<s>
In	O
boolean	O
logic	O
,	O
a	O
disjunctive	B-Application
normal	I-Application
form	I-Application
(	O
DNF	O
)	O
is	O
a	O
canonical	O
normal	O
form	O
of	O
a	O
logical	O
formula	O
consisting	O
of	O
a	O
disjunction	O
of	O
conjunctions	O
;	O
it	O
can	O
also	O
be	O
described	O
as	O
an	O
OR	O
of	O
ANDs	O
,	O
a	O
sum	O
of	O
products	O
,	O
or	O
(	O
in	O
philosophical	O
logic	O
)	O
a	O
cluster	O
concept	O
.	O
</s>
<s>
As	O
a	O
normal	O
form	O
,	O
it	O
is	O
useful	O
in	O
automated	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
A	O
DNF	O
formula	O
is	O
in	O
full	B-Application
disjunctive	I-Application
normal	I-Application
form	I-Application
if	O
each	O
of	O
its	O
variables	O
appears	O
exactly	O
once	O
in	O
every	O
conjunction	O
.	O
</s>
<s>
As	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
(	O
CNF	O
)	O
,	O
the	O
only	O
propositional	O
operators	O
in	O
DNF	O
are	O
and	O
(	O
)	O
,	O
or	O
(	O
)	O
,	O
and	O
not	O
(	O
)	O
.	O
</s>
<s>
All	O
logical	O
formulas	O
can	O
be	O
converted	O
into	O
an	O
equivalent	O
disjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Every	O
particular	O
Boolean	O
function	O
can	O
be	O
represented	O
by	O
one	O
and	O
only	O
one	O
full	B-Application
disjunctive	I-Application
normal	I-Application
form	I-Application
,	O
one	O
of	O
the	O
canonical	O
forms	O
.	O
</s>
<s>
In	O
contrast	O
,	O
two	O
different	O
plain	O
disjunctive	B-Application
normal	I-Application
forms	I-Application
may	O
denote	O
the	O
same	O
Boolean	O
function	O
;	O
see	O
the	O
illustrations	O
.	O
</s>
<s>
The	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
on	O
conjunctive	B-Application
normal	I-Application
form	I-Application
formulas	O
is	O
NP-hard	O
;	O
by	O
the	O
duality	O
principle	O
,	O
so	O
is	O
the	O
falsifiability	O
problem	O
on	O
DNF	O
formulas	O
.	O
</s>
<s>
An	O
important	O
variation	O
used	O
in	O
the	O
study	O
of	O
computational	B-General_Concept
complexity	I-General_Concept
is	O
k-DNF	O
.	O
</s>
