Hey, It is well-known that negative datatypes can be used to encode recursion, without actually explicitly using recursion. As a little exercise, I set out to define the fixpoint combinator using negative datatypes. I think the result is kinda cool :) Comments are welcome :) Edsko {- Definition of the fixpoint combinator without using recursion Thanks to Dimitri Vytiniotis for an explanation of the basic principle. -} module Y where {-# NOINLINE app #-} data Fn a = Fn (Fn a -> Fn a) | Value a -- Application app :: Fn a -> Fn a -> Fn a app (Fn f) x = f x -- \x -> f (x x) delta :: Fn a -> Fn a delta f = Fn (\x -> f `app` (x `app` x)) -- Y combinator: \f -> (\x -> f (x x)) (\x -> f (x x)) y :: Fn a -> Fn a y f = delta f `app` delta f -- Lifting a function to Fn lift :: (a -> a) -> Fn a lift f = Fn (\(Value x) -> Value (f x)) -- Inverse of lift unlift :: Fn a -> (a -> a) unlift f = \x -> case f `app` Value x of Value y -> y -- Fixpoint combinator fix :: ((a -> a) -> (a -> a)) -> (a -> a) fix f = unlift (y (Fn (\rec -> lift (f (unlift rec))))) -- Example: factorial facR f n = if n == 1 then 1 else n * f (n - 1) fac = fix facR