Notes on Free monads

The following article is just a few notes on the nature of the Free monad.

> {-# LANGUAGE DeriveFunctor #-}
> {-# LANGUAGE GeneralizedNewtypeDeriving #-}
> {-# LANGUAGE UndecidableInstances #-}
>
> module FreeMaybe where
>
> import Control.Monad (join)
> import Control.Monad.Writer.Class

There can be just two values of type Maybe a: Nothing and Just a. Now let’s look at the free monad of Maybe a, Free Maybe a:

> data Free f a = Pure a | Free (f (Free f a))
>
> instance Functor f => Functor (Free f) where
>     fmap f (Pure a)   = Pure (f a)
>     fmap f (Free ffa) = Free $ fmap (fmap f) ffa
>
> instance Functor f => Monad (Free f) where
>     return = Pure
>     Pure a >>= f = f a
>     Free ffa >>= f = Free $ fmap (>>= f) ffa
>
> instance (Show a, Show (f (Free f a))) => Show (Free f a) where
>     showsPrec d (Pure a) = showParen (d > 10) $
>         showString "Pure " . showsPrec 11 a
>     showsPrec d (Free m) = showParen (d > 10) $
>         showString "Free " . showsPrec 11 m

There are four “shapes” that values of Free Maybe a can take:

Pure a
Free Nothing
Free (Just (Free (Just (... (Free Nothing)))))
Free (Just (Free (Just (... (Free (Pure a))))))

In terms of whether a Free Maybe a represents an a or not, Free Maybe a is equivalent to Maybe a. However, Maybe a is right adjoint to Free Maybe a, meaning that it forgets the structure of Free Maybe a – namely, which of the four shapes above the value was, and how many occurences of Free (Just there were.

Why would you ever use Free Maybe a? Precisely if you cared about the number of Justs. Now, say we had a functor that carried other information:

> data Info a = Info { infoExtra :: String, infoData :: a }
>     deriving (Show, Functor)

Then Free Info a is isomorphic to if infoExtra had been [String]:

> main :: IO ()
> main = do
>     print $ Free (Info "Hello" (Free (Info "World" (Pure "!"))))

Which results in:

>>> main
Free (Info {infoExtra = "Hello",
            infoData = Free (Info {infoExtra = "World", infoData = Pure "!"})})
it :: ()

But now it’s also a Monad, even though we never defined a Monad instance for Info:

> main :: IO ()
> main = do
>     print $ do
>         x <- Free (Info "Hello" (Pure "!"))
>         y <- Free (Info "World" (Pure "!"))
>         return $ x ++ y

This outputs:

>>> foo
Free (Info {infoExtra = "Hello",
            infoData = Free (Info {infoExtra = "World", infoData = Pure "!!"})})
it :: ()

This works because the Free monad simply accumulates the states of the various functor values, without “combining” them as a real monadic join would have done. Free Info a has left it up to us to do that joining later.