Foldr/Foldl for free when Tree is implementing Foldable foldMap?

Question

I am a beginner at Haskell and learning from "Learn You a Haskell".

There's something I don't understand about the Tree implementation of Foldable.

instance F.Foldable Tree where  
    foldMap f Empty = mempty  
    foldMap f (Node x l r) = F.foldMap f l `mappend`  
                             f x           `mappend`  
                             F.foldMap f r  

Quote from LYAH: "So if we just implement foldMap for some type, we get foldr and foldl on that type for free!".

Can someone explain this? I don't understand how and why do I get foldr and foldl for free now...

Answer 1

foldr can always be defined as:

foldr f z t = appEndo (foldMap (Endo . f) t) z

where appEndo and Endo are just newtype unwrappers/wrappers. In fact, this code got pulled straight from the Foldable typeclass. So, by defining foldMap, you automatically get foldr.

Answer 2

We begin with the type of foldMap:

foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

foldMap works by mapping the a -> m function over the data structure and then running through it smashing the elements into a single accumulated value with mappend.

Next, we note that, given some type b, the b -> b functions form a monoid, with (.) as its binary operation (i.e. mappend) and id as the identity element (i.e. mempty. In case you haven't met it yet, id is defined as id x = x). If we were to specialise foldMap for that monoid, we would get the following type:

foldEndo :: Foldable t => (a -> (b -> b)) -> t a -> (b -> b)

(I called the function foldEndo because an endofunction is a function from one type to the same type.)

Now, if we look at the signature of the list foldr

foldr :: (a -> b -> b) -> b -> [a] -> b

we can see that foldEndo matches it, except for the generalisation to any Foldable and for some reordering of the arguments.

Before we get to an implementation, there is a technical complication in that b -> b can't be directly made an instance of Monoid. To solve that, we use the Endo newtype wrapper from Data.Monoid instead:

newtype Endo a = Endo { appEndo :: a -> a }

instance Monoid (Endo a) where
        mempty = Endo id
        Endo f `mappend` Endo g = Endo (f . g)

Written in terms of Endo, foldEndo is just specialised foldMap:

foldEndo :: Foldable t => (a -> Endo b) -> t a -> Endo b

So we will jump directly to foldr, and define it in terms of foldMap.

foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
foldr f z t = appEndo (foldMap (Endo . f) t) z

Which is the default definition you can find in Data.Foldable. The trickiest bit is probably Endo . f; if you have trouble with that, think of f not as a binary operator, but as a function of one argument with type a -> (b -> b); we then wrap the resulting endofunction with Endo.

As for foldl, the derivation is essentially the same, except that we use a different monoid of endofunctions, with flip (.) as the binary operation (i.e. we compose the functions in the opposite direction).