Apparently one can define the Y combinator with lambda expressions in Haskell. The problem with `fix f = (\x -> f (x x))(\x -> f (x x))`

is that one needs a solution to the type equation `b = b -> a`

. Fortunately this can be done with Haskell’s data types.

> newtype Mu a = Roll { unroll :: Mu a -> a }

Haskell allows non-monotonic data types, so this is a legal definition. The resulting type `Mu a`

is isomorphic to `Mu a -> a`

. Using this isomorphism one can define the standard Y combinator.

#ifndef __GLASGOW_HASKELL__ > fix f = (\x -> f ((unroll x) x)) (Roll (\x -> f ((unroll x) x))) #endif

Don’t try this with GHC. Due to a mis-feature, GHC’s inliner will continuously expand this expression. If you are desperate you can try Michael Shulman’s solution.

#ifdef __GLASGOW_HASKELL__ > fix f = fixH (Roll fixH) > where > {-# NOINLINE fixH #-} > fixH x = f ((unroll x) x) #endif

Of course, this is just an academic exercise. To actually define a fixpoint combinator in Haskell, one would use recursive definitions.

> fix2 f = f (fix2 f)