<s>
In	O
mathematical	O
logic	O
and	O
type	O
theory	O
,	O
the	O
λ-cube	B-Application
(	O
also	O
written	O
lambda	B-Application
cube	I-Application
)	O
is	O
a	O
framework	O
introduced	O
by	O
Henk	O
Barendregt	O
to	O
investigate	O
the	O
different	O
dimensions	O
in	O
which	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
is	O
a	O
generalization	O
of	O
the	O
simply	O
typed	O
λ-calculus	O
.	O
</s>
<s>
The	O
respective	O
dimensions	O
of	O
the	O
λ-cube	B-Application
correspond	O
to	O
:	O
</s>
<s>
The	O
λ-cube	B-Application
can	O
be	O
generalized	O
into	O
the	O
concept	O
of	O
a	O
pure	O
type	O
system	O
.	O
</s>
<s>
The	O
simplest	O
system	O
found	O
in	O
the	O
λ-cube	B-Application
is	O
the	O
simply	O
typed	O
lambda	O
calculus	O
,	O
also	O
called	O
λ	O
→	O
.	O
</s>
<s>
The	O
terms	O
beginning	O
with	O
a	O
are	O
called	O
polymorphic	O
,	O
as	O
they	O
can	O
be	O
applied	O
to	O
different	O
types	O
to	O
get	O
different	O
functions	O
,	O
similarly	O
to	O
polymorphic	O
functions	O
in	O
ML-like	B-Language
languages	I-Language
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
polymorphic	O
identity	O
of	O
OCaml	B-Language
has	O
type	O
meaning	O
it	O
can	O
take	O
an	O
argument	O
of	O
any	O
type	O
'	O
a	O
and	O
return	O
an	O
element	O
of	O
that	O
type	O
.	O
</s>
<s>
The	O
previous	O
type	O
constructor	O
roughly	O
corresponds	O
to	O
the	O
following	O
definition	O
of	O
a	O
tree	O
with	O
labeled	O
leaves	O
in	O
OCaml	B-Language
:	O
</s>
<s>
In	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
,	O
denoted	O
as	O
λC	O
in	O
the	O
cube	O
or	O
as	O
λPω	O
,	O
these	O
four	O
features	O
cohabit	O
,	O
so	O
that	O
both	O
types	O
and	O
terms	O
can	O
depend	O
on	O
types	O
and	O
terms	O
.	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
has	O
both	O
the	O
predicate	O
expressiveness	O
of	O
λP	O
and	O
the	O
computational	O
power	O
of	O
λω	O
,	O
hence	O
why	O
λC	O
is	O
also	O
called	O
λPω	O
,	O
so	O
it	O
is	O
very	O
powerful	O
,	O
both	O
on	O
the	O
logical	O
side	O
and	O
on	O
the	O
computational	O
side	O
.	O
</s>
<s>
The	O
ML-like	B-Language
languages	I-Language
,	O
from	O
a	O
typing	O
point	O
of	O
view	O
,	O
lie	O
somewhere	O
between	O
λ	O
→	O
and	O
λ2	O
,	O
as	O
they	O
admit	O
a	O
restricted	O
kind	O
of	O
polymorphic	O
types	O
,	O
that	O
is	O
the	O
types	O
in	O
prenex	O
normal	O
form	O
.	O
</s>
<s>
Conversely	O
,	O
the	O
systems	O
of	O
the	O
lambda	B-Application
cube	I-Application
can	O
be	O
expressed	O
as	O
pure	O
type	O
systems	O
with	O
two	O
sorts	O
,	O
the	O
only	O
axiom	O
,	O
and	O
a	O
set	O
of	O
rules	O
such	O
that	O
.	O
</s>
<s>
Via	O
the	O
Curry-Howard	O
isomorphism	O
,	O
there	O
is	O
a	O
one-to-one	O
correspondence	O
between	O
the	O
systems	O
in	O
the	O
lambda	B-Application
cube	I-Application
and	O
logical	O
systems	O
,	O
namely	O
:	O
</s>
<s>
the	O
only	O
connectives	O
are	O
and	O
)	O
,	O
however	O
one	O
can	O
define	O
other	O
connectives	O
such	O
as	O
or	O
in	O
an	O
impredicative	O
way	O
in	O
second	O
and	O
higher	B-Algorithm
order	I-Algorithm
logics	I-Algorithm
.	O
</s>
<s>
In	O
the	O
weak	O
higher	B-Algorithm
order	I-Algorithm
logics	I-Algorithm
,	O
there	O
are	O
variables	O
for	O
higher	O
order	O
predicates	O
,	O
but	O
no	O
quantification	O
on	O
those	O
can	O
be	O
done	O
.	O
</s>
<s>
Further	O
extensions	O
to	O
allow	O
the	O
definition	O
of	O
purely	O
functional	O
objects	O
;	O
these	O
systems	O
were	O
generally	O
developed	O
after	O
the	O
lambda	B-Application
cube	I-Application
paper	O
was	O
published	O
.	O
</s>
<s>
The	O
framework	O
of	O
pure	O
type	O
systems	O
generalizes	O
the	O
lambda	B-Application
cube	I-Application
in	O
the	O
sense	O
that	O
all	O
corners	O
of	O
the	O
cube	O
,	O
as	O
well	O
as	O
many	O
other	O
systems	O
can	O
be	O
represented	O
as	O
instances	O
of	O
this	O
general	O
framework	O
.	O
</s>
<s>
This	O
framework	O
predates	O
the	O
lambda	B-Application
cube	I-Application
by	O
a	O
couple	O
of	O
years	O
.	O
</s>
