<s>
An	O
aperiodic	O
finite-state	B-Architecture
automaton	I-Architecture
(	O
also	O
called	O
a	O
counter-free	B-General_Concept
automaton	I-General_Concept
)	O
is	O
a	O
finite-state	B-Architecture
automaton	I-Architecture
whose	O
transition	O
monoid	O
is	O
aperiodic	O
.	O
</s>
<s>
A	O
regular	B-General_Concept
language	I-General_Concept
is	O
star-free	O
if	O
and	O
only	O
if	O
it	O
is	O
accepted	O
by	O
an	O
automaton	O
with	O
a	O
finite	O
and	O
aperiodic	O
transition	O
monoid	O
.	O
</s>
<s>
This	O
result	O
of	O
algebraic	O
automata	B-Application
theory	I-Application
is	O
due	O
to	O
Marcel-Paul	O
Schützenberger	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
minimum	B-General_Concept
automaton	I-General_Concept
of	O
a	O
star-free	O
language	O
is	O
always	O
counter-free	O
(	O
however	O
,	O
a	O
star-free	O
language	O
may	O
also	O
be	O
recognized	O
by	O
other	O
automata	O
which	O
are	O
not	O
aperiodic	O
)	O
.	O
</s>
<s>
A	O
counter-free	B-General_Concept
language	I-General_Concept
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
for	O
which	O
there	O
is	O
an	O
integer	O
n	O
such	O
that	O
for	O
all	O
words	O
x	O
,	O
y	O
,	O
z	O
and	O
integers	O
m	O
≥	O
n	O
we	O
have	O
xymz	O
in	O
L	O
if	O
and	O
only	O
if	O
xynz	O
in	O
L	O
.	O
Another	O
way	O
to	O
state	O
Schützenberger	O
's	O
theorem	O
is	O
that	O
star-free	O
languages	O
and	O
counter-free	B-General_Concept
languages	I-General_Concept
are	O
the	O
same	O
thing	O
.	O
</s>
