Deriving free monads from value-level functors

Or, I have finally cured insomnia. Call CNN.


This might be the least interesting article title on the Internet. However, it is something I have been looking into as I work on my language, psilo.

So I have a few overlapping concerns:

  1. psilo, like Haskell, doesn't semantically support imperative programming (ie, recipe-style "statement; statement; ..." programming). However, I'm not a sadist: I want the programmer to be able to write more intuitive programs if she can earn it.

  2. I'm a fan of the free monad pattern in Haskell but I don't want to support typeclasses. Thus I am interested in implementing free monad infrastructure using nothing but algebraic data types.

  3. Since delimited continuations are tantamount to monads, I want the user to be able to write what amounts to an abstract syntax tree type and a few functions in continuation passing style and in return receive a monad, do notation, and composable imperative programming constructs.

This post is a simple exploration of implementing the Functor typeclass as an algebraic data type, and then deriving a free monad from it.

1 Functors and Free Monads

{-# LANGUAGE RankNTypes #-}

data F f = F {
    _map :: forall a b. (a -> b) -> f a -> f b

An instance of F should contain a function called _map constrained by whichever base type is being functor-ized.

data Mu f a
    = Term { retract :: a }
    | Cont (f (Mu f a))

Mu is the free monad. I like calling it Mu for two reasons: because calling it "free" is confusing and not informative; and because Mu behaves like the type-level analog to the Y-combinator, called Mu.

unit :: a -> Mu f a
unit = Term

yield :: a -> Mu f a
yield = Term

These two are different names for the same thing, mostly for aesthetic reasons. You'll see.

bind :: F f -> Mu f a -> (a -> Mu f b) -> Mu f b
bind i arg fn = case arg of
   Term t      -> fn t
   Cont k      -> Cont (map (bind' fn) k)

   where map   = _map i
         bind' = flip (bind i)

This is the cool part. bind is defined for any type which is wrapped by Mu, and for a type to be wrapped by Mu it must instantiate our F type. Thus we can ask for an "instance" argument, called i here.

sequence' i x   = case x of
    []      -> unit []
    m:ms    -> m              >>= \x  ->
               sequence' i ms >>= \xs ->
               unit (x:xs)
        (>>=) = bind i

For kicks I have re-written the sequence function available in Haskell. psilo will essentially use this or an equivalent function to implement do notation under the covers.

2 An example: re-creating the Maybe monad.

Apple's Swift programming language seems to have ignited more widespread interest in optional types. So, just to show there is nothing up my sleeves, I am re-creating the Maybe monad with the name Optional.

data Optional s
    = Nil
    | Some s
    deriving (Show)

Now for the fun part. I will create an F instance for Optional along with some convenience functions to use in monadic computations. While I don't do it here, this could be derived automatically.

fOptional :: F Optional
fOptional = F {
    _map = \f x -> case x of
        Nil     -> Nil
        Some s  -> Some $ f s

nil = Cont Nil
some = Term

In ~12 lines of real code, I have created a Maybe clone and proven it is a functor. As a result all the remaining code necessary to compose a monad has been derived automatically.

Since this was a free monad, the only remaining code is that to "run" the monadic computation built up using unit and bind.

runOptional :: Mu Optional a -> Optional a
runOptional (Term t) = Some t
runOptional (Cont k) = case k of
    Nil     -> Nil
    Some v  -> runOptional v

3 Tests!

Without further ado, here is some example code written in our Optional monad.

testOptional1 = some 5 >>= \a ->
                some 6 >>= \b ->
                yield $ a * b
    where (>>=) = bind fOptional

testOptional2 = some 5 >>= \a ->
                nil    >>= \b ->
                some 6 >>= \c ->
                yield $ a * c
    where (>>=) = bind fOptional

Try it out for yourself to see the results.