Codata and ListT Done Right

Posted on March 31, 2021

When deciding how to implement ListT (the monad transformer) in Haskell, one is tempted to do something like:

newtype ListT m a = ListT {runListT :: m [a]}

The problem with this version of ListT is that it does not form a Monad unless the base monad is commutative (i.e. the order of operations does not matter). This rules out potentially useful instances such as ListT IO which you might like to use as a simple streaming interface.

How can we come up with a version of ListT that works? The one above was the only obvious version.

A Quick Detour, Codata

Data can be thought of as a way of constructing something. A function from a value to a type.

data A = MkA Int

Here, the constructor MkA has the type Int -> A

Codata, on the other hand, is a way of deconstructing something. That is, we flip the arrows! (This is what “co” tends to mean). So we are looking for something with the type A -> Int instead. Data types in Haskell are actually both data and codata. We even have special syntax for codata. It looks like this:

data A = A
    { getA :: Int
    }

Here, getA has the type A -> Int.

It’s also trivial to convert between a data and codata in Haskell. Basically all we have to do is create a destructor that witnesses every constructor.

For example:

data A = X Int | Y Char

can be expressed as

data CoA = CoA
    { getA :: Either Int Char  -- Left witnesses X, Right witnesses Y
    }

Side note, Either Int Char is the same thing as A, so this feels like cheating. Remember, data is codata in Haskell

Back to List. Let’s quickly define it:

data List a = Cons a (List a) | Nil

Can we “convert” this to codata?

data CoList a = CoList { getList :: Maybe (a, CoList a) }

Here, Nothing witnesses the Nil, and Just witnesses the Cons. Note that we can’t just use the List definition here like we did above because List is a recursive type.

ListT, with Codata

So, now that we have a codata version of List (which is the same things as List in Haskell, because data is codata), can we come up with a better ListT? Sure, we can do the same trivial thing we tried to do with our regular List definition, just wrap it in the base monad!

data ListT m a = ListT { runListT :: m (Maybe (a, ListT m a)) }

Which is exactly ListT done right.

Proving that this forms a monad is left as an exercise to the reader.

It also fits our intuition of what ListT might be for streaming. Each element is produced in the base monad along with the rest of the list.

This corresponds closely with streaming or iterator patterns in other languages. e.g. Rust has the Iterator trait which has the sole method fn next(&mut self) -> Option<Self::Item> which is the same as our ListT IO. The IO is implied, the return type is an Option of the next element, and instead of returning the rest of the list, the list is mutated in place.

So, next time you are struggling to find the correct path forward for your data types, consider thinking about it with the arrows flipped!