<s>
The	O
SKI	B-Application
combinator	I-Application
calculus	I-Application
is	O
a	O
combinatory	B-Application
logic	I-Application
system	I-Application
and	O
a	O
computational	O
system	O
.	O
</s>
<s>
Instead	O
,	O
it	O
is	O
important	O
in	O
the	O
mathematical	O
theory	O
of	O
algorithms	O
because	O
it	O
is	O
an	O
extremely	O
simple	O
Turing	B-Algorithm
complete	I-Algorithm
language	O
.	O
</s>
<s>
It	O
can	O
be	O
likened	O
to	O
a	O
reduced	O
version	O
of	O
the	O
untyped	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
All	O
operations	O
in	O
lambda	B-Language
calculus	I-Language
can	O
be	O
encoded	O
via	O
abstraction	B-Application
elimination	I-Application
into	O
the	O
SKI	B-Application
calculus	I-Application
as	O
binary	O
trees	O
whose	O
leaves	O
are	O
one	O
of	O
the	O
three	O
symbols	O
S	O
,	O
K	O
,	O
and	O
I	O
(	O
called	O
combinators	B-Application
)	O
.	O
</s>
<s>
The	O
"	O
function	O
"	O
,	O
"	O
argument	O
"	O
and	O
the	O
"	O
value	O
"	O
are	O
either	O
combinators	B-Application
or	O
binary	O
trees	O
.	O
</s>
<s>
From	O
these	O
definitions	O
it	O
can	O
be	O
shown	O
that	O
SKI	B-Application
calculus	I-Application
is	O
not	O
the	O
minimum	O
system	O
that	O
can	O
fully	O
perform	O
the	O
computations	O
of	O
lambda	B-Language
calculus	I-Language
,	O
as	O
all	O
occurrences	O
of	O
I	O
in	O
any	O
expression	O
can	O
be	O
replaced	O
by	O
(	O
SKK	O
)	O
or	O
(	O
SKS	O
)	O
or	O
(	O
SK	O
whatever	O
)	O
and	O
the	O
resulting	O
expression	O
will	O
yield	O
the	O
same	O
result	O
.	O
</s>
<s>
Since	O
I	O
is	O
optional	O
,	O
the	O
system	O
is	O
also	O
referred	O
as	O
SK	B-Application
calculus	I-Application
or	O
SK	B-Application
combinator	I-Application
calculus	I-Application
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
define	O
a	O
complete	O
system	O
using	O
only	O
one	O
(	O
improper	O
)	O
combinator	B-Application
.	O
</s>
<s>
An	O
example	O
is	O
Chris	O
Barker	O
's	O
iota	B-Application
combinator	B-Application
,	O
which	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
S	O
and	O
K	O
as	O
follows	O
:	O
</s>
<s>
It	O
is	O
possible	O
to	O
reconstruct	O
S	O
,	O
K	O
,	O
and	O
I	O
from	O
the	O
iota	B-Application
combinator	B-Application
.	O
</s>
<s>
This	O
is	O
known	O
as	O
U	B-Application
combinator	I-Application
.	O
</s>
<s>
which	O
gives	O
us	O
one	O
possible	O
encoding	O
of	O
the	O
Y	O
combinator	B-Application
.	O
</s>
<s>
SKI	B-Application
combinator	I-Application
calculus	I-Application
can	O
also	O
implement	O
Boolean	O
logic	O
in	O
the	O
form	O
of	O
an	O
if-then-else	O
structure	O
.	O
</s>
<s>
The	O
first	O
works	O
just	O
like	O
one	O
of	O
our	O
basic	O
combinators	B-Application
:	O
</s>
<s>
As	O
the	O
SKI	B-Application
calculus	I-Application
is	O
complete	O
,	O
it	O
is	O
also	O
possible	O
to	O
express	O
NOT	O
,	O
OR	O
and	O
AND	O
as	O
prefix	O
operators	O
:	O
</s>
<s>
The	O
combinators	B-Application
K	O
and	O
S	O
correspond	O
to	O
two	O
well-known	O
axioms	O
of	O
sentential	O
logic	O
:	O
</s>
<s>
In	O
order	O
for	O
combinatory	B-Application
logic	I-Application
to	O
have	O
as	O
a	O
model	O
:	O
</s>
<s>
This	O
connection	O
between	O
the	O
types	O
of	O
combinators	B-Application
and	O
the	O
corresponding	O
logical	O
axioms	O
is	O
an	O
instance	O
of	O
the	O
Curry	O
–	O
Howard	O
isomorphism	O
.	O
</s>
