<s>
In	O
theoretical	O
computer	O
science	O
and	O
formal	O
language	O
theory	O
,	O
a	O
regular	B-General_Concept
language	I-General_Concept
(	O
also	O
called	O
a	O
rational	B-General_Concept
language	I-General_Concept
)	O
is	O
a	O
formal	O
language	O
that	O
can	O
be	O
defined	O
by	O
a	O
regular	B-Language
expression	I-Language
,	O
in	O
the	O
strict	O
sense	O
in	O
theoretical	O
computer	O
science	O
(	O
as	O
opposed	O
to	O
many	O
modern	O
regular	B-Language
expression	I-Language
engines	O
,	O
which	O
are	O
augmented	O
with	O
features	O
that	O
allow	O
the	O
recognition	O
of	O
non-regular	O
languages	O
)	O
.	O
</s>
<s>
Alternatively	O
,	O
a	O
regular	B-General_Concept
language	I-General_Concept
can	O
be	O
defined	O
as	O
a	O
language	O
recognized	B-Application
by	O
a	O
finite	B-Architecture
automaton	I-Architecture
.	O
</s>
<s>
The	O
equivalence	O
of	O
regular	B-Language
expressions	I-Language
and	O
finite	B-Architecture
automata	I-Architecture
is	O
known	O
as	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
(	O
after	O
American	O
mathematician	O
Stephen	O
Cole	O
Kleene	O
)	O
.	O
</s>
<s>
In	O
the	O
Chomsky	O
hierarchy	O
,	O
regular	B-General_Concept
languages	I-General_Concept
are	O
the	O
languages	O
generated	O
by	O
Type-3	O
grammars	O
.	O
</s>
<s>
The	O
collection	O
of	O
regular	B-General_Concept
languages	I-General_Concept
over	O
an	O
alphabet	O
Σ	O
is	O
defined	O
recursively	O
as	O
follows	O
:	O
</s>
<s>
The	O
empty	O
language	O
Ø	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
For	O
each	O
a	O
∈	O
Σ	O
(	O
a	O
belongs	O
to	O
Σ	O
)	O
,	O
the	O
singleton	O
language	O
 { a&hairsp; } 	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
If	O
A	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
,	O
A*	O
(	O
Kleene	O
star	O
)	O
is	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
If	O
A	O
and	O
B	O
are	O
regular	B-General_Concept
languages	I-General_Concept
,	O
then	O
A	O
∪	O
B	O
(	O
union	O
)	O
and	O
A	O
•	O
B	O
(	O
concatenation	O
)	O
are	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
See	O
regular	B-Language
expression	I-Language
for	O
syntax	O
and	O
semantics	O
of	O
regular	B-Language
expressions	I-Language
.	O
</s>
<s>
Intuitively	O
,	O
it	O
cannot	O
be	O
recognized	B-Application
with	O
a	O
finite	B-Architecture
automaton	I-Architecture
,	O
since	O
a	O
finite	B-Architecture
automaton	I-Architecture
has	O
finite	O
memory	O
and	O
it	O
cannot	O
remember	O
the	O
exact	O
number	O
of	O
a	O
's	O
.	O
</s>
<s>
A	O
regular	B-General_Concept
language	I-General_Concept
satisfies	O
the	O
following	O
equivalent	O
properties	O
:	O
</s>
<s>
(	O
This	O
number	O
equals	O
the	O
number	O
of	O
states	O
of	O
the	O
minimal	B-General_Concept
deterministic	I-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
accepting	O
L	O
.	O
)	O
</s>
<s>
Properties	O
10	O
.	O
and	O
11	O
.	O
are	O
purely	O
algebraic	O
approaches	O
to	O
define	O
regular	B-General_Concept
languages	I-General_Concept
;	O
a	O
similar	O
set	O
of	O
statements	O
can	O
be	O
formulated	O
for	O
a	O
monoid	O
M	O
⊆	O
Σ*	O
.	O
</s>
<s>
as	O
an	O
alternative	O
definition	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Some	O
of	O
the	O
equivalences	O
above	O
,	O
particularly	O
those	O
among	O
the	O
first	O
four	O
formalisms	O
,	O
are	O
called	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
in	O
textbooks	O
.	O
</s>
<s>
One	O
textbook	O
calls	O
the	O
equivalence	O
of	O
regular	B-Language
expressions	I-Language
and	O
NFAs	O
(	O
"	O
1.	O
"	O
</s>
<s>
above	O
)	O
"	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
"	O
.	O
</s>
<s>
Another	O
textbook	O
calls	O
the	O
equivalence	O
of	O
regular	B-Language
expressions	I-Language
and	O
DFAs	B-General_Concept
(	O
"	O
1.	O
"	O
</s>
<s>
above	O
)	O
"	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
"	O
.	O
</s>
<s>
Two	O
other	O
textbooks	O
first	O
prove	O
the	O
expressive	O
equivalence	O
of	O
NFAs	O
and	O
DFAs	B-General_Concept
(	O
"	O
2.	O
"	O
</s>
<s>
and	O
then	O
state	O
"	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
"	O
as	O
the	O
equivalence	O
between	O
regular	B-Language
expressions	I-Language
and	O
finite	B-Architecture
automata	I-Architecture
(	O
the	O
latter	O
said	O
to	O
describe	O
"	O
recognizable	O
languages	O
"	O
)	O
.	O
</s>
<s>
above	O
)	O
with	O
DFAs	B-General_Concept
and	O
NFAs	O
,	O
calls	O
the	O
languages	O
generated	O
by	O
(	O
any	O
of	O
)	O
these	O
"	O
regular	O
"	O
,	O
after	O
which	O
it	O
introduces	O
regular	B-Language
expressions	I-Language
which	O
it	O
terms	O
to	O
describe	O
"	O
rational	B-General_Concept
languages	I-General_Concept
"	O
,	O
and	O
finally	O
states	O
"	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
"	O
as	O
the	O
coincidence	O
of	O
regular	O
and	O
rational	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Other	O
authors	O
simply	O
define	O
"	O
rational	O
expression	O
"	O
and	O
"	O
regular	B-Language
expressions	I-Language
"	O
as	O
synonymous	O
and	O
do	O
the	O
same	O
with	O
"	O
rational	B-General_Concept
languages	I-General_Concept
"	O
and	O
"	O
regular	B-General_Concept
languages	I-General_Concept
"	O
.	O
</s>
<s>
Noam	O
Chomsky	O
,	O
in	O
his	O
1959	O
seminal	O
article	O
,	O
used	O
the	O
term	O
"	O
regular	O
"	O
in	O
a	O
different	O
meaning	O
at	O
first	O
(	O
referring	O
to	O
what	O
is	O
called	O
"	O
Chomsky	O
normal	O
form	O
"	O
today	O
)	O
,	O
but	O
noticed	O
that	O
his	O
"	O
finite	B-Architecture
state	I-Architecture
languages	I-Architecture
"	O
were	O
equivalent	O
to	O
Kleene	O
's	O
"	O
regular	O
events	O
"	O
.	O
</s>
<s>
The	O
regular	B-General_Concept
languages	I-General_Concept
are	O
closed	O
under	O
various	O
operations	O
,	O
that	O
is	O
,	O
if	O
the	O
languages	O
K	O
and	O
L	O
are	O
regular	O
,	O
so	O
is	O
the	O
result	O
of	O
the	O
following	O
operations	O
:	O
</s>
<s>
the	O
trio	O
operations	O
:	O
string	O
homomorphism	O
,	O
inverse	O
string	O
homomorphism	O
,	O
and	O
intersection	O
with	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
As	O
a	O
consequence	O
they	O
are	O
closed	O
under	O
arbitrary	O
finite	B-Architecture
state	I-Architecture
transductions	I-Architecture
,	O
like	O
quotient	O
K	O
/	O
L	O
with	O
a	O
regular	B-General_Concept
language	I-General_Concept
.	O
</s>
<s>
Even	O
more	O
,	O
regular	B-General_Concept
languages	I-General_Concept
are	O
closed	O
under	O
quotients	O
with	O
arbitrary	O
languages	O
:	O
If	O
L	O
is	O
regular	O
then	O
L	O
/	O
K	O
is	O
regular	O
for	O
any	O
K	O
.	O
</s>
<s>
Given	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
to	O
recognize	O
L	O
,	O
an	O
automaton	O
for	O
LR	O
can	O
be	O
obtained	O
by	O
reversing	O
all	O
transitions	O
and	O
interchanging	O
starting	O
and	O
finishing	O
states	O
.	O
</s>
<s>
Given	O
two	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
A	O
and	O
B	O
,	O
it	O
is	O
decidable	O
whether	O
they	O
accept	O
the	O
same	O
language	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
using	O
the	O
above	O
closure	O
properties	O
,	O
the	O
following	O
problems	O
are	O
also	O
decidable	O
for	O
arbitrarily	O
given	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
A	O
and	O
B	O
,	O
with	O
accepted	O
languages	O
LA	O
and	O
LB	O
,	O
respectively	O
:	O
</s>
<s>
For	O
regular	B-Language
expressions	I-Language
,	O
the	O
universality	O
problem	O
is	O
NP-complete	O
already	O
for	O
a	O
singleton	O
alphabet	O
.	O
</s>
<s>
If	O
regular	B-Language
expressions	I-Language
are	O
extended	O
to	O
allow	O
also	O
a	O
squaring	O
operator	O
,	O
with	O
"	O
A2	O
"	O
denoting	O
the	O
same	O
as	O
"	O
AA	O
"	O
,	O
still	O
just	O
regular	B-General_Concept
languages	I-General_Concept
can	O
be	O
described	O
,	O
but	O
the	O
universality	O
problem	O
has	O
an	O
exponential	O
space	O
lower	O
bound	O
,	O
and	O
is	O
in	O
fact	O
complete	O
for	O
exponential	O
space	O
with	O
respect	O
to	O
polynomial-time	O
reduction	O
.	O
</s>
<s>
For	O
a	O
fixed	O
finite	O
alphabet	O
,	O
the	O
theory	O
of	O
the	O
set	O
of	O
all	O
languages	O
—	O
together	O
with	O
strings	O
,	O
membership	O
of	O
a	O
string	O
in	O
a	O
language	O
,	O
and	O
for	O
each	O
character	O
,	O
a	O
function	O
to	O
append	O
the	O
character	O
to	O
a	O
string	O
(	O
and	O
no	O
other	O
operations	O
)	O
—	O
is	O
decidable	O
,	O
and	O
its	O
minimal	O
elementary	O
substructure	O
consists	O
precisely	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
In	O
computational	O
complexity	O
theory	O
,	O
the	O
complexity	O
class	O
of	O
all	O
regular	B-General_Concept
languages	I-General_Concept
is	O
sometimes	O
referred	O
to	O
as	O
REGULAR	O
or	O
REG	B-Language
and	O
equals	O
DSPACE(O(1 )	O
)	O
,	O
the	O
decision	O
problems	O
that	O
can	O
be	O
solved	O
in	O
constant	O
space	O
(	O
the	O
space	O
used	O
is	O
independent	O
of	O
the	O
input	O
size	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
REGULAR	O
does	O
not	O
contain	O
AC0	O
,	O
because	O
the	O
nonregular	O
language	O
of	O
palindromes	O
,	O
or	O
the	O
nonregular	O
language	O
can	O
both	O
be	O
recognized	B-Application
in	O
AC0	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
DSPACE(o(loglogn )	O
)	O
equals	O
the	O
class	O
of	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
To	O
locate	O
the	O
regular	B-General_Concept
languages	I-General_Concept
in	O
the	O
Chomsky	O
hierarchy	O
,	O
one	O
notices	O
that	O
every	O
regular	B-General_Concept
language	I-General_Concept
is	O
context-free	O
.	O
</s>
<s>
Other	O
approaches	O
include	O
using	O
the	O
closure	O
properties	O
of	O
regular	B-General_Concept
languages	I-General_Concept
or	O
quantifying	O
Kolmogorov	O
complexity	O
.	O
</s>
<s>
These	O
are	O
regular	B-General_Concept
languages	I-General_Concept
,	O
as	O
one	O
can	O
create	O
a	O
regular	B-Language
expression	I-Language
that	O
is	O
the	O
union	O
of	O
every	O
word	O
in	O
the	O
language	O
.	O
</s>
<s>
Star-free	O
languages	O
,	O
those	O
that	O
can	O
be	O
described	O
by	O
a	O
regular	B-Language
expression	I-Language
constructed	O
from	O
the	O
empty	O
symbol	O
,	O
letters	O
,	O
concatenation	O
and	O
all	O
boolean	O
operators	O
(	O
see	O
algebra	O
of	O
sets	O
)	O
including	O
complementation	O
but	O
not	O
the	O
Kleene	O
star	O
:	O
this	O
class	O
includes	O
all	O
finite	O
languages	O
.	O
</s>
<s>
In	O
general	O
,	O
for	O
every	O
regular	B-General_Concept
language	I-General_Concept
there	O
exists	O
a	O
constant	O
such	O
that	O
for	O
all	O
,	O
the	O
number	O
of	O
words	O
of	O
length	O
is	O
asymptotically	O
.	O
</s>
<s>
The	O
zeta	O
function	O
of	O
a	O
regular	B-General_Concept
language	I-General_Concept
is	O
not	O
in	O
general	O
rational	O
,	O
but	O
that	O
of	O
an	O
arbitrary	O
cyclic	B-General_Concept
language	I-General_Concept
is	O
.	O
</s>
<s>
The	O
notion	O
of	O
a	O
regular	B-General_Concept
language	I-General_Concept
has	O
been	O
generalized	O
to	O
infinite	O
words	O
(	O
see	O
ω-automata	B-General_Concept
)	O
and	O
to	O
trees	O
(	O
see	O
tree	B-Application
automaton	I-Application
)	O
.	O
</s>
<s>
Rational	B-Application
set	I-Application
generalizes	O
the	O
notion	O
(	O
of	O
regular/rational	O
language	O
)	O
to	O
monoids	O
that	O
are	O
not	O
necessarily	O
free	O
.	O
</s>
<s>
Likewise	O
,	O
the	O
notion	O
of	O
a	O
recognizable	O
language	O
(	O
by	O
a	O
finite	B-Architecture
automaton	I-Architecture
)	O
has	O
namesake	O
as	O
recognizable	B-Application
set	I-Application
over	O
a	O
monoid	O
that	O
is	O
not	O
necessarily	O
free	O
.	O
</s>
<s>
Howard	O
Straubing	O
notes	O
in	O
relation	O
to	O
these	O
facts	O
that	O
“	O
The	O
term	O
"	O
regular	B-General_Concept
language	I-General_Concept
"	O
is	O
a	O
bit	O
unfortunate	O
.	O
</s>
<s>
Papers	O
influenced	O
by	O
Eilenberg	O
's	O
monograph	O
often	O
use	O
either	O
the	O
term	O
"	O
recognizable	O
language	O
"	O
,	O
which	O
refers	O
to	O
the	O
behavior	O
of	O
automata	O
,	O
or	O
"	O
rational	B-General_Concept
language	I-General_Concept
"	O
,	O
which	O
refers	O
to	O
important	O
analogies	O
between	O
regular	B-Language
expressions	I-Language
and	O
rational	O
power	O
series	O
.	O
</s>
<s>
This	O
approach	O
gives	O
rise	O
to	O
weighted	O
rational	O
expressions	O
and	O
weighted	B-General_Concept
automata	I-General_Concept
.	O
</s>
<s>
In	O
this	O
algebraic	O
context	O
,	O
the	O
regular	B-General_Concept
languages	I-General_Concept
(	O
corresponding	O
to	O
Boolean-weighted	O
rational	O
expressions	O
)	O
are	O
usually	O
called	O
rational	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Also	O
in	O
this	O
context	O
,	O
Kleene	B-General_Concept
's	I-General_Concept
theorem	I-General_Concept
finds	O
a	O
generalization	O
called	O
the	O
Kleene-Schützenberger	O
theorem	O
.	O
</s>
