<s>
In	O
mathematical	O
logic	O
,	O
an	O
uninterpreted	B-Language
function	I-Language
or	O
function	B-Language
symbol	I-Language
is	O
one	O
that	O
has	O
no	O
other	O
property	O
than	O
its	O
name	O
and	O
n-ary	O
form	O
.	O
</s>
<s>
Function	B-Language
symbols	I-Language
are	O
used	O
,	O
together	O
with	O
constants	O
and	O
variables	O
,	O
to	O
form	O
terms	O
.	O
</s>
<s>
The	O
theory	O
of	O
uninterpreted	B-Language
functions	I-Language
is	O
also	O
sometimes	O
called	O
the	O
free	B-Language
theory	I-Language
,	O
because	O
it	O
is	O
freely	O
generated	O
,	O
and	O
thus	O
a	O
free	O
object	O
,	O
or	O
the	O
empty	B-Language
theory	I-Language
,	O
being	O
the	O
theory	O
having	O
an	O
empty	O
set	O
of	O
sentences	O
(	O
in	O
analogy	O
to	O
an	O
initial	O
algebra	O
)	O
.	O
</s>
<s>
The	O
satisfiability	O
problem	O
for	O
free	B-Language
theories	I-Language
is	O
solved	O
by	O
syntactic	O
unification	B-Algorithm
;	O
algorithms	O
for	O
the	O
latter	O
are	O
used	O
by	O
interpreters	O
for	O
various	O
computer	O
languages	O
,	O
such	O
as	O
Prolog	B-Language
.	O
</s>
<s>
Syntactic	O
unification	B-Algorithm
is	O
also	O
used	O
in	O
algorithms	O
for	O
the	O
satisfiability	O
problem	O
for	O
certain	O
other	O
equational	O
theories	O
,	O
see	O
Unification	B-Algorithm
(	O
computer	O
science	O
)	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
uninterpreted	B-Language
functions	I-Language
for	O
SMT-LIB	O
,	O
if	O
this	O
input	O
is	O
given	O
to	O
an	O
SMT	B-Application
solver	I-Application
:	O
</s>
<s>
the	O
SMT	B-Application
solver	I-Application
would	O
return	O
"	O
This	O
input	O
is	O
satisfiable	O
"	O
.	O
</s>
<s>
That	O
happens	O
because	O
f	O
is	O
an	O
uninterpreted	B-Language
function	I-Language
(	O
i.e.	O
,	O
all	O
that	O
is	O
known	O
about	O
f	O
is	O
its	O
signature	O
)	O
,	O
so	O
it	O
is	O
possible	O
that	O
f(10 )	O
=	O
1	O
.	O
</s>
<s>
the	O
SMT	B-Application
solver	I-Application
would	O
return	O
"	O
This	O
input	O
is	O
unsatisfiable	O
"	O
.	O
</s>
<s>
The	O
decision	O
problem	O
for	O
free	B-Language
theories	I-Language
is	O
particularly	O
important	O
,	O
because	O
many	O
theories	O
can	O
be	O
reduced	O
by	O
it	O
.	O
</s>
<s>
Free	B-Language
theories	I-Language
can	O
be	O
solved	O
by	O
searching	O
for	O
common	O
subexpressions	O
to	O
form	O
the	O
congruence	O
closure	O
.	O
</s>
<s>
Solvers	O
include	O
satisfiability	B-Application
modulo	I-Application
theories	I-Application
solvers	O
.	O
</s>
