# Free Monads

# Free monads split monadic computations into data structures and interpreters

For instance, a computation involving commands to read and write from the prompt:

First we describe the "commands" of our computation as a Functor data type

{-# LANGUAGE DeriveFunctor #-}

data TeletypeF next
    = PrintLine String next
    | ReadLine (String -> next)
    deriving Functor

Then we use Free to create the "Free Monad over TeletypeF" and build some basic operations.

import Control.Monad.Free (Free, liftF, iterM)

type Teletype = Free TeletypeF

printLine :: String -> Teletype ()
printLine str = liftF (PrintLine str ())

readLine :: Teletype String
readLine = liftF (ReadLine id)

Since Free f is a Monad whenever f is a Functor, we can use the standard Monad combinators (including do notation) to build Teletype computations.

import Control.Monad -- we can use the standard combinators

echo :: Teletype ()
echo = readLine >>= printLine

mockingbird :: Teletype a
mockingbird = forever echo

Finally, we write an "interpreter" turning Teletype a values into something we know how to work with like IO a

interpretTeletype :: Teletype a -> IO a
interpretTeletype = foldFree run where
  run :: TeletypeF a -> IO a
  run (PrintLine str x) = putStrLn *> return x
  run (ReadLine f) = fmap f getLine

Which we can use to "run" the Teletype a computation in IO

> interpretTeletype mockingbird
this will go on forever
this will go on forever

# The Freer monad

There's an alternative formulation of the free monad called the Freer (or Prompt, or Operational) monad. The Freer monad doesn't require a Functor instance for its underlying instruction set, and it has a more recognisably list-like structure than the standard free monad.

The Freer monad represents programs as a sequence of atomic instructions belonging to the instruction set i :: * -> *. Each instruction uses its parameter to declare its return type. For example, the set of base instructions for the State monad are as follows:

data StateI s a where
    Get :: StateI s s  -- the Get instruction returns a value of type 's'
    Put :: s -> StateI s ()  -- the Put instruction contains an 's' as an argument and returns ()

Sequencing these instructions takes place with the :>>= constructor. :>>= takes a single instruction returning an a and prepends it to the rest of the program, piping its return value into the continuation. In other words, given an instruction returning an a, and a function to turn an a into a program returning a b, :>>= will produce a program returning a b.

data Freer i a where
    Return :: a -> Freer i a
    (:>>=) :: i a -> (a -> Freer i b) -> Freer i b

Note that a is existentially quantified in the :>>= constructor. The only way for an interpreter to learn what a is is by pattern matching on the GADT i.

**Aside**: The co-Yoneda lemma tells us that `Freer` is isomorphic to `Free`. Recall the definition of the `CoYoneda` functor:

data CoYoneda i b where
  CoYoneda :: i a -> (a -> b) -> CoYoneda i b

Freer i is equivalent to Free (CoYoneda i). If you take Free's constructors and set f ~ CoYoneda i, you get:

Pure :: a -> Free (CoYoneda i) a
Free :: CoYoneda i (Free (CoYoneda i) b) -> Free (CoYonda i) b ~
        i a -> (a -> Free (CoYoneda i) b) -> Free (CoYoneda i) b

from which we can recover `Freer i`'s constructors by just setting `Freer i ~ Free (CoYoneda i)`.

Because CoYoneda i is a Functor for any i, Freer is a Monad for any i, even if i isn't a Functor.

instance Monad (Freer i) where
    return = Return
    Return x >>= f = f x
    (i :>>= g) >>= f = i :>>= fmap (>>= f) g  -- using `(->) r`'s instance of Functor, so fmap = (.)

Interpreters can be built for Freer by mapping instructions to some handler monad.

foldFreer :: Monad m => (forall x. i x -> m x) -> Freer i a -> m a
foldFreer eta (Return x) = return x
foldFreer eta (i :>>= f) = eta i >>= (foldFreer eta . f)

For example, we can interpret the Freer (StateI s) monad using the regular State s monad as a handler:

runFreerState :: Freer (StateI s) a -> s -> (a, s)
runFreerState = State.runState . foldFreer toState
    where toState :: StateI s a -> State s a
          toState Get = State.get
          toState (Put x) = State.put x

# Free Monads are like fixed points

Compare the definition of Free to that of Fix:

data Free f a = Return a
              | Free (f (Free f a))

newtype Fix f = Fix { unFix :: f (Fix f) }

In particular, compare the type of the Free constructor with the type of the Fix constructor. Free layers up a functor just like Fix, except that Free has an additional Return a case.

# How do foldFree and iterM work?

There are some functions to help tear down Free computations by interpreting them into another monad m: iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> (Free f a -> m a) and foldFree :: Monad m => (forall x. f x -> m x) -> (Free f a -> m a). What are they doing?

First let's see what it would take to tear down an interpret a Teletype a function into IO manually. We can see Free f a as being defined

data Free f a 
    = Pure a 
    | Free (f (Free f a))

The Pure case is easy:

interpretTeletype :: Teletype a -> IO a
interpretTeletype (Pure x) = return x
interpretTeletype (Free teletypeF) = _

Now, how to interpret a Teletype computation that was built with the Free constructor? We'd like to arrive at a value of type IO a by examining teletypeF :: TeletypeF (Teletype a). To start with, we'll write a function runIO :: TeletypeF a -> IO a which maps a single layer of the free monad to an IO action:

runIO :: TeletypeF a -> IO a
runIO (PrintLine msg x) = putStrLn msg *> return x
runIO (ReadLine k) = fmap k getLine

Now we can use runIO to fill in the rest of interpretTeletype. Recall that teletypeF :: TeletypeF (Teletype a) is a layer of the TeletypeF functor which contains the rest of the Free computation. We'll use runIO to interpret the outermost layer (so we have runIO teletypeF :: IO (Teletype a)) and then use the IO monad's >>= combinator to interpret the returned Teletype a.

interpretTeletype :: Teletype a -> IO a
interpretTeletype (Pure x) = return x
interpretTeletype (Free teletypeF) = runIO teletypeF >>= interpretTeletype

The definition of foldFree is just that of interpretTeletype, except that the runIO function has been factored out. As a result, foldFree works independently of any particular base functor and of any target monad.

foldFree :: Monad m => (forall x. f x -> m x) -> Free f a -> m a
foldFree eta (Pure x) = return x
foldFree eta (Free fa) = eta fa >>= foldFree eta

foldFree has a rank-2 type: eta is a natural transformation. We could have given foldFree a type of Monad m => (f (Free f a) -> m (Free f a)) -> Free f a -> m a, but that gives eta the liberty of inspecting the Free computation inside the f layer. Giving foldFree this more restrictive type ensures that eta can only process a single layer at a time.

iterM does give the folding function the ability to examine the subcomputation. The (monadic) result of the previous iteration is available to the next, inside f's parameter. iterM is analogous to a paramorphism whereas foldFree is like a catamorphism.

iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a
iterM phi (Pure x) = return x
iterM phi (Free fa) = phi (fmap (iterM phi) fa)