<s>
The	O
Bird	B-Application
–	I-Application
Meertens	I-Application
formalism	I-Application
(	O
BMF	O
)	O
is	O
a	O
calculus	O
for	O
deriving	B-Application
programs	I-Application
from	O
specifications	B-Application
(	O
in	O
a	O
functional-programming	B-Language
setting	O
)	O
by	O
a	O
process	O
of	O
equational	O
reasoning	O
.	O
</s>
<s>
It	O
was	O
devised	O
by	O
Richard	O
Bird	O
and	O
Lambert	O
Meertens	O
as	O
part	O
of	O
their	O
work	O
within	O
IFIP	B-Language
Working	I-Language
Group	I-Language
2.1	I-Language
.	O
</s>
<s>
Facetiously	O
it	O
is	O
also	O
referred	O
to	O
as	O
Squiggol	B-Application
,	O
as	O
a	O
nod	O
to	O
ALGOL	B-Language
,	O
which	O
was	O
also	O
in	O
the	O
remit	O
of	O
WG	O
2.1	O
,	O
and	O
because	O
of	O
the	O
"	O
squiggly	O
"	O
symbols	O
it	O
uses	O
.	O
</s>
<s>
A	O
less-used	O
variant	O
name	O
,	O
but	O
actually	O
the	O
first	O
one	O
suggested	O
,	O
is	O
SQUIGOL	B-Application
.	O
</s>
<s>
Likewise	O
,	O
reduce	B-Application
is	O
a	O
function	O
that	O
collapses	O
a	O
list	O
into	O
a	O
single	O
value	O
by	O
repeated	O
application	O
of	O
a	O
binary	O
operator	O
.	O
</s>
<s>
point-free	B-Application
style	I-Application
.	O
</s>
<s>
Similarly	O
,	O
writing	O
for	O
functional	B-Application
composition	I-Application
and	O
for	O
conjunction	O
,	O
it	O
is	O
easy	O
to	O
write	O
a	O
function	O
testing	O
that	O
all	O
elements	O
of	O
a	O
list	O
satisfy	O
a	O
predicate	O
p	O
,	O
simply	O
as	O
:	O
</s>
<s>
Bird	O
(	O
1989	O
)	O
transforms	O
inefficient	O
easy-to-understand	O
expressions	O
(	O
"	O
specifications	B-Application
"	O
)	O
into	O
efficient	O
involved	O
expressions	O
(	O
"	O
programs	O
"	O
)	O
by	O
algebraic	O
manipulation	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
specification	B-Application
""	O
is	O
an	O
almost	O
literal	O
translation	O
of	O
the	O
maximum	B-Language
segment	I-Language
sum	I-Language
problem	I-Language
,	O
but	O
running	O
that	O
functional	B-Language
program	I-Language
on	O
a	O
list	O
of	O
size	O
will	O
take	O
time	O
in	O
general	O
.	O
</s>
<s>
From	O
this	O
,	O
Bird	O
computes	O
an	O
equivalent	O
functional	B-Language
program	I-Language
that	O
runs	O
in	O
time	O
,	O
and	O
is	O
in	O
fact	O
a	O
functional	O
version	O
of	O
Kadane	B-Language
's	I-Language
algorithm	I-Language
.	O
</s>
<s>
As	O
above	O
,	O
function	B-Application
composition	I-Application
is	O
denoted	O
by	O
""	O
,	O
which	O
has	O
lowest	O
binding	O
precedence	O
.	O
</s>
<s>
The	O
homomorphism	B-Application
lemma	I-Application
states	O
that	O
h	O
is	O
a	O
homomorphism	O
if	O
and	O
only	O
if	O
there	O
exists	O
an	O
operator	O
and	O
a	O
function	O
f	O
such	O
that	O
.	O
</s>
<s>
A	O
point	O
of	O
great	O
interest	O
for	O
this	O
lemma	O
is	O
its	O
application	O
to	O
the	O
derivation	O
of	O
highly	O
parallel	B-Operating_System
implementations	O
of	O
computations	O
.	O
</s>
<s>
Indeed	O
,	O
it	O
is	O
trivial	O
to	O
see	O
that	O
has	O
a	O
highly	O
parallel	B-Operating_System
implementation	O
,	O
and	O
so	O
does	O
—	O
most	O
obviously	O
as	O
a	O
binary	O
tree	O
.	O
</s>
<s>
Thus	O
for	O
any	O
list	O
homomorphism	O
h	O
,	O
there	O
exists	O
a	O
parallel	B-Operating_System
implementation	O
.	O
</s>
<s>
In	O
any	O
application	O
where	O
the	O
list	O
is	O
enormous	O
and	O
the	O
result	O
is	O
a	O
very	O
simple	O
type	O
–	O
say	O
an	O
integer	O
–	O
the	O
benefits	O
of	O
parallelisation	B-Operating_System
are	O
considerable	O
.	O
</s>
<s>
This	O
is	O
the	O
basis	O
of	O
the	O
map-reduce	B-Operating_System
approach	O
.	O
</s>
