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## Exploring linear Traversable using generics

8 July 2021 — by Sjoerd Visscher

The design of a library like linear-base, Tweag’s standard library for Haskell’s linear types, requires many choices. One of these choices for linear-base is what a linear Traversable type class should look like.

One way I use to approach such problems is to implement instances for the types in GHC.Generics. GHC.Generics provides a way to implement datatype-generic functions by translating a datatype to and from a limited set of primitive datatypes representing sums, products, constants, etc. Therefore, a type class implemented for this set can be given an instance at a large class of algebraic datatypes. So this is a nice way to apply structural reasoning to a problem.

## The problem

Here is the signature of fmap in linear-base, as it applies, for instance, to lists:

-- The syntax for linear functions is %1->.
-- Linear functions consume their input exactly once.
fmap :: (a %1-> b) -> t a %1-> t b

Correspondingly, it seems pretty clear that the type of linear traverse should be:

traverse :: Applicative f => (a %1-> f b) -> t a %1-> f (t b)

But which Applicative type class should we use? In the context of linear types, there are three different ones! The one from the prelude:

Prelude.pure :: a -> f b
Prelude.fmap :: (a -> b) -> (f a -> f b)
Prelude.liftA2 :: (a -> b -> c) -> (f a -> f b -> f c)

And two linear ones: a linear data Applicative:

Data.pure :: a -> f b
Data.fmap :: (a %1-> b) -> (f a %1-> f b)
Data.liftA2 :: (a %1-> b %1-> c) -> (f a %1-> f b %1-> f c)

and a linear control Applicative:

Control.pure :: a %1-> f b
Control.fmap :: (a %1-> b) %1-> (f a %1-> f b)
Control.liftA2 :: (a %1-> b %1-> c) %1-> (f a %1-> f b %1-> f c)

Note that I’m leaving out <*>. Its type signature hides a nice symmetry, and can be defined as liftA2 (\$) in all cases.

Another thing to note is that any linear function is also a regular function, which means that any linear control Applicative is also a linear data Applicative, so the latter is a superclass of the former. On the other hand, neither linear Applicative is a subclass (or superclass) of the prelude Applicative, since they can be given linear functions as input, while prelude Applicative requires regular functions as input (subtyping goes the other way for inputs).

## Instances of GHC.Generics datatypes

### V1

Let’s start by writing an instance for V1, which represents empty data types. Such instances are usually either easy or impossible to implement. In this case we’re lucky since callers of traverse are supposed to provide a value of type V1 a, which doesn’t exist, so pattern matching on it (using the extensions LambdaCase and EmptyCase) is all we need to do. There’s also no need for any Applicative method, so we don’t learn anything either.

-- data V1 a
instance Traversable V1 where
traverse _ = \case

### U1

Next is U1, which represents constructors with no fields. We get a value of type U1 a and need to produce a value of type f (U1 b). We can produce values of type U1 b for any b out of thin air, and by applying pure, linear or not, we get the desired f (U1 b). So this works for all 3 Applicatives. But note that the linearity forces us to pattern match on the input to prove that we completely consume it. Just traverse _ _ = pure U1 does not work.

-- data U1 a = U1
instance Traversable U1 where
traverse _ U1 = pure U1

### Par1

The constructor Par1 represents uses of the type parameter. It is declared as

newtype Par1 a = Par1 a

and traverse at this type is:

traverse :: (a %1-> f b) -> (Par1 a %1-> f (Par1 b))

We’re going to implement this in roughly the following way:

traverse g (Par1 a) = let fb = g a in fmap Par1 fb

Since we’re promising to use Par1 a linearly, every step needs to be linear. We pattern match to get an a, apply the input function of type a %1-> f b and get fb :: f b linearly. But we need f (Par1 b), so we apply fmap Par1. But fmap from the Prelude isn’t linear, it does not promise to use fb exactly once, so we can’t use it here for linear Traversable! That means that the prelude Applicative is also out.

We’re down to either the linear data Applicative or the linear control Applicative. Thanks to the superclass relation between the two we can use the Data one here in both cases:

instance Traversable Par1 where
traverse g (Par1 a) = Data.fmap Par1 (g a)

### :+:, :.:, M1, Rec1

The instances for the datatypes representing multiple constructors (sums) :+:, composition of type constructors :.:, meta information M1 and reference to other type constructors R1 just use fmap and don’t provide any new information.

-- data (l :+: r) a = L1 (l a) | R1 (r a)
instance (Traversable l, Traversable r) => Traversable (l :+: r) where
traverse f (L1 la) = Data.fmap L1 (traverse f la)
traverse f (R1 ra) = Data.fmap R1 (traverse f ra)
-- newtype (s :.: t) a = Comp1 (s (t a))
instance (Traversable s, Traversable t) => Traversable (s :.: t) where
traverse f (Comp1 sta) = Data.fmap Comp1 (traverse (traverse f) sta)
-- newtype M1 i c t a = M1 (t a)
instance Traversable t => Traversable (M1 i c t) where
traverse f (M1 ta) = Data.fmap M1 (traverse f ta)
-- newtype Rec1 t a = Rec1 (t a)
instance Traversable t => Traversable (Rec1 t) where
traverse f (Rec1 ta) = Data.fmap Rec1 (traverse f ta)

### :*:

In the case of the product (:*:), we need to combine two Applicative values using liftA2. Again we have to do this linearly so the prelude Applicative does not work, but the two linear ones do.

-- data (l :*: r) a = l a :*: r a
instance (Traversable l, Traversable r) => Traversable (l :*: r) where
traverse f (la :*: ra) = Data.liftA2 (:*:) (traverse f la) (traverse f ra)

### K1

We have one more datatype left to do, K1 for constants. We have an input value of type K1 i c a and need to produce a value of type f (K1 i c b). We can pattern match on the input to get the constant of type c out, and apply the K1 constructor again to produce a value of type K1 i c b. Next we can apply pure to get the desired result, but we need to do so linearly and Data.pure is not linear! So we’re left with just the linear control Applicative, and this is indeed what linear-base provides.

-- newtype K1 i c a = K1 c
instance Traversable (K1 i c) where
traverse _ (K1 c) = Control.pure (K1 c)

## An alternative

We couldn’t use Data.pure because it doesn’t consume its input linearly, and we can use the constant just once. But what if we could use the constant more than once? This is exactly what Movable from linear-base provides!

class Movable a where
move :: a %1-> Ur a

Here Ur a (Ur stands for unrestricted) gives unlimited access to values of type a, even in linear context. So if we have an instance of Movable for the constant we can use Data.pure, and therefore it seems useful to also have a version of Traversable that uses the linear data Applicative.

instance Moveable c => Traversable (K1 i c) where
traverse _ (K1 c) = move c & \case (Ur c') -> Data.pure (K1 c')

This would lead to instances like:

instance Movable a => Traversable (Either a)
instance Movable a => Traversable ((,) a)

Is having two different Traversable classes worth it? It seems so, for at least one particular reason: Foldable can be shown to be derived from Traversable using the Const Applicative. But Const is not a linear control Applicative! It is only a linear data Applicative, because its pure implementation throws away the argument, so it is not linear. There is an issue open to add these classes to linear-base.

## Conclusion

One final thought (and you might have been wondering about this) now that we have instances of Traversable for all GHC.Generics datatypes: can we generically derive instances of linear Traversable? Well, there is one catch. The to1 and from1 methods from the Generic1 class that are needed are not linear! They should be in principle, since instances are not supposed to throw stuff out nor duplicate anything either. But we can’t be sure, so for now we can only provide generic instances using Unsafe.toLinear with a big fat warning sign. (We’re still working out the best way to do so.)

I hope you enjoyed this exploration of linear traversals over the zoo of GHC.Generics datatypes as much as I did. I encourage you to play with linear types a bit, it can be really surprising what works and what doesn’t work linearly!

Sjoerd Visscher

Sjoerd Visscher is a software engineer who loves beautiful code and making users happy.

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