Church Numerals in haskell

Question

I am trying to print church numerals in haskell using the definions:

0 := λfx.x
1 := λfx.f x

Haskell code:

c0 = \f x -> x
c1 = \f x -> f x

When I enter it in the haskell console I get an error which says

    test> c1

    <interactive>:1:0:
    No instance for (Show ((t -> t1) -> t -> t1))
      arising from a use of `print' at <interactive>:1:0-1
    Possible fix:
      add an instance declaration for (Show ((t -> t1) -> t -> t1))
    In a stmt of an interactive GHCi command: print it

I am not able to exactly figure out what error says.

Thank you!

Answer 1

The problem here is that, by default, it's not possible to print values in Haskell. The default way to print things--used by the print function and by the GHCi REPL, among others--is the show function, defined by the type class Show.

The error you're getting, then, is informing you that you've evaluated an expression of a type that doesn't have an instance of Show defined. Modulo some verbiage, this is all the error message is saying:

No instance for (Show ((t -> t1) -> t -> t1))

The type ((t -> t1) -> t -> t1) is what was inferred for the expression you evaluated. This is a valid type for the Church numeral 1, though the "correct" type for a Church numeral should actually be (a -> a) -> a -> a.

  arising from a use of `print' at <interactive>:1:0-1

It's implicitly using the print function to display the values. Normally this would tell you where in your program the error was found, but in this case it says <interactive>:1:0-1 because the error was caused by an expression in the REPL.

Possible fix:
  add an instance declaration for (Show ((t -> t1) -> t -> t1))

This is just suggesting that you could fix the error by defining the instance it was expecting.

---

Now, you probably want to actually print your Church numerals, not just know why you can't. Unfortunately, this isn't as simple as adding the instance it asked for: If you write an instance for (a -> a) -> a -> a, Haskell interprets this as an instance for any specific a, whereas the correct interpretation of a Church numeral is a polymorphic function that works on any arbitrary a.

In other words, you want your show function to be something like this:

showChurch n = show $ n (+1) 0

If you really want to, you may implement the Show instance like this:

instance (Show a, Num a) => Show ((a -> a) -> a -> a) where
    show n = show $ n (+1) 0

and add {-#LANGUAGE FlexibleInstances#-} to the first line of the file. Or you may implement something similar to convert them to a regular number

> churchToInt c1
1
> showChurch c1
"1"

etc.