	B-General_Concept
DFA	I-General_Concept
identification	O
problem	O
is	O
reduced	O
to	O
deciding	O
the	O
satisfiability	O
of	O
a	O
Boolean	O
formula	O
.	O
</s>
<s>
The	O
main	O
idea	O
is	O
to	O
build	O
a	O
augmented	O
prefix-tree	O
acceptor	O
(	O
a	O
trie	B-General_Concept
containing	O
all	O
input	O
words	O
with	O
corresponding	O
labels	O
)	O
based	O
on	O
the	O
input	O
sets	O
and	O
reduce	O
the	O
problem	O
of	O
finding	O
a	O
DFA	B-General_Concept
with	O
states	B-Application
to	O
coloring	O
the	O
tree	O
vertices	O
with	O
states	B-Application
in	O
such	O
a	O
way	O
that	O
when	O
vertices	O
with	O
one	O
color	O
are	O
merged	O
to	O
one	O
state	O
,	O
the	O
generated	O
automaton	O
is	O
deterministic	O
and	O
complies	O
with	O
and	O
.	O
</s>
<s>
Though	O
this	O
approach	O
allows	O
finding	O
the	O
minimal	B-General_Concept
DFA	I-General_Concept
,	O
it	O
suffers	O
from	O
exponential	O
blow-up	O
of	O
execution	O
time	O
when	O
the	O
size	O
of	O
input	O
data	O
increases	O
.	O
</s>
<s>
Therefore	O
,	O
Heule	O
and	O
Verwer	O
's	O
initial	O
algorithm	O
has	O
later	O
been	O
augmented	O
with	O
making	O
several	O
steps	O
of	O
the	O
EDSM	O
algorithm	O
prior	O
to	O
SAT	B-Algorithm
solver	O
execution	O
:	O
the	O
DFASAT	O
algorithm	O
.	O
</s>
<s>
by	O
means	O
of	O
new	O
symmetry	O
breaking	O
predicates	O
based	O
on	O
the	O
breadth-first	B-Algorithm
search	I-Algorithm
algorithm	O
:	O
</s>
<s>
the	O
sought	O
DFA	B-General_Concept
's	O
states	B-Application
are	O
constrained	O
to	O
be	O
numbered	O
according	O
to	O
the	O
BFS	B-Algorithm
algorithm	I-Algorithm
launched	O
from	O
the	O
initial	O
state	O
.	O
</s>
<s>
are	O
almost	O
exactly	O
equivalent	B-General_Concept
to	O
DFAs	B-General_Concept
.	O
</s>
<s>
is	O
a	O
finite	O
set	O
of	O
states	B-Application
;	O
</s>
<s>
is	O
a	O
function	O
called	O
the	O
transition	B-Application
function	I-Application
,	O
R	O
is	O
a	O
right	O
movement	O
(	O
a	O
right	O
shift	O
)	O
;	O
</s>
<s>
The	O
machine	O
always	O
accepts	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
the	O
one	O
element	O
set	O
of	O
final	O
states	B-Application
:	O
.	O
</s>
