# A one-combinator base for Concatenative Combinatory Logic

Let's use the small base `{s',k}` as described in [1]

          [B] [A] k  = A
  [C] [B] [A] [X] s' = [[C] B] X [C] A

Following an approach similar to what we described in [2], we'll look
for a combinator `base` such that

         [base] base = k
  [[base] base] base = s' = [k] base

If such combinator exists, it is a base for the Concatenative Combinatory Logic.

Let's define the combinator `triple`:

and let's assume that `base` has the form:

[𝛾] [𝛽] [𝛼] triple

We want to have `[k] base = s'`:

[k] base = [k] [𝛾] [𝛽] [𝛼] triple
= [𝛾] [𝛽] [𝛼] k
= [𝛾] 𝛼
= s'

To satisfy this equation we can simply set:

𝛼 = i
𝛾 = s'

We want to have `[base] base = k`

[base] base = [[𝛾] [𝛽] [𝛼] triple] [𝛾] [𝛽] [𝛼] triple
= [𝛾] [𝛽] [𝛼] [𝛾] [𝛽] [𝛼] triple
= [𝛾] [𝛽] [𝛾] [𝛽] [𝛼] 𝛼
= k

Replacing in it the values for 𝛼 and 𝛾, we have:

k = [𝛾] [𝛽] [𝛾] [𝛽] [𝛼] 𝛼
= [s'] [𝛽] [s'] [𝛽] [i] i
= [s'] [𝛽] [s'] [𝛽] i
= [s'] [𝛽] [s'] 𝛽

To satisfy this equation, we can set:

𝛽 = [[[k] k] k] k

In fact:

[s'] [𝛽] [s'] 𝛽 = [s'] [𝛽] [s'] [[[k] k] k] k
= [s'] [𝛽] [[k] k] k
= [s'] [k] k
= k

Which gives us a definition for the single-combinator base:

base = [s'] [[[[k] k] k] k] [i] triple

## Bibliography

[1]  *The Theory of Concatenative Combinators*,
     Brent Kerby (bkerby at byu dot net).
     Completed June 19, 2002. Updated February 5, 2007.
     ([link](http://tunes.org/~iepos/joy.html))

[2] *A one-combinator base for Combinatory Logic*,
    [e-mail].
    February 2022.
    ([link](https://hackmd.io/@qeHlwm2zQ62-hoUHOp_E5w/SyCXlWjCY))