<s>
nondeterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
(	O
NFA	B-General_Concept
)	O
where	O
each	O
transition	O
is	O
labeled	O
with	O
any	O
regular	B-Language
expression	I-Language
.	O
</s>
<s>
The	O
GNFA	B-General_Concept
reads	O
blocks	O
of	O
symbols	O
from	O
the	O
input	O
which	O
constitute	O
a	O
string	O
as	O
defined	O
by	O
the	O
regular	B-Language
expression	I-Language
on	O
the	O
transition	O
.	O
</s>
<s>
There	O
are	O
several	O
differences	O
between	O
a	O
standard	O
finite	O
state	O
machine	O
and	O
a	O
generalized	B-General_Concept
nondeterministic	I-General_Concept
finite	I-General_Concept
state	I-General_Concept
machine	I-General_Concept
.	O
</s>
<s>
A	O
GNFA	B-General_Concept
must	O
have	O
only	O
one	O
start	O
state	O
and	O
one	O
accept	O
state	O
,	O
and	O
these	O
cannot	O
be	O
the	O
same	O
state	O
,	O
whereas	O
an	O
NFA	B-General_Concept
or	O
DFA	B-General_Concept
both	O
may	O
have	O
several	O
accept	O
states	O
,	O
and	O
the	O
start	O
state	O
can	O
be	O
an	O
accept	O
state	O
.	O
</s>
<s>
A	O
GNFA	B-General_Concept
must	O
have	O
only	O
one	O
transition	O
between	O
any	O
two	O
states	O
,	O
whereas	O
a	O
NFA	B-General_Concept
or	O
DFA	B-General_Concept
both	O
allow	O
for	O
numerous	O
transitions	O
between	O
states	O
.	O
</s>
<s>
In	O
a	O
GNFA	B-General_Concept
,	O
a	O
state	O
has	O
a	O
single	O
transition	O
to	O
every	O
state	O
in	O
the	O
machine	O
,	O
although	O
often	O
it	O
is	O
a	O
convention	O
to	O
ignore	O
the	O
transitions	O
that	O
are	O
labelled	O
with	O
the	O
empty	O
set	O
when	O
drawing	O
generalized	B-General_Concept
nondeterministic	I-General_Concept
finite	I-General_Concept
state	I-General_Concept
machines	I-General_Concept
.	O
</s>
<s>
where	O
R	O
is	O
the	O
collection	O
of	O
all	O
regular	B-Language
expressions	I-Language
over	O
the	O
alphabet	O
Σ	O
.	O
</s>
<s>
The	O
transition	O
function	O
takes	O
as	O
its	O
argument	O
a	O
pair	O
of	O
two	O
states	O
and	O
outputs	O
a	O
regular	B-Language
expression	I-Language
(	O
the	O
label	O
of	O
the	O
transition	O
)	O
.	O
</s>
<s>
This	O
differs	O
from	O
other	O
finite	O
state	O
machines	O
,	O
which	O
take	O
as	O
input	O
a	O
single	O
state	O
and	O
an	O
input	O
from	O
the	O
alphabet	O
(	O
or	O
the	O
empty	O
string	O
in	O
the	O
case	O
of	O
nondeterministic	B-General_Concept
finite	I-General_Concept
state	I-General_Concept
machines	I-General_Concept
)	O
and	O
outputs	O
the	O
next	O
state	O
(	O
or	O
the	O
set	O
of	O
possible	O
states	O
in	O
the	O
case	O
of	O
nondeterministic	B-General_Concept
finite	I-General_Concept
state	I-General_Concept
machines	I-General_Concept
)	O
.	O
</s>
<s>
A	O
DFA	B-General_Concept
or	O
NFA	B-General_Concept
can	O
easily	O
be	O
converted	O
into	O
a	O
GNFA	B-General_Concept
and	O
then	O
the	O
GNFA	B-General_Concept
can	O
be	O
easily	O
converted	O
into	O
a	O
regular	B-Language
expression	I-Language
by	O
repeatedly	O
collapsing	O
parts	O
of	O
it	O
to	O
single	O
edges	O
until	O
S	O
=	O
{	O
s	O
,	O
a}	O
.	O
</s>
<s>
Similarly	O
,	O
GNFAs	B-General_Concept
can	O
be	O
reduced	O
to	O
NFAs	O
by	O
changing	O
regular	B-Language
expression	I-Language
operators	O
into	O
new	O
edges	O
until	O
each	O
edge	O
is	O
labelled	O
with	O
a	O
regular	B-Language
expression	I-Language
matching	O
a	O
single	O
string	O
of	O
length	O
at	O
most	O
1	O
.	O
</s>
<s>
NFAs	O
,	O
in	O
turn	O
,	O
can	O
be	O
reduced	O
to	O
DFAs	B-General_Concept
using	O
the	O
powerset	B-General_Concept
construction	I-General_Concept
.	O
</s>
<s>
This	O
shows	O
that	O
GNFAs	B-General_Concept
recognize	O
the	O
same	O
set	O
of	O
formal	O
languages	O
as	O
DFAs	B-General_Concept
and	O
NFAs	O
.	O
</s>
