<s>
Anti-unification	B-Application
is	O
the	O
process	O
of	O
constructing	O
a	O
generalization	O
common	O
to	O
two	O
given	O
symbolic	O
expressions	O
.	O
</s>
<s>
As	O
in	O
unification	B-Algorithm
,	O
several	O
frameworks	O
are	O
distinguished	O
depending	O
on	O
which	O
expressions	O
(	O
also	O
called	O
terms	O
)	O
are	O
allowed	O
,	O
and	O
which	O
expressions	O
are	O
considered	O
equal	O
.	O
</s>
<s>
If	O
variables	O
representing	O
functions	O
are	O
allowed	O
in	O
an	O
expression	O
,	O
the	O
process	O
is	O
called	O
"	O
higher-order	O
anti-unification	B-Application
"	O
,	O
otherwise	O
"	O
first-order	O
anti-unification	B-Application
"	O
.	O
</s>
<s>
If	O
the	O
generalization	O
is	O
required	O
to	O
have	O
an	O
instance	O
literally	O
equal	O
to	O
each	O
input	O
expression	O
,	O
the	O
process	O
is	O
called	O
"	O
syntactical	O
anti-unification	B-Application
"	O
,	O
otherwise	O
"	O
E-anti-unification	O
"	O
,	O
or	O
"	O
anti-unification	B-Application
modulo	O
theory	O
"	O
.	O
</s>
<s>
An	O
anti-unification	B-Application
algorithm	O
should	O
compute	O
for	O
given	O
expressions	O
a	O
complete	O
and	O
minimal	O
generalization	O
set	O
,	O
that	O
is	O
,	O
a	O
set	O
covering	O
all	O
generalizations	O
and	O
containing	O
no	O
redundant	O
members	O
,	O
respectively	O
.	O
</s>
<s>
For	O
first-order	O
syntactical	O
anti-unification	B-Application
,	O
Gordon	O
Plotkin	O
gave	O
an	O
algorithm	O
that	O
computes	O
a	O
complete	O
and	O
minimal	O
singleton	O
generalization	O
set	O
containing	O
the	O
so-called	O
"	O
least	O
general	O
generalization	O
"	O
(	O
lgg	O
)	O
.	O
</s>
<s>
Anti-unification	B-Application
should	O
not	O
be	O
confused	O
with	O
dis-unification	B-Application
.	O
</s>
<s>
For	O
higher-order	O
anti-unification	B-Application
,	O
it	O
is	O
convenient	O
to	O
choose	O
V	O
disjoint	O
from	O
the	O
set	O
of	O
lambda-term	O
bound	O
variables	O
.	O
</s>
<s>
A	O
set	O
T	O
of	O
terms	O
such	O
that	O
V	O
⊆	O
T	O
.	O
For	O
first-order	O
and	O
higher-order	O
anti-unification	B-Application
,	O
T	O
is	O
usually	O
the	O
set	O
of	O
first-order	O
terms	O
(	O
terms	O
built	O
from	O
variable	O
and	O
function	O
symbols	O
)	O
and	O
lambda	O
terms	O
(	O
terms	O
containing	O
some	O
higher-order	O
variables	O
)	O
,	O
respectively	O
.	O
</s>
<s>
For	O
higher-order	O
anti-unification	B-Application
,	O
usually	O
if	O
and	O
are	O
alpha	O
equivalent	O
.	O
</s>
<s>
For	O
first-order	O
E-anti-unification	O
,	O
reflects	O
the	O
background	O
knowledge	O
about	O
certain	O
function	O
symbols	O
;	O
for	O
example	O
,	O
if	O
is	O
considered	O
commutative	O
,	O
if	O
results	O
from	O
by	O
swapping	O
the	O
arguments	O
of	O
at	O
some	O
(	O
possibly	O
all	O
)	O
occurrences	O
.	O
</s>
<s>
An	O
anti-unification	B-Application
problem	O
is	O
a	O
pair	O
of	O
terms	O
.	O
</s>
<s>
For	O
a	O
given	O
anti-unification	B-Application
problem	O
,	O
a	O
set	O
of	O
anti-unifiers	O
is	O
called	O
complete	O
if	O
each	O
generalization	O
subsumes	O
some	O
term	O
;	O
the	O
set	O
is	O
called	O
minimal	O
if	O
none	O
of	O
its	O
members	O
subsumes	O
another	O
one	O
.	O
</s>
<s>
The	O
framework	O
of	O
first-order	O
syntactical	O
anti-unification	B-Application
is	O
based	O
on	O
being	O
the	O
set	O
of	O
first-order	O
terms	O
(	O
over	O
some	O
given	O
set	O
of	O
variables	O
,	O
of	O
constants	O
and	O
of	O
-ary	O
function	O
symbols	O
)	O
and	O
on	O
being	O
syntactic	O
equality	O
.	O
</s>
<s>
In	O
this	O
framework	O
,	O
each	O
anti-unification	B-Application
problem	O
has	O
a	O
complete	O
,	O
and	O
obviously	O
minimal	O
,	O
singleton	O
solution	O
set	O
.	O
</s>
<s>
The	O
lgg	O
is	O
unique	O
up	O
to	O
variants	O
:	O
if	O
and	O
are	O
both	O
complete	O
and	O
minimal	O
solution	O
sets	O
of	O
the	O
same	O
syntactical	O
anti-unification	B-Application
problem	O
,	O
then	O
and	O
for	O
some	O
terms	O
and	O
,	O
that	O
are	O
renamings	O
of	O
each	O
other	O
.	O
</s>
<s>
Plotkin	O
used	O
his	O
algorithm	O
to	O
compute	O
the	O
"	O
relative	O
least	O
general	O
generalization	O
(	O
rlgg	O
)	O
"	O
of	O
two	O
clause	O
sets	O
in	O
first-order	O
logic	O
,	O
which	O
was	O
the	O
basis	O
of	O
the	O
Golem	B-Application
approach	O
to	O
inductive	B-Application
logic	I-Application
programming	I-Application
.	O
</s>
