Is there a function to flatten a nested list of elements?

Question

Yes, it’s concat from the Standard Prelude, given by

concat :: [[a]] -> [a]
concat xss = foldr (++) [] xss

If you want to turn [[[a]]] into [a], you must use it twice:

Prelude> (concat . concat) [[[1,2],[3]],[[4]]]
[1,2,3,4]

Since nobody else has given this, it is possible to define a function which will flatten lists of an arbitrary depth by using MultiParamTypeClasses. I haven't actually found it useful, but hopefully it could be considered an interesting hack. I got the idea from Oleg's polyvariadic function implementation.

{-# LANGUAGE MultiParamTypeClasses, OverlappingInstances, FlexibleInstances #-}

module Flatten where

class Flatten i o where
  flatten :: [i] -> [o]

instance Flatten a a where
  flatten = id

instance Flatten i o => Flatten [i] o where 
  flatten = concatMap flatten

Now if you load it and run in ghci:

*Flatten> let g = [1..5]
*Flatten> flatten g :: [Integer]
[1,2,3,4,5]
*Flatten> let h = [[1,2,3],[4,5]]
*Flatten> flatten h :: [Integer]
[1,2,3,4,5]
*Flatten> let i = [[[1,2],[3]],[],[[4,5],[6]]]
*Flatten> :t i
i :: [[[Integer]]]
*Flatten> flatten i :: [Integer]
[1,2,3,4,5,6]

Note that it's usually necessary to provide the result type annotation, because otherwise ghc can't figure out where to stop recursively applying the flatten class method. If you use a function with a monomorphic type that's sufficient however.

*Flatten> :t sum
sum :: Num a => [a] -> a
*Flatten> sum $ flatten g

<interactive>:1:7:
    No instance for (Flatten Integer a0)
      arising from a use of `flatten'
    Possible fix: add an instance declaration for (Flatten Integer a0)
    In the second argument of `($)', namely `flatten g'
    In the expression: sum $ flatten g
    In an equation for `it': it = sum $ flatten g
*Flatten> let sumInt = sum :: [Integer] -> Integer
*Flatten> sumInt $ flatten g
15
*Flatten> sumInt $ flatten h
15

As others have pointed out, concat :: [[a]] -> [a] is the function you are looking for, and it can't flatten nested lists of arbitrary depth. You need to call it multiple times to flatten it down to the desired level.

The operation does generalize to other monads, though. It is then known as join, and has the type Monad m => m (m a) -> m a.

Prelude Control.Monad> join [[1, 2], [3, 4]]
[1,2,3,4]    
Prelude Control.Monad> join (Just (Just 3))
Just 3
Prelude Control.Monad.Reader> join (+) 21
42

import Data.List
let flatten = intercalate []

flatten $ flatten [[[1,2],[3]],[[4]]]
[1,2,3,4]