<s>
In	O
mathematical	O
logic	O
,	O
a	O
formula	O
is	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
(	O
NNF	O
)	O
if	O
the	O
negation	O
operator	O
(	O
,	O
)	O
is	O
only	O
applied	O
to	O
variables	O
and	O
the	O
only	O
other	O
allowed	O
Boolean	O
operators	O
are	O
conjunction	O
(	O
,	O
)	O
and	O
disjunction	O
(	O
,	O
)	O
.	O
</s>
<s>
Negation	B-Application
normal	I-Application
form	I-Application
is	O
not	O
a	O
canonical	O
form	O
:	O
for	O
example	O
,	O
and	O
are	O
equivalent	O
,	O
and	O
are	O
both	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Transformation	O
into	O
negation	B-Application
normal	I-Application
form	I-Application
can	O
increase	O
the	O
size	O
of	O
a	O
formula	O
only	O
linearly	O
:	O
the	O
number	O
of	O
occurrences	O
of	O
atomic	O
formulas	O
remains	O
the	O
same	O
,	O
the	O
total	O
number	O
of	O
occurrences	O
of	O
and	O
is	O
unchanged	O
,	O
and	O
the	O
number	O
of	O
occurrences	O
of	O
in	O
the	O
normal	O
form	O
is	O
bounded	O
by	O
the	O
length	O
of	O
the	O
original	O
formula	O
.	O
</s>
<s>
A	O
formula	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
can	O
be	O
put	O
into	O
the	O
stronger	O
conjunctive	B-Application
normal	I-Application
form	I-Application
or	O
disjunctive	B-Application
normal	I-Application
form	I-Application
by	O
applying	O
distributivity	O
.	O
</s>
<s>
In	O
the	O
classical	O
propositional	O
logic	O
,	O
transformation	O
to	O
negation	B-Application
normal	I-Application
form	I-Application
does	O
not	O
impact	O
computational	O
properties	O
:	O
the	O
satisfiability	B-Algorithm
problem	I-Algorithm
continues	O
to	O
be	O
NP-complete	O
,	O
and	O
the	O
validity	O
problem	O
continues	O
to	O
be	O
co-NP-complete	O
.	O
</s>
<s>
For	O
formulas	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
,	O
the	O
validity	O
problem	O
is	O
solvable	O
in	O
polynomial	O
time	O
,	O
and	O
for	O
formulas	O
in	O
disjunctive	B-Application
normal	I-Application
form	I-Application
,	O
the	O
satisfiability	B-Algorithm
problem	I-Algorithm
is	O
solvable	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
following	O
formulae	O
are	O
all	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
:	O
</s>
<s>
The	O
first	O
example	O
is	O
also	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
and	O
the	O
last	O
two	O
are	O
in	O
both	O
conjunctive	B-Application
normal	I-Application
form	I-Application
and	O
disjunctive	B-Application
normal	I-Application
form	I-Application
,	O
but	O
the	O
second	O
example	O
is	O
in	O
neither	O
.	O
</s>
<s>
The	O
following	O
formulae	O
are	O
not	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
:	O
</s>
<s>
They	O
are	O
however	O
respectively	O
equivalent	O
to	O
the	O
following	O
formulae	O
in	O
negation	B-Application
normal	I-Application
form	I-Application
:	O
</s>
