<s>
In	O
computational	O
complexity	O
theory	O
,	O
a	O
branch	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
states	O
necessary	O
and	O
sufficient	O
conditions	O
under	O
which	O
a	O
finite	O
set	O
S	O
of	O
relations	O
over	O
the	O
Boolean	O
domain	O
yields	O
polynomial-time	O
or	O
NP-complete	O
problems	O
when	O
the	O
relations	O
of	O
S	O
are	O
used	O
to	O
constrain	O
some	O
of	O
the	O
propositional	O
variables	O
.	O
</s>
<s>
Special	O
cases	O
of	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
include	O
the	O
NP-completeness	O
of	O
SAT	O
(	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
)	O
and	O
its	O
two	O
popular	O
variants	O
1-in-3	O
SAT	O
and	O
not-all-equal	B-Application
3SAT	I-Application
(	O
often	O
denoted	O
by	O
NAE-3SAT	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
these	O
two	O
variants	O
of	O
SAT	O
,	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
shows	O
that	O
their	O
monotone	O
versions	O
(	O
where	O
negations	O
of	O
variables	O
are	O
not	O
allowed	O
)	O
are	O
also	O
NP-complete	O
.	O
</s>
<s>
Schaefer	O
defines	O
a	O
decision	O
problem	O
that	O
he	O
calls	O
the	O
Generalized	O
Satisfiability	B-Algorithm
problem	I-Algorithm
for	O
S	O
(	O
denoted	O
by	O
SAT(S )	O
)	O
,	O
where	O
is	O
a	O
finite	O
set	O
of	O
relations	O
over	O
propositional	O
variables	O
.	O
</s>
<s>
A	O
finite	O
set	O
of	O
relations	O
S	O
over	O
the	O
Boolean	O
domain	O
defines	O
a	O
polynomial	O
time	O
computable	O
satisfiability	B-Algorithm
problem	I-Algorithm
if	O
any	O
one	O
of	O
the	O
following	O
conditions	O
holds	O
:	O
</s>
<s>
all	O
relations	O
are	O
equivalent	O
to	O
a	O
conjunction	O
of	O
Horn	B-Application
clauses	I-Application
;	O
</s>
<s>
all	O
relations	O
are	O
equivalent	O
to	O
a	O
conjunction	O
of	O
dual-Horn	B-Application
clauses	I-Application
;	O
</s>
<s>
In	O
modern	O
terms	O
,	O
the	O
problem	O
SAT(S )	O
is	O
viewed	O
as	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
over	O
the	O
Boolean	O
domain	O
.	O
</s>
<s>
For	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
,	O
the	O
most	O
important	O
concept	O
in	O
universal	O
algebra	O
is	O
that	O
of	O
a	O
polymorphism	O
.	O
</s>
<s>
This	O
definition	O
allows	O
for	O
the	O
algebraic	O
formulation	O
of	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
.	O
</s>
<s>
If	O
Γ	O
 '	O
⊆	O
≪Γ≫	O
for	O
some	O
finite	O
constraint	O
sets	O
Γ	O
and	O
Γ	O
 '	O
,	O
then	O
CSP(Γ' )	O
reduces	B-Algorithm
to	O
CSP(Γ )	O
.	O
</s>
<s>
Moreover	O
,	O
if	O
Pol(Γ )	O
⊆	O
Pol(Γ' )	O
for	O
two	O
finite	O
relation	O
sets	O
Γ	O
and	O
Γ	O
 '	O
,	O
then	O
Γ	O
 '	O
⊆	O
≪Γ≫	O
and	O
CSP(Γ' )	O
reduces	B-Algorithm
to	O
CSP(Γ )	O
.	O
</s>
<s>
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
was	O
recently	O
generalized	O
to	O
a	O
larger	O
class	O
of	O
relations	O
.	O
</s>
<s>
A	O
Mal'tsev	O
operation	O
m	O
is	O
a	O
ternary	O
operation	O
that	O
satisfies	O
An	O
example	O
of	O
a	O
Mal'tsev	O
operation	O
is	O
the	O
Minority	O
operation	O
given	O
in	O
the	O
modern	O
,	O
algebraic	O
formulation	O
of	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
above	O
.	O
</s>
