Discussion: should we make liftA2 an Applicative method?

David Feuer david.feuer at gmail.com
Sat Jan 14 21:49:02 UTC 2017

Right now, we define

liftA2 :: Applicative f
  => (a -> b -> c) -> f a -> f b -> f c
liftA2 f x y = f <$> x <*> y

For some functors, like IO, this definition is just dandy. But for others,
it's not so hot. For ZipList, for example, we get

liftA2 f (ZipList xs) (ZipList ys) =
  ZipList $ zipWith id (map f xs) ys

In this particular case, rewrite rules will likely save the day, but for
many similar types they won't. If we defined a custom liftA2, it would be
the obviously-efficient

liftA2 f (ZipList xs) (ZipList ys) =
  ZipList $ zipWith f xs ys

The fmap problem shows up a lot in Traversable instances. Consider a binary
leaf tree:

data Tree a = Bin (Tree a) (Tree a) | Leaf a

The obvious way to write the Traversable instance today is

instance Traversable Tree where
  traverse _f Tip = pure Tip
  traverse f (Bin p q) = Bin <$> traverse f p <*> traverse f q

In this definition, every single internal node has an fmap! We could end up
allocating a lot more intermediate structure than we need. It's possible to
work around this by reassociating. But it's complicated (see
Control.Lens.Traversal.confusing[1]), it's expensive, and it can break
things in the presence of infinite structures with lazy applicatives (see
Dan Doel's blog post on free monoids[2] for a discussion of a somewhat
related issue). With liftA2 as a method, we don't need to reassociate!

traverse f (Bin p q) = liftA2 Bin (traverse f p) (traverse f q)

The complication with Traversable instances boils down to an efficiency
asymmetry in <*> association. According to the "composition" law,

(.) <$> u <*> v <*> w = u <*> (v <*> w)

But the version on the left has an extra fmap, which may not be cheap. With
liftA2 in the class, we get a more balanced law:

If for all x and y, p (q x y) = f x . g y, then liftA2 p (liftA2 q u v) =
liftA2 f u . liftA2 g v


[2] http://comonad.com/reader/2015/free-monoids-in-haskell/
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