<s>
In	O
computational	O
complexity	O
,	O
not-all-equal	B-Application
3-satisfiability	I-Application
(	O
NAE3SAT	O
)	O
is	O
an	O
NP-complete	O
variant	O
of	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
often	O
used	O
in	O
proofs	O
of	O
NP-completeness	O
.	O
</s>
<s>
The	O
NP-completeness	O
of	O
NAE3SAT	O
can	O
be	O
proven	O
by	O
a	O
reduction	B-Algorithm
from	O
3-satisfiability	O
(	O
3SAT	O
)	O
.	O
</s>
<s>
First	O
the	O
nonsymmetric	O
3SAT	O
is	O
reduced	O
to	O
the	O
symmetric	O
NAE4SAT	O
by	O
adding	O
a	O
common	O
dummy	O
literal	O
to	O
every	O
clause	O
,	O
then	O
NAE4SAT	O
is	O
reduced	O
to	O
NAE3SAT	O
by	O
splitting	O
clauses	O
as	O
in	O
the	O
reduction	B-Algorithm
of	O
general	O
-satisfiability	O
to	O
3SAT	O
.	O
</s>
<s>
NAE3SAT	O
remains	O
NP-complete	O
when	O
all	O
clauses	O
are	O
monotone	O
(	O
meaning	O
that	O
variables	O
are	O
never	O
negated	O
)	O
,	O
by	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
.	O
</s>
