<s>
In	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
in	O
general	O
,	O
a	O
fixed	O
point	O
of	O
a	O
function	O
is	O
a	O
value	O
that	O
is	O
mapped	O
to	O
itself	O
by	O
the	O
function	O
.	O
</s>
<s>
In	O
combinatory	B-Application
logic	I-Application
for	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
fixed-point	B-Application
combinator	I-Application
(	O
or	O
fixpoint	B-Application
combinator	I-Application
)	O
is	O
a	O
higher-order	B-Language
function	I-Language
that	O
returns	O
some	O
fixed	O
point	O
of	O
its	O
argument	O
function	O
,	O
if	O
one	O
exists	O
.	O
</s>
<s>
In	O
the	O
classical	O
untyped	O
lambda	B-Language
calculus	I-Language
,	O
every	O
function	O
has	O
a	O
fixed	O
point	O
.	O
</s>
<s>
In	O
functional	B-Language
programming	I-Language
,	O
the	O
Y	O
combinator	B-Application
can	O
be	O
used	O
to	O
formally	O
define	O
recursive	O
functions	O
in	O
a	O
programming	O
language	O
that	O
does	O
not	O
support	O
recursion	O
.	O
</s>
<s>
This	O
combinator	B-Application
may	O
be	O
used	O
in	O
implementing	O
Curry	O
's	O
paradox	O
.	O
</s>
<s>
The	O
heart	O
of	O
Curry	O
's	O
paradox	O
is	O
that	O
untyped	O
lambda	B-Language
calculus	I-Language
is	O
unsound	O
as	O
a	O
deductive	O
system	O
,	O
and	O
the	O
Y	O
combinator	B-Application
demonstrates	O
this	O
by	O
allowing	O
an	O
anonymous	O
expression	O
to	O
represent	O
zero	O
,	O
or	O
even	O
many	O
values	O
.	O
</s>
<s>
Applied	O
to	O
a	O
function	O
with	O
one	O
variable	O
,	O
the	O
Y	O
combinator	B-Application
usually	O
does	O
not	O
terminate	O
.	O
</s>
<s>
More	O
interesting	O
results	O
are	O
obtained	O
by	O
applying	O
the	O
Y	O
combinator	B-Application
to	O
functions	O
of	O
two	O
or	O
more	O
variables	O
.	O
</s>
<s>
The	O
resulting	O
function	O
behaves	O
like	O
a	O
while	O
or	O
a	O
for	O
loop	O
in	O
an	O
imperative	B-Application
language	I-Application
.	O
</s>
<s>
Used	O
in	O
this	O
way	O
,	O
the	O
Y	O
combinator	B-Application
implements	O
simple	O
recursion	O
.	O
</s>
<s>
In	O
the	O
lambda	B-Language
calculus	I-Language
,	O
it	O
is	O
not	O
possible	O
to	O
refer	O
to	O
the	O
definition	O
of	O
a	O
function	O
inside	O
its	O
own	O
body	O
by	O
name	O
.	O
</s>
<s>
The	O
Y	O
combinator	B-Application
demonstrates	O
this	O
style	O
of	O
programming	O
.	O
</s>
<s>
An	O
example	O
implementation	O
of	O
Y	O
combinator	B-Application
in	O
two	O
languages	O
is	O
presented	O
below	O
.	O
</s>
<s>
The	O
Y	O
combinator	B-Application
is	O
an	O
implementation	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
Fixed-point	B-Application
combinators	I-Application
may	O
also	O
be	O
easily	O
defined	O
in	O
other	O
functional	O
and	O
imperative	B-Application
languages	I-Application
.	O
</s>
<s>
The	O
implementation	O
in	O
lambda	B-Language
calculus	I-Language
is	O
more	O
difficult	O
due	O
to	O
limitations	O
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
The	O
fixed-point	B-Application
combinator	I-Application
may	O
be	O
used	O
in	O
a	O
number	O
of	O
different	O
areas	O
:	O
</s>
<s>
Fixed-point	B-Application
combinators	I-Application
may	O
be	O
applied	O
to	O
a	O
range	O
of	O
different	O
functions	O
,	O
but	O
normally	O
will	O
not	O
terminate	O
unless	O
there	O
is	O
an	O
extra	O
parameter	O
.	O
</s>
<s>
In	O
the	O
untyped	O
lambda	B-Language
calculus	I-Language
,	O
the	O
function	O
to	O
apply	O
the	O
fixed-point	B-Application
combinator	I-Application
to	O
may	O
be	O
expressed	O
using	O
an	O
encoding	O
,	O
like	O
Church	B-Application
encoding	I-Application
.	O
</s>
<s>
In	O
this	O
case	O
particular	O
lambda	B-General_Concept
terms	O
(	O
which	O
define	O
functions	O
)	O
are	O
considered	O
as	O
values	O
.	O
</s>
<s>
"	O
Running	O
"	O
(	O
beta	O
reducing	O
)	O
the	O
fixed-point	B-Application
combinator	I-Application
on	O
the	O
encoding	O
gives	O
a	O
lambda	B-General_Concept
term	O
for	O
the	O
result	O
which	O
may	O
then	O
be	O
interpreted	O
as	O
fixed-point	O
value	O
.	O
</s>
<s>
Alternately	O
,	O
a	O
function	O
may	O
be	O
considered	O
as	O
a	O
lambda	B-General_Concept
term	O
defined	O
purely	O
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
These	O
different	O
approaches	O
affect	O
how	O
a	O
mathematician	O
and	O
a	O
programmer	O
may	O
regard	O
a	O
fixed-point	B-Application
combinator	I-Application
.	O
</s>
<s>
A	O
lambda	B-Language
calculus	I-Language
mathematician	O
may	O
see	O
the	O
Y	O
combinator	B-Application
applied	O
to	O
a	O
function	O
as	O
being	O
an	O
expression	O
satisfying	O
the	O
fixed-point	O
equation	O
,	O
and	O
therefore	O
a	O
solution	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
person	O
only	O
wanting	O
to	O
apply	O
a	O
fixed-point	B-Application
combinator	I-Application
to	O
some	O
general	O
programming	O
task	O
may	O
see	O
it	O
only	O
as	O
a	O
means	O
of	O
implementing	O
recursion	O
.	O
</s>
<s>
This	O
is	O
true	O
in	O
general	O
mathematics	O
and	O
it	O
must	O
be	O
true	O
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
This	O
means	O
that	O
in	O
lambda	B-Language
calculus	I-Language
,	O
applying	O
a	O
fixed-point	B-Application
combinator	I-Application
to	O
a	O
function	O
gives	O
you	O
an	O
expression	O
whose	O
value	O
is	O
the	O
fixed	O
point	O
of	O
the	O
function	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
a	O
value	O
in	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
,	O
it	O
may	O
not	O
correspond	O
to	O
any	O
value	O
in	O
the	O
domain	O
of	O
the	O
function	O
,	O
so	O
in	O
a	O
practical	O
sense	O
it	O
is	O
not	O
necessarily	O
a	O
fixed	O
point	O
of	O
the	O
function	O
,	O
and	O
only	O
in	O
the	O
lambda	B-Language
calculus	I-Language
domain	O
is	O
it	O
a	O
fixed	O
point	O
of	O
the	O
equation	O
.	O
</s>
<s>
Division	O
of	O
signed	O
numbers	O
may	O
be	O
implemented	O
in	O
the	O
Church	B-Application
encoding	I-Application
,	O
so	O
f	O
may	O
be	O
represented	O
by	O
a	O
lambda	B-General_Concept
term	O
.	O
</s>
<s>
However	O
,	O
the	O
lambda	B-General_Concept
term	O
for	O
the	O
solution	O
for	O
the	O
above	O
equation	O
is	O
weirder	O
than	O
that	O
.	O
</s>
<s>
The	O
lambda	B-General_Concept
term	O
represents	O
the	O
state	O
where	O
x	O
could	O
be	O
either	O
i	O
or	O
-i	O
,	O
as	O
one	O
value	O
.	O
</s>
<s>
For	O
the	O
lambda	B-Language
calculus	I-Language
mathematician	O
,	O
this	O
is	O
a	O
consequence	O
of	O
the	O
definition	O
of	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
For	O
the	O
programmer	O
,	O
it	O
means	O
that	O
the	O
beta	O
reduction	O
of	O
the	O
lambda	B-General_Concept
term	O
will	O
loop	O
forever	O
,	O
never	O
reaching	O
a	O
normal	O
form	O
.	O
</s>
<s>
The	O
fixed-point	B-Application
combinator	I-Application
may	O
be	O
defined	O
in	O
mathematics	O
and	O
then	O
implemented	O
in	O
other	O
languages	O
.	O
</s>
<s>
Lambda	B-Language
calculus	I-Language
and	O
programming	O
languages	O
regard	O
function	O
identity	O
as	O
an	O
intensional	O
property	O
.	O
</s>
<s>
A	O
lambda	B-Language
calculus	I-Language
function	O
(	O
or	O
term	O
)	O
is	O
an	O
implementation	O
of	O
a	O
mathematical	O
function	O
.	O
</s>
<s>
In	O
the	O
lambda	B-Language
calculus	I-Language
there	O
are	O
a	O
number	O
of	O
combinators	B-Application
(	O
implementations	O
)	O
that	O
satisfy	O
the	O
mathematical	O
definition	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
.	O
</s>
<s>
Combinatory	B-Application
logic	I-Application
is	O
a	O
higher-order	B-Language
functions	I-Language
theory	O
.	O
</s>
<s>
A	O
combinator	B-Application
is	O
a	O
closed	O
lambda	B-General_Concept
expression	I-General_Concept
,	O
meaning	O
that	O
it	O
has	O
no	O
free	O
variables	O
.	O
</s>
<s>
The	O
combinators	B-Application
may	O
be	O
combined	O
to	O
direct	O
values	O
to	O
their	O
correct	O
places	O
in	O
the	O
expression	O
without	O
ever	O
naming	O
them	O
as	O
variables	O
.	O
</s>
<s>
Fixed-point	B-Application
combinators	I-Application
can	O
be	O
used	O
to	O
implement	O
recursive	O
definition	O
of	O
functions	O
.	O
</s>
<s>
Strongly	O
normalizing	O
type	O
systems	O
such	O
as	O
the	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
disallow	O
non-termination	O
and	O
hence	O
fixed-point	B-Application
combinators	I-Application
often	O
cannot	O
be	O
assigned	O
a	O
type	O
or	O
require	O
complex	O
type	O
system	O
features	O
.	O
</s>
<s>
Furthermore	O
fixed-point	B-Application
combinators	I-Application
are	O
often	O
inefficient	O
compared	O
to	O
other	O
strategies	O
for	O
implementing	O
recursion	O
,	O
as	O
they	O
require	O
more	O
function	O
reductions	O
and	O
construct	O
and	O
take	O
apart	O
a	O
tuple	O
for	O
each	O
group	O
of	O
mutually	B-Algorithm
recursive	I-Algorithm
definitions	O
.	O
</s>
<s>
The	O
factorial	O
function	O
provides	O
a	O
good	O
example	O
of	O
how	O
the	O
fixed-point	B-Application
combinator	I-Application
may	O
be	O
applied	O
.	O
</s>
<s>
The	O
result	O
demonstrates	O
simple	O
recursion	O
,	O
as	O
would	O
be	O
implemented	O
in	O
a	O
single	O
loop	O
in	O
an	O
imperative	B-Application
language	I-Application
.	O
</s>
<s>
The	O
definition	O
of	O
numbers	O
used	O
is	O
explained	O
in	O
Church	B-Application
encoding	I-Application
.	O
</s>
<s>
The	O
lambda	B-General_Concept
term	O
may	O
not	O
,	O
in	O
general	O
,	O
β-reduce	O
to	O
the	O
term	O
.	O
</s>
<s>
An	O
expression	O
for	O
y	O
may	O
be	O
derived	O
using	O
rules	O
from	O
the	O
definition	O
of	O
a	O
let	B-Application
expression	I-Application
.	O
</s>
<s>
Curry	O
's	O
Y	O
combinator	B-Application
may	O
be	O
readily	O
obtained	O
from	O
the	O
definition	O
of	O
y	O
.	O
</s>
<s>
A	O
lambda	B-Language
abstraction	I-Language
does	O
not	O
support	O
reference	O
to	O
the	O
variable	O
name	O
in	O
the	O
applied	O
expression	O
,	O
so	O
x	O
must	O
be	O
passed	O
in	O
as	O
a	O
parameter	O
to	O
x	O
.	O
</s>
<s>
The	O
let	B-Application
expression	I-Application
may	O
be	O
regarded	O
as	O
the	O
definition	O
of	O
the	O
function	O
y	O
,	O
where	O
z	O
is	O
the	O
parameter	O
.	O
</s>
<s>
This	O
is	O
possibly	O
the	O
simplest	O
implementation	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
However	O
,	O
one	O
beta	O
reduction	O
gives	O
the	O
more	O
symmetrical	O
form	O
of	O
Curry	O
's	O
Y	O
combinator	B-Application
:	O
</s>
<s>
See	O
also	O
Translating	O
between	O
let	O
and	O
lambda	B-General_Concept
expressions	I-General_Concept
.	O
</s>
<s>
In	O
untyped	O
lambda	B-Language
calculus	I-Language
fixed-point	B-Application
combinators	I-Application
are	O
not	O
especially	O
rare	O
.	O
</s>
<s>
In	O
2005	O
Mayer	O
Goldberg	O
showed	O
that	O
the	O
set	O
of	O
fixed-point	B-Application
combinators	I-Application
of	O
untyped	O
lambda	B-Language
calculus	I-Language
is	O
recursively	O
enumerable	O
.	O
</s>
<s>
Additional	O
combinators	B-Application
(	O
B	B-Application
,	I-Application
C	I-Application
,	I-Application
K	I-Application
,	I-Application
W	I-Application
system	I-Application
)	O
allow	O
for	O
a	O
much	O
shorter	O
definition	O
.	O
</s>
<s>
The	O
following	O
fixed-point	B-Application
combinator	I-Application
is	O
simpler	O
than	O
the	O
Y	O
combinator	B-Application
,	O
and	O
β-reduces	O
into	O
the	O
Y	O
combinator	B-Application
;	O
it	O
is	O
sometimes	O
cited	O
as	O
the	O
Y	O
combinator	B-Application
itself	O
:	O
</s>
<s>
Another	O
common	O
fixed-point	B-Application
combinator	I-Application
is	O
the	O
Turing	O
fixed-point	B-Application
combinator	I-Application
(	O
named	O
after	O
its	O
discoverer	O
,	O
Alan	O
Turing	O
)	O
:	O
</s>
<s>
The	O
analog	O
for	O
mutual	B-Algorithm
recursion	I-Algorithm
is	O
a	O
polyvariadic	O
fix-point	O
combinator	B-Application
,	O
which	O
may	O
be	O
denoted	O
Y*	O
.	O
</s>
<s>
In	O
a	O
strict	B-Application
programming	I-Application
language	I-Application
the	O
Y	O
combinator	B-Application
will	O
expand	O
until	O
stack	O
overflow	O
,	O
or	O
never	O
halt	O
in	O
case	O
of	O
tail	O
call	O
optimization	O
.	O
</s>
<s>
The	O
Z	O
combinator	B-Application
will	O
work	O
in	O
strict	B-Application
languages	I-Application
(	O
also	O
called	O
eager	O
languages	O
,	O
where	O
applicative	O
evaluation	O
order	O
is	O
applied	O
)	O
.	O
</s>
<s>
The	O
Z	O
combinator	B-Application
has	O
the	O
next	O
argument	O
defined	O
explicitly	O
,	O
preventing	O
the	O
expansion	O
of	O
Z	O
g	O
in	O
the	O
right-hand	O
side	O
of	O
the	O
definition	O
:	O
</s>
<s>
and	O
in	O
lambda	B-Language
calculus	I-Language
it	O
is	O
an	O
eta-expansion	O
of	O
the	O
Y	O
combinator	B-Application
:	O
</s>
<s>
In	O
untyped	O
lambda	B-Language
calculus	I-Language
there	O
are	O
terms	O
that	O
have	O
the	O
same	O
Böhm	B-Application
tree	I-Application
as	O
a	O
fixed-point	B-Application
combinator	I-Application
,	O
that	O
is	O
they	O
have	O
the	O
same	O
infinite	O
extension	O
λx.x	O
(	O
x	O
(	O
x	O
...	O
)	O
)	O
.	O
</s>
<s>
These	O
are	O
called	O
non-standard	O
fixed-point	B-Application
combinators	I-Application
.	O
</s>
<s>
Any	O
fixed-point	B-Application
combinator	I-Application
is	O
also	O
a	O
non-standard	O
one	O
,	O
but	O
not	O
all	O
non-standard	O
fixed-point	B-Application
combinators	I-Application
are	O
fixed-point	B-Application
combinators	I-Application
because	O
some	O
of	O
them	O
fail	O
to	O
satisfy	O
the	O
equation	O
that	O
defines	O
the	O
"	O
standard	O
"	O
ones	O
.	O
</s>
<s>
These	O
strange	O
combinators	B-Application
are	O
called	O
strictly	O
non-standard	O
fixed-point	B-Application
combinators	I-Application
;	O
an	O
example	O
is	O
the	O
following	O
combinator	B-Application
:	O
</s>
<s>
The	O
set	O
of	O
non-standard	O
fixed-point	B-Application
combinators	I-Application
is	O
not	O
recursively	O
enumerable	O
.	O
</s>
<s>
(	O
The	O
Y	O
combinator	B-Application
is	O
a	O
particular	O
implementation	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
in	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
Its	O
structure	O
is	O
determined	O
by	O
the	O
limitations	O
of	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
It	O
is	O
not	O
necessary	O
or	O
helpful	O
to	O
use	O
this	O
structure	O
in	O
implementing	O
the	O
fixed-point	B-Application
combinator	I-Application
in	O
other	O
languages	O
.	O
)	O
</s>
<s>
Simple	O
examples	O
of	O
fixed-point	B-Application
combinators	I-Application
implemented	O
in	O
some	O
programming	O
paradigms	O
are	O
given	O
below	O
.	O
</s>
<s>
In	O
a	O
language	O
that	O
supports	O
lazy	O
evaluation	O
,	O
like	O
in	O
Haskell	B-Language
,	O
it	O
is	O
possible	O
to	O
define	O
a	O
fixed-point	B-Application
combinator	I-Application
using	O
the	O
defining	O
equation	O
of	O
the	O
fixed-point	B-Application
combinator	I-Application
which	O
is	O
conventionally	O
named	O
fix	O
.	O
</s>
<s>
Since	O
Haskell	B-Language
has	O
lazy	O
datatypes	O
,	O
this	O
combinator	B-Application
can	O
also	O
be	O
used	O
to	O
define	O
fixed	O
points	O
of	O
data	O
constructors	O
(	O
and	O
not	O
only	O
to	O
implement	O
recursive	O
functions	O
)	O
.	O
</s>
<s>
In	O
a	O
strict	O
functional	B-Language
language	I-Language
,	O
as	O
illustrated	O
below	O
with	O
OCaml	B-Language
,	O
the	O
argument	O
to	O
f	O
is	O
expanded	O
beforehand	O
,	O
yielding	O
an	O
infinite	O
call	O
sequence	O
,	O
</s>
<s>
In	O
a	O
multi-paradigm	O
functional	B-Language
language	I-Language
(	O
one	O
decorated	O
with	O
imperative	B-Application
features	O
)	O
,	O
such	O
as	O
Lisp	B-Language
,	O
Landin	O
(	O
1963	O
)	O
suggests	O
the	O
use	O
of	O
a	O
variable	O
assignment	O
to	O
create	O
a	O
fixed-point	B-Application
combinator	I-Application
,	O
as	O
in	O
the	O
below	O
example	O
using	O
Scheme	B-Language
:	O
</s>
<s>
Using	O
a	O
lambda	B-Language
calculus	I-Language
with	O
axioms	O
for	O
assignment	O
statements	O
,	O
it	O
can	O
be	O
shown	O
that	O
Y	O
!	O
</s>
<s>
satisfies	O
the	O
same	O
fixed-point	O
law	O
as	O
the	O
call-by-value	O
Y	O
combinator	B-Application
:	O
</s>
<s>
In	O
more	O
idiomatic	O
modern	O
Lisp	B-Language
usage	O
,	O
this	O
would	O
typically	O
be	O
handled	O
via	O
a	O
lexically	O
scoped	O
label	O
(	O
a	O
let	B-Application
expression	I-Application
)	O
,	O
as	O
lexical	O
scope	O
was	O
not	O
introduced	O
to	O
Lisp	B-Language
until	O
the	O
1970s	O
:	O
</s>
<s>
This	O
example	O
is	O
a	O
slightly	O
interpretive	O
implementation	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
.	O
</s>
<s>
In	O
System	O
F	O
(	O
polymorphic	O
lambda	B-Language
calculus	I-Language
)	O
a	O
polymorphic	O
fixed-point	B-Application
combinator	I-Application
has	O
type	O
;	O
</s>
<s>
where	O
a	O
is	O
a	O
type	B-Application
variable	I-Application
.	O
</s>
<s>
In	O
the	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
extended	O
with	O
recursive	O
data	O
types	O
,	O
fixed-point	B-Application
operators	I-Application
can	O
be	O
written	O
,	O
but	O
the	O
type	O
of	O
a	O
"	O
useful	O
"	O
fixed-point	B-Application
operator	I-Application
(	O
one	O
whose	O
application	O
always	O
returns	O
)	O
may	O
be	O
restricted	O
.	O
</s>
<s>
In	O
the	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
,	O
the	O
fixed-point	B-Application
combinator	I-Application
Y	O
cannot	O
be	O
assigned	O
a	O
type	O
because	O
at	O
some	O
point	O
it	O
would	O
deal	O
with	O
the	O
self-application	O
sub-term	O
by	O
the	O
application	O
rule	O
:	O
</s>
<s>
No	O
fixed-point	B-Application
combinator	I-Application
can	O
in	O
fact	O
be	O
typed	O
;	O
in	O
those	O
systems	O
,	O
any	O
support	O
for	O
recursion	O
must	O
be	O
explicitly	O
added	O
to	O
the	O
language	O
.	O
</s>
<s>
In	O
programming	O
languages	O
that	O
support	O
recursive	O
data	O
types	O
,	O
it	O
is	O
possible	O
to	O
type	O
the	O
Y	O
combinator	B-Application
by	O
appropriately	O
accounting	O
for	O
the	O
recursion	O
at	O
the	O
type	O
level	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
following	O
Haskell	B-Language
code	O
,	O
we	O
have	O
In	O
and	O
out	O
being	O
the	O
names	O
of	O
the	O
two	O
directions	O
of	O
the	O
isomorphism	O
,	O
with	O
types	O
:	O
</s>
<s>
Or	O
equivalently	O
in	O
OCaml	B-Language
:	O
</s>
<s>
Because	O
fixed-point	B-Application
combinators	I-Application
can	O
be	O
used	O
to	O
implement	O
recursion	O
,	O
it	O
is	O
possible	O
to	O
use	O
them	O
to	O
describe	O
specific	O
types	O
of	O
recursive	O
computations	O
,	O
such	O
as	O
those	O
in	O
fixed-point	O
iteration	O
,	O
iterative	B-Algorithm
methods	I-Algorithm
,	O
recursive	B-Application
join	I-Application
in	O
relational	B-Application
databases	I-Application
,	O
data-flow	O
analysis	O
,	O
FIRST	O
and	O
FOLLOW	O
sets	O
of	O
non-terminals	O
in	O
a	O
context-free	O
grammar	O
,	O
transitive	O
closure	O
,	O
and	O
other	O
types	O
of	O
closure	O
operations	O
.	O
</s>
<s>
In	O
contrast	O
to	O
universal	O
quantification	O
over	O
all	O
,	O
a	O
fixed-point	B-Application
combinator	I-Application
constructs	O
one	O
value	O
that	O
is	O
a	O
fixed	O
point	O
of	O
.	O
</s>
<s>
The	O
remarkable	O
property	O
of	O
a	O
fixed-point	B-Application
combinator	I-Application
is	O
that	O
it	O
constructs	O
a	O
fixed	O
point	O
for	O
an	O
arbitrary	O
given	O
function	O
.	O
</s>
<s>
In	O
lambda	B-Language
calculus	I-Language
,	O
from	O
a	O
computational	O
point	O
of	O
view	O
,	O
applying	O
a	O
fixed-point	B-Application
combinator	I-Application
to	O
an	O
identity	O
function	O
or	O
an	O
idempotent	O
function	O
typically	O
results	O
in	O
non-terminating	O
computation	O
.	O
</s>
<s>
Fixed-point	B-Application
combinators	I-Application
do	O
not	O
necessarily	O
exist	O
in	O
more	O
restrictive	O
models	O
of	O
computation	O
.	O
</s>
<s>
For	O
instance	O
,	O
they	O
do	O
not	O
exist	O
in	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
The	O
Y	O
combinator	B-Application
allows	O
recursion	O
to	O
be	O
defined	O
as	O
a	O
set	O
of	O
rewrite	O
rules	O
,	O
without	O
requiring	O
native	O
recursion	O
support	O
in	O
the	O
language	O
.	O
</s>
<s>
In	O
programming	O
languages	O
that	O
support	O
anonymous	B-General_Concept
functions	I-General_Concept
,	O
fixed-point	B-Application
combinators	I-Application
allow	O
the	O
definition	O
and	O
use	O
of	O
anonymous	O
recursive	O
functions	O
,	O
i.e.	O
</s>
<s>
In	O
this	O
setting	O
,	O
the	O
use	O
of	O
fixed-point	B-Application
combinators	I-Application
is	O
sometimes	O
called	O
anonymous	B-Language
recursion	I-Language
.	O
</s>
