<s>
In	O
the	O
theory	O
of	O
computation	O
and	O
automata	B-Application
theory	I-Application
,	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
or	O
subset	B-General_Concept
construction	I-General_Concept
is	O
a	O
standard	O
method	O
for	O
converting	O
a	O
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	O
)	O
into	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
DFA	B-General_Concept
)	O
which	O
recognizes	O
the	O
same	O
formal	O
language	O
.	O
</s>
<s>
It	O
is	O
important	O
in	O
theory	O
because	O
it	O
establishes	O
that	O
NFAs	O
,	O
despite	O
their	O
additional	O
flexibility	O
,	O
are	O
unable	O
to	O
recognize	O
any	O
language	O
that	O
cannot	O
be	O
recognized	O
by	O
some	O
DFA	B-General_Concept
.	O
</s>
<s>
It	O
is	O
also	O
important	O
in	O
practice	O
for	O
converting	O
easier-to-construct	O
NFAs	O
into	O
more	O
efficiently	O
executable	O
DFAs	B-General_Concept
.	O
</s>
<s>
However	O
,	O
if	O
the	O
NFA	O
has	O
n	O
states	O
,	O
the	O
resulting	O
DFA	B-General_Concept
may	O
have	O
up	O
to	O
2n	O
states	O
,	O
an	O
exponentially	O
larger	O
number	O
,	O
which	O
sometimes	O
makes	O
the	O
construction	O
impractical	O
for	O
large	O
NFAs	O
.	O
</s>
<s>
The	O
construction	O
,	O
sometimes	O
called	O
the	O
Rabin	O
–	O
Scott	O
powerset	B-General_Concept
construction	I-General_Concept
(	O
or	O
subset	B-General_Concept
construction	I-General_Concept
)	O
to	O
distinguish	O
it	O
from	O
similar	O
constructions	O
for	O
other	O
types	O
of	O
automata	O
,	O
was	O
first	O
published	O
by	O
Michael	O
O	O
.	O
Rabin	O
and	O
Dana	O
Scott	O
in	O
1959	O
.	O
</s>
<s>
To	O
simulate	O
the	O
operation	O
of	O
a	O
DFA	B-General_Concept
on	O
a	O
given	O
input	O
string	O
,	O
one	O
needs	O
to	O
keep	O
track	O
of	O
a	O
single	O
state	O
at	O
any	O
time	O
:	O
the	O
state	O
that	O
the	O
automaton	O
will	O
reach	O
after	O
seeing	O
a	O
prefix	O
of	O
the	O
input	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
sets	O
of	O
reachable	O
NFA	O
states	O
play	O
the	O
same	O
role	O
in	O
the	O
NFA	O
simulation	O
as	O
single	O
DFA	B-General_Concept
states	O
play	O
in	O
the	O
DFA	B-General_Concept
simulation	O
,	O
and	O
in	O
fact	O
the	O
sets	O
of	O
NFA	O
states	O
appearing	O
in	O
this	O
simulation	O
may	O
be	O
re-interpreted	O
as	O
being	O
states	O
of	O
a	O
DFA	B-General_Concept
.	O
</s>
<s>
The	O
powerset	B-General_Concept
construction	I-General_Concept
applies	O
most	O
directly	O
to	O
an	O
NFA	O
that	O
does	O
not	O
allow	O
state	O
transformations	O
without	O
consuming	O
input	O
symbols	O
(	O
aka	O
:	O
"	O
ε-moves	O
"	O
)	O
.	O
</s>
<s>
Such	O
an	O
automaton	O
may	O
be	O
defined	O
as	O
a	O
5-tuple	B-Application
,	O
in	O
which	O
is	O
the	O
set	O
of	O
states	O
,	O
is	O
the	O
set	O
of	O
input	O
symbols	O
,	O
is	O
the	O
transition	O
function	O
(	O
mapping	O
a	O
state	O
and	O
an	O
input	O
symbol	O
to	O
a	O
set	O
of	O
states	O
)	O
,	O
is	O
the	O
initial	O
state	O
,	O
and	O
is	O
the	O
set	O
of	O
accepting	O
states	O
.	O
</s>
<s>
The	O
corresponding	O
DFA	B-General_Concept
has	O
states	O
corresponding	O
to	O
subsets	O
of	O
.	O
</s>
<s>
The	O
initial	O
state	O
of	O
the	O
DFA	B-General_Concept
is	O
,	O
the	O
(	O
one-element	O
)	O
set	O
of	O
initial	O
states	O
.	O
</s>
<s>
The	O
transition	O
function	O
of	O
the	O
DFA	B-General_Concept
maps	O
a	O
state	O
(	O
representing	O
a	O
subset	O
of	O
)	O
and	O
an	O
input	O
symbol	O
to	O
the	O
set	O
,	O
the	O
set	O
of	O
all	O
states	O
that	O
can	O
be	O
reached	O
by	O
an	O
-transition	O
from	O
a	O
state	O
in	O
.	O
</s>
<s>
A	O
state	O
of	O
the	O
DFA	B-General_Concept
is	O
an	O
accepting	O
state	O
if	O
and	O
only	O
if	O
at	O
least	O
one	O
member	O
of	O
is	O
an	O
accepting	O
state	O
of	O
the	O
NFA	O
.	O
</s>
<s>
In	O
the	O
simplest	O
version	O
of	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
,	O
the	O
set	O
of	O
all	O
states	O
of	O
the	O
DFA	B-General_Concept
is	O
the	O
powerset	O
of	O
,	O
the	O
set	O
of	O
all	O
possible	O
subsets	O
of	O
.	O
</s>
<s>
However	O
,	O
many	O
states	O
of	O
the	O
resulting	O
DFA	B-General_Concept
may	O
be	O
useless	O
as	O
they	O
may	O
be	O
unreachable	O
from	O
the	O
initial	O
state	O
.	O
</s>
<s>
Van	O
Noord	O
recognizes	O
three	O
possible	O
ways	O
of	O
incorporating	O
this	O
closure	O
computation	O
in	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
:	O
</s>
<s>
Compute	O
the	O
ε-closure	O
of	O
the	O
entire	O
automaton	O
as	O
a	O
preprocessing	O
step	O
,	O
producing	O
an	O
equivalent	O
NFA	O
without	O
ε-moves	O
,	O
then	O
apply	O
the	O
regular	O
powerset	B-General_Concept
construction	I-General_Concept
.	O
</s>
<s>
This	O
version	O
,	O
also	O
discussed	O
by	O
Hopcroft	O
and	O
Ullman	O
,	O
is	O
straightforward	O
to	O
implement	O
,	O
but	O
impractical	O
for	O
automata	O
with	O
large	O
numbers	O
of	O
ε-moves	O
,	O
as	O
commonly	O
arise	O
in	O
natural	B-Language
language	I-Language
processing	I-Language
application	O
.	O
</s>
<s>
If	O
NFAs	O
are	O
defined	O
to	O
allow	O
for	O
multiple	O
initial	O
states	O
,	O
the	O
initial	O
state	O
of	O
the	O
corresponding	O
DFA	B-General_Concept
is	O
the	O
set	O
of	O
all	O
initial	O
states	O
of	O
the	O
NFA	O
,	O
or	O
(	O
if	O
the	O
NFA	O
also	O
has	O
ε-moves	O
)	O
the	O
set	O
of	O
all	O
states	O
reachable	O
from	O
initial	O
states	O
by	O
ε-moves	O
.	O
</s>
<s>
The	O
initial	O
state	O
of	O
the	O
DFA	B-General_Concept
constructed	O
from	O
this	O
NFA	O
is	O
the	O
set	O
of	O
all	O
NFA	O
states	O
that	O
are	O
reachable	O
from	O
state	O
1	O
by	O
ε-moves	O
;	O
that	O
is	O
,	O
it	O
is	O
the	O
set	O
 { 1 , 2 , 3 } 	O
.	O
</s>
<s>
Therefore	O
,	O
( {	O
1	O
,	O
2	O
,	O
3}	O
,	O
0	O
)	O
={2,4},	O
and	O
by	O
the	O
same	O
reasoning	O
the	O
full	O
DFA	B-General_Concept
constructed	O
from	O
the	O
NFA	O
is	O
as	O
shown	O
below	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
in	O
this	O
example	O
,	O
there	O
are	O
five	O
states	O
reachable	O
from	O
the	O
start	O
state	O
of	O
the	O
DFA	B-General_Concept
;	O
the	O
remaining	O
11	O
sets	O
in	O
the	O
powerset	O
of	O
the	O
set	O
of	O
NFA	O
states	O
are	O
not	O
reachable	O
.	O
</s>
<s>
Because	O
the	O
DFA	B-General_Concept
states	O
consist	O
of	O
sets	O
of	O
NFA	O
states	O
,	O
an	O
-state	O
NFA	O
may	O
be	O
converted	O
to	O
a	O
DFA	B-General_Concept
with	O
at	O
most	O
states	O
.	O
</s>
<s>
For	O
every	O
,	O
there	O
exist	O
-state	O
NFAs	O
such	O
that	O
every	O
subset	O
of	O
states	O
is	O
reachable	O
from	O
the	O
initial	O
subset	O
,	O
so	O
that	O
the	O
converted	O
DFA	B-General_Concept
has	O
exactly	O
states	O
,	O
giving	O
Θ( )	O
worst-case	B-General_Concept
time	O
complexity	O
.	O
</s>
<s>
It	O
can	O
be	O
represented	O
by	O
an	O
-state	O
NFA	O
,	O
but	O
it	O
requires	O
DFA	B-General_Concept
states	O
,	O
one	O
for	O
each	O
-character	O
suffix	O
of	O
the	O
input	O
;	O
cf	O
.	O
</s>
<s>
Brzozowski	O
's	O
algorithm	O
for	O
DFA	B-General_Concept
minimization	O
uses	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
,	O
twice	O
.	O
</s>
<s>
It	O
converts	O
the	O
input	O
DFA	B-General_Concept
into	O
an	O
NFA	O
for	O
the	O
reverse	O
language	O
,	O
by	O
reversing	O
all	O
its	O
arrows	O
and	O
exchanging	O
the	O
roles	O
of	O
initial	O
and	O
accepting	O
states	O
,	O
converts	O
the	O
NFA	O
back	O
into	O
a	O
DFA	B-General_Concept
using	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
,	O
and	O
then	O
repeats	O
its	O
process	O
.	O
</s>
<s>
Its	O
worst-case	B-General_Concept
complexity	I-General_Concept
is	O
exponential	O
,	O
unlike	O
some	O
other	O
known	O
DFA	B-General_Concept
minimization	O
algorithms	O
,	O
but	O
in	O
many	O
examples	O
it	O
performs	O
more	O
quickly	O
than	O
its	O
worst-case	B-General_Concept
complexity	I-General_Concept
would	O
suggest	O
.	O
</s>
<s>
Safra	O
's	O
construction	O
,	O
which	O
converts	O
a	O
non-deterministic	O
Büchi	B-General_Concept
automaton	I-General_Concept
with	O
states	O
into	O
a	O
deterministic	O
Muller	B-General_Concept
automaton	I-General_Concept
or	O
into	O
a	O
deterministic	O
Rabin	B-General_Concept
automaton	I-General_Concept
with	O
2O( log )	O
states	O
,	O
uses	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
as	O
part	O
of	O
its	O
machinery	O
.	O
</s>
