# Freer, yet Too Costly Higher-order Effects

I’m throwing in the towel. For now at least.

As of today I have free, higher-order effects working. Unfortunately, they are not fast. I don’t think this is a fundamental limitation, merely that whatever code I’ve written isn’t amenable to GHC’s optimization process.

I’ve been hammering on this for about 50 hours now. It’s been driving me slowly crazy, and promised myself I’d stop if I hadn’t solved it by now. That being said, before putting this project to rest, I wanted to do a quick write-up detailing what I’ve learned, how everything fits together, and where I’m hoping someone will pick up the ball. Here’s the repository.

## Higher Order Effects

In the freer-simple model, effects are first-order—meaning they are unable to embed Eff computations inside of them. This is occasionally annoying, primarily when trying to write bracket-like effects.

You can sort of work around the problem by encoding your scoped computation as an interpretation of an effect, but this often comes at the cost of fixing the interpretations of the other effects you’re dealing with.

This fundamental limitation comes from the fact that freer-simple effects have kind * -> *. There’s nowhere in here to stick the Eff stack you’re working in. You can kind of hack it in, but it never plays nicely.

The solution is given in the paper Effect Handlers in Scope, and implemented in the fused-effects package. The idea is to parameterize each of your effects with a monad—ie. they have kind (* -> *) -> * -> *. This parameter gets instantiated at the entire Eff stack, as seen by the type of send :: Member e r => e (Eff r) a -> Eff r a

While it’s an obvious insight, actually getting everything to play nicely is tricky. The primary issue is how do you push interpretations through these additional monadic contexts? For example, let’s consider a bracket-esque effect:

data Bracket m x where
Bracket :: m a          -- ^ allocate
-> (a -> m ())  -- ^ deallocate
-> (a -> m x)   -- ^ use
-> Bracket m x

Assume we want to push a State effect through these ms. What are the correct semantics for how the state is accumulated? Should any state changed in the deallocate block count outside of the bracket? What should happen in the use case if an exception is thrown and the rest of the block is short-circuited?

Not only are we concerned with the semantics, but also the actual mechanism of propagating this state throughout the computation.

Effect Handlers in Scope introduces weave in a typeclass, which is responsible for this state-propagation behavior. Statefulness for an effect is encoded as some arbitrarily chosen functor f, and weave describes how the effect should move that state through the effect. Behold:

class Effect e where
weave
:: Functor f
=> f ()
-> (forall x. f (m x) -> n (f x))
-> e m a
-> e n (n (f a))

The f () parameter is the current “state of the world”, and the rank-2 thing is this distribution law. You can intuit the m parameter being an effect stack with all of the effects present, and the n parameter as being the same effect stack—but with the top of it taken off. To clarify, we could monomorphize it thusly:

weave
:: Functor f
=> f ()
-> (forall x. f (Eff (g ': r) x) -> Eff r (f x))
-> e (Eff (g ': r)) a
-> e (Eff r) (Eff r (f a))

This janky return type: e (Eff r) (Eff r (f a)) comes from the fact that Effect Handlers in Scope describes a traditional “free” (as opposed to freer) approach to a free monad. The last parameter of an effect is actually a continuation for the next piece of the computation. By mangling it, we’re ensuring the caller (which will be library code) respects the evaluation semantics.

weave implicitly defines the evaluation semantics of an effect—it pins how state propagates through them, which in turn defines which pieces of the effect are observable to the outside.

## Freeing the Higher Orders

Warning: This next section describes a janky-ass solution to the problem. It’s clearly a hack and clearly not the right answer. But maybe by writing out what I did, someone with a keen eye can point out where I went wrong.

So this is all well and good. It works, but requires a lot of boilerplate. As presented in the paper, a new effect requires:

• A Functor instance
• An MFunctor instance (providing hoist :: forall x. (f x -> g x) -> e f a -> e g a)
• An Effect instance as above, and an additional method not described here

If you’re following in the fused-effects tradition, for each interpretation you additionally need a new Carrier type, with its own Functor, Applicative and Monad instances, and then another typeclass tying the effect to its carrier.

fused-effects improves the $O(n^2)$ MTL instance problem to $O(n)$—albeit with a big constant factor :(

This is a huge amount of work! I’ve said it before and I’ll say it again: ain’t nobody got time for that. If it feels like too much work, people aren’t going to do it. A solution that depends on humans not being lazy isn’t one that’s going to take off.

So wouldn’t it be nice if we could just all of this effect stuff for free?

Here’s where I admittedly went a little off the rails. The first step towards getting a freer Functor-less Monad instance for Eff is to define it in terms of its final encoding. I made the obvious changes to last time without thinking too much about it:

newtype Freer f a = Freer
{ runFreer
:: forall m
=> (forall x. f (Freer f) x -> m x)
-> m a
}

I have no idea if this is right, but at least it gives a Monad instance for free. One limitation you’ll notice is that the continuation in runFreer is a natural transformation, and thus it’s unable to change its return type.

That means interpretations like runError :: Eff (Error e ': r) a -> Eff r (Either e a) are surprisingly difficult to implement. More on this later—I just wanted to point out this flaw.

From here I followed Oleg and implemented Eff as a type synonym, making sure to tie the knot and instantiate the m parameter correctly:

type Eff r = Freer (Union r (Eff r))

But how can we get a free implementation for weave?

It’s this mental thing I came up with, which is sort of like Coyoneda but for weave:

data Yo e m a where
Yo :: (Monad n, Functor f)
=> e m a
-> f ()
-> (forall x. f (m x) ~> n (f x))
-> (f a -> b)
-> Yo e n b

In retrospect, I would not spend any more time on this approach—I’d just make people give an instance of weave for higher-order effects, and machinery to derive it automatically for first-order ones.

But then how can we get an MFunctor instance for free? You can’t just derive it generically—lots of higher-order effects want existentials, and thus can’t have Generic instances.

This Yo thing mirrors the definition of weave pretty closely. The idea is that it can accumulate arbitrarily many weaves into a single Yo, and then dispatch them all simultaneously.

Some interesting points to note are that the state functor f is existentialized, and that there is this final f a -> b parameter to make it play nicely with the Union (more on this in a second.) We can implement weave now by replacing the existing state functor with a Compose of the new one and the old one.

weave
=> f ()
-> (forall x. f (m x) -> n (f x))
-> Union r m a
-> Union r n (f a)
weave s' distrib (Union w (Yo e s nt f)) =
Union w $Yo e (Compose$ s <$s') (fmap Compose . distrib . fmap nt . getCompose) (fmap f . getCompose) We can also use Yo to get a free MFunctor instance: instance MFunctor (Yo e) where hoist f (Yo e s nt z) = Yo e s (f . nt) z OK, all of this works I guess. But what’s with this weird f a -> b thing in Yo that I mentioned earlier? Well recall the type of runFreer, when instantiated at Union: runFreer :: forall m . Monad m => (forall x. Union r (Eff r) x -> m x) -> m a The only way we can produce an m a is via this rank-2 thing, which is a natural transformation from Union r (Eff r) to m. In other words, it’s not allowed to change the type. We can’t just stuff the f into the result and return an m (f a) instead—this thing doesn’t form a Monad! Fuck! All of this comes to a head when we ask ourselves how to actually get the state out of such a contraption. For example, when we call runState we want the resulting state at the end of the day! The trick is the same one I used in the last post—we’re able to instantiate the m inside runFreer at whatever we like, so we just choose StateT s (Eff r) and then run that thing afterwards. Again, this is very clearly a hack. Because weave is given freely, interpretations must eventually actually decide what that thing should look like. Some combinators can help; for example, this is the interface I came up with for implementing runBracket: runBracket :: Member (Lift IO) r => (Eff r ~> IO) -> Eff (Bracket ': r) a -> Eff r a runBracket finish = deep$ \start continue -> \case
Bracket alloc dealloc use -> sendM $X.bracket (finish$ start alloc)
(finish . continue dealloc)
(finish . continue use)

The deep combinator gives you start and continue (which are the cleverly disguised results of weave), and asks you to give a natural transformation from your effect into Eff r.

The actual implementation of deep isn’t going to win any awards in understandability, or in inline-ability:

deep
:: (∀ m f y
. Functor f
=> (forall x. m x -> (Eff r (f x)))
-> (∀ i o. (i -> m o) -> f i -> Eff r (f o))
-> e m y
-> Eff r (f y)
)
-> Eff (e ': r) a
-> Eff r a
deep transform (Freer m) = m $\u -> case decomp u of Left x -> liftEff$ hoist (deep transform) x
Right (Yo eff state nt f) -> fmap f $transform (deep transform . nt . (<$ state))
(\ff -> deep transform . nt . fmap ff)
eff

Notice that whoever implements transform needs to give an equivalent of an implementation of weave anyway. Except that instead of only writing it once per effect, they need to write it per interpretation.

We can also give an implementation for runState in terms of a StateT s:

runState :: forall s r a. s -> Eff (State s ': r) a -> Eff r (s, a)
runState s = flip runStateT s . go
where
go :: forall x. Eff (State s ': r) x -> StateT s (Eff r) x
go (Freer m) = m $\u -> case decomp u of -- W T F Left x -> StateT$ \s' ->
liftEff . weave (s', ())
(uncurry (flip runStateT))
$hoist go x Right (Yo Get state nt f) -> fmap f$ do
s' <- get
go $nt$ pure s' <$state Right (Yo (Put s') state nt f) -> fmap f$ do
put s'
go $nt$ pure () <$state This is also completely insane. Notice the aptly-named section W T F, where for no reason I can discern other than satisfying the typechecker, we convert from a StateT s to a (s, ()) and back again. But why?? Because this is what weave wants—and we need to satisfy weave because it’s the only way to change the type of a Union—and we need to do that in order to reinterpret everything as a StateT s. There is a similarly WTF implementation for runError. But worse is the combinator I wrote that generalizes this pattern for running an effect in terms of an underlying monad transformer: shundle :: ∀ a f t e r . ( MonadTrans t , ∀ m. Monad m => Monad (t m) , Functor f ) => (∀ x. Eff r (f x) -> t (Eff r) x) -> (∀ x. t (Eff r) x -> Eff r (f x)) -> (∀ x. f (t (Eff r) x) -> Eff r (f x)) -> f () -> (∀ m tk y . Functor tk => (∀ x. f () -> tk (m x) -> Eff r (f (tk x))) -> tk () -> e m y -> t (Eff r) (tk y) ) -> Eff (e ': r) a -> Eff r (f a) shundle intro finish dist tk zonk = finish . go where go :: ∀ x. Eff (e ': r) x -> t (Eff r) x go (Freer m) = m$ \u ->
case decomp u of
Left x -> intro . liftEff . weave tk dist $hoist go x Right (Yo e sf nt f) -> fmap f$
zonk (\r -> shundle intro finish dist r zonk . nt) sf e

Don’t ask why it’s called shundle or why it has a variable called zonk. It was funny to me at the time and I needed whatever smiles I could get to continue making progress. Believe it or not, this abomination does indeed generalize runState:

statefully
:: (∀ m. e m ~> StateT s (Eff r))
-> s
-> Eff (e ': r) a
-> Eff r (s, a)
statefully f s =
shundle
(StateT . const)
(flip runStateT s)
(uncurry $flip runStateT) (s, ())$ \_ tk -> fmap (<$tk) . f runState :: s -> Eff (State s ': r) a -> Eff r (s, a) runState = statefully$ \case
Get    -> get
Put s' -> put s'

runState is actually quite reasonable! At this point I was willing to concede “worse is better”—that most of the time people only really care about StateT-esque state in their effects. And really the only thing they’ve been missing is bracket. And we now have bracket, and an easy way of doing StateT-esque state.

So I was going to hand it in to one library or another—if not for full marks, then at least for the participation trophy.

But then I ran my benchmark, and saw that it performs 10x slower than freer-simple. Even for effect stacks that don’t need to weave.

SHAME.

SORROW.

And I guess this is where I leave it. The machinery is clearly wrong, but amazingly it actually does what it says on the tin. Unfortunately it does it so slowly that I think the people who complain about the performance of free monads might actually have a point this time.

I’ve got my heart crossed that someone will swoop in here and say “here’s what you’ve done wrong” and it will be a minor change and then everything will optimize away and we will make merry and the freer monad revolution will be complete.

I’m very tired.

## What a Real Solution Would Look Like

I think I have enough familiarity with the problem at this point to know what a solution would look like—even if I can’t find it myself:

• runFreer could produce a f a if you asked for it
• Weave would be a typeclass that effects authors would need to implement. But they could derive it for free if the m parameter was unused. This would be the only instance necessary to implement by hand.
• The library would provide a handleRelayS-esque interface for defining interpreters.
• By the time the user’s code ran for the interpretation, every monadic value in their effect would be bindable without any further effort—ala cata.

I’d also be happy if an existing solution (probably fused-effects) were given the tools to satisfy the above list. In particular, the handleRelayS thing.