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Fun with the function arrow monad

One of the things that surprised me while studying Haskell is that (->) a is a monad. It seems reasonable that functions form a monad, but I don’t believe I would have thought of it myself. How can this monad be used? An example, namely, join (***), was considered in a previous post. But when I think monads in Haskell I, think of do notation. In this post we will define the (->) a monad ourselves and explored a bit of the weirdness of its do notation.

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}

module Arrow where

import Prelude hiding (Functor(..), Applicative(..), Monad(..))

We hide Functor, Applicative, and Monad because we will implement our own hierarchy, similar to the one before the functor-applicative-monad proposal. FlexibleInstances and UndecidableInstances will be necessary for later shenanigans.

First, we define Functor and an instance of it for our monad. fmap is just (.), pretty lame.

class Functor f where
  fmap :: (a -> b) -> f a -> f b

instance Functor ((->) a) where
  fmap = (.)

This can be easily excused, however. Lists are another type constructor that have this problem (fmap is just map, boring!), yet their monad instance simulates nondeterministic computations (awesome!).

Let’s continue with the Applicative instance:

class Functor f => Applicative f where
  pure :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b

instance {-# OVERLAPPING #-} Applicative ((->) a) where
  pure = const
  f <*> g = \x -> f x (g x)

Things have certainly got a lot more interesting. (<*>) and pure are the S and K combinators from the SKI combinator calculus!

(The reason why the applicative instance of (->) a is OVERLAPPING will become clear later.)

With them, we can define the I combinator:

k :: a -> b -> a
k = pure

i :: a -> a
i = k <*> k

With applicative functors out of our way, we define a monad instance for (->) a which will give us access to that sweet do notation. Hopefully, things are about to get exciting.

class Functor m => Monad m where
  return :: a -> m a
  (>>=) :: m a -> (a -> m b) -> m b

instance Monad ((->) a) where
  return = const
  f >>= g = \x -> g (f x) x

Oh, no! Things have got kind of boring again! (>>=) is just the S combinator with shuffled parameter order! And indeed, (>>=) could have been implemented in terms of S and K:

instance Monad ((->) a) where
  return = const
  (>>=) = s (k (s (k (s (k (s s (s k))))) (s (s (k s) k)))) k

Yet another possible implementation (thanks to Brad Neimann):

instance Monad ((->) a) where
  return = const
  f >>= g = flip g <*> f

With the monad instance, we finally acquired do notation, which allows us to write some weird things:

s :: (a -> b -> c) -> (a -> b) -> a -> c
s x y = do
  x <- x
  y <- y
  return (x y)

which desugars to:

s :: (a -> b -> c) -> (a -> b) -> a -> c
s x y = x >>= \x -> y >>= \y -> return (x y)

which is just a default implementation of (<*>) for any given monad:

instance {-# OVERLAPPABLE #-} (Functor m, Monad m) => Applicative m where
  pure = return
  f <*> x = f >>= \f -> x >>= \x -> return (f x)

This applicative instance is the reason for the extensions and the OVERLAPPING in the applicative instance of (->) a.

I didn’t manage to think of any useful examples of do notation for the (->) a monad. If you know of any, please let me know on Twitter. I would love to see more of this unexpected monad.

Special thanks to the wonderful folks at the Functional Programming Zulip.