<s>
In	O
computer	O
science	O
,	O
function-level	B-Application
programming	I-Application
refers	O
to	O
one	O
of	O
the	O
two	O
contrasting	O
programming	O
paradigms	O
identified	O
by	O
John	O
Backus	O
in	O
his	O
work	O
on	O
programs	O
as	O
mathematical	O
objects	O
,	O
the	O
other	O
being	O
value-level	B-Application
programming	I-Application
.	O
</s>
<s>
He	O
designed	O
FP	B-Language
to	O
be	O
the	O
first	O
programming	O
language	O
to	O
specifically	O
support	O
the	O
function-level	B-Application
programming	I-Application
style	O
.	O
</s>
<s>
A	O
function-level	B-Application
program	O
is	O
variable-free	O
(	O
cf	O
.	O
</s>
<s>
point-free	B-Application
programming	I-Application
)	O
,	O
since	O
program	O
variables	O
,	O
which	O
are	O
essential	O
in	O
value-level	B-Application
definitions	O
,	O
are	O
not	O
needed	O
in	O
function-level	B-Application
programs	O
.	O
</s>
<s>
In	O
the	O
function-level	B-Application
style	O
of	O
programming	O
,	O
a	O
program	O
is	O
built	O
directly	O
from	O
programs	O
that	O
are	O
given	O
at	O
the	O
outset	O
,	O
by	O
combining	O
them	O
with	O
program-forming	O
operations	O
or	O
functionals	O
.	O
</s>
<s>
Thus	O
,	O
in	O
contrast	O
with	O
the	O
value-level	B-Application
approach	O
that	O
applies	O
the	O
given	O
programs	O
to	O
values	O
to	O
form	O
a	O
succession	O
of	O
values	O
culminating	O
in	O
the	O
desired	O
result	O
value	O
,	O
the	O
function-level	B-Application
approach	O
applies	O
program-forming	O
operations	O
to	O
the	O
given	O
programs	O
to	O
form	O
a	O
succession	O
of	O
programs	O
culminating	O
in	O
the	O
desired	O
result	O
program	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
function-level	B-Application
approach	O
to	O
programming	O
invites	O
study	O
of	O
the	O
space	O
of	O
programs	O
under	O
program-forming	O
operations	O
,	O
looking	O
to	O
derive	O
useful	O
algebraic	O
properties	O
of	O
these	O
program-forming	O
operations	O
.	O
</s>
<s>
The	O
function-level	B-Application
approach	O
offers	O
the	O
possibility	O
of	O
making	O
the	O
set	O
of	O
programs	O
a	O
mathematical	O
space	O
by	O
emphasizing	O
the	O
algebraic	O
properties	O
of	O
the	O
program-forming	O
operations	O
over	O
the	O
space	O
of	O
programs	O
.	O
</s>
<s>
Another	O
potential	O
advantage	O
of	O
the	O
function-level	B-Application
view	O
is	O
the	O
ability	O
to	O
use	O
only	O
strict	O
functions	O
and	O
thereby	O
have	O
bottom-up	O
semantics	O
,	O
which	O
are	O
the	O
simplest	O
kind	O
of	O
all	O
.	O
</s>
<s>
Yet	O
another	O
is	O
the	O
existence	O
of	O
function-level	B-Application
definitions	O
that	O
are	O
not	O
the	O
lifted	O
(	O
that	O
is	O
,	O
lifted	O
from	O
a	O
lower	O
value-level	B-Application
to	O
a	O
higher	O
function-level	B-Application
)	O
image	O
of	O
any	O
existing	O
value-level	B-Application
one	O
:	O
these	O
(	O
often	O
terse	O
)	O
function-level	B-Application
definitions	O
represent	O
a	O
more	O
powerful	O
style	O
of	O
programming	O
not	O
available	O
at	O
the	O
value-level	B-Application
.	O
</s>
<s>
When	O
Backus	O
studied	O
and	O
publicized	O
his	O
function-level	B-Application
style	O
of	O
programming	O
,	O
his	O
message	O
was	O
mostly	O
misunderstood	O
as	O
supporting	O
the	O
traditional	O
functional	B-Language
programming	I-Language
style	O
languages	O
instead	O
of	O
his	O
own	O
FP	B-Language
and	O
its	O
successor	O
FL	B-Language
.	O
</s>
<s>
Backus	O
calls	O
functional	B-Language
programming	I-Language
applicative	B-Application
programming	I-Application
;	O
his	O
function-level	B-Application
programming	I-Application
is	O
a	O
particular	O
,	O
constrained	O
type	O
.	O
</s>
<s>
A	O
key	O
distinction	O
from	O
functional	B-Language
languages	I-Language
is	O
that	O
Backus	O
 '	O
language	O
has	O
the	O
following	O
hierarchy	O
of	O
types	O
:	O
</s>
<s>
and	O
...	O
the	O
only	O
way	O
to	O
generate	O
new	O
functions	O
is	O
to	O
use	O
one	O
of	O
the	O
functional	B-Language
forms	I-Language
,	O
which	O
are	O
fixed	O
:	O
you	O
cannot	O
build	O
your	O
own	O
functional	B-Language
form	I-Language
(	O
at	O
least	O
not	O
within	O
FP	B-Language
;	O
you	O
can	O
within	O
FFP	B-Language
(	O
Formal	B-Language
FP	I-Language
)	O
)	O
.	O
</s>
<s>
This	O
restriction	O
means	O
that	O
functions	O
in	O
FP	B-Language
are	O
a	O
module	O
(	O
generated	O
by	O
the	O
built-in	O
functions	O
)	O
over	O
the	O
algebra	O
of	O
functional	B-Language
forms	I-Language
,	O
and	O
are	O
thus	O
algebraically	O
tractable	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
general	O
question	O
of	O
equality	O
of	O
two	O
functions	O
is	O
equivalent	O
to	O
the	O
halting	O
problem	O
,	O
and	O
is	O
undecidable	O
,	O
but	O
equality	O
of	O
two	O
functions	O
in	O
FP	B-Language
is	O
just	O
equality	O
in	O
the	O
algebra	O
,	O
and	O
thus	O
(	O
Backus	O
imagines	O
)	O
easier	O
.	O
</s>
<s>
Even	O
today	O
,	O
many	O
users	O
of	O
lambda	B-Language
style	I-Language
languages	O
often	O
misinterpret	O
Backus	O
 '	O
function-level	B-Application
approach	O
as	O
a	O
restrictive	O
variant	O
of	O
the	O
lambda	B-Language
style	I-Language
,	O
which	O
is	O
a	O
de	O
facto	O
value-level	B-Application
style	O
.	O
</s>
<s>
In	O
fact	O
,	O
Backus	O
would	O
not	O
have	O
disagreed	O
with	O
the	O
'	O
restrictive	O
 '	O
accusation	O
:	O
he	O
argued	O
that	O
it	O
was	O
precisely	O
due	O
to	O
such	O
restrictions	O
that	O
a	O
well-formed	O
mathematical	O
space	O
could	O
arise	O
,	O
in	O
a	O
manner	O
analogous	O
to	O
the	O
way	O
structured	B-Language
programming	I-Language
limits	O
programming	O
to	O
a	O
restricted	O
version	O
of	O
all	O
the	O
control-flow	O
possibilities	O
available	O
in	O
plain	O
,	O
unrestricted	O
unstructured	B-Application
programs	I-Application
.	O
</s>
<s>
The	O
value-free	O
style	O
of	O
FP	B-Language
is	O
closely	O
related	O
to	O
the	O
equational	O
logic	O
of	O
a	O
cartesian-closed	B-Application
category	I-Application
.	O
</s>
<s>
The	O
canonical	O
function-level	B-Application
programming	I-Application
language	O
is	O
FP	B-Language
.	O
</s>
<s>
Others	O
include	O
FL	B-Language
,	O
and	O
J	B-Language
.	O
</s>
