The Meaning of Monad in MonadTrans

At work, someone noticed that they got a compiler warning for a derived instance of MonadTrans.

newtype FooT m a = FooT { unFooT :: StateT Int m a }
    deriving newtype
        (Functor, Applicative, Monad, MonadTrans)

GHC complained about a redundant Monad constraint. After passing -ddump-deriv, I saw that GHC was pasting in basically this instance:

instance MonadTrans FooT where
    lift :: Monad m => m a -> t m a
    lift = coerce (lift :: m a -> StateT Int m a)

The problem was that the Monad m constraint there is redundant - we’re not actually using it. However, it’s a mirror for the definition of the class method.

In transformers < 0.6, the definition of MonadTrans class looked like this:

class MonadTrans t where
    lift :: Monad m => m a -> t m a

In transformers-0.6, a quantified superclass constraint was added to MonadTrans:

class (forall m. Monad m => Monad (t m)) => MonadTrans t where
    lift :: Monad m => m a -> t m a

I’m having a bit of semantic satiation with the word Monad, which isn’t an unfamiliar phenomenon for a Haskell developer. However, while explaining this behavior, I found it to be a very subtle distinction in what these constraints fundamentally mean.

What is a Constraint?

A Constraint is a thing in Haskell that GHC needs to solve in order to make your code work. Solving a constraint is similar to proving a proposition in constructive logic - GHC needs to find evidence that the claim holds, in the form of a type class instance.

When we write:

foo :: Num a => a -> a
foo x = x * 2

We’re saying:

I have a polymorphic function foo which can operate on types, if those types are instances of the class Num.

If is the big thing here - it’s a way of making a conditional expression. For a totally polymorphic function, like id :: a -> a, there are no conditions. You can call it with any type you want. But a conditional polymorphic function expresses some requirements, or constraints, upon the input.

If you ask for constraints you don’t need, then you can get a warning by enabling -Wredundant-constraints.

woops :: (Bounded a, Num a) => a -> a
woops x = x + 5

GHC will happily warn us that we don’t actually use the Bounded a constraint, and it’s redundant. We should delete it. Indeed, there are many Num that aren’t Bounded, and by requiring Bounded, we are reducing the potential types we could call this function on for no reason.

Constraints Liberate

(from Constraints Liberate, Liberties Constrain)

A constraint is a perspective on what is happening - it is the perspective of the caller of a function. It’s almost like I see a function type:

someCoolFunction :: _ => a -> b

And think - “Ah hah! I can call this at any type a and b that I want!” Only to find that there’s a bunch of constraints on a and b, and now I am constrained in the types I can call this function at.

However, a constraint feels very different from the implementer of a function. Let’s look at the classic identity function:

id :: a -> a
id a = a

As an implementer, I have quite a few constraints! Indeed, I can’t really do anything here. I can write equivalent functions, or much slower versions of this function, or I can escape hatch with unsafe behavior - but my options are really pretty limited.

id' :: a -> a
id' a = repeat a !! 1000

id'' :: a -> a
id'' a = let y = a in y

id''' :: a -> a
id''' a = iterate id' a !! 1000

However, a Constraint means that I now have some extra power.

foo :: Num a => a -> a

With this signature, I now have access to the Num type class methods, as well as any other function that is polymorphic over Num. The constraint is a liberty - I have gained the power to do stuff with the input.

The Two Monads

Back to MonadTrans -

    (forall m. Monad m => Monad (t m))
    MonadTrans t
    lift :: Monad m => m a -> t m a

Method Constraint

Let’s talk about that lift constraint.

    lift :: Monad m => m a -> t m a

This constraint means that the input to lift must prove that it is a Monad. This means that, as implementers of lift, we can use the methods on Monad in order to make lift work out. We often don’t need it - consider these instances.

newtype IdentityT m a = IdentityT (m a)

instance MonadTrans IdentityT where
    lift action = IdentityT action

IdentityT can use the constructor directly, and does not require any methods at all to work with action.

newtype ReaderT r m a = ReaderT (r -> m a)

instance MonadTrans (ReaderT r) where
    lift action = ReaderT $ \_ -> action

ReaderT uses the constructor and throws away the r.

newtype ExceptT e m a = ExceptT (m (Either e a))

instance MonadTrans (ExceptT e) where
    lift action = ExceptT $ fmap Right action

Ah, now we’re using the Functor method fmap in order to make the inner action fit. We’re given an action :: m a, and we need an m (Either e a). And we’ve got Right :: a -> Either e a and fmap to make it work. We are allowed to call fmap here because Monad implies Applicative implies Functor, and we’ve been given the Monad m constraint as part of the method.

Superclass Constraint

Let’s talk about the quantified superclass constraint (wow, what a fancy phrase).

    (forall m. Monad m => Monad (t m))

This superclass constraint means that the type t m a is a Monad if m is a Monad, and this is true for all m, not just a particular one. Prior to this, if you wanted to write a do block thta was arbitrary in a monad transformer, you’d have to write:

ohno :: (Monad (t m), Monad m, MonadTrans t) => m a -> t m a
ohno action = do
    lift action
    lift action

What’s more annoying is that, if you had a few different underlying type parameters, you’d need to request the Monad (t m) for each one - Monad (t m), Monad (t n), Monad (t f). Boring and redundant. Obviously, if m is a Monad, and t is a MonadTransformer, then t m must also be a Monad - otherwise it’s not a valid MonadTransformer!

So this Constraint is slightly different. The first perspective on Constraint is the user of a function:

I am constrained in the types I can use a function with

The second perspective on Constraint is the implementer of a function:

By constraining my inputs, I gain knowledge and power over them

But this superclass constraint is a bit different. It doesn’t seem to be about requiring anything from our users. It also doesn’t seem to be about allowing more power for implementers.

Instead, it’s a form of evidence propagation. We’re saying:

GHC, if you know that m is a Monad, then you may also infer that t m is a Monad.

Type classes form a compositional tool for logic programming. Constraints like these are a conditional proposition that allow GHC to see more options for solving and satisfying problems.

The Koan of Constraint

Four blind monks touch a Constraint, trying to identify what it is.

The first exclaims “This is a tool for limiting people.” The second laughs and says, “No, this is a tool for empowering people!” The third shakes his head solemnly and retorts, “No, this is a tool for clarifying wisdom.”

The fourth says “I cannot satisfy this.”