<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
formal	B-Architecture
methods	I-Architecture
,	O
a	O
SAT	B-Application
solver	I-Application
is	O
a	O
computer	B-Application
program	I-Application
which	O
aims	O
to	O
solve	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
On	O
input	O
a	O
formula	O
over	O
Boolean	O
variables	O
,	O
such	O
as	O
"	O
(	O
x	O
or	O
y	O
)	O
and	O
(	O
x	O
or	O
not	O
y	O
)	O
"	O
,	O
a	O
SAT	B-Application
solver	I-Application
outputs	O
whether	O
the	O
formula	O
is	O
satisfiable	O
,	O
meaning	O
that	O
there	O
are	O
possible	O
values	O
of	O
x	O
and	O
y	O
which	O
make	O
the	O
formula	O
true	O
,	O
or	O
unsatisfiable	O
,	O
meaning	O
that	O
there	O
are	O
no	O
such	O
values	O
of	O
x	O
and	O
y	O
.	O
</s>
<s>
Since	O
the	O
introduction	O
of	O
algorithms	O
for	O
SAT	O
in	O
the	O
1960s	O
,	O
modern	O
SAT	B-Application
solvers	I-Application
have	O
grown	O
into	O
complex	O
software	O
artifacts	O
involving	O
a	O
large	O
number	O
of	O
heuristics	B-Algorithm
and	O
program	O
optimizations	O
to	O
work	O
efficiently	O
.	O
</s>
<s>
By	O
a	O
result	O
known	O
as	O
the	O
Cook	O
–	O
Levin	O
theorem	O
,	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
is	O
an	O
NP-complete	O
problem	O
in	O
general	O
.	O
</s>
<s>
SAT	B-Application
solvers	I-Application
often	O
begin	O
by	O
converting	O
a	O
formula	O
to	O
conjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
They	O
are	O
often	O
based	O
on	O
core	O
algorithms	O
such	O
as	O
the	O
DPLL	B-Application
algorithm	I-Application
,	O
but	O
incorporate	O
a	O
number	O
of	O
extensions	O
and	O
features	O
.	O
</s>
<s>
Most	O
SAT	B-Application
solvers	I-Application
include	O
time-outs	O
,	O
so	O
they	O
will	O
terminate	O
in	O
reasonable	O
time	O
even	O
if	O
they	O
cannot	O
find	O
a	O
solution	O
with	O
an	O
output	O
such	O
as	O
"	O
unknown	O
"	O
.	O
</s>
<s>
Often	O
,	O
SAT	B-Application
solvers	I-Application
do	O
not	O
just	O
provide	O
an	O
answer	O
,	O
but	O
can	O
provide	O
further	O
information	O
including	O
an	O
example	O
assignment	O
(	O
values	O
for	O
x	O
,	O
y	O
,	O
etc	O
.	O
)	O
</s>
<s>
Modern	O
SAT	B-Application
solvers	I-Application
have	O
had	O
a	O
significant	O
impact	O
on	O
fields	O
including	O
software	O
verification	O
,	O
program	O
analysis	O
,	O
constraint	B-Application
solving	I-Application
,	O
artificial	B-Application
intelligence	I-Application
,	O
electronic	O
design	O
automation	O
,	O
and	O
operations	O
research	O
.	O
</s>
<s>
Powerful	O
solvers	O
are	O
readily	O
available	O
as	O
free	B-License
and	I-License
open-source	I-License
software	I-License
and	O
are	O
built	O
into	O
some	O
programming	O
languages	O
such	O
as	O
exposing	O
SAT	B-Application
solvers	I-Application
as	O
constraints	O
in	O
constraint	B-Application
logic	I-Application
programming	I-Application
.	O
</s>
<s>
A	O
DPLL	O
SAT	B-Application
solver	I-Application
employs	O
a	O
systematic	O
backtracking	O
search	O
procedure	O
to	O
explore	O
the	O
(	O
exponentially	O
sized	O
)	O
space	O
of	O
variable	O
assignments	O
looking	O
for	O
satisfying	O
assignments	O
.	O
</s>
<s>
The	O
basic	O
search	O
procedure	O
was	O
proposed	O
in	O
two	O
seminal	O
papers	O
in	O
the	O
early	O
1960s	O
(	O
see	O
references	O
below	O
)	O
and	O
is	O
now	O
commonly	O
referred	O
to	O
as	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
"	O
DPLL	O
"	O
or	O
"	O
DLL	O
"	O
)	O
.	O
</s>
<s>
Many	O
modern	O
approaches	O
to	O
practical	O
SAT	O
solving	O
are	O
derived	O
from	O
the	O
DPLL	B-Application
algorithm	I-Application
and	O
share	O
the	O
same	O
structure	O
.	O
</s>
<s>
Often	O
they	O
only	O
improve	O
the	O
efficiency	O
of	O
certain	O
classes	O
of	O
SAT	B-Algorithm
problems	I-Algorithm
such	O
as	O
instances	O
that	O
appear	O
in	O
industrial	O
applications	O
or	O
randomly	O
generated	O
instances	O
.	O
</s>
<s>
Algorithms	O
that	O
are	O
not	O
part	O
of	O
the	O
DPLL	O
family	O
include	O
stochastic	O
local	B-Application
search	I-Application
algorithms	O
.	O
</s>
<s>
One	O
example	O
is	O
WalkSAT	B-Application
.	O
</s>
<s>
In	O
contrast	O
,	O
randomized	O
algorithms	O
like	O
the	O
PPSZ	O
algorithm	O
by	O
Paturi	O
,	O
Pudlak	O
,	O
Saks	O
,	O
and	O
Zane	O
set	O
variables	O
in	O
a	O
random	O
order	O
according	O
to	O
some	O
heuristics	B-Algorithm
,	O
for	O
example	O
bounded-width	O
resolution	O
.	O
</s>
<s>
If	O
the	O
heuristic	B-Algorithm
ca	O
n't	O
find	O
the	O
correct	O
setting	O
,	O
the	O
variable	O
is	O
assigned	O
randomly	O
.	O
</s>
<s>
Modern	O
SAT	B-Application
solvers	I-Application
(	O
developed	O
in	O
the	O
2000s	O
)	O
come	O
in	O
two	O
flavors	O
:	O
"	O
conflict-driven	O
"	O
and	O
"	O
look-ahead	O
"	O
.	O
</s>
<s>
Conflict-driven	O
solvers	O
,	O
such	O
as	O
conflict-driven	B-Application
clause	I-Application
learning	I-Application
(	O
CDCL	B-Application
)	O
,	O
augment	O
the	O
basic	O
DPLL	O
search	O
algorithm	O
with	O
efficient	O
conflict	O
analysis	O
,	O
clause	O
learning	O
,	O
non-chronological	O
backtracking	O
(	O
a.k.a.	O
</s>
<s>
backjumping	B-Application
)	O
,	O
as	O
well	O
as	O
"	O
two-watched-literals	O
"	O
unit	O
propagation	O
,	O
adaptive	O
branching	O
,	O
and	O
random	O
restarts	O
.	O
</s>
<s>
Well	O
known	O
implementations	O
include	O
Chaff	B-Application
and	O
GRASP	B-Application
.	O
</s>
<s>
Look-ahead	O
solvers	O
have	O
especially	O
strengthened	O
reductions	O
(	O
going	O
beyond	O
unit-clause	O
propagation	O
)	O
and	O
the	O
heuristics	B-Algorithm
,	O
and	O
they	O
are	O
generally	O
stronger	O
than	O
conflict-driven	O
solvers	O
on	O
hard	O
instances	O
(	O
while	O
conflict-driven	O
solvers	O
can	O
be	O
much	O
better	O
on	O
large	O
instances	O
which	O
actually	O
have	O
an	O
easy	O
instance	O
inside	O
)	O
.	O
</s>
<s>
A	O
modern	O
Parallel	B-Operating_System
SAT	B-Application
solver	I-Application
is	O
ManySAT	O
.	O
</s>
<s>
Google	O
's	O
CP-SAT	O
solver	O
,	O
part	O
of	O
OR-Tools	B-Application
,	O
won	O
gold	O
medals	O
at	O
the	O
Minizinc	O
constraint	O
programming	O
competitions	O
in	O
2018	O
,	O
2019	O
,	O
2020	O
,	O
and	O
2021	O
.	O
</s>
<s>
Particularly	O
in	O
hardware	B-General_Concept
design	I-General_Concept
and	O
verification	O
applications	O
,	O
satisfiability	O
and	O
other	O
logical	O
properties	O
of	O
a	O
given	O
propositional	O
formula	O
are	O
sometimes	O
decided	O
based	O
on	O
a	O
representation	O
of	O
the	O
formula	O
as	O
a	O
binary	B-Application
decision	I-Application
diagram	I-Application
(	O
BDD	O
)	O
.	O
</s>
<s>
Different	O
SAT	B-Application
solvers	I-Application
will	O
find	O
different	O
instances	O
easy	O
or	O
hard	O
,	O
and	O
some	O
excel	O
at	O
proving	O
unsatisfiability	O
,	O
and	O
others	O
at	O
finding	O
solutions	O
.	O
</s>
<s>
Parallel	B-Operating_System
SAT	B-Application
solvers	I-Application
come	O
in	O
three	O
categories	O
:	O
portfolio	O
,	O
divide-and-conquer	B-Algorithm
and	O
parallel	B-Operating_System
local	B-Application
search	I-Application
algorithms	O
.	O
</s>
<s>
With	O
parallel	B-Operating_System
portfolios	O
,	O
multiple	O
different	O
SAT	B-Application
solvers	I-Application
run	O
concurrently	O
.	O
</s>
<s>
Each	O
of	O
them	O
solves	O
a	O
copy	O
of	O
the	O
SAT	O
instance	O
,	O
whereas	O
divide-and-conquer	B-Algorithm
algorithms	I-Algorithm
divide	O
the	O
problem	O
between	O
the	O
processors	O
.	O
</s>
<s>
Different	O
approaches	O
exist	O
to	O
parallelize	O
local	B-Application
search	I-Application
algorithms	O
.	O
</s>
<s>
The	O
International	O
SAT	B-Application
Solver	I-Application
Competition	O
has	O
a	O
parallel	B-Operating_System
track	O
reflecting	O
recent	O
advances	O
in	O
parallel	B-Operating_System
SAT	O
solving	O
.	O
</s>
<s>
In	O
2016	O
,	O
2017	O
and	O
2018	O
,	O
the	O
benchmarks	O
were	O
run	O
on	O
a	O
shared-memory	B-Operating_System
system	O
with	O
24	O
processing	B-General_Concept
cores	I-General_Concept
,	O
therefore	O
solvers	O
intended	O
for	O
distributed	B-Operating_System
memory	I-Operating_System
or	O
manycore	B-General_Concept
processors	I-General_Concept
might	O
have	O
fallen	O
short	O
.	O
</s>
<s>
In	O
general	O
there	O
is	O
no	O
SAT	B-Application
solver	I-Application
that	O
performs	O
better	O
than	O
all	O
other	O
solvers	O
on	O
all	O
SAT	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
These	O
limitations	O
motivate	O
the	O
parallel	B-Operating_System
portfolio	O
approach	O
.	O
</s>
<s>
All	O
solvers	O
in	O
a	O
parallel	B-Operating_System
portfolio	O
run	O
on	O
different	O
processors	O
to	O
solve	O
of	O
the	O
same	O
problem	O
.	O
</s>
<s>
Diversifying	O
their	O
seeds	B-Algorithm
is	O
a	O
simple	O
way	O
to	O
diversify	O
a	O
portfolio	O
.	O
</s>
<s>
Other	O
diversification	O
strategies	O
involve	O
enabling	O
,	O
disabling	O
or	O
diversifying	O
certain	O
heuristics	B-Algorithm
in	O
the	O
sequential	O
solver	O
.	O
</s>
<s>
One	O
drawback	O
of	O
parallel	B-Operating_System
portfolios	O
is	O
the	O
amount	O
of	O
duplicate	O
work	O
.	O
</s>
<s>
If	O
clause	O
learning	O
is	O
used	O
in	O
the	O
sequential	O
solvers	O
,	O
sharing	O
learned	O
clauses	O
between	O
parallel	B-Operating_System
running	O
solvers	O
can	O
reduce	O
duplicate	O
work	O
and	O
increase	O
performance	O
.	O
</s>
<s>
Yet	O
,	O
even	O
merely	O
running	O
a	O
portfolio	O
of	O
the	O
best	O
solvers	O
in	O
parallel	B-Operating_System
makes	O
a	O
competitive	O
parallel	B-Operating_System
solver	O
.	O
</s>
<s>
It	O
was	O
designed	O
to	O
find	O
a	O
lower	O
bound	O
for	O
the	O
performance	O
a	O
parallel	B-Operating_System
SAT	B-Application
solver	I-Application
should	O
be	O
able	O
to	O
deliver	O
.	O
</s>
<s>
Despite	O
the	O
large	O
amount	O
of	O
duplicate	O
work	O
due	O
to	O
lack	O
of	O
optimizations	O
,	O
it	O
performed	O
well	O
on	O
a	O
shared	B-Operating_System
memory	I-Operating_System
machine	O
.	O
</s>
<s>
HordeSat	O
is	O
a	O
parallel	B-Operating_System
portfolio	O
solver	O
for	O
large	O
clusters	O
of	O
computing	O
nodes	O
.	O
</s>
<s>
In	O
recent	O
years	O
parallel	B-Operating_System
portfolio	O
SAT	B-Application
solvers	I-Application
have	O
dominated	O
the	O
parallel	B-Operating_System
track	O
of	O
the	O
International	O
SAT	B-Application
Solver	I-Application
Competitions	O
.	O
</s>
<s>
In	O
contrast	O
to	O
parallel	B-Operating_System
portfolios	O
,	O
parallel	B-Operating_System
divide-and-conquer	B-Algorithm
tries	O
to	O
split	O
the	O
search	O
space	O
between	O
the	O
processing	O
elements	O
.	O
</s>
<s>
Divide-and-conquer	B-Algorithm
algorithms	I-Algorithm
,	O
such	O
as	O
the	O
sequential	O
DPLL	O
,	O
already	O
apply	O
the	O
technique	O
of	O
splitting	O
the	O
search	O
space	O
,	O
hence	O
their	O
extension	O
towards	O
a	O
parallel	B-Operating_System
algorithm	I-Operating_System
is	O
straight	O
forward	O
.	O
</s>
<s>
Thus	O
the	O
DPLL	B-Application
algorithm	I-Application
typically	O
does	O
not	O
process	O
each	O
part	O
of	O
the	O
search	O
space	O
in	O
the	O
same	O
amount	O
of	O
time	O
,	O
yielding	O
a	O
challenging	O
load	B-Application
balancing	I-Application
problem	O
.	O
</s>
<s>
Due	O
to	O
non-chronological	O
backtracking	O
,	O
parallelization	O
of	O
conflict-driven	B-Application
clause	I-Application
learning	I-Application
is	O
more	O
difficult	O
.	O
</s>
<s>
There	O
are	O
three	O
heuristics	B-Algorithm
involved	O
in	O
the	O
cube	O
phase	O
.	O
</s>
<s>
The	O
variables	O
in	O
the	O
cubes	O
are	O
chosen	O
by	O
the	O
decision	O
heuristic	B-Algorithm
.	O
</s>
<s>
The	O
direction	O
heuristic	B-Algorithm
decides	O
which	O
variable	O
assignment	O
(	O
true	O
or	O
false	O
)	O
to	O
explore	O
first	O
.	O
</s>
<s>
The	O
cutoff	O
heuristic	B-Algorithm
decides	O
when	O
to	O
stop	O
expanding	O
a	O
cube	O
and	O
instead	O
forward	O
it	O
to	O
a	O
sequential	O
conflict-driven	O
solver	O
.	O
</s>
<s>
Treengeling	O
is	O
an	O
example	O
for	O
a	O
parallel	B-Operating_System
solver	O
that	O
applies	O
the	O
Cube-and-Conquer	O
paradigm	O
.	O
</s>
<s>
Since	O
its	O
introduction	O
in	O
2012	O
it	O
has	O
had	O
multiple	O
successes	O
at	O
the	O
International	O
SAT	B-Application
Solver	I-Application
Competition	O
.	O
</s>
<s>
One	O
strategy	O
towards	O
a	O
parallel	B-Operating_System
local	B-Application
search	I-Application
algorithm	O
for	O
SAT	O
solving	O
is	O
trying	O
multiple	O
variable	O
flips	O
concurrently	O
on	O
different	O
processing	O
units	O
.	O
</s>
<s>
Another	O
is	O
to	O
apply	O
the	O
aforementioned	O
portfolio	O
approach	O
,	O
however	O
clause	O
sharing	O
is	O
not	O
possible	O
since	O
local	B-Application
search	I-Application
solvers	O
do	O
not	O
produce	O
clauses	O
.	O
</s>
