<s>
In	O
Boolean	O
logic	O
,	O
a	O
formula	O
is	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
(	O
CNF	O
)	O
or	O
clausal	B-Application
normal	I-Application
form	I-Application
if	O
it	O
is	O
a	O
conjunction	O
of	O
one	O
or	O
more	O
clauses	B-Application
,	O
where	O
a	O
clause	B-Application
is	O
a	O
disjunction	O
of	O
literals	O
;	O
otherwise	O
put	O
,	O
it	O
is	O
a	O
product	O
of	O
sums	O
or	O
an	O
AND	O
of	O
ORs	O
.	O
</s>
<s>
As	O
a	O
canonical	O
normal	O
form	O
,	O
it	O
is	O
useful	O
in	O
automated	B-Application
theorem	I-Application
proving	I-Application
and	O
circuit	O
theory	O
.	O
</s>
<s>
All	O
conjunctions	O
of	O
literals	O
and	O
all	O
disjunctions	O
of	O
literals	O
are	O
in	O
CNF	O
,	O
as	O
they	O
can	O
be	O
seen	O
as	O
conjunctions	O
of	O
one-literal	O
clauses	B-Application
and	O
conjunctions	O
of	O
a	O
single	O
clause	B-Application
,	O
respectively	O
.	O
</s>
<s>
As	O
in	O
the	O
disjunctive	B-Application
normal	I-Application
form	I-Application
(	O
DNF	B-Application
)	O
,	O
the	O
only	O
propositional	O
connectives	O
a	O
formula	O
in	O
CNF	O
can	O
contain	O
are	O
and	O
,	O
or	O
,	O
and	O
not	O
.	O
</s>
<s>
In	O
automated	B-Application
theorem	I-Application
proving	I-Application
,	O
the	O
notion	O
"	O
clausal	B-Application
normal	I-Application
form	I-Application
"	O
is	O
often	O
used	O
in	O
a	O
narrower	O
sense	O
,	O
meaning	O
a	O
particular	O
representation	O
of	O
a	O
CNF	O
formula	O
as	O
a	O
set	O
of	O
sets	O
of	O
literals	O
.	O
</s>
<s>
All	O
of	O
the	O
following	O
formulas	O
in	O
the	O
variables	O
,	O
and	O
are	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
:	O
</s>
<s>
For	O
clarity	O
,	O
the	O
disjunctive	O
clauses	B-Application
are	O
written	O
inside	O
parentheses	O
above	O
.	O
</s>
<s>
In	O
disjunctive	B-Application
normal	I-Application
form	I-Application
with	O
parenthesized	O
conjunctive	O
clauses	B-Application
,	O
the	O
last	O
case	O
is	O
the	O
same	O
,	O
but	O
the	O
next	O
to	O
last	O
is	O
.	O
</s>
<s>
The	O
constants	O
true	O
and	O
false	O
are	O
denoted	O
by	O
the	O
empty	O
conjunct	O
and	O
one	O
clause	B-Application
consisting	O
of	O
the	O
empty	O
disjunct	O
,	O
but	O
are	O
normally	O
written	O
explicitly	O
.	O
</s>
<s>
The	O
following	O
formulas	O
are	O
not	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
:	O
</s>
<s>
Every	O
formula	O
can	O
be	O
equivalently	O
written	O
as	O
a	O
formula	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Since	O
all	O
propositional	O
formulas	O
can	O
be	O
converted	O
into	O
an	O
equivalent	O
formula	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
,	O
proofs	O
are	O
often	O
based	O
on	O
the	O
assumption	O
that	O
all	O
formulae	O
are	O
CNF	O
.	O
</s>
<s>
For	O
example	O
,	O
translating	O
the	O
following	O
non-CNF	O
formula	O
into	O
CNF	O
produces	O
a	O
formula	O
with	O
clauses	B-Application
:	O
</s>
<s>
This	O
formula	O
contains	O
clauses	B-Application
;	O
each	O
clause	B-Application
contains	O
either	O
or	O
for	O
each	O
.	O
</s>
<s>
There	O
exist	O
transformations	O
into	O
CNF	O
that	O
avoid	O
an	O
exponential	O
increase	O
in	O
size	O
by	O
preserving	O
satisfiability	B-Algorithm
rather	O
than	O
equivalence	O
.	O
</s>
<s>
An	O
alternative	O
translation	O
,	O
the	O
Tseitin	O
transformation	O
,	O
includes	O
also	O
the	O
clauses	B-Application
.	O
</s>
<s>
With	O
these	O
clauses	B-Application
,	O
the	O
formula	O
implies	O
;	O
this	O
formula	O
is	O
often	O
regarded	O
to	O
"	O
define	O
"	O
to	O
be	O
a	O
name	O
for	O
.	O
</s>
<s>
In	O
first	O
order	O
logic	O
,	O
conjunctive	B-Application
normal	I-Application
form	I-Application
can	O
be	O
taken	O
further	O
to	O
yield	O
the	O
clausal	B-Application
normal	I-Application
form	I-Application
of	O
a	O
logical	O
formula	O
,	O
which	O
can	O
be	O
then	O
used	O
to	O
perform	O
first-order	O
resolution	O
.	O
</s>
<s>
An	O
important	O
set	O
of	O
problems	O
in	O
computational	O
complexity	O
involves	O
finding	O
assignments	O
to	O
the	O
variables	O
of	O
a	O
boolean	O
formula	O
expressed	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
,	O
such	O
that	O
the	O
formula	O
is	O
true	O
.	O
</s>
<s>
3-SAT	B-Algorithm
is	O
NP-complete	O
(	O
like	O
any	O
other	O
k-SAT	O
problem	O
with	O
k>2	O
)	O
while	O
2-SAT	B-Application
is	O
known	O
to	O
have	O
solutions	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
task	O
of	O
converting	O
a	O
formula	O
into	O
a	O
DNF	B-Application
,	O
preserving	O
satisfiability	B-Algorithm
,	O
is	O
NP-hard	O
;	O
dually	O
,	O
converting	O
into	O
CNF	O
,	O
preserving	O
validity	O
,	O
is	O
also	O
NP-hard	O
;	O
hence	O
equivalence-preserving	O
conversion	O
into	O
DNF	B-Application
or	O
CNF	O
is	O
again	O
NP-hard	O
.	O
</s>
<s>
Typical	O
problems	O
in	O
this	O
case	O
involve	O
formulas	O
in	O
"	O
3CNF	O
"	O
:	O
conjunctive	B-Application
normal	I-Application
form	I-Application
with	O
no	O
more	O
than	O
three	O
variables	O
per	O
conjunct	O
.	O
</s>
<s>
Convert	O
to	O
negation	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Repeatedly	O
replace	O
with	O
,	O
where	O
is	O
a	O
new	O
-ary	O
function	O
symbol	O
,	O
a	O
so-called	O
"	O
Skolem	B-Application
function	I-Application
"	O
.	O
</s>
<s>
This	O
is	O
the	O
only	O
step	O
that	O
preserves	O
only	O
satisfiability	B-Algorithm
rather	O
than	O
equivalence	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
formula	O
saying	O
"	O
Anyone	O
who	O
loves	O
all	O
animals	O
,	O
is	O
in	O
turn	O
loved	O
by	O
someone	O
"	O
is	O
converted	O
into	O
CNF	O
(	O
and	O
subsequently	O
into	O
clause	B-Application
form	O
in	O
the	O
last	O
line	O
)	O
as	O
follows	O
(	O
highlighting	O
replacement	O
rule	O
redexes	O
in	O
)	O
:	O
</s>
<s>
Informally	O
,	O
the	O
Skolem	B-Application
function	I-Application
can	O
be	O
thought	O
of	O
as	O
yielding	O
the	O
person	O
by	O
whom	O
is	O
loved	O
,	O
while	O
yields	O
the	O
animal	O
(	O
if	O
any	O
)	O
that	O
does	O
n't	O
love	O
.	O
</s>
