Y Combinator in Haskell


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.


> fix f = (\x -> f ((unroll x) x)) (Roll (\x -> f ((unroll x) x)))


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.


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


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)


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