<s>
Computable	B-Algorithm
topology	I-Algorithm
is	O
a	O
discipline	O
in	O
mathematics	O
that	O
studies	O
the	O
topological	O
and	O
algebraic	O
structure	O
of	O
computation	O
.	O
</s>
<s>
Computable	B-Algorithm
topology	I-Algorithm
is	O
not	O
to	O
be	O
confused	O
with	O
algorithmic	O
or	O
computational	B-Algorithm
topology	I-Algorithm
,	O
which	O
studies	O
the	O
application	O
of	O
computation	O
to	O
topology	B-Architecture
.	O
</s>
<s>
As	O
shown	O
by	O
Alan	O
Turing	O
and	O
Alonzo	O
Church	O
,	O
the	O
λ-calculus	B-Language
is	O
strong	O
enough	O
to	O
describe	O
all	O
mechanically	O
computable	O
functions	O
(	O
see	O
Church	O
–	O
Turing	O
thesis	O
)	O
.	O
</s>
<s>
Lambda-calculus	B-Language
is	O
thus	O
effectively	O
a	O
programming	O
language	O
,	O
from	O
which	O
other	O
languages	O
can	O
be	O
built	O
.	O
</s>
<s>
For	O
this	O
reason	O
when	O
considering	O
the	O
topology	B-Architecture
of	O
computation	O
it	O
is	O
common	O
to	O
focus	O
on	O
the	O
topology	B-Architecture
of	O
λ-calculus	B-Language
.	O
</s>
<s>
Note	O
that	O
this	O
is	O
not	O
necessarily	O
a	O
complete	O
description	O
of	O
the	O
topology	B-Architecture
of	O
computation	O
,	O
since	O
functions	O
which	O
are	O
equivalent	O
in	O
the	O
Church-Turing	O
sense	O
may	O
still	O
have	O
different	O
topologies	B-Architecture
.	O
</s>
<s>
The	O
topology	B-Architecture
of	O
λ-calculus	B-Language
is	O
the	O
Scott	O
topology	B-Architecture
,	O
and	O
when	O
restricted	O
to	O
continuous	O
functions	O
the	O
type	O
free	O
λ-calculus	B-Language
amounts	O
to	O
a	O
topological	O
space	O
reliant	O
on	O
the	O
tree	B-Data_Structure
topology	I-Data_Structure
.	O
</s>
<s>
Both	O
the	O
Scott	O
and	O
Tree	B-Data_Structure
topologies	I-Data_Structure
exhibit	O
continuity	O
with	O
respect	O
to	O
the	O
binary	O
operators	O
of	O
application	O
(	O
f	O
applied	O
to	O
a	O
=	O
fa	O
)	O
and	O
abstraction	O
((λx.t(x )	O
)	O
a	O
=	O
t(a )	O
)	O
with	O
a	O
modular	O
equivalence	O
relation	O
based	O
on	O
a	O
congruency	O
.	O
</s>
<s>
The	O
λ-algebra	O
describing	O
the	O
algebraic	O
structure	O
of	O
the	O
lambda-calculus	B-Language
is	O
found	O
to	O
be	O
an	O
extension	O
of	O
the	O
combinatory	B-Application
algebra	I-Application
,	O
with	O
an	O
element	O
introduced	O
to	O
accommodate	O
abstraction	O
.	O
</s>
<s>
Type	O
free	O
λ-calculus	B-Language
treats	O
functions	O
as	O
rules	O
and	O
does	O
not	O
differentiate	O
functions	O
and	O
the	O
objects	O
which	O
they	O
are	O
applied	O
to	O
,	O
meaning	O
λ-calculus	B-Language
is	O
type	O
free	O
.	O
</s>
<s>
A	O
by-product	O
of	O
type	O
free	O
λ-calculus	B-Language
is	O
an	O
effective	O
computability	O
equivalent	O
to	O
general	O
recursion	O
and	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
The	O
set	O
of	O
λ-terms	O
can	O
be	O
considered	O
a	O
functional	O
topology	B-Architecture
in	O
which	O
a	O
function	O
space	O
can	O
be	O
embedded	O
,	O
meaning	O
λ	O
mappings	O
within	O
the	O
space	O
X	O
are	O
such	O
that	O
λ:X	O
→	O
X	O
.	O
</s>
<s>
Introduced	O
November	O
1969	O
,	O
Dana	O
Scott	O
's	O
untyped	O
set	O
theoretic	O
model	O
constructed	O
a	O
proper	O
topology	B-Architecture
for	O
any	O
λ-calculus	B-Language
model	O
whose	O
function	O
space	O
is	O
limited	O
to	O
continuous	O
functions	O
.	O
</s>
<s>
The	O
result	O
of	O
a	O
Scott	O
continuous	O
λ-calculus	B-Language
topology	B-Architecture
is	O
a	O
function	O
space	O
built	O
upon	O
a	O
programming	O
semantic	O
allowing	O
fixed	O
point	O
combinatorics	O
,	O
such	O
as	O
the	O
Y	O
combinator	B-Application
,	O
and	O
data	O
types	O
.	O
</s>
<s>
By	O
1971	O
,	O
λ-calculus	B-Language
was	O
equipped	O
to	O
define	O
any	O
sequential	O
computation	O
and	O
could	O
be	O
easily	O
adapted	O
to	O
parallel	O
computations	O
.	O
</s>
<s>
The	O
reducibility	O
of	O
all	O
computations	O
to	O
λ-calculus	B-Language
allows	O
these	O
λ-topological	O
properties	O
to	O
become	O
adopted	O
by	O
all	O
programming	O
languages	O
.	O
</s>
<s>
Based	O
on	O
the	O
operators	O
within	O
lambda	B-Language
calculus	I-Language
,	O
application	O
and	O
abstraction	O
,	O
it	O
is	O
possible	O
to	O
develop	O
an	O
algebra	O
whose	O
group	O
structure	O
uses	O
application	O
and	O
abstraction	O
as	O
binary	O
operators	O
.	O
</s>
<s>
Abstraction	O
incorporates	O
undefined	O
variables	O
by	O
denoting	O
λx.t(x )	O
as	O
the	O
function	O
assigning	O
the	O
variable	O
a	O
to	O
the	O
lambda	O
term	O
with	O
value	O
t(a )	O
via	O
the	O
operation	O
( ( λ	O
x.t(x )	O
)	O
a	O
=	O
t(a )	O
)	O
.	O
</s>
<s>
Lastly	O
,	O
an	O
equivalence	O
relation	O
emerges	O
which	O
identifies	O
λ-terms	O
modulo	O
convertible	O
terms	O
,	O
an	O
example	O
being	O
beta	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
The	O
Scott	O
topology	B-Architecture
is	O
essential	O
in	O
understanding	O
the	O
topological	O
structure	O
of	O
computation	O
as	O
expressed	O
through	O
the	O
λ-calculus	B-Language
.	O
</s>
<s>
Scott	O
found	O
that	O
after	O
constructing	O
a	O
function	O
space	O
using	O
λ-calculus	B-Language
one	O
obtains	O
a	O
Kolmogorov	O
space	O
,	O
a	O
topological	O
space	O
which	O
exhibits	O
pointwise	B-Algorithm
convergence	I-Algorithm
,	O
in	O
short	O
the	O
product	O
topology	B-Architecture
.	O
</s>
<s>
It	O
is	O
the	O
ability	O
of	O
self	O
homeomorphism	O
as	O
well	O
as	O
the	O
ability	O
to	O
embed	O
every	O
space	O
into	O
such	O
a	O
space	O
,	O
denoted	O
Scott	O
continuous	O
,	O
as	O
previously	O
described	O
which	O
allows	O
Scott	O
's	O
topology	B-Architecture
to	O
be	O
applicable	O
to	O
logic	O
and	O
recursive	O
function	O
theory	O
.	O
</s>
<s>
Scott	O
approaches	O
his	O
derivation	O
using	O
a	O
complete	O
lattice	O
,	O
resulting	O
in	O
a	O
topology	B-Architecture
dependent	O
on	O
the	O
lattice	O
structure	O
.	O
</s>
<s>
For	O
this	O
reason	O
a	O
more	O
general	O
understanding	O
of	O
the	O
computational	B-Algorithm
topology	I-Algorithm
is	O
provided	O
through	O
complete	O
partial	O
orders	O
.	O
</s>
<s>
We	O
will	O
re-iterate	O
to	O
familiarize	O
ourselves	O
with	O
the	O
notation	O
to	O
be	O
used	O
during	O
the	O
discussion	O
of	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
We	O
are	O
now	O
able	O
to	O
define	O
the	O
Scott	O
topology	B-Architecture
over	O
a	O
cpo	O
(	O
D	O
,	O
≤	O
)	O
.	O
</s>
<s>
Ap	O
is	O
continuous	O
with	O
respect	O
to	O
the	O
Scott	O
topology	B-Architecture
on	O
the	O
product	O
(	O
)	O
:	O
</s>
<s>
Proof	O
:	O
λx.f(x )	O
=	O
f	O
is	O
continuous	O
.	O
</s>
<s>
It	O
has	O
now	O
been	O
shown	O
application	O
is	O
continuous	O
under	O
the	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
In	O
order	O
to	O
demonstrate	O
the	O
Scott	O
topology	B-Architecture
is	O
a	O
suitable	O
fit	O
for	O
λ-calculus	B-Language
it	O
is	O
necessary	O
to	O
prove	O
abstraction	O
remains	O
continuous	O
over	O
the	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
Once	O
completed	O
it	O
will	O
have	O
been	O
shown	O
that	O
the	O
mathematical	O
foundation	O
of	O
λ-calculus	B-Language
is	O
a	O
well	O
defined	O
and	O
suitable	O
candidate	O
functional	O
paradigm	O
for	O
the	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
It	O
has	O
not	O
been	O
demonstrated	O
how	O
and	O
why	O
the	O
λ-calculus	B-Language
defines	O
the	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
Böhm	B-Application
trees	I-Application
,	O
easily	O
represented	O
graphically	O
,	O
express	O
the	O
computational	O
behavior	O
of	O
a	O
lambda	O
term	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
predict	O
the	O
functionality	O
of	O
a	O
given	O
lambda	O
expression	O
from	O
reference	O
to	O
its	O
correlating	O
Böhm	B-Application
tree	I-Application
.	O
</s>
<s>
Böhm	B-Application
trees	I-Application
can	O
be	O
seen	O
somewhat	O
analogous	O
to	O
where	O
the	O
Böhm	B-Application
tree	I-Application
of	O
a	O
given	O
set	O
is	O
similar	O
to	O
the	O
continued	O
fraction	O
of	O
a	O
real	O
number	O
,	O
and	O
what	O
is	O
more	O
,	O
the	O
Böhm	B-Application
tree	I-Application
corresponding	O
to	O
a	O
sequence	O
in	O
normal	O
form	O
is	O
finite	O
,	O
similar	O
to	O
the	O
rational	O
subset	O
of	O
the	O
Reals	O
.	O
</s>
<s>
Böhm	B-Application
trees	I-Application
are	O
defined	O
by	O
a	O
mapping	O
of	O
elements	O
within	O
a	O
sequence	O
of	O
numbers	O
with	O
ordering	O
( ≤	O
,	O
lh	O
)	O
and	O
a	O
binary	O
operator	O
*	O
to	O
a	O
set	O
of	O
symbols	O
.	O
</s>
<s>
The	O
Böhm	B-Application
tree	I-Application
is	O
then	O
a	O
relation	O
among	O
a	O
set	O
of	O
symbols	O
through	O
a	O
partial	O
mapping	O
ψ	O
.	O
</s>
<s>
Informally	O
Böhm	B-Application
trees	I-Application
may	O
be	O
conceptualized	O
as	O
follows	O
:	O
</s>
<s>
y	O
|	O
n	O
∈	O
y	O
are	O
variables	O
and	O
denoting	O
BT(M )	O
as	O
the	O
Böhm	B-Application
tree	I-Application
for	O
a	O
lambda	O
term	O
M	O
we	O
then	O
have	O
:	O
</s>
<s>
The	O
Böhm	B-Application
tree	I-Application
of	O
a	O
λ	O
term	O
M	O
,	O
denoted	O
BT(M )	O
,	O
is	O
the	O
Σ	O
labelled	O
tree	O
defined	O
as	O
follows	O
:	O
</s>
<s>
We	O
may	O
now	O
move	O
on	O
to	O
show	O
that	O
Böhm	B-Application
trees	I-Application
act	O
as	O
suitable	O
mappings	O
from	O
the	O
tree	B-Data_Structure
topology	I-Data_Structure
to	O
the	O
scott	O
topology	B-Architecture
.	O
</s>
<s>
Allowing	O
one	O
to	O
see	O
computational	O
constructs	O
,	O
be	O
it	O
within	O
the	O
Scott	O
or	O
tree	B-Data_Structure
topology	I-Data_Structure
,	O
as	O
Böhm	B-Application
tree	I-Application
formations	O
.	O
</s>
<s>
It	O
is	O
found	O
that	O
Böhm	B-Application
tree	I-Application
's	O
allow	O
for	O
a	O
continuous	O
mapping	O
from	O
the	O
tree	B-Data_Structure
topology	I-Data_Structure
to	O
the	O
Scott	O
topology	B-Architecture
.	O
</s>
<s>
An	O
equivalent	O
definition	O
would	O
be	O
to	O
say	O
the	O
open	O
sets	O
of	O
Ɣ	O
are	O
the	O
image	O
of	O
the	O
inverse	O
Böhm	B-Application
tree	I-Application
(	O
O	O
)	O
where	O
O	O
is	O
Scott	O
open	O
in	O
B	O
.	O
</s>
<s>
The	O
applicability	O
of	O
the	O
Bömh	O
trees	O
and	O
the	O
tree	B-Data_Structure
topology	I-Data_Structure
has	O
many	O
interesting	O
consequences	O
to	O
λ-terms	O
expressed	O
topologically	O
:	O
</s>
<s>
Application	O
and	O
abstraction	O
,	O
similar	O
to	O
the	O
Scott	O
topology	B-Architecture
,	O
are	O
continuous	O
on	O
the	O
tree	B-Data_Structure
topology	I-Data_Structure
.	O
</s>
<s>
New	O
methods	O
of	O
interpretation	O
of	O
the	O
λ-calculus	B-Language
are	O
not	O
only	O
interesting	O
in	O
themselves	O
but	O
allow	O
new	O
modes	O
of	O
thought	O
concerning	O
the	O
behaviors	O
of	O
computer	O
science	O
.	O
</s>
<s>
A	O
combinatory	B-Application
algebra	I-Application
allows	O
for	O
the	O
application	O
operator	O
and	O
acts	O
as	O
a	O
useful	O
starting	O
point	O
but	O
remains	O
insufficient	O
for	O
the	O
λ-calculus	B-Language
in	O
being	O
unable	O
to	O
express	O
abstraction	O
.	O
</s>
<s>
The	O
λ	O
algebra	O
becomes	O
a	O
combinatory	B-Application
algebra	I-Application
M	O
combined	O
with	O
a	O
syntactic	O
operator	O
λ*	O
that	O
transforms	O
a	O
term	O
B(x,y )	O
,	O
with	O
constants	O
in	O
M	O
,	O
into	O
C( )	O
≡	O
λ*	O
x.B(x, )	O
.	O
</s>
<s>
The	O
construction	O
of	O
the	O
λ-algebra	O
through	O
the	O
introduction	O
of	O
an	O
abstraction	B-Language
operator	I-Language
proceeds	O
as	O
follows	O
:	O
</s>
<s>
We	O
must	O
construct	O
an	O
algebra	O
which	O
allows	O
for	O
solutions	O
to	O
equations	O
such	O
as	O
axy	O
=	O
xyy	O
such	O
that	O
a	O
=	O
λ	O
xy.xyy	O
there	O
is	O
need	O
for	O
the	O
combinatory	B-Application
algebra	I-Application
.	O
</s>
<s>
Relevant	O
attributes	O
of	O
the	O
combinatory	B-Application
algebra	I-Application
are	O
:	O
</s>
<s>
Within	O
combinatory	B-Application
algebra	I-Application
there	O
exists	O
applicative	O
structures	O
.	O
</s>
<s>
An	O
applicative	O
structure	O
W	O
is	O
a	O
combinatory	B-Application
algebra	I-Application
provided	O
:	O
</s>
<s>
W	O
exhibits	O
combinatory	O
completeness	O
(	O
see	O
completeness	B-Application
of	I-Application
the	I-Application
S-K	I-Application
basis	I-Application
)	O
.	O
</s>
<s>
The	O
combinatory	B-Application
algebra	I-Application
is	O
:	O
</s>
<s>
Combinatory	B-Application
algebras	I-Application
remain	O
unable	O
to	O
act	O
as	O
the	O
algebraic	O
structure	O
for	O
λ-calculus	B-Language
,	O
the	O
lack	O
of	O
recursion	O
being	O
a	O
major	O
disadvantage	O
.	O
</s>
<s>
However	O
the	O
existence	O
of	O
an	O
applicative	O
term	O
)	O
provides	O
a	O
good	O
starting	O
point	O
to	O
build	O
a	O
λ-calculus	B-Language
algebra	O
.	O
</s>
<s>
include	O
λx.A(x, )	O
.	O
</s>
<s>
We	O
begin	O
by	O
exploiting	O
the	O
fact	O
that	O
within	O
a	O
combinatory	B-Application
algebra	I-Application
M	O
,	O
with	O
A(x, )	O
within	O
the	O
set	O
of	O
terms	O
,	O
then	O
:	O
</s>
<s>
λx	B-Language
.	O
</s>
<s>
(	O
λy.yx	O
)	O
=	O
λx.x	O
in	O
λ(W )	O
.	O
</s>
<s>
Though	O
arduous	O
,	O
the	O
foundation	O
has	O
been	O
set	O
for	O
a	O
proper	O
algebraic	O
framework	O
for	O
which	O
the	O
λ-calculus	B-Language
,	O
and	O
therefore	O
computation	O
,	O
may	O
be	O
investigated	O
in	O
a	O
group	O
theoretic	O
manner	O
.	O
</s>
