<s>
In	O
theoretical	O
computer	O
science	O
,	O
in	O
particular	O
in	O
formal	O
language	O
theory	O
,	O
Kleene	B-General_Concept
's	I-General_Concept
algorithm	I-General_Concept
transforms	O
a	O
given	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	O
)	O
into	O
a	O
regular	B-Language
expression	I-Language
.	O
</s>
<s>
Together	O
with	O
other	O
conversion	O
algorithms	O
,	O
it	O
establishes	O
the	O
equivalence	O
of	O
several	O
description	O
formats	O
for	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
A	O
presentation	O
of	O
the	O
algorithm	O
in	O
the	O
case	O
of	O
deterministic	B-General_Concept
finite	I-General_Concept
automata	I-General_Concept
(	O
DFAs	B-General_Concept
)	O
is	O
given	O
in	O
Hopcroft	O
and	O
Ullman	O
(	O
1979	O
)	O
.	O
</s>
<s>
Each	O
set	O
R	O
is	O
represented	O
by	O
a	O
regular	B-Language
expression	I-Language
;	O
the	O
algorithm	O
computes	O
them	O
step	O
by	O
step	O
for	O
k	O
=	O
-1	O
,	O
0	O
,	O
...	O
,	O
n	O
.	O
Since	O
there	O
is	O
no	O
state	O
numbered	O
higher	O
than	O
n	O
,	O
the	O
regular	B-Language
expression	I-Language
R	O
represents	O
the	O
set	O
of	O
all	O
strings	O
that	O
take	O
M	O
from	O
its	O
start	O
state	O
q0	O
to	O
qj	O
.	O
</s>
<s>
If	O
F	O
=	O
{	O
q1	O
,...,	O
qf	O
}	O
is	O
the	O
set	O
of	O
accept	O
states	O
,	O
the	O
regular	B-Language
expression	I-Language
R	O
|	O
...	O
|	O
R	O
represents	O
the	O
language	O
accepted	O
by	O
M	O
.	O
</s>
<s>
The	O
initial	O
regular	B-Language
expressions	I-Language
,	O
for	O
k	O
=	O
-1	O
,	O
are	O
computed	O
as	O
follows	O
for	O
i≠j	O
:	O
</s>
<s>
Another	O
way	O
to	O
understand	O
the	O
operation	O
of	O
the	O
algorithm	O
is	O
as	O
an	O
"	O
elimination	O
method	O
"	O
,	O
where	O
the	O
states	O
from	O
0	O
to	O
n	O
are	O
successively	O
removed	O
:	O
when	O
state	O
k	O
is	O
removed	O
,	O
the	O
regular	B-Language
expression	I-Language
R	O
,	O
which	O
describes	O
the	O
words	O
that	O
label	O
a	O
path	O
from	O
state	O
i>k	O
to	O
state	O
j>k	O
,	O
is	O
rewritten	O
into	O
R	O
so	O
as	O
to	O
take	O
into	O
account	O
the	O
possibility	O
of	O
going	O
via	O
the	O
"	O
eliminated	O
"	O
state	O
k	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
length	O
of	O
the	O
regular	B-Language
expression	I-Language
representing	O
the	O
language	O
accepted	O
by	O
M	O
is	O
at	O
most	O
(	O
4n+1( 6s+7	O
)	O
f	O
-	O
f	O
-	O
3	O
)	O
symbols	O
,	O
where	O
f	O
denotes	O
the	O
number	O
of	O
final	O
states	O
.	O
</s>
<s>
This	O
exponential	O
blowup	O
is	O
inevitable	O
,	O
because	O
there	O
exist	O
families	O
of	O
DFAs	B-General_Concept
for	O
which	O
any	O
equivalent	O
regular	B-Language
expression	I-Language
must	O
be	O
of	O
exponential	O
size	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
size	O
of	O
the	O
regular	B-Language
expression	I-Language
obtained	O
by	O
running	O
the	O
algorithm	O
can	O
be	O
very	O
different	O
depending	O
on	O
the	O
order	O
in	O
which	O
the	O
states	O
are	O
considered	O
by	O
the	O
procedure	O
,	O
i.e.	O
,	O
the	O
order	O
in	O
which	O
they	O
are	O
numbered	O
from	O
0	O
to	O
n	O
.	O
</s>
<s>
Kleene	O
algebra	O
equalities	O
are	O
used	O
to	O
simplify	O
the	O
regular	B-Language
expressions	I-Language
as	O
much	O
as	O
possible	O
.	O
</s>
<s>
Since	O
q0	O
is	O
the	O
start	O
state	O
and	O
q1	O
is	O
the	O
only	O
accept	O
state	O
,	O
the	O
regular	B-Language
expression	I-Language
R	O
denotes	O
the	O
set	O
of	O
all	O
strings	O
accepted	O
by	O
the	O
automaton	O
.	O
</s>
