<s>
A	O
Hindley	B-Algorithm
–	I-Algorithm
Milner	I-Algorithm
(	O
HM	O
)	O
type	O
system	O
is	O
a	O
classical	O
type	O
system	O
for	O
the	O
lambda	B-Language
calculus	I-Language
with	O
parametric	O
polymorphism	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
Damas	B-Algorithm
–	I-Algorithm
Milner	I-Algorithm
or	O
Damas	B-Algorithm
–	I-Algorithm
Hindley	I-Algorithm
–	I-Algorithm
Milner	I-Algorithm
.	O
</s>
<s>
HM	O
is	O
preferably	O
used	O
for	O
functional	B-Language
languages	I-Language
.	O
</s>
<s>
It	O
was	O
first	O
implemented	O
as	O
part	O
of	O
the	O
type	O
system	O
of	O
the	O
programming	O
language	O
ML	B-Language
.	O
</s>
<s>
Since	O
then	O
,	O
HM	O
has	O
been	O
extended	O
in	O
various	O
ways	O
,	O
most	O
notably	O
with	O
type	O
class	O
constraints	O
like	O
those	O
in	O
Haskell	B-Language
.	O
</s>
<s>
As	O
a	O
type	O
inference	O
method	O
,	O
Hindley	B-Algorithm
–	I-Algorithm
Milner	I-Algorithm
is	O
able	O
to	O
deduce	O
the	O
types	O
of	O
variables	O
,	O
expressions	O
and	O
functions	O
from	O
programs	O
written	O
in	O
an	O
entirely	O
untyped	O
style	O
.	O
</s>
<s>
Being	O
scope	B-Language
sensitive	O
,	O
it	O
is	O
not	O
limited	O
to	O
deriving	O
the	O
types	O
only	O
from	O
a	O
small	O
portion	O
of	O
source	O
code	O
,	O
but	O
rather	O
from	O
complete	O
programs	O
or	O
modules	O
.	O
</s>
<s>
Being	O
able	O
to	O
cope	O
with	O
parametric	O
types	O
,	O
too	O
,	O
it	O
is	O
core	O
to	O
the	O
type	O
systems	O
of	O
many	O
functional	B-Language
programming	I-Language
languages	I-Language
.	O
</s>
<s>
It	O
was	O
first	O
applied	O
in	O
this	O
manner	O
in	O
the	O
ML	B-Language
programming	I-Language
language	I-Language
.	O
</s>
<s>
The	O
origin	O
is	O
the	O
type	O
inference	O
algorithm	O
for	O
the	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
that	O
was	O
devised	O
by	O
Haskell	B-Language
Curry	O
and	O
Robert	O
Feys	O
in	O
1958	O
.	O
</s>
<s>
In	O
the	O
simply	O
typed	O
lambda	B-Language
calculus	I-Language
,	O
types	O
are	O
either	O
atomic	O
type	O
constants	O
or	O
function	O
types	O
of	O
form	O
.	O
</s>
<s>
Contrary	O
to	O
this	O
,	O
the	O
untyped	O
lambda	B-Language
calculus	I-Language
is	O
neutral	O
to	O
typing	O
at	O
all	O
,	O
and	O
many	O
of	O
its	O
functions	O
can	O
be	O
meaningfully	O
applied	O
to	O
all	O
type	O
of	O
arguments	O
.	O
</s>
<s>
Contrary	O
to	O
the	O
type	O
systems	O
used	O
for	O
example	O
in	O
Pascal	B-Application
(	O
1970	O
)	O
or	O
C	B-Language
(	O
1972	O
)	O
,	O
which	O
only	O
support	O
monomorphic	O
types	O
,	O
HM	O
is	O
designed	O
with	O
emphasis	O
on	O
parametric	O
polymorphism	O
.	O
</s>
<s>
The	O
successors	O
of	O
the	O
languages	O
mentioned	O
,	O
like	O
C++	B-Language
(	O
1985	O
)	O
,	O
focused	O
on	O
different	O
types	O
of	O
polymorphism	O
,	O
namely	O
subtyping	O
in	O
connection	O
with	O
object-oriented	B-Language
programming	I-Language
and	O
overloading	O
.	O
</s>
<s>
While	O
subtyping	O
is	O
incompatible	O
with	O
HM	O
,	O
a	O
variant	O
of	O
systematic	O
overloading	O
is	O
available	O
in	O
the	O
HM-based	O
type	O
system	O
of	O
Haskell	B-Language
.	O
</s>
<s>
When	O
extending	O
the	O
type	O
inference	O
for	O
the	O
simply-typed	O
lambda	B-Language
calculus	I-Language
towards	O
polymorphism	O
,	O
one	O
has	O
to	O
define	O
when	O
deriving	O
an	O
instance	O
of	O
a	O
value	O
is	O
admissible	O
.	O
</s>
<s>
Unfortunately	O
,	O
type	O
inference	O
in	O
polymorphic	O
lambda	B-Language
calculus	I-Language
is	O
not	O
decidable	O
.	O
</s>
<s>
The	O
presentation	O
here	O
of	O
such	O
a	O
syntax	O
is	O
not	O
too	O
formal	O
,	O
in	O
that	O
it	O
is	O
written	O
down	O
not	O
to	O
study	O
the	O
surface	O
grammar	O
,	O
but	O
rather	O
the	O
depth	B-Application
grammar	I-Application
,	O
and	O
leaves	O
some	O
syntactical	O
details	O
open	O
.	O
</s>
<s>
The	O
expressions	O
to	O
be	O
typed	O
are	O
exactly	O
those	O
of	O
the	O
lambda	B-Language
calculus	I-Language
extended	O
with	O
a	O
let-expression	O
as	O
shown	O
in	O
the	O
adjacent	O
table	O
.	O
</s>
<s>
A	O
function	O
which	O
returns	O
the	O
cardinality	B-Application
of	O
a	O
set	O
would	O
be	O
a	O
value	O
of	O
this	O
type	O
.	O
</s>
<s>
For	O
instance	O
,	O
one	O
does	O
not	O
have	O
clauses	O
with	O
free	O
variables	O
in	O
Prolog	B-Language
.	O
</s>
<s>
Likewise	O
in	O
Haskell	B-Language
,	O
where	O
all	O
type	O
variables	O
implicitly	O
occur	O
quantified	O
,	O
i.e.	O
</s>
<s>
a	O
Haskell	B-Language
type	O
a	O
->	O
a	O
means	O
here	O
.	O
</s>
<s>
The	O
free	O
monotype	O
variable	O
originates	O
from	O
the	O
type	O
of	O
the	O
variable	O
bound	O
in	O
the	O
surrounding	O
scope	B-Language
.	O
</s>
<s>
But	O
such	O
a	O
scoping	B-Language
cannot	O
be	O
expressed	O
in	O
HM	O
.	O
</s>
<s>
An	O
effective	O
method	O
is	O
known	O
to	O
"	O
make	O
two	O
terms	O
equal	O
"	O
by	O
substitution	O
,	O
Robinson	O
's	O
Unification	B-Algorithm
in	O
combination	O
with	O
the	O
so-called	O
Union-Find	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Assuming	O
an	O
implementation	O
of	O
union-find	B-Algorithm
at	O
hand	O
,	O
one	O
can	O
formulate	O
the	O
unification	B-Algorithm
of	O
two	O
monotypes	O
as	O
follows	O
:	O
</s>
<s>
Overall	O
,	O
the	O
algorithm	O
now	O
proceeds	O
by	O
always	O
making	O
the	O
most	O
general	O
choice	O
leaving	O
the	O
specialization	O
to	O
the	O
unification	B-Algorithm
,	O
which	O
by	O
itself	O
produces	O
the	O
most	O
general	O
result	O
.	O
</s>
<s>
Efficiency	O
is	O
slightly	O
reduced	O
because	O
the	O
binding	O
of	O
type	O
variables	O
in	O
the	O
context	O
has	O
to	O
be	O
maintained	O
to	O
allow	O
computation	O
of	O
and	O
enable	O
an	O
occurs	B-Application
check	I-Application
to	O
prevent	O
the	O
building	O
of	O
recursive	O
types	O
during	O
.	O
</s>
<s>
of	O
producing	O
a	O
substitution	O
which	O
is	O
the	O
most	B-Algorithm
general	I-Algorithm
unifier	I-Algorithm
.	O
</s>
<s>
To	O
make	O
programming	B-Algorithm
practical	I-Algorithm
recursive	O
functions	O
are	O
needed	O
.	O
</s>
<s>
are	O
not	O
directly	O
available	O
,	O
but	O
can	O
instead	O
be	O
expressed	O
with	O
a	O
fixed	B-Application
point	I-Application
combinator	I-Application
.	O
</s>
<s>
Type	O
theory	O
connects	O
lambda	B-Language
calculus	I-Language
with	O
computation	O
and	O
logic	O
.	O
</s>
<s>
The	O
logic	O
collapses	B-General_Concept
because	O
the	O
type	O
becomes	O
inhabited	O
.	O
</s>
<s>
in	O
C++	B-Language
.	O
</s>
<s>
While	O
ad	O
hoc	O
overloading	O
has	O
been	O
avoided	O
in	O
functional	B-Language
programming	I-Language
for	O
the	O
computation	O
costs	O
both	O
in	O
type	O
checking	O
and	O
inference	O
,	O
a	O
means	O
to	O
systematise	O
overloading	O
has	O
been	O
introduced	O
that	O
resembles	O
both	O
in	O
form	O
and	O
naming	O
to	O
object	B-Language
oriented	I-Language
programming	I-Language
,	O
but	O
works	O
one	O
level	O
upwards	O
.	O
</s>
<s>
The	O
quicksort	O
example	O
mentioned	O
in	O
the	O
introduction	O
uses	O
the	O
overloading	O
in	O
the	O
orders	O
,	O
having	O
the	O
following	O
type	O
annotation	O
in	O
Haskell	B-Language
:	O
</s>
<s>
Unfortunately	O
,	O
unification	B-Algorithm
is	O
no	O
longer	O
decidable	O
in	O
the	O
presence	O
of	O
meta	O
types	O
,	O
rendering	O
type	O
inference	O
impossible	O
in	O
this	O
extend	O
of	O
generality	O
.	O
</s>
<s>
Research	O
in	O
second	O
order	O
lambda	B-Language
calculus	I-Language
,	O
one	O
step	O
upwards	O
,	O
showed	O
that	O
type	O
inference	O
is	O
undecidable	O
in	O
this	O
generality	O
.	O
</s>
<s>
Parts	O
of	O
one	O
extra	O
level	O
has	O
been	O
introduced	O
into	O
Haskell	B-Language
named	O
kind	O
,	O
where	O
it	O
is	O
used	O
helping	O
to	O
type	O
monads	O
.	O
</s>
<s>
and	O
showed	O
that	O
an	O
algebraic	O
take	O
on	O
the	O
formalization	O
of	O
subtyping	O
allowed	O
generating	O
compact	O
principal	O
typing	O
schemes	O
for	O
an	O
ML-like	O
language	O
(	O
called	O
MLsub	O
)	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
type	O
inference	O
has	O
proven	O
more	O
difficult	O
in	O
the	O
context	O
of	O
object-oriented	B-Language
programming	I-Language
languages	I-Language
,	O
</s>
<s>
Consequently	O
,	O
type	O
systems	O
with	O
subtyping	O
enabling	O
object-oriented	B-Language
programming	I-Language
,	O
such	O
as	O
Cardelli	O
's	O
system	O
,	O
do	O
not	O
support	O
HM-style	O
type	O
inference	O
.	O
</s>
