<s>
In	O
mathematical	O
logic	O
,	O
a	O
formula	O
of	O
first-order	O
logic	O
is	O
in	O
Skolem	B-Application
normal	I-Application
form	I-Application
if	O
it	O
is	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
with	O
only	O
universal	O
first-order	O
quantifiers	O
.	O
</s>
<s>
Every	O
first-order	O
formula	O
may	O
be	O
converted	O
into	O
Skolem	B-Application
normal	I-Application
form	I-Application
while	O
not	O
changing	O
its	O
satisfiability	O
via	O
a	O
process	O
called	O
Skolemization	B-Application
(	O
sometimes	O
spelled	O
Skolemnization	B-Application
)	O
.	O
</s>
<s>
Reduction	O
to	O
Skolem	B-Application
normal	I-Application
form	I-Application
is	O
a	O
method	O
for	O
removing	O
existential	B-Algorithm
quantifiers	I-Algorithm
from	O
formal	O
logic	O
statements	O
,	O
often	O
performed	O
as	O
the	O
first	O
step	O
in	O
an	O
automated	B-Application
theorem	I-Application
prover	I-Application
.	O
</s>
<s>
The	O
simplest	O
form	O
of	O
Skolemization	B-Application
is	O
for	O
existentially	B-Algorithm
quantified	I-Algorithm
variables	O
that	O
are	O
not	O
inside	O
the	O
scope	B-Language
of	O
a	O
universal	O
quantifier	O
.	O
</s>
<s>
More	O
generally	O
,	O
Skolemization	B-Application
is	O
performed	O
by	O
replacing	O
every	O
existentially	B-Algorithm
quantified	I-Algorithm
variable	O
with	O
a	O
term	O
whose	O
function	O
symbol	O
is	O
new	O
.	O
</s>
<s>
If	O
the	O
formula	O
is	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
,	O
then	O
are	O
the	O
variables	O
that	O
are	O
universally	O
quantified	O
and	O
whose	O
quantifiers	O
precede	O
that	O
of	O
.	O
</s>
<s>
In	O
general	O
,	O
they	O
are	O
the	O
variables	O
that	O
are	O
quantified	O
universally	O
(	O
we	O
assume	O
we	O
get	O
rid	O
of	O
existential	B-Algorithm
quantifiers	I-Algorithm
in	O
order	O
,	O
so	O
all	O
existential	B-Algorithm
quantifiers	I-Algorithm
before	O
have	O
been	O
removed	O
)	O
and	O
such	O
that	O
occurs	O
in	O
the	O
scope	B-Language
of	O
their	O
quantifiers	O
.	O
</s>
<s>
The	O
function	O
introduced	O
in	O
this	O
process	O
is	O
called	O
a	O
Skolem	B-Application
function	I-Application
(	O
or	O
Skolem	B-Application
constant	I-Application
if	O
it	O
is	O
of	O
zero	O
arity	O
)	O
and	O
the	O
term	O
is	O
called	O
a	O
Skolem	B-Application
term	I-Application
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
formula	O
is	O
not	O
in	O
Skolem	B-Application
normal	I-Application
form	I-Application
because	O
it	O
contains	O
the	O
existential	B-Algorithm
quantifier	I-Algorithm
.	O
</s>
<s>
Skolemization	B-Application
replaces	O
with	O
,	O
where	O
is	O
a	O
new	O
function	O
symbol	O
,	O
and	O
removes	O
the	O
quantification	O
over	O
The	O
resulting	O
formula	O
is	O
.	O
</s>
<s>
The	O
Skolem	B-Application
term	I-Application
contains	O
,	O
but	O
not	O
,	O
because	O
the	O
quantifier	O
to	O
be	O
removed	O
is	O
in	O
the	O
scope	B-Language
of	O
,	O
but	O
not	O
in	O
that	O
of	O
;	O
since	O
this	O
formula	O
is	O
in	O
prenex	B-Application
normal	I-Application
form	I-Application
,	O
this	O
is	O
equivalent	O
to	O
saying	O
that	O
,	O
in	O
the	O
list	O
of	O
quantifiers	O
,	O
precedes	O
while	O
does	O
not	O
.	O
</s>
<s>
Skolemization	B-Application
works	O
by	O
applying	O
a	O
second-order	O
equivalence	O
together	O
with	O
the	O
definition	O
of	O
first-order	O
satisfiability	O
.	O
</s>
<s>
The	O
equivalence	O
provides	O
a	O
way	O
for	O
"	O
moving	O
"	O
an	O
existential	B-Algorithm
quantifier	I-Algorithm
before	O
a	O
universal	O
one	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
sentence	O
"	O
for	O
every	O
there	B-Algorithm
exists	I-Algorithm
a	O
such	O
that	O
"	O
is	O
converted	O
into	O
the	O
equivalent	O
form	O
"	O
there	B-Algorithm
exists	I-Algorithm
a	O
function	O
mapping	O
every	O
into	O
a	O
such	O
that	O
,	O
for	O
every	O
it	O
holds	O
"	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
first-order	O
formula	O
is	O
satisfiable	O
if	O
there	B-Algorithm
exists	I-Algorithm
a	O
model	O
and	O
an	O
evaluation	O
of	O
the	O
free	O
variables	O
of	O
the	O
formula	O
that	O
evaluate	O
the	O
formula	O
to	O
true	O
.	O
</s>
<s>
The	O
model	O
contains	O
the	O
evaluation	O
of	O
all	O
function	O
symbols	O
;	O
therefore	O
,	O
Skolem	B-Application
functions	I-Application
are	O
implicitly	O
existentially	B-Algorithm
quantified	I-Algorithm
.	O
</s>
<s>
In	O
the	O
example	O
above	O
,	O
is	O
satisfiable	O
if	O
and	O
only	O
if	O
there	B-Algorithm
exists	I-Algorithm
a	O
model	O
,	O
which	O
contains	O
an	O
evaluation	O
for	O
,	O
such	O
that	O
is	O
true	O
for	O
some	O
evaluation	O
of	O
its	O
free	O
variables	O
(	O
none	O
in	O
this	O
case	O
)	O
.	O
</s>
<s>
Since	O
first-order	O
models	O
contain	O
the	O
evaluation	O
of	O
all	O
function	O
symbols	O
,	O
any	O
Skolem	B-Application
function	I-Application
that	O
contains	O
is	O
implicitly	O
existentially	B-Algorithm
quantified	I-Algorithm
by	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
after	O
replacing	O
existential	B-Algorithm
quantifiers	I-Algorithm
over	O
variables	O
by	O
existential	B-Algorithm
quantifiers	I-Algorithm
over	O
functions	O
at	O
the	O
front	O
of	O
the	O
formula	O
,	O
the	O
formula	O
still	O
may	O
be	O
treated	O
as	O
a	O
first-order	O
one	O
by	O
removing	O
these	O
existential	B-Algorithm
quantifiers	I-Algorithm
.	O
</s>
<s>
This	O
final	O
step	O
of	O
treating	O
as	O
may	O
be	O
completed	O
because	O
functions	O
are	O
implicitly	O
existentially	B-Algorithm
quantified	I-Algorithm
by	O
in	O
the	O
definition	O
of	O
first-order	O
satisfiability	O
.	O
</s>
<s>
Correctness	O
of	O
Skolemization	B-Application
may	O
be	O
shown	O
on	O
the	O
example	O
formula	O
as	O
follows	O
.	O
</s>
<s>
This	O
formula	O
is	O
satisfied	O
by	O
a	O
model	O
if	O
and	O
only	O
if	O
,	O
for	O
each	O
possible	O
value	O
for	O
in	O
the	O
domain	O
of	O
the	O
model	O
,	O
there	B-Algorithm
exists	I-Algorithm
a	O
value	O
for	O
in	O
the	O
domain	O
of	O
the	O
model	O
that	O
makes	O
true	O
.	O
</s>
<s>
By	O
the	O
axiom	O
of	O
choice	O
,	O
there	B-Algorithm
exists	I-Algorithm
a	O
function	O
such	O
that	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
satisfiable	O
,	O
then	O
there	B-Algorithm
exists	I-Algorithm
a	O
model	O
that	O
satisfies	O
it	O
;	O
this	O
model	O
includes	O
an	O
evaluation	O
for	O
the	O
function	O
such	O
that	O
,	O
for	O
every	O
value	O
of	O
,	O
the	O
formula	O
holds	O
.	O
</s>
<s>
One	O
of	O
the	O
uses	O
of	O
Skolemization	B-Application
is	O
automated	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
method	O
of	O
analytic	O
tableaux	O
,	O
whenever	O
a	O
formula	O
whose	O
leading	O
quantifier	O
is	O
existential	O
occurs	O
,	O
the	O
formula	O
obtained	O
by	O
removing	O
that	O
quantifier	O
via	O
Skolemization	B-Application
may	O
be	O
generated	O
.	O
</s>
<s>
This	O
form	O
of	O
Skolemization	B-Application
is	O
an	O
improvement	O
over	O
"	O
classical	O
"	O
Skolemization	B-Application
in	O
that	O
only	O
variables	O
that	O
are	O
free	O
in	O
the	O
formula	O
are	O
placed	O
in	O
the	O
Skolem	B-Application
term	I-Application
.	O
</s>
<s>
This	O
is	O
an	O
improvement	O
because	O
the	O
semantics	O
of	O
tableaux	O
may	O
implicitly	O
place	O
the	O
formula	O
in	O
the	O
scope	B-Language
of	O
some	O
universally	O
quantified	O
variables	O
that	O
are	O
not	O
in	O
the	O
formula	O
itself	O
;	O
these	O
variables	O
are	O
not	O
in	O
the	O
Skolem	B-Application
term	I-Application
,	O
while	O
they	O
would	O
be	O
there	O
according	O
to	O
the	O
original	O
definition	O
of	O
Skolemization	B-Application
.	O
</s>
<s>
Another	O
improvement	O
that	O
may	O
be	O
used	O
is	O
applying	O
the	O
same	O
Skolem	B-Application
function	I-Application
symbol	O
for	O
formulae	O
that	O
are	O
identical	O
up	O
to	O
variable	O
renaming	O
.	O
</s>
<s>
(	O
For	O
an	O
example	O
see	O
drinker	B-Algorithm
paradox	I-Algorithm
.	O
)	O
</s>
<s>
An	O
important	O
result	O
in	O
model	O
theory	O
is	O
the	O
Lowenheim-Skolem	O
theorem	O
,	O
which	O
can	O
be	O
proven	O
via	O
Skolemizing	O
the	O
theory	O
and	O
closing	O
under	O
the	O
resulting	O
Skolem	B-Application
functions	I-Application
.	O
</s>
<s>
In	O
general	O
,	O
if	O
is	O
a	O
theory	O
and	O
for	O
each	O
formula	O
with	O
free	O
variables	O
there	O
is	O
a	O
function	O
symbol	O
that	O
is	O
provably	O
a	O
Skolem	B-Application
function	I-Application
for	O
,	O
then	O
is	O
called	O
a	O
Skolem	B-Application
theory	I-Application
.	O
</s>
<s>
Every	O
Skolem	B-Application
theory	I-Application
is	O
model	O
complete	O
,	O
i.e.	O
</s>
<s>
Given	O
a	O
model	O
M	O
of	O
a	O
Skolem	B-Application
theory	I-Application
T	O
,	O
the	O
smallest	O
substructure	O
containing	O
a	O
certain	O
set	O
A	O
is	O
called	O
the	O
Skolem	B-Application
hull	I-Application
of	O
A	O
.	O
</s>
<s>
The	O
Skolem	B-Application
hull	I-Application
of	O
A	O
is	O
an	O
atomic	O
prime	O
model	O
over	O
A	O
.	O
</s>
<s>
Skolem	B-Application
normal	I-Application
form	I-Application
is	O
named	O
after	O
the	O
late	O
Norwegian	O
mathematician	O
Thoralf	O
Skolem	O
.	O
</s>
