<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
permutation	B-General_Concept
automaton	I-General_Concept
,	O
or	O
pure-group	B-General_Concept
automaton	I-General_Concept
,	O
is	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
such	O
that	O
each	O
input	O
symbol	O
permutes	B-Algorithm
the	O
set	O
of	O
states	O
.	O
</s>
<s>
Formally	O
,	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
may	O
be	O
defined	O
by	O
the	O
tuple	O
(	O
Q	O
,	O
Σ	O
,	O
δ	O
,	O
q0	O
,	O
F	O
)	O
,	O
</s>
<s>
is	O
a	O
permutation	B-General_Concept
automaton	I-General_Concept
if	O
and	O
only	O
if	O
,	O
for	O
every	O
two	O
distinct	O
states	O
and	O
in	O
Q	O
and	O
every	O
input	O
symbol	O
in	O
Σ	O
,	O
δ(qi,x )	O
≠	O
δ(qj,x )	O
.	O
</s>
<s>
A	O
formal	O
language	O
is	O
p-regular	B-General_Concept
(	O
also	O
:	O
a	O
pure-group	B-General_Concept
language	I-General_Concept
)	O
if	O
it	O
is	O
accepted	O
by	O
a	O
permutation	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
the	O
set	O
of	O
strings	O
of	O
even	O
length	O
forms	O
a	O
p-regular	B-General_Concept
language	I-General_Concept
:	O
it	O
may	O
be	O
accepted	O
by	O
a	O
permutation	B-General_Concept
automaton	I-General_Concept
with	O
two	O
states	O
in	O
which	O
every	O
transition	O
replaces	O
one	O
state	O
by	O
the	O
other	O
.	O
</s>
<s>
The	O
pure-group	B-General_Concept
languages	I-General_Concept
were	O
the	O
first	O
interesting	O
family	O
of	O
regular	B-General_Concept
languages	I-General_Concept
for	O
which	O
the	O
star	O
height	O
problem	O
was	O
proved	O
to	O
be	O
computable	O
.	O
</s>
<s>
Another	O
mathematical	O
problem	O
on	O
regular	B-General_Concept
languages	I-General_Concept
is	O
the	O
separating	O
words	O
problem	O
,	O
which	O
asks	O
for	O
the	O
size	O
of	O
a	O
smallest	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
that	O
distinguishes	O
between	O
two	O
given	O
words	O
of	O
length	O
at	O
most	O
n	O
by	O
accepting	O
one	O
word	O
and	O
rejecting	O
the	O
other	O
.	O
</s>
<s>
The	O
problem	O
was	O
later	O
studied	O
for	O
the	O
restriction	O
to	O
permutation	B-General_Concept
automata	I-General_Concept
.	O
</s>
