Question

I would like to create three Haskell functions: a, b, and c.

Each function is to have one argument. The argument is one of the three functions.

I would like function a to have this behavior:

• if the argument is function a then return function a.
• if the argument is function b then return function b.
• if the argument is function c then return function a.

Here’s a recap of the behavior I desire for function a:

a a = a
a b = c
a c = a 

And here’s the behavior I desire for the other two functions:

b a = a
b b = a
b c = c

c a = c
c b = b
c c = c

Once created, I would like to be able to compose the functions in various ways, for example:

  a (c b)
= a (b)
= c

How do I create these functions?

Answer 1

Since you have given no criteria for how you are going to observe the results, then a = b = c = id satisfies your criteria. But of course that is not what you want. But the idea is important: it doesn’t just matter what behavior you want your functions to have, but how you are going to observe that behavior.

There is a most general model if you allow some freedom in the notation, and you get this by using an algebraic data type:

data F = A | B | C
    deriving (Eq, Show) -- ability to compare for equality and print

infixl 1 % 
(%) :: F -> F -> F
A % A = A
A % B = C
A % C = A
B % A = A
...

and so on. Instead of saying a b, you have to say A % B, but that is the only difference. You can compose them:

  A % (C % B)
= A % B
= B

and you can turn them into functions by partially applying (%):

a :: F -> F
a = (A %)

But you cannot compare this a, as ehird says. This model is equivalent to the one you specified, it just looks a little different.