<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
more	O
precisely	O
,	O
in	O
the	O
theory	O
of	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
DFA	B-General_Concept
)	O
,	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
or	O
reset	B-General_Concept
sequence	I-General_Concept
is	O
a	O
word	O
in	O
the	O
input	O
alphabet	O
of	O
the	O
DFA	B-General_Concept
that	O
sends	O
any	O
state	O
of	O
the	O
DFA	B-General_Concept
to	O
one	O
and	O
the	O
same	O
state	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
an	O
ensemble	O
of	O
copies	O
of	O
the	O
DFA	B-General_Concept
are	O
each	O
started	O
in	O
different	O
states	O
,	O
and	O
all	O
of	O
the	O
copies	O
process	O
the	O
synchronizing	B-General_Concept
word	I-General_Concept
,	O
they	O
will	O
all	O
end	O
up	O
in	O
the	O
same	O
state	O
.	O
</s>
<s>
Not	O
every	O
DFA	B-General_Concept
has	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
;	O
for	O
instance	O
,	O
a	O
DFA	B-General_Concept
with	O
two	O
states	O
,	O
one	O
for	O
words	O
of	O
even	O
length	O
and	O
one	O
for	O
words	O
of	O
odd	O
length	O
,	O
can	O
never	O
be	O
synchronized	O
.	O
</s>
<s>
Given	O
a	O
DFA	B-General_Concept
,	O
the	O
problem	O
of	O
determining	O
if	O
it	O
has	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
can	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
a	O
theorem	O
due	O
to	O
Ján	O
Černý	O
.	O
</s>
<s>
A	O
simple	O
approach	O
considers	O
the	O
power	O
set	O
of	O
states	O
of	O
the	O
DFA	B-General_Concept
,	O
and	O
builds	O
a	O
directed	O
graph	O
where	O
nodes	O
belong	O
to	O
the	O
power	O
set	O
,	O
and	O
a	O
directed	O
edge	O
describes	O
the	O
action	O
of	O
the	O
transition	O
function	O
.	O
</s>
<s>
A	O
path	O
from	O
the	O
node	O
of	O
all	O
states	O
to	O
a	O
singleton	O
state	O
shows	O
the	O
existence	O
of	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
.	O
</s>
<s>
A	O
polynomial	O
algorithm	O
results	O
however	O
,	O
due	O
to	O
a	O
theorem	O
of	O
Černý	O
that	O
exploits	O
the	O
substructure	O
of	O
the	O
problem	O
,	O
and	O
shows	O
that	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
exists	O
if	O
and	O
only	O
if	O
every	O
pair	O
of	O
states	O
has	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
.	O
</s>
<s>
The	O
problem	O
of	O
estimating	O
the	O
length	O
of	O
synchronizing	B-General_Concept
words	I-General_Concept
has	O
a	O
long	O
history	O
and	O
was	O
posed	O
independently	O
by	O
several	O
authors	O
,	O
but	O
it	O
is	O
commonly	O
known	O
as	O
the	O
Černý	O
conjecture	O
.	O
</s>
<s>
In	O
1969	O
,	O
Ján	O
Černý	O
conjectured	O
that	O
(	O
n1	O
)	O
2	O
is	O
the	O
upper	O
bound	O
for	O
the	O
length	O
of	O
the	O
shortest	O
synchronizing	B-General_Concept
word	I-General_Concept
for	O
any	O
n-state	O
complete	O
DFA	B-General_Concept
(	O
a	O
DFA	B-General_Concept
with	O
complete	O
state	O
transition	O
graph	O
)	O
.	O
</s>
<s>
If	O
this	O
is	O
true	O
,	O
it	O
would	O
be	O
tight	O
:	O
in	O
his	O
1964	O
paper	O
,	O
Černý	O
exhibited	O
a	O
class	O
of	O
automata	O
(	O
indexed	O
by	O
the	O
number	O
n	O
of	O
states	O
)	O
for	O
which	O
the	O
shortest	O
reset	B-General_Concept
words	I-General_Concept
have	O
this	O
length	O
.	O
</s>
<s>
For	O
n-state	O
DFAs	B-General_Concept
over	O
a	O
k-letter	O
input	O
alphabet	O
,	O
an	O
algorithm	O
by	O
David	O
Eppstein	O
finds	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
of	O
length	O
at	O
most	O
11n3/48	O
+	O
O(n2 )	O
,	O
and	O
runs	O
in	O
time	O
complexity	O
O( n3+kn2	O
)	O
.	O
</s>
<s>
This	O
algorithm	O
does	O
not	O
always	O
find	O
the	O
shortest	O
possible	O
synchronizing	B-General_Concept
word	I-General_Concept
for	O
a	O
given	O
automaton	O
;	O
as	O
Eppstein	O
also	O
shows	O
,	O
the	O
problem	O
of	O
finding	O
the	O
shortest	O
synchronizing	B-General_Concept
word	I-General_Concept
is	O
NP-complete	O
.	O
</s>
<s>
However	O
,	O
for	O
a	O
special	O
class	O
of	O
automata	O
in	O
which	O
all	O
state	O
transitions	O
preserve	O
the	O
cyclic	O
order	O
of	O
the	O
states	O
,	O
he	O
describes	O
a	O
different	O
algorithm	O
with	O
time	O
O(kn2 )	O
that	O
always	O
finds	O
the	O
shortest	O
synchronizing	B-General_Concept
word	I-General_Concept
,	O
proves	O
that	O
these	O
automata	O
always	O
have	O
a	O
synchronizing	B-General_Concept
word	I-General_Concept
of	O
length	O
at	O
most	O
(	O
n1	O
)	O
2	O
(	O
the	O
bound	O
given	O
in	O
Černý	O
's	O
conjecture	O
)	O
,	O
and	O
exhibits	O
examples	O
of	O
automata	O
with	O
this	O
special	O
form	O
whose	O
shortest	O
synchronizing	B-General_Concept
word	I-General_Concept
has	O
length	O
exactly	O
(	O
n1	O
)	O
2	O
.	O
</s>
<s>
The	O
road	O
coloring	O
problem	O
is	O
the	O
problem	O
of	O
labeling	O
the	O
edges	O
of	O
a	O
regular	O
directed	O
graph	O
with	O
the	O
symbols	O
of	O
a	O
k-letter	O
input	O
alphabet	O
(	O
where	O
k	O
is	O
the	O
outdegree	O
of	O
each	O
vertex	O
)	O
in	O
order	O
to	O
form	O
a	O
synchronizable	O
DFA	B-General_Concept
.	O
</s>
<s>
It	O
was	O
conjectured	O
in	O
1970	O
by	O
Benjamin	O
Weiss	O
and	O
Roy	O
Adler	O
that	O
any	O
strongly	O
connected	O
and	O
aperiodic	B-Algorithm
regular	O
digraph	O
can	O
be	O
labeled	O
in	O
this	O
way	O
;	O
their	O
conjecture	O
was	O
proven	O
in	O
2007	O
by	O
Avraham	O
Trahtman	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
corresponds	O
to	O
a	O
transformation	O
semigroup	O
with	O
a	O
distinguished	O
generator	O
set	O
.	O
</s>
