<s>
Datalog	B-Language
is	O
a	O
declarative	B-Language
logic	B-Language
programming	I-Language
language	I-Language
.	O
</s>
<s>
While	O
it	O
is	O
syntactically	O
a	O
subset	O
of	O
Prolog	B-Language
,	O
Datalog	B-Language
generally	O
uses	O
a	O
bottom-up	O
rather	O
than	O
top-down	O
evaluation	O
model	O
.	O
</s>
<s>
This	O
difference	O
yields	O
significantly	O
different	O
behavior	O
and	O
properties	O
from	O
Prolog	B-Language
.	O
</s>
<s>
It	O
is	O
often	O
used	O
as	O
a	O
query	B-Language
language	I-Language
for	O
deductive	B-General_Concept
databases	I-General_Concept
.	O
</s>
<s>
Datalog	B-Language
has	O
been	O
applied	O
to	O
problems	O
in	O
data	B-General_Concept
integration	I-General_Concept
,	O
networking	B-Architecture
,	O
program	O
analysis	O
,	O
and	O
more	O
.	O
</s>
<s>
A	O
Datalog	B-Language
program	O
consists	O
of	O
facts	O
,	O
which	O
are	O
statements	O
that	O
are	O
held	O
to	O
be	O
true	O
,	O
and	O
rules	O
,	O
which	O
say	O
how	O
to	O
deduce	O
new	O
facts	O
from	O
known	O
facts	O
.	O
</s>
<s>
Some	O
Datalog	B-Language
implementations	O
do	O
n't	O
deduce	O
all	O
possible	O
facts	O
,	O
but	O
instead	O
answer	O
queries	O
:	O
</s>
<s>
This	O
query	B-Application
asks	O
:	O
Who	O
are	O
all	O
the	O
X	O
that	O
xerces	O
is	O
an	O
ancestor	O
of	O
?	O
</s>
<s>
The	O
non-recursive	O
subset	O
of	O
Datalog	B-Language
is	O
closely	O
related	O
to	O
query	B-Language
languages	I-Language
for	O
relational	B-Application
databases	I-Application
,	O
such	O
as	O
SQL	B-Language
.	O
</s>
<s>
The	O
following	O
table	B-Application
maps	O
between	O
Datalog	B-Language
,	O
relational	O
algebra	O
,	O
and	O
SQL	B-Language
concepts	O
:	O
</s>
<s>
More	O
formally	O
,	O
non-recursive	O
Datalog	B-Language
corresponds	O
precisely	O
to	O
unions	O
of	O
conjunctive	O
queries	O
,	O
or	O
equivalently	O
,	O
negation-free	O
relational	O
algebra	O
.	O
</s>
<s>
A	O
Datalog	B-Language
program	O
consists	O
of	O
a	O
list	O
of	O
rules	O
(	O
Horn	B-Application
clauses	I-Application
)	O
.	O
</s>
<s>
If	O
constant	O
and	O
variable	O
are	O
two	O
countable	O
sets	O
of	O
constants	O
and	O
variables	O
respectively	O
and	O
relation	O
is	O
a	O
countable	O
set	O
of	O
predicate	B-Algorithm
symbols	I-Algorithm
,	O
then	O
the	O
following	O
BNF	O
grammar	O
expresses	O
the	O
structure	O
of	O
a	O
Datalog	B-Language
program	O
:	O
</s>
<s>
Every	O
Datalog	B-Language
program	O
must	O
satisfy	O
the	O
condition	O
that	O
every	O
variable	O
that	O
appears	O
in	O
the	O
head	O
of	O
a	O
rule	O
also	O
appears	O
in	O
the	O
body	O
(	O
this	O
condition	O
is	O
sometimes	O
called	O
the	O
)	O
.	O
</s>
<s>
Note	O
that	O
under	O
this	O
definition	O
,	O
Datalog	B-Language
does	O
include	O
negation	O
nor	O
aggregates	O
;	O
see	O
for	O
more	O
information	O
about	O
those	O
constructs	O
.	O
</s>
<s>
The	O
set	O
of	O
facts	O
is	O
called	O
the	O
or	O
of	O
the	O
Datalog	B-Language
program	O
.	O
</s>
<s>
The	O
set	O
of	O
tuples	B-Application
computed	O
by	O
evaluating	O
the	O
Datalog	B-Language
program	O
is	O
called	O
the	O
or	O
.	O
</s>
<s>
Many	O
implementations	O
of	O
logic	B-Language
programming	I-Language
extend	O
the	O
above	O
grammar	O
to	O
allow	O
writing	O
facts	O
without	O
the	O
:-	O
,	O
like	O
so	O
:	O
</s>
<s>
There	O
are	O
three	O
widely-used	O
approaches	O
to	O
the	O
semantics	O
of	O
Datalog	B-Language
programs	O
:	O
model-theoretic	O
,	O
fixed-point	O
,	O
and	O
proof-theoretic	O
.	O
</s>
<s>
The	O
Herbrand	O
base	O
of	O
a	O
Datalog	B-Language
program	O
is	O
the	O
set	O
of	O
all	O
ground	O
atoms	O
that	O
can	O
be	O
made	O
with	O
the	O
constants	O
appearing	O
in	O
the	O
program	O
.	O
</s>
<s>
The	O
of	O
a	O
Datalog	B-Language
program	O
is	O
the	O
smallest	O
subset	O
of	O
the	O
Herbrand	O
base	O
such	O
that	O
,	O
for	O
each	O
ground	O
instance	O
of	O
each	O
rule	O
in	O
the	O
program	O
,	O
if	O
the	O
atoms	O
in	O
the	O
body	O
of	O
the	O
rule	O
are	O
in	O
the	O
set	O
,	O
then	O
so	O
is	O
the	O
head	O
.	O
</s>
<s>
The	O
proof-theoretic	O
semantics	O
defines	O
the	O
meaning	O
of	O
a	O
Datalog	B-Language
program	O
to	O
be	O
the	O
set	O
of	O
facts	O
with	O
corresponding	O
proof	O
trees	O
.	O
</s>
<s>
One	O
might	O
be	O
interested	O
in	O
knowing	O
whether	O
or	O
not	O
a	O
particular	O
ground	O
atom	O
appears	O
in	O
the	O
minimal	O
Herbrand	O
model	O
of	O
a	O
Datalog	B-Language
program	O
,	O
perhaps	O
without	O
caring	O
much	O
about	O
the	O
rest	O
of	O
the	O
model	O
.	O
</s>
<s>
This	O
reading	O
informs	O
the	O
SLD	B-Application
resolution	I-Application
algorithm	O
,	O
which	O
forms	O
the	O
basis	O
for	O
the	O
evaluation	O
of	O
Prolog	B-Language
.	O
</s>
<s>
There	O
are	O
many	O
different	O
ways	O
to	O
evaluate	O
a	O
Datalog	B-Language
program	O
,	O
with	O
different	O
performance	O
characteristics	O
.	O
</s>
<s>
Bottom-up	O
evaluation	O
strategies	O
start	O
with	O
the	O
facts	O
in	O
the	O
program	O
and	O
repeatedly	O
apply	O
the	O
rules	O
until	O
the	O
either	O
some	O
goal	O
or	O
query	B-Application
is	O
established	O
,	O
or	O
until	O
the	O
complete	O
minimal	O
model	O
of	O
the	O
program	O
is	O
produced	O
.	O
</s>
<s>
Naïve	O
evaluation	O
mirrors	O
the	O
fixpoint	O
semantics	O
for	O
Datalog	B-Language
programs	O
.	O
</s>
<s>
Naïve	O
and	O
semi-naïve	O
evaluation	O
both	O
evaluate	O
recursive	O
Datalog	B-Language
rules	O
by	O
repeatedly	O
applying	O
them	O
to	O
a	O
set	O
of	O
known	O
facts	O
until	O
a	O
fixed	O
point	O
is	O
reached	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
each	O
non-recursive	O
Datalog	B-Language
rule	O
corresponds	O
precisely	O
to	O
a	O
conjunctive	O
query	B-Application
.	O
</s>
<s>
Index	B-Data_Structure
selection	O
and	O
data	O
structures	O
(	O
hash	B-Error_Name
table	B-Application
,	O
B-tree	B-Architecture
,	O
etc	O
.	O
)	O
</s>
<s>
Many	O
such	O
techniques	O
are	O
implemented	O
in	O
modern	O
bottom-up	O
Datalog	B-Language
engines	O
such	O
as	O
Soufflé	O
.	O
</s>
<s>
Some	O
Datalog	B-Language
engines	O
integrate	O
SQL	B-Language
databases	I-Language
directly	O
.	O
</s>
<s>
Bottom-up	O
evaluation	O
of	O
Datalog	B-Language
is	O
also	O
amenable	O
to	O
parallelization	B-Operating_System
.	O
</s>
<s>
Parallel	O
Datalog	B-Language
engines	O
are	O
generally	O
divided	O
into	O
two	O
paradigms	O
:	O
</s>
<s>
In	O
the	O
shared-memory	O
,	O
multi-core	O
setting	O
,	O
Datalog	B-Language
engines	O
execute	O
on	O
a	O
single	O
node	O
.	O
</s>
<s>
Coordination	O
between	O
threads	O
may	O
be	O
achieved	O
using	O
locking	B-Operating_System
or	O
lock-free	B-Operating_System
data	I-Operating_System
structures	I-Operating_System
.	O
</s>
<s>
Examples	O
include	O
Datalog	B-Language
engines	O
using	O
OpenMP	B-Application
.	O
</s>
<s>
In	O
the	O
shared-nothing	B-Operating_System
setting	O
,	O
Datalog	B-Language
engines	O
execute	O
on	O
a	O
cluster	B-Architecture
of	O
nodes	O
.	O
</s>
<s>
Such	O
engines	O
generally	O
operate	O
by	O
splitting	O
relations	O
into	O
disjoint	O
subsets	O
based	O
on	O
a	O
hash	B-Error_Name
function	I-Error_Name
,	O
performing	O
computations	O
(	O
joins	O
)	O
on	O
each	O
node	O
,	O
and	O
then	O
exchanging	O
newly-generated	O
tuples	B-Application
over	O
the	O
network	O
.	O
</s>
<s>
Examples	O
include	O
Datalog	B-Language
engines	O
based	O
on	O
MPI	B-Application
,	O
Hadoop	B-Application
,	O
and	O
Spark	B-Language
.	O
</s>
<s>
SLD	B-Application
resolution	I-Application
is	O
sound	O
and	O
complete	O
for	O
Datalog	B-Language
programs	O
.	O
</s>
<s>
Top-down	O
evaluation	O
strategies	O
begin	O
with	O
a	O
query	B-Application
or	O
goal	O
.	O
</s>
<s>
Bottom-up	O
evaluation	O
strategies	O
can	O
answer	O
queries	O
by	O
computing	O
the	O
entire	O
minimal	O
model	O
and	O
matching	O
the	O
query	B-Application
against	O
it	O
,	O
but	O
this	O
can	O
be	O
inefficient	O
if	O
the	O
answer	O
only	O
depends	O
on	O
a	O
small	O
subset	O
of	O
the	O
entire	O
model	O
.	O
</s>
<s>
The	O
magic	B-Language
sets	I-Language
algorithm	I-Language
takes	O
a	O
Datalog	B-Language
program	O
and	O
a	O
query	B-Application
,	O
and	O
produces	O
a	O
more	O
efficient	O
program	O
that	O
computes	O
the	O
same	O
answer	O
to	O
the	O
query	B-Application
while	O
still	O
using	O
bottom-up	O
evaluation	O
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
magic	B-Language
sets	I-Language
algorithm	I-Language
has	O
been	O
shown	O
to	O
produce	O
programs	O
that	O
,	O
when	O
evaluated	O
using	O
semi-naïve	O
evaluation	O
,	O
are	O
as	O
efficient	O
as	O
top-down	O
evaluation	O
.	O
</s>
<s>
The	O
decision	O
problem	O
formulation	O
of	O
Datalog	B-Language
evaluation	O
is	O
as	O
follows	O
:	O
Given	O
a	O
Datalog	B-Language
program	O
split	O
into	O
a	O
set	O
of	O
facts	O
(	O
EDB	O
)	O
and	O
a	O
set	O
of	O
rules	O
,	O
and	O
an	O
interpretation	O
,	O
is	O
in	O
the	O
minimal	O
model	O
of	O
?	O
</s>
<s>
In	O
this	O
formulation	O
,	O
there	O
are	O
three	O
variations	O
of	O
the	O
computational	O
complexity	O
of	O
evaluating	O
Datalog	B-Language
programs	O
:	O
</s>
<s>
With	O
respect	O
to	O
data	O
complexity	O
,	O
the	O
decision	O
problem	O
for	O
Datalog	B-Language
is	O
P-complete	O
.	O
</s>
<s>
In	O
particular	O
,	O
evaluating	O
Datalog	B-Language
programs	O
always	O
terminates	O
;	O
Datalog	B-Language
is	O
not	O
Turing-complete	B-Algorithm
.	O
</s>
<s>
Some	O
extensions	O
to	O
Datalog	B-Language
do	O
not	O
preserve	O
these	O
complexity	O
bounds	O
.	O
</s>
<s>
Extensions	O
implemented	O
in	O
some	O
Datalog	B-Language
engines	O
Datalog	B-Language
engines	O
,	O
such	O
as	O
algebraic	O
data	O
types	O
,	O
can	O
even	O
make	O
the	O
resulting	O
language	O
Turing-complete	B-Algorithm
.	O
</s>
<s>
Several	O
extensions	O
have	O
been	O
made	O
to	O
Datalog	B-Language
,	O
e.g.	O
,	O
to	O
support	O
negation	O
,	O
aggregate	O
functions	O
,	O
inequalities	O
,	O
to	O
allow	O
object-oriented	B-Language
programming	I-Language
,	O
or	O
to	O
allow	O
disjunctions	O
as	O
heads	O
of	O
clauses	O
.	O
</s>
<s>
These	O
extensions	O
have	O
significant	O
impacts	O
on	O
the	O
language	O
's	O
semantics	O
and	O
on	O
the	O
implementation	O
of	O
a	O
corresponding	O
interpreter	B-Application
.	O
</s>
<s>
Datalog	B-Language
is	O
a	O
syntactic	O
subset	O
of	O
Prolog	B-Language
,	O
disjunctive	O
Datalog	B-Language
,	O
answer	B-Application
set	I-Application
programming	I-Application
,	O
Datalogℤ	O
,	O
and	O
constraint	B-Application
logic	I-Application
programming	I-Application
.	O
</s>
<s>
When	O
evaluated	O
as	O
an	O
answer	B-Application
set	I-Application
program	I-Application
,	O
a	O
Datalog	B-Language
program	O
yields	O
a	O
single	O
answer	B-Application
set	I-Application
,	O
which	O
is	O
exactly	O
its	O
minimal	O
model	O
.	O
</s>
<s>
Many	O
implementations	O
of	O
Datalog	B-Language
extend	O
Datalog	B-Language
with	O
additional	O
features	O
;	O
see	O
for	O
more	O
information	O
.	O
</s>
<s>
Datalog	B-Language
can	O
be	O
extended	O
to	O
support	O
aggregate	O
functions	O
.	O
</s>
<s>
Notable	O
Datalog	B-Language
engines	O
that	O
implement	O
aggregation	O
include	O
:	O
</s>
<s>
Adding	O
negation	O
to	O
Datalog	B-Language
complicates	O
its	O
semantics	O
,	O
leading	O
to	O
whole	O
new	O
languages	O
and	O
strategies	O
for	O
evaluation	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
language	O
that	O
results	O
from	O
adding	O
negation	O
with	O
the	O
stable	B-Application
model	I-Application
semantics	I-Application
is	O
exactly	O
answer	B-Application
set	I-Application
programming	I-Application
.	O
</s>
<s>
Stratified	O
negation	O
can	O
be	O
added	O
to	O
Datalog	B-Language
while	O
retaining	O
its	O
model-theoretic	O
and	O
fixed	O
point	O
semantics	O
.	O
</s>
<s>
Notable	O
Datalog	B-Language
engines	O
that	O
implement	O
stratified	O
negation	O
include	O
:	O
</s>
<s>
Unlike	O
in	O
Prolog	B-Language
,	O
statements	O
of	O
a	O
Datalog	B-Language
program	O
can	O
be	O
stated	O
in	O
any	O
order	O
.	O
</s>
<s>
Datalog	B-Language
does	O
not	O
have	O
Prolog	B-Language
's	O
cut	B-Application
operator	O
.	O
</s>
<s>
This	O
makes	O
Datalog	B-Language
a	O
fully	O
declarative	B-Language
language	I-Language
.	O
</s>
<s>
disallows	O
complex	O
terms	O
as	O
arguments	O
of	O
predicates	B-Algorithm
,	O
e.g.	O
,	O
p(x, y )	O
is	O
admissible	O
but	O
not	O
p(f(x )	O
,	O
y	O
)	O
,	O
</s>
<s>
requires	O
that	O
every	O
variable	O
that	O
appears	O
in	O
the	O
head	O
of	O
a	O
clause	B-Application
also	O
appear	O
in	O
a	O
literal	O
in	O
the	O
body	O
of	O
the	O
clause	B-Application
.	O
</s>
<s>
This	O
article	O
deals	O
primarily	O
with	O
Datalog	B-Language
without	O
negation	O
(	O
see	O
also	O
)	O
.	O
</s>
<s>
However	O
,	O
stratified	O
negation	O
is	O
a	O
common	O
addition	O
to	O
Datalog	B-Language
;	O
the	O
following	O
list	O
contrasts	O
Prolog	B-Language
with	O
Datalog	B-Language
with	O
stratified	O
negation	O
.	O
</s>
<s>
also	O
disallows	O
complex	O
terms	O
as	O
arguments	O
of	O
predicates	B-Algorithm
,	O
</s>
<s>
requires	O
that	O
every	O
variable	O
that	O
appears	O
in	O
the	O
head	O
of	O
a	O
clause	B-Application
also	O
appear	O
in	O
a	O
positive	O
(	O
i.e.	O
,	O
not	O
negated	O
)	O
atom	O
in	O
the	O
body	O
of	O
the	O
clause	B-Application
,	O
</s>
<s>
requires	O
that	O
every	O
variable	O
appearing	O
in	O
a	O
negative	O
literal	O
in	O
the	O
body	O
of	O
a	O
clause	B-Application
also	O
appear	O
in	O
some	O
positive	O
literal	O
in	O
the	O
body	O
of	O
the	O
clause	B-Application
.	O
</s>
<s>
The	O
for	O
Datalog	B-Language
asks	O
,	O
given	O
a	O
Datalog	B-Language
program	O
,	O
whether	O
it	O
is	O
,	O
i.e.	O
,	O
the	O
maximal	O
recursion	O
depth	O
reached	O
when	O
evaluating	O
the	O
program	O
on	O
an	O
input	O
database	O
can	O
be	O
bounded	O
by	O
some	O
constant	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
this	O
question	O
asks	O
whether	O
the	O
Datalog	B-Language
program	O
could	O
be	O
rewritten	O
as	O
a	O
nonrecursive	O
Datalog	B-Language
program	O
.	O
</s>
<s>
Solving	O
the	O
boundedness	O
problem	O
on	O
arbitrary	O
Datalog	B-Language
programs	O
is	O
undecidable	O
,	O
but	O
it	O
can	O
be	O
made	O
decidable	O
by	O
restricting	O
to	O
some	O
fragments	O
of	O
Datalog	B-Language
.	O
</s>
<s>
Systems	O
that	O
implement	O
languages	O
inspired	O
by	O
Datalog	B-Language
,	O
whether	O
compilers	B-Language
,	O
interpreters	B-Application
,	O
libraries	B-Library
,	O
or	O
embedded	O
DSLs	O
,	O
are	O
referred	O
to	O
as	O
.	O
</s>
<s>
Datalog	B-Language
engines	O
often	O
implement	O
extensions	O
of	O
Datalog	B-Language
,	O
extending	O
it	O
with	O
additional	O
data	O
types	O
,	O
foreign	B-Application
function	I-Application
interfaces	I-Application
,	O
or	O
support	O
for	O
user-defined	O
lattices	O
.	O
</s>
<s>
Such	O
extensions	O
may	O
allow	O
for	O
writing	O
non-terminating	B-Application
or	O
otherwise	O
ill-defined	O
programs	O
.	O
</s>
<s>
Datalog	B-Language
is	O
quite	O
limited	O
in	O
its	O
expressivity	O
.	O
</s>
<s>
It	O
is	O
not	O
Turing-complete	B-Algorithm
,	O
and	O
does	O
n't	O
include	O
basic	O
data	O
types	O
such	O
as	O
integers	O
or	O
strings	O
.	O
</s>
<s>
This	O
parsimony	O
is	O
appealing	O
from	O
a	O
theoretical	O
standpoint	O
,	O
but	O
it	O
means	O
Datalog	B-Language
per	O
se	O
is	O
rarely	O
used	O
as	O
a	O
programming	O
language	O
.	O
</s>
<s>
Most	O
Datalog	B-Language
engines	O
implement	O
substantial	O
extensions	O
of	O
Datalog	B-Language
.	O
</s>
<s>
However	O
,	O
Datalog	B-Language
has	O
a	O
strong	O
influence	O
on	O
such	O
implementations	O
,	O
and	O
many	O
authors	O
do	O
n't	O
bother	O
to	O
distinguish	O
them	O
from	O
Datalog	B-Language
as	O
presented	O
in	O
this	O
article	O
.	O
</s>
<s>
Accordingly	O
,	O
the	O
applications	O
discussed	O
in	O
this	O
section	O
include	O
applications	O
of	O
realistic	O
implementations	O
of	O
Datalog-based	O
languages	O
.	O
</s>
<s>
Datalog	B-Language
has	O
been	O
applied	O
to	O
problems	O
in	O
data	B-General_Concept
integration	I-General_Concept
,	O
information	B-General_Concept
extraction	I-General_Concept
,	O
networking	B-Architecture
,	O
security	O
,	O
cloud	B-Architecture
computing	I-Architecture
and	O
machine	O
learning	O
.	O
</s>
<s>
Google	B-Application
has	O
developed	O
an	O
extension	O
to	O
Datalog	B-Language
for	O
big	B-Application
data	I-Application
processing	O
.	O
</s>
<s>
Datalog	B-Language
has	O
seen	O
application	O
in	O
static	O
program	O
analysis	O
.	O
</s>
<s>
The	O
Soufflé	O
dialect	O
has	O
been	O
used	O
to	O
write	O
pointer	O
analyses	O
for	O
Java	B-Language
and	O
a	O
control-flow	O
analysis	O
for	O
Scheme	B-Language
.	O
</s>
<s>
Datalog	B-Language
has	O
been	O
integrated	O
with	O
SMT	B-Application
solvers	I-Application
to	O
make	O
it	O
easier	O
to	O
write	O
certain	O
static	O
analyses	O
.	O
</s>
<s>
The	O
Flix	B-Language
dialect	O
is	O
also	O
suited	O
to	O
writing	O
static	O
program	O
analyses	O
.	O
</s>
<s>
Some	O
widely	O
used	O
database	B-Application
systems	I-Application
include	O
ideas	O
and	O
algorithms	O
developed	O
for	O
Datalog	B-Language
.	O
</s>
<s>
For	O
example	O
,	O
the	O
SQL:1999	B-Language
standard	O
includes	O
recursive	B-Language
queries	I-Language
,	O
and	O
the	O
Magic	B-Language
Sets	I-Language
algorithm	I-Language
(	O
initially	O
developed	O
for	O
the	O
faster	O
evaluation	O
of	O
Datalog	B-Language
queries	O
)	O
is	O
implemented	O
in	O
IBM	O
's	O
DB2	B-Application
.	O
</s>
<s>
The	O
origins	O
of	O
Datalog	B-Language
date	O
back	O
to	O
the	O
beginning	O
of	O
logic	B-Language
programming	I-Language
,	O
but	O
it	O
became	O
prominent	O
as	O
a	O
separate	O
area	O
around	O
1977	O
when	O
Hervé	O
Gallaire	O
and	O
Jack	O
Minker	O
organized	O
a	O
workshop	O
on	O
logic	B-Language
and	O
databases	B-Application
.	O
</s>
<s>
David	O
Maier	O
is	O
credited	O
with	O
coining	O
the	O
term	O
Datalog	B-Language
.	O
</s>
