<s>
In	O
machine	O
learning	O
,	O
first-order	B-Application
inductive	I-Application
learner	I-Application
(	O
FOIL	O
)	O
is	O
a	O
rule-based	O
learning	O
algorithm	O
.	O
</s>
<s>
Developed	O
in	O
1990	O
by	O
Ross	O
Quinlan	O
,	O
FOIL	O
learns	O
function-free	O
Horn	B-Application
clauses	I-Application
,	O
a	O
subset	O
of	O
first-order	O
predicate	B-Algorithm
calculus	O
.	O
</s>
<s>
Given	O
positive	O
and	O
negative	O
examples	O
of	O
some	O
concept	O
and	O
a	O
set	O
of	O
background-knowledge	O
predicates	B-Algorithm
,	O
FOIL	O
inductively	O
generates	O
a	O
logical	O
concept	O
definition	O
or	O
rule	O
for	O
the	O
concept	O
.	O
</s>
<s>
The	O
induced	O
rule	O
must	O
not	O
involve	O
any	O
constants	O
(color(X , red )	O
becomes	O
color(X,Y )	O
,	O
red(Y )	O
)	O
or	O
function	O
symbols	O
,	O
but	O
may	O
allow	O
negated	O
predicates	B-Algorithm
;	O
recursive	O
concepts	O
are	O
also	O
learnable	O
.	O
</s>
<s>
Like	O
the	O
ID3	B-Algorithm
algorithm	I-Algorithm
,	O
FOIL	O
hill	O
climbs	O
using	O
a	O
metric	O
based	O
on	O
information	O
theory	O
to	O
construct	O
a	O
rule	O
that	O
covers	O
the	O
data	O
.	O
</s>
<s>
Unlike	O
ID3	O
,	O
however	O
,	O
FOIL	O
uses	O
a	O
separate-and-conquer	O
method	O
rather	O
than	O
divide-and-conquer	B-Algorithm
,	O
focusing	O
on	O
creating	O
one	O
rule	O
at	O
a	O
time	O
and	O
collecting	O
uncovered	O
examples	O
for	O
the	O
next	O
iteration	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
This	O
can	O
be	O
extended	O
by	O
conjoining	O
Body	O
with	O
any	O
of	O
the	O
literals	O
father(X,X )	O
,	O
father(Y,Z )	O
,	O
parent(U,Y )	O
,	O
or	O
many	O
others	O
–	O
to	O
create	O
this	O
literal	O
,	O
the	O
algorithm	O
must	O
choose	O
both	O
a	O
predicate	B-Algorithm
name	O
and	O
a	O
set	O
of	O
variables	O
for	O
the	O
predicate	B-Algorithm
(	O
at	O
least	O
one	O
of	O
which	O
is	O
required	O
to	O
be	O
present	O
already	O
in	O
an	O
unnegated	O
literal	O
of	O
the	O
clause	O
)	O
.	O
</s>
<s>
On	O
the	O
next	O
iteration	O
of	O
FOIL	O
after	O
parent(X,Z )	O
has	O
been	O
added	O
,	O
the	O
algorithm	O
will	O
consider	O
all	O
combinations	O
of	O
predicate	B-Algorithm
names	O
and	O
variables	O
such	O
that	O
at	O
least	O
one	O
variable	O
in	O
the	O
new	O
literal	O
is	O
present	O
in	O
the	O
existing	O
clause	O
.	O
</s>
<s>
Constraints	O
on	O
the	O
search	O
space	O
are	O
allowed	O
,	O
as	O
are	O
predicates	B-Algorithm
that	O
are	O
defined	O
on	O
a	O
rule	O
rather	O
than	O
on	O
a	O
set	O
of	O
examples	O
(	O
called	O
intensional	O
predicates	B-Algorithm
)	O
;	O
most	O
importantly	O
a	O
potentially	O
incorrect	O
hypothesis	O
is	O
allowed	O
as	O
an	O
initial	O
approximation	O
to	O
the	O
predicate	B-Algorithm
to	O
be	O
learned	O
.	O
</s>
<s>
The	O
main	O
goal	O
of	O
FOCL	O
is	O
to	O
incorporate	O
the	O
methods	O
of	O
explanation-based	B-General_Concept
learning	I-General_Concept
(	O
EBL	O
)	O
into	O
the	O
empirical	O
methods	O
of	O
FOIL	O
.	O
</s>
<s>
Even	O
when	O
no	O
additional	O
knowledge	O
is	O
provided	O
to	O
FOCL	O
over	O
FOIL	O
,	O
however	O
,	O
it	O
utilizes	O
an	O
iterative	O
widening	O
search	O
strategy	O
similar	O
to	O
depth-first	B-Algorithm
search	I-Algorithm
:	O
first	O
FOCL	O
attempts	O
to	O
learn	O
a	O
clause	O
by	O
introducing	O
no	O
free	O
variables	O
.	O
</s>
<s>
If	O
this	O
fails	O
(	O
no	O
positive	O
gain	O
)	O
,	O
one	O
additional	O
free	O
variable	O
per	O
failure	O
is	O
allowed	O
until	O
the	O
number	O
of	O
free	O
variables	O
exceeds	O
the	O
maximum	O
used	O
for	O
any	O
predicate	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
a	O
predicate	B-Algorithm
livesAt(X,Y )	O
may	O
have	O
types	O
livesAt(person, location )	O
.	O
</s>
<s>
Additional	O
predicates	B-Algorithm
may	O
need	O
to	O
be	O
introduced	O
,	O
though	O
–	O
without	O
types	O
,	O
nextDoor(X,Y )	O
could	O
determine	O
whether	O
person	O
X	O
and	O
person	O
Y	O
live	O
next	O
door	O
to	O
each	O
other	O
,	O
or	O
whether	O
two	O
locations	O
are	O
next	O
door	O
to	O
each	O
other	O
.	O
</s>
<s>
With	O
types	O
,	O
two	O
different	O
predicates	B-Algorithm
nextDoor(person, person )	O
and	O
nextDoor(location, location )	O
would	O
need	O
to	O
exist	O
for	O
this	O
functionality	O
to	O
be	O
maintained	O
.	O
</s>
<s>
However	O
,	O
this	O
typing	O
mechanism	O
eliminates	O
the	O
need	O
for	O
predicates	B-Algorithm
such	O
as	O
isPerson(X )	O
or	O
isLocation(Y )	O
,	O
and	O
need	O
not	O
consider	O
livesAt(A,B )	O
when	O
A	O
and	O
B	O
are	O
defined	O
to	O
be	O
person	O
variables	O
,	O
reducing	O
the	O
search	O
space	O
.	O
</s>
<s>
Additionally	O
,	O
typing	O
can	O
improve	O
the	O
accuracy	O
of	O
the	O
resulting	O
rule	O
by	O
eliminating	O
from	O
consideration	O
impossible	O
literals	O
such	O
as	O
livesAt(A,B )	O
which	O
may	O
nevertheless	O
appear	O
to	O
have	O
a	O
high	O
information	B-Algorithm
gain	I-Algorithm
.	O
</s>
<s>
Rather	O
than	O
implementing	O
trivial	O
predicates	B-Algorithm
such	O
as	O
equals(X,X )	O
or	O
between(X,X,Y )	O
,	O
FOCL	O
introduces	O
implicit	O
constraints	O
on	O
variables	O
,	O
further	O
reducing	O
search	O
space	O
.	O
</s>
<s>
Some	O
predicates	B-Algorithm
must	O
have	O
all	O
variables	O
unique	O
,	O
others	O
must	O
have	O
commutativity	O
(adjacent(X , Y )	O
is	O
equivalent	O
to	O
adjacent(Y,X )	O
)	O
,	O
still	O
others	O
may	O
require	O
that	O
a	O
particular	O
variable	O
be	O
present	O
in	O
the	O
current	O
clause	O
,	O
and	O
many	O
other	O
potential	O
constraints	O
.	O
</s>
<s>
Operational	O
rules	O
are	O
those	O
rules	O
which	O
are	O
defined	O
extensionally	O
,	O
or	O
as	O
a	O
list	O
of	O
tuples	O
for	O
which	O
a	O
predicate	B-Algorithm
is	O
true	O
.	O
</s>
<s>
Allowing	O
for	O
partial	O
definitions	O
reduces	O
the	O
amount	O
of	O
work	O
needed	O
as	O
the	O
algorithm	O
need	O
not	O
generate	O
these	O
partial	O
definitions	O
for	O
itself	O
,	O
and	O
the	O
incorrect	O
rules	O
do	O
not	O
add	O
significantly	O
to	O
the	O
work	O
needed	O
since	O
they	O
are	O
discarded	O
if	O
they	O
are	O
not	O
judged	O
to	O
provide	O
positive	O
information	B-Algorithm
gain	I-Algorithm
.	O
</s>
<s>
Non-operational	O
rules	O
are	O
advantageous	O
as	O
the	O
individual	O
rules	O
which	O
they	O
combine	O
may	O
not	O
provide	O
information	B-Algorithm
gain	I-Algorithm
on	O
their	O
own	O
,	O
but	O
are	O
useful	O
when	O
taken	O
in	O
conjunction	O
.	O
</s>
<s>
If	O
a	O
literal	O
with	O
the	O
most	O
information	B-Algorithm
gain	I-Algorithm
in	O
an	O
iteration	O
of	O
FOCL	O
is	O
non-operational	O
,	O
it	O
is	O
operationalized	O
and	O
its	O
definition	O
is	O
added	O
to	O
the	O
clause	O
under	O
construction	O
.	O
</s>
