Intro to type-level programming in Haskell - Part 1
I heard a lot lately about using types in Haskell to describe function arguments in more details (e.g. function takes a list that is non-empty) and thus achieve higher compile-time safety. It sounded cool so I decided to research more about it and I created this blog post as a memo of what I learned.
Resources I used:
- Tutorial on GADTs and DataKinds by andre.tips
- explains GADTs well, but DataKinds not so much. I would recommend to start with this one. [The page seems to be down, so I linked to the Wayback Machine url.]
- Basic Type Level Programming in Haskell by Matt Parsons
- pretty extensive, but not so intuitive. Still the most complete intro (and pretty much the only one) to the topic. Most of my examples below are taken from it.
Why type-level programming?
As I wrote above, my current understanding is that with it we can add more info into the type signature of a function, making it “safer” in the compile time. E.g. instead of just saying “this function takes a list” we can say “this function takes a non-empty list”, or “this function takes Int which is > 10 and < 127” (although this last one might be solved just by creating an appropriate type, e.g. using TH?).
TODO: I would love to learn about more examples where this is used.
Example: A function that accepts only a non-empty list
We want to be able to tell from its type whether a list is empty or not. To do that we will create a new type which will have that information stored in it.
First, let’s see how we would create a “normal” list type on our own:
This is a usual recursive definition of list (
Cons stands for Constructor). Once there is
an instance of this type we can tell which type of elements the list contains (e.g.
List (Maybe String)), but nothing more than that. We cannot deduce from its type (which means
in compile time) whether it is empty or not.
We could of course check it in runtime and throw an exception if the list is empty, but we want to be stricter than that. We want to ensure that program cannot even be compiled if an empty list is provided where it shouldn’t be.
What are we missing?
The problem with the “normal” list we defined above is that we are missing information in its type.
We have only one piece of
information and that is type of the elements within the list - we use type parameter
a to declare
that. We can also use it to define functions that work only for a specific
a. E.g. here is a
function that works only on a list of integers:
This is guaranteed in compile-time. If we call this function with a list of e.g.
compiler will throw an error at us.
Using type params to encode extra information
So here’s an idea - why don’t we just use the same mechanism again (having a type parameter) to know whether a list is empty or not. If we added another type parameter to keep track of that, our type (disregarding data constructors for now) would look like this:
Just as type param
a means any type (e.g.
MyType), the same applies for
E.g. we could have
List Int Double or
List String Bool or
List Int SomeCustomType.
Just as we restricted function
sumListElements above to work only when
Int, we can use
empty in the same way.
Let’s say we want to implement a safe version of
head function - that means it accepts only a
non-empty list as an argument, otherwise it won’t compile. We will call it
Ok, we introduced
empty as another type parameter, but the question we are facing now is what
do we do with it? Which concrete types will take its place and how?
Lets introduce two new types:
The interesting thing here is that we have only type constructors for these types and no data constructors. That means we cannot create instances (values) of these types, but that is ok! We need these types only at the type level, in function signatures. Such types are also called uninhabited or empty.
Now let’s imagine we have a way to correctly assign
NonEmpty to empty and non-empty
lists’ types. Then we could define
safeHead as follows:
We wouldn’t even have to define a case for
End since the type guarantees it can’t ever happen.
Assigning correct type to
The main question that is left is how do we produce such lists with an extra type parameter, and
how do we make sure which type
empty takes when? This is what we will look at now.
This is how our type
List looks once we have added
empty as a second type parameter:
Except adding that extra type parameter, nothing else changed. When I first saw this,
I was confused by
the fact there is a type parameter on the left side that doesn’t appear anywhere on the right side
as a data.
How is that possible, why would that make sense? (Ok, there is
empty on the right side here, but
only as a part of a type designation and not as a part of data constructor. Which means there will
never be anything of type
empty in some value of this type).
But turns out it does make sense, since we use it as a designation at the type level only, to show that an underlying value has a certain property (empty or non-empty in this case). Such types, which have a type parameter(s) on the left side that don’t appear on the right are also called phantom types.
Let’s see now what happens if we create an instance of our new
List and test its type in GHCi:
As we can see, GHCi concluded that
a is a string in
nonEmptyList, but could not deduce anything
empty in either case, since it is not used anywhere. So how can we solve that
and make sure that
Before we continue, let’s check types of our data constructors,
Cons (since they are
functions as well, we can do that):
We can see they have no power to change or specify
empty type param in any way.
End will leave
it unspecified, while
Cons will preserve it from the input list. Also, we have no way to change
these type signatures as they are automatically derived from
List type definition.
This is exactly where GADTs come in. GADTs (Generalized Algebraic Data Types) is a Haskell extension that lets us explicitly define the type signatures of data constructors.
Before seeing GADTs in action, let’s first remind ourselves of the standard,
non-GADT definition of
List we used above:
Now let’s rearrange it a bit and add types of data constructors in comments so we can more easily reason about them:
As we mentioned, the types in the comments are automatically derived and we cannot control them. But that is exactly what we want to do, and GADTs let us achieve that using the following syntax:
We can see it is very similar to our “rearranged”
List definition above! What GADTs let us do
is write by ourselves types of data constructors (which were in the previous definition in the
comments), giving us control to specify them as we wish!
The difference in the syntax is that we have to add
where after the type name and then
for each data constructor we specify its type signature.
Now we finally have the power to control
empty type parameter (in
List a empty)! We specified
End will mark list as
Cons will mark it as
NonEmpty. And this is exactly
what we wanted to do, because if we used
End we know the list is empty, while if we used
we know there is at least one element in it, which makes it non-empty.
Let’s see it in action! Using it stays the same as without GADTs, just that this time there will
empty type param which assumes an appropriate type:
Wohoo, this works now! We see we can construct values of this type and we will always know whether
it is empty or not.
safeHead function we defined above will work on these without any problems.
Can we just use smart constructors instead of GADTs?
One possible “downside” of GADTs is that it is a language extension we have to enable, thus making our codebase a bit heavier (longer compilation time?) and less “standard”.
Sometimes we can avoid using GADTs with smart constructors. Let’s see what that is and how it would work in this case.
Smart constructor is simply a function that is used to create a certain value instead of using its data constructor directly. We typically do that (hide data constructors and expose smart constructor functions) when we want to have extra control over the value creation. E.g. we want to make some extra checks, or make sure an invalid value isn’t provided etc.
For example, we could provide a following smart constructor to create an empty list:
And this works! By defining the type signature of
createEmptyList we made sure that
always assume the type of
Empty when this function is called. Since
End has a type signature
End :: List a empty, we just “casted” type param
empty here into a specific type.
Let’s try to do the same for the other data constructor, adding an element to the list:
What we are trying to achieve here is make sure that whenever an element is added to the list,
NonEmpty, and we again use type signature for that, to provide that extra
But if we try to compile this, we get the following error:
Couldn't match type ‘empty’
To understand the problem, let’s remind ourselves of
Cons :: a -> List a empty -> List a empty.
The problem is in that
empty to stay the same, so whatever type it is in the
input list, it must stay the same in the newly constructed list. Although we specified we want
to change it to
NonEmpty in the
addElemToList’s type signature,
Cons is not flexible enough
to do that and this is why we got an error during compilation.
Although smart constructors might be a solution in some simpler cases (e.g. when we have “flat” data
and we are merely “casting” general type params into the specific ones, such as we did with
in this case where we have a recursive data structure it wasn’t enough because the initial
data constructor was too rigid.
List Int Double?
List Int Double means nothing, it doesn’t make sense. We can only construct and know how
to work with lists whose
empty type parameter is either
But the problem is although it doesn’t make sense, we can still write things like this and it will happily compile:
There is no way to execute this function since there is no way to construct such a list where
empty type param is
Bool, but strange stuff can appear in our codebase and we cannot detect
it in compile time.
Here is a more “real world” example when this could be a problem: let’s say you are
NonEmpty types for list as we explained above,
but you are also using
No types for something else in your codebase. And then your
colleague starts implementing some new functionality for your lists, and by mistake he starts using
No in the place of
empty. And there is nothing to stop him until he actually tries to
connect everything together and run the code!
The problem we see is there is no “safety” at the type level, we cannot say “
empty can be only
this kind of type”. But, there is a mechanism that can help us.
Not all types are used in the same way
Just a short observation before we continue. I wanted to put attention to the fact that we are now differentiating between two possible uses of a type:
- type is used to produce values (store data) - e.g.
Maybe Bool, …
- type is used only at the type level as a designation of something, never producing an
actual value - e.g.
Despite these very different uses, we currently don’t have a way to differentiate between such types - we declare them both in the same way and Haskell can’t tell how are we going to use them later.
In standard Haskell each type has a kind, which can be thought of as a “type of a type”. E.g.:
And that is it, all kinds are expressed with
*s and automatically derived for us.
a type that has values.
But as we saw earlier, this is not enough for us. We also want to cover that other use case so we can say “here goes only type(s) that tell us whether a list is empty or not.”
And this is exactly what
DataKinds extension allows us to do, it lets us define other kinds
Now let’s see what happened here. We made a new type
ListStatus and then we use its data
constructors (prefixed with
') in the place of types in GADT for
The thing with
DataKinds is the following: for every type we create it additionaly creates for us
new types, named after data constructors we used and prefixed with
'. It also creates a new kind,
which is named after the type’s name. Specifically for this case:
DataKindscreated for us two extra types,
- Kinds of these new types are
- These types cannot have values, they can be used only at the type level
I was really confused the first time I realized this. This extension just like that creates extra types for us, without even asking us about it, for every type we create!
Since we are using only types of
ListStatus kind in
List’s data constructors’ signatures,
Haskell inferred from that that
empty type param must be of kind
ListStatus and won’t let us
use anythng else. If we try to create a function which takes
List a Bool, we will receive the
Expected kind ‘ListStatus’, but ‘Bool’ has kind ‘*’
Which is exactly what we wanted! With this we achieved kind safety, besides the usual type safety in the compile time.
To make things even more explicit, we can turn on
KindSignatures extension which lets us
explicitly define kinds of type parameters in a type:
Now everybody can see that
a is a “standard” type that has values, while
empty can be only
'NonEmpty. We didn’t have to write this explicit version as Haskell can infer it on
its own, but it is a matter of style and documentation. We can also omit
' in front of types and
Haskell in a lot of cases can infer by itself if it is a type or data constructor. I found it easier
to have everything explicit for now, a lot is going on behind the scenes so this made it clearer
And that is it for this first part! We learned about type-level programming, how to use GADTs and data kinds and saw everything together in action. Hope you found it useful, please let me know in the comments if you have any questions, I said something wrong or I can explain something better.
In the Part 2 we will go even deeper and take a look at some more cool examples that build on top
of this one! Here’s a teaser question: with our
List a empty that we developed above, how would
safeTail function which works only on non-empty lists,
analogous to what we have done with
safeHead? Can you do it, what is its return type?