Title: Combinatory logic from scratch
Date: 2008-11-28 01:56:32

Cause it's sooooo sexy, let's speak about Combinatory Logic!

- Rule 1: You don't talk about Combinatory Logic
- Rule 2: You don't talk about Combinatory Logic
- Rule 3: Combinatory Logic is based on Lambda Calculus
  (see Wikipedia for both)
- Rule 4: A combinator is a Lambda expression taking One and only One combinator as parameter, and returning a Combinator.

As i'm speaking to developers, I'LL use the `C#` Lambda
syntax which is: `(parameter) => statement` Let's now try our first
Combinator, named the Identity Combinator `I = (a) => a`; I named it `I`,
it takes one parameter, localy named `a` and return the parameter as is.

Important point: How to build combinator taking more than one parameter
? In *C#* you should use `(a, b, c) => blah blah...` but the *Rule 4*
forbid us to give more than one paraneter, so let's cheat, imagine : `K =
(x) => (y) => x`; `K` is a combinator taking `x`, returning a
combinator taking `y` and returning `x`. so we have `K(x) = (y) => x` and
`K(x)(y) = x`! So `K` take two arguments, `x` and `y`, and returns `x`, but! `K`
can take only one argument, look at the `K(x) = (y) => x` ...

Let's try with three arguments:

    S = (x) => (y) => (z) => x(z)(y(z))

can be called with one,

    S(x) returns (y) => (z) => x(z)(y(z))

two,

    S(x)(y) returns (z) => x(z)(y(z))

or three arguments:

    s(x)(y)(z) returns x(z)(y(z))

In combinatory logic, they write:

- `I a = a`
- `K x y = x`
- `S x y z = x(z)(y(z))`

then they say that in fact, `I` can be build from `S` and `K`:

    I = SKK

Ok but what does that means?

Where are arguments? it's easy: `I = S(K)(K)`; `S` can take 2
parameters `S(x)(y)` returns `(z) => x(z)(y(z))`, so: `I = (z) =>
K(z)(K(z))` We have to execute it from left to right, remember, `K(a)(b)`
returns `a`, so (with `a == z` and `b == K(z))`: `I = (z) => z`;

Do you want more?

Let's try to understand

    B = S (K S) K x y z

`B` stands for Barbara, from "Syllogism Barbara" (Wikipedia explains:

- All men are animals.
- All animals are mortal.
-All men are mortal.

So before all, write `B` as we understand it, and for readability
reasons, currently executed combinator and it's arguments will be emphased:

- B = __S (K(S)) K (x)__ (y) (z)

We have to execute it from left to right, and we have a `S` with three parameters:
`S(a)(b)(c)` returns `a(c)(b(c))`:

- B = __K (S) (x)__ (K(x)) (y) (z)

From left to right we have a `K` with two parameters, `S` and `x`, it will
return `S`:

- B = __S (K(x)) (y) (z)__

Calling `S` with three parameters `(K(x))`, `(y), and `(z)` returns
`(K(x))(z)((y)(z))`:

- B = **K (x) (z)** ((y)(z))

Calling `K` with two parameters `(x)`, and `(z)`, it returns `x`:

- B = x((y)(z))

Which can be simplified to:

- B = x(y(z))

It's time to try it !

    :::csharp
    delegate C C(C c);
    static void Main(string[] args)
    {
        C K = (a) => (b) => a;
        C S = (a) => (b) => (c) => a(c)(b(c));
        C I = S(K)(K);
        C B = S(K(S))(K);
    }

It works! Enjoy!! Next time, we will try a Swap combinator, a
Combinator reducing to himself and progressing step to the Y
Combinator! *dramatic chord*
