<s>
In	O
logic	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
sometimes	O
called	O
propositional	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
and	O
abbreviated	O
SATISFIABILITY	O
,	O
SAT	O
or	O
B-SAT	O
)	O
is	O
the	O
problem	O
of	O
determining	O
if	O
there	B-Algorithm
exists	I-Algorithm
an	O
interpretation	O
that	O
satisfies	O
a	O
given	O
Boolean	O
formula	O
.	O
</s>
<s>
There	O
is	O
no	O
known	O
algorithm	O
that	O
efficiently	O
solves	O
each	O
SAT	B-Algorithm
problem	I-Algorithm
,	O
and	O
it	O
is	O
generally	O
believed	O
that	O
no	O
such	O
algorithm	O
exists	O
;	O
yet	O
this	O
belief	O
has	O
not	O
been	O
proved	O
mathematically	O
,	O
and	O
resolving	O
the	O
question	O
of	O
whether	O
SAT	O
has	O
a	O
polynomial-time	O
algorithm	O
is	O
equivalent	O
to	O
the	O
P	O
versus	O
NP	O
problem	O
,	O
which	O
is	O
a	O
famous	O
open	O
problem	O
in	O
the	O
theory	O
of	O
computing	O
.	O
</s>
<s>
Nevertheless	O
,	O
as	O
of	O
2007	O
,	O
heuristic	O
SAT-algorithms	O
are	O
able	O
to	O
solve	O
problem	O
instances	O
involving	O
tens	O
of	O
thousands	O
of	O
variables	O
and	O
formulas	O
consisting	O
of	O
millions	O
of	O
symbols	O
,	O
which	O
is	O
sufficient	O
for	O
many	O
practical	O
SAT	B-Algorithm
problems	I-Algorithm
from	O
,	O
e.g.	O
,	O
artificial	B-Application
intelligence	I-Application
,	O
circuit	O
design	O
,	O
and	O
automatic	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
The	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	O
)	O
is	O
,	O
given	O
a	O
formula	O
,	O
to	O
check	O
whether	O
it	O
is	O
satisfiable	O
.	O
</s>
<s>
This	O
decision	O
problem	O
is	O
of	O
central	O
importance	O
in	O
many	O
areas	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
including	O
theoretical	O
computer	B-General_Concept
science	I-General_Concept
,	O
complexity	O
theory	O
,	O
algorithmics	B-Data_Structure
,	O
cryptography	O
and	O
artificial	B-Application
intelligence	I-Application
.	I-Application
</s>
<s>
A	O
clause	O
is	O
called	O
a	O
Horn	B-Application
clause	I-Application
if	O
it	O
contains	O
at	O
most	O
one	O
positive	O
literal	O
.	O
</s>
<s>
A	O
formula	O
is	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
(	O
CNF	O
)	O
if	O
it	O
is	O
a	O
conjunction	O
of	O
clauses	O
(	O
or	O
a	O
single	O
clause	O
)	O
.	O
</s>
<s>
The	O
formula	O
is	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
;	O
its	O
first	O
and	O
third	O
clauses	O
are	O
Horn	B-Application
clauses	I-Application
,	O
but	O
its	O
second	O
clause	O
is	O
not	O
.	O
</s>
<s>
For	O
some	O
versions	O
of	O
the	O
SAT	B-Algorithm
problem	I-Algorithm
,	O
it	O
is	O
useful	O
to	O
define	O
the	O
notion	O
of	O
a	O
generalized	O
conjunctive	B-Application
normal	I-Application
form	I-Application
formula	O
,	O
viz	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
R( ¬x	O
,	O
a	O
,	O
b	O
)	O
is	O
a	O
generalized	O
clause	O
,	O
and	O
R( ¬x	O
,	O
a	O
,	O
b	O
)	O
∧	O
R(b,y,c )	O
∧	O
R( c	O
,	O
d	O
,	O
¬z	O
)	O
is	O
a	O
generalized	O
conjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Using	O
the	O
laws	O
of	O
Boolean	O
algebra	O
,	O
every	O
propositional	O
logic	O
formula	O
can	O
be	O
transformed	O
into	O
an	O
equivalent	O
conjunctive	B-Application
normal	I-Application
form	I-Application
,	O
which	O
may	O
,	O
however	O
,	O
be	O
exponentially	O
longer	O
.	O
</s>
<s>
However	O
,	O
with	O
use	O
of	O
the	O
Tseytin	O
transformation	O
,	O
we	O
may	O
find	O
an	O
equisatisfiable	O
conjunctive	B-Application
normal	I-Application
form	I-Application
formula	O
with	O
length	O
linear	O
in	O
the	O
size	O
of	O
the	O
original	O
propositional	O
logic	O
formula	O
.	O
</s>
<s>
The	O
proof	O
shows	O
how	O
every	O
decision	O
problem	O
in	O
the	O
complexity	O
class	O
NP	O
can	O
be	O
reduced	B-Algorithm
to	O
the	O
SAT	B-Algorithm
problem	I-Algorithm
for	O
CNF	O
formulas	O
,	O
sometimes	O
called	O
CNFSAT	O
.	O
</s>
<s>
Like	O
the	O
satisfiability	B-Algorithm
problem	I-Algorithm
for	O
arbitrary	O
formulas	O
,	O
determining	O
the	O
satisfiability	O
of	O
a	O
formula	O
in	O
conjunctive	B-Application
normal	I-Application
form	I-Application
where	O
each	O
clause	O
is	O
limited	O
to	O
at	O
most	O
three	O
literals	O
is	O
NP-complete	O
also	O
;	O
this	O
problem	O
is	O
called	O
3-SAT	O
,	O
3CNFSAT	O
,	O
or	O
3-satisfiability	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
from	O
3-SAT	O
to	O
the	O
other	O
problem	O
.	O
</s>
<s>
Difficulty	O
is	O
measured	O
in	O
number	O
recursive	O
calls	O
made	O
by	O
a	O
DPLL	B-Application
algorithm	I-Application
.	O
</s>
<s>
Conjunctive	B-Application
normal	I-Application
form	I-Application
(	O
in	O
particular	O
with	O
3	O
literals	O
per	O
clause	O
)	O
is	O
often	O
considered	O
the	O
canonical	O
representation	O
for	O
SAT	O
formulas	O
.	O
</s>
<s>
As	O
shown	O
above	O
,	O
the	O
general	O
SAT	B-Algorithm
problem	I-Algorithm
reduces	O
to	O
3-SAT	O
,	O
the	O
problem	O
of	O
determining	O
satisfiability	O
for	O
formulas	O
in	O
this	O
form	O
.	O
</s>
<s>
SAT	O
is	O
trivial	O
if	O
the	O
formulas	O
are	O
restricted	O
to	O
those	O
in	O
disjunctive	B-Application
normal	I-Application
form	I-Application
,	O
that	O
is	O
,	O
they	O
are	O
a	O
disjunction	O
of	O
conjunctions	O
of	O
literals	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
they	O
are	O
restricted	O
to	O
being	O
in	O
full	B-Application
disjunctive	I-Application
normal	I-Application
form	I-Application
,	O
in	O
which	O
every	O
variable	O
appears	O
exactly	O
once	O
in	O
every	O
conjunction	O
,	O
they	O
can	O
be	O
checked	O
in	O
constant	O
time	O
(	O
each	O
conjunction	O
represents	O
one	O
satisfying	O
assignment	O
)	O
.	O
</s>
<s>
But	O
it	O
can	O
take	O
exponential	O
time	O
and	O
space	O
to	O
convert	O
a	O
general	O
SAT	B-Algorithm
problem	I-Algorithm
to	O
disjunctive	B-Application
normal	I-Application
form	I-Application
;	O
for	O
an	O
example	O
exchange	O
"	O
∧	O
"	O
and	O
"	O
∨	O
"	O
in	O
the	O
above	O
exponential	O
blow-up	O
example	O
for	O
conjunctive	B-Application
normal	I-Application
forms	I-Application
.	O
</s>
<s>
Given	O
a	O
conjunctive	B-Application
normal	I-Application
form	I-Application
with	O
three	O
literals	O
per	O
clause	O
,	O
the	O
problem	O
is	O
to	O
determine	O
whether	O
there	B-Algorithm
exists	I-Algorithm
a	O
truth	O
assignment	O
to	O
the	O
variables	O
so	O
that	O
each	O
clause	O
has	O
exactly	O
one	O
TRUE	O
literal	O
(	O
and	O
thus	O
exactly	O
two	O
FALSE	O
literals	O
)	O
.	O
</s>
<s>
Formally	O
,	O
a	O
one-in-three	O
3-SAT	O
problem	O
is	O
given	O
as	O
a	O
generalized	O
conjunctive	B-Application
normal	I-Application
form	I-Application
with	O
all	O
generalized	O
clauses	O
using	O
a	O
ternary	O
operator	O
R	O
that	O
is	O
TRUE	O
just	O
if	O
exactly	O
one	O
of	O
its	O
arguments	O
is	O
.	O
</s>
<s>
When	O
all	O
literals	O
of	O
a	O
one-in-three	O
3-SAT	O
formula	O
are	O
positive	O
,	O
the	O
satisfiability	B-Algorithm
problem	I-Algorithm
is	O
called	O
one-in-three	O
positive	O
3-SAT	O
.	O
</s>
<s>
One-in-three	O
3-SAT	O
was	O
proved	O
to	O
be	O
NP-complete	O
by	O
Thomas	O
Jerome	O
Schaefer	O
as	O
a	O
special	O
case	O
of	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
,	O
which	O
asserts	O
that	O
any	O
problem	O
generalizing	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
in	O
a	O
certain	O
way	O
is	O
either	O
in	O
the	O
class	O
P	O
or	O
is	O
NP-complete	O
.	O
</s>
<s>
Schaefer	O
gives	O
a	O
construction	O
allowing	O
an	O
easy	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
from	O
3-SAT	O
to	O
one-in-three	O
3-SAT	O
.	O
</s>
<s>
Given	O
a	O
conjunctive	B-Application
normal	I-Application
form	I-Application
with	O
three	O
literals	O
per	O
clause	O
,	O
the	O
problem	O
is	O
to	O
determine	O
if	O
an	O
assignment	O
to	O
the	O
variables	O
exists	O
such	O
that	O
in	O
no	O
clause	O
all	O
three	O
literals	O
have	O
the	O
same	O
truth	O
value	O
.	O
</s>
<s>
This	O
problem	O
is	O
NP-complete	O
,	O
too	O
,	O
even	O
if	O
no	O
negation	O
symbols	O
are	O
admitted	O
,	O
by	O
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
.	O
</s>
<s>
A	O
3-SAT	O
formula	O
is	O
Linear	B-Algorithm
SAT	I-Algorithm
(	O
LSAT	O
)	O
if	O
each	O
clause	O
(	O
viewed	O
as	O
a	O
set	O
of	O
literals	O
)	O
intersects	O
at	O
most	O
one	O
other	O
clause	O
,	O
and	O
,	O
moreover	O
,	O
if	O
two	O
clauses	O
intersect	O
,	O
then	O
they	O
have	O
exactly	O
one	O
literal	O
in	O
common	O
.	O
</s>
<s>
SAT	O
is	O
easier	O
if	O
the	O
number	O
of	O
literals	O
in	O
a	O
clause	O
is	O
limited	O
to	O
at	O
most	O
2	O
,	O
in	O
which	O
case	O
the	O
problem	O
is	O
called	O
2-SAT	B-Application
.	O
</s>
<s>
If	O
additionally	O
all	O
OR	O
operations	O
in	O
literals	O
are	O
changed	O
to	O
XOR	O
operations	O
,	O
the	O
result	O
is	O
called	O
exclusive-or	O
2-satisfiability	B-Application
,	O
which	O
is	O
a	O
problem	O
complete	O
for	O
the	O
complexity	O
class	O
SL	O
=	O
L	O
.	O
</s>
<s>
The	O
problem	O
of	O
deciding	O
the	O
satisfiability	O
of	O
a	O
given	O
conjunction	O
of	O
Horn	B-Application
clauses	I-Application
is	O
called	O
Horn-satisfiability	O
,	O
or	O
HORN-SAT	O
.	O
</s>
<s>
It	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
by	O
a	O
single	O
step	O
of	O
the	O
Unit	O
propagation	O
algorithm	O
,	O
which	O
produces	O
the	O
single	O
minimal	O
model	O
of	O
the	O
set	O
of	O
Horn	B-Application
clauses	I-Application
(	O
w.r.t.	O
</s>
<s>
It	O
can	O
be	O
seen	O
as	O
P	O
's	O
version	O
of	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Also	O
,	O
deciding	O
the	O
truth	O
of	O
quantified	O
Horn	B-Application
formulas	I-Application
can	O
be	O
done	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Horn	B-Application
clauses	I-Application
are	O
of	O
interest	O
because	O
they	O
are	O
able	O
to	O
express	O
implication	O
of	O
one	O
variable	O
from	O
a	O
set	O
of	O
other	O
variables	O
.	O
</s>
<s>
For	O
example	O
,	O
(	O
x1	O
∨	O
¬x2	O
)	O
∧	O
( ¬x1	O
∨	O
x2	O
∨	O
x3	O
)	O
∧	O
¬x1	O
is	O
not	O
a	O
Horn	B-Application
formula	I-Application
,	O
but	O
can	O
be	O
renamed	O
to	O
the	O
Horn	B-Application
formula	I-Application
(	O
x1	O
∨	O
¬x2	O
)	O
∧	O
( ¬x1	O
∨	O
x2	O
∨	O
¬y3	O
)	O
∧	O
¬x1	O
by	O
introducing	O
y3	O
as	O
negation	O
of	O
x3	O
.	O
</s>
<s>
In	O
contrast	O
,	O
no	O
renaming	O
of	O
(	O
x1	O
∨	O
¬x2	O
∨	O
¬x3	O
)	O
∧	O
( ¬x1	O
∨	O
x2	O
∨	O
x3	O
)	O
∧	O
¬x1	O
leads	O
to	O
a	O
Horn	B-Application
formula	I-Application
.	O
</s>
<s>
Checking	O
the	O
existence	O
of	O
such	O
a	O
replacement	O
can	O
be	O
done	O
in	O
linear	O
time	O
;	O
therefore	O
,	O
the	O
satisfiability	O
of	O
such	O
formulae	O
is	O
in	O
P	O
as	O
it	O
can	O
be	O
solved	O
by	O
first	O
performing	O
this	O
replacement	O
and	O
then	O
checking	O
the	O
satisfiability	O
of	O
the	O
resulting	O
Horn	B-Application
formula	I-Application
.	O
</s>
<s>
Solving	O
an	O
XOR-SAT	O
exampleby	O
Gaussian	B-Algorithm
elimination	I-Algorithm
Given	O
formula	O
("⊕"	O
means	O
XOR	O
,	O
the	O
is	O
optional	O
)	O
(	O
a⊕c⊕d	O
)	O
∧	O
(	O
b⊕¬c⊕d	O
)	O
∧	O
(	O
a⊕b⊕¬d	O
)	O
∧	O
(	O
a⊕¬b⊕¬c	O
)	O
Equation	O
system	O
(	O
"	O
1	O
"	O
means	O
TRUE	O
,	O
"	O
0	O
"	O
means	O
FALSE	O
)	O
Each	O
clause	O
leads	O
to	O
one	O
equation	O
.	O
</s>
<s>
This	O
is	O
in	O
P	O
,	O
since	O
an	O
XOR-SAT	O
formula	O
can	O
also	O
be	O
viewed	O
as	O
a	O
system	O
of	O
linear	O
equations	O
mod	O
2	O
,	O
and	O
can	O
be	O
solved	O
in	O
cubic	O
time	O
by	O
Gaussian	B-Algorithm
elimination	I-Algorithm
;	O
see	O
the	O
box	O
for	O
an	O
example	O
.	O
</s>
<s>
Schaefer	B-Application
's	I-Application
dichotomy	I-Application
theorem	I-Application
states	O
that	O
,	O
for	O
any	O
restriction	O
to	O
Boolean	O
functions	O
that	O
can	O
be	O
used	O
to	O
form	O
these	O
subformulae	O
,	O
the	O
corresponding	O
satisfiability	B-Algorithm
problem	I-Algorithm
is	O
in	O
P	O
or	O
NP-complete	O
.	O
</s>
<s>
An	O
extension	O
that	O
has	O
gained	O
significant	O
popularity	O
since	O
2003	O
is	O
satisfiability	B-Application
modulo	I-Application
theories	I-Application
(	O
SMT	O
)	O
that	O
can	O
enrich	O
CNF	O
formulas	O
with	O
linear	O
constraints	O
,	O
arrays	O
,	O
all-different	O
constraints	O
,	O
uninterpreted	B-Language
functions	I-Language
,	O
etc	O
.	O
</s>
<s>
The	O
satisfiability	B-Algorithm
problem	I-Algorithm
becomes	O
more	O
difficult	O
if	O
both	O
"	O
for	O
all	O
"	O
( ∀	O
)	O
and	O
"	O
there	B-Algorithm
exists	I-Algorithm
"	O
( ∃	O
)	O
quantifiers	B-Language
are	O
allowed	O
to	O
bind	O
the	O
Boolean	O
variables	O
.	O
</s>
<s>
SAT	O
itself	O
(	O
tacitly	O
)	O
uses	O
only	O
∃	B-Algorithm
quantifiers	B-Language
.	O
</s>
<s>
If	O
only	O
∀	O
quantifiers	B-Language
are	O
allowed	O
instead	O
,	O
the	O
so-called	O
tautology	O
problem	O
is	O
obtained	O
,	O
which	O
is	O
co-NP-complete	O
.	O
</s>
<s>
If	O
both	O
quantifiers	B-Language
are	O
allowed	O
,	O
the	O
problem	O
is	O
called	O
the	O
quantified	O
Boolean	O
formula	O
problem	O
(	O
QBF	O
)	O
,	O
which	O
can	O
be	O
shown	O
to	O
be	O
PSPACE-complete	O
.	O
</s>
<s>
Using	O
highly	O
parallel	O
P	B-Application
systems	I-Application
,	O
QBF-SAT	O
problems	O
can	O
be	O
solved	O
in	O
linear	O
time	O
.	O
</s>
<s>
It	O
is	O
complete	O
for	O
US	O
,	O
the	O
complexity	O
class	O
describing	O
problems	O
solvable	O
by	O
a	O
non-deterministic	O
polynomial	O
time	O
Turing	B-Architecture
machine	I-Architecture
that	O
accepts	O
when	O
there	O
is	O
exactly	O
one	O
nondeterministic	O
accepting	O
path	O
and	O
rejects	O
otherwise	O
.	O
</s>
<s>
UNAMBIGUOUS-SAT	O
is	O
the	O
name	O
given	O
to	O
the	O
satisfiability	B-Algorithm
problem	I-Algorithm
when	O
the	O
input	O
is	O
restricted	O
to	O
formulas	O
having	O
at	O
most	O
one	O
satisfying	O
assignment	O
.	O
</s>
<s>
MAX-SAT	B-Application
,	O
the	O
maximum	B-Application
satisfiability	I-Application
problem	I-Application
,	O
is	O
an	O
FNP	O
generalization	O
of	O
SAT	O
.	O
</s>
<s>
It	O
has	O
efficient	O
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
but	O
is	O
NP-hard	O
to	O
solve	O
exactly	O
.	O
</s>
<s>
Worse	O
still	O
,	O
it	O
is	O
APX-complete	B-Algorithm
,	O
meaning	O
there	O
is	O
no	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
for	O
this	O
problem	O
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
Other	O
generalizations	O
include	O
satisfiability	O
for	O
first	O
-	O
and	O
second-order	O
logic	O
,	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
,	O
0-1	O
integer	O
programming	O
.	O
</s>
<s>
Since	O
the	O
SAT	B-Algorithm
problem	I-Algorithm
is	O
NP-complete	O
,	O
only	O
algorithms	O
with	O
exponential	O
worst-case	O
complexity	O
are	O
known	O
for	O
it	O
.	O
</s>
<s>
Examples	O
of	O
such	O
problems	O
in	O
electronic	O
design	O
automation	O
(	O
EDA	O
)	O
include	O
formal	B-Application
equivalence	I-Application
checking	I-Application
,	O
model	B-Application
checking	I-Application
,	O
formal	O
verification	O
of	O
pipelined	B-Architecture
microprocessors	I-Architecture
,	O
automatic	O
test	O
pattern	O
generation	O
,	O
routing	B-Algorithm
of	O
FPGAs	B-Architecture
,	O
planning	B-Application
,	O
and	O
scheduling	O
problems	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Major	O
techniques	O
used	O
by	O
modern	O
SAT	O
solvers	O
include	O
the	O
Davis	B-Application
–	I-Application
Putnam	I-Application
–	I-Application
Logemann	I-Application
–	I-Application
Loveland	I-Application
algorithm	I-Application
(	O
or	O
DPLL	O
)	O
,	O
conflict-driven	B-Application
clause	I-Application
learning	I-Application
(	O
CDCL	B-Application
)	O
,	O
and	O
stochastic	O
local	B-Application
search	I-Application
algorithms	O
such	O
as	O
WalkSAT	B-Application
.	O
</s>
<s>
Modern	O
SAT	O
solvers	O
are	O
also	O
having	O
significant	O
impact	O
on	O
the	O
fields	O
of	O
software	O
verification	O
,	O
constraint	B-Application
solving	I-Application
in	O
artificial	B-Application
intelligence	I-Application
,	O
and	O
operations	O
research	O
,	O
among	O
others	O
.	O
</s>
