How I Learned to Stop Worrying and Love the Type System

haskell, technical

This post marks my intended shift to more technical topics; the target audience is people with some programming experience. However, if you’re otherwise interested, feel free to skip the technical bits and see what you think!

I

A pretty common “soft” question in software interviews is “what’s your favorite programming language, and why?” I have no idea what the interviewer intends to tease out with such a question, but historically my answer has always been along the lines of:

“C++. Why? Because when I get something working, I feel like a god. It just puts so much power at my fingertips.”

How’s that for being overly controlling? I actually enjoyed the tedium of writing copy constructors, keeping track of the ownership of my pointers, and especially in writing fancy templated data structures. If you haven’t had the pleasure of writing a fancy templated data structure, they often look something like this:

template<class TItem>
class ScopedPool {
public:
  template<class... TCtorArgs>
  class Object {
  public:
    explicit Object(
      ScopedPool<TItem> &pool,
      TCtorArgs&&... ctorArgs)
    {
      ptr_ =
        objs.empty()
        ? std::make_unique<TItem>(std::forward<TCtorArgs>(ctorArgs)...)
        : std::move(objs.top());

Yes, once upon a time, I actually wrote that unholy thing, and perhaps more surprisingly, I even knew what it did. I’ve been programming for about a decade, and things like this were the only challenges I’d face when working on projects; gnarly pieces of software that really made me think.

Then, about a year ago, I discovered Haskell and dove in. It felt like power-lifting for my brain, and all of a sudden, programming was fun again. Even the most trivial programs gave me a hard time in Haskell. I’m not here to discuss Haskell in general today; it just happens to be why I’ve been thinking about type systems so much lately.

For me, the most interesting feature of Haskell is that if your program type-checks, there’s a good chance it will do what you want. There are no segfaults, no race conditions, and no memory stomps to worry about. Also, the type-checker is really, really picky, which probably helps. In my opinion, this kind of really picky type-checking is exactly what modern programming needs.

Of course, C++ and Haskell’s type-systems are both Turing-complete, so technically they are equally powerful. And as we all know, being technically right is the best kind of being right. That being said, if you are the type of person who is going to try make this argument, try to suspend your disbelief, and hopefully I’ll convince you that I might be onto something here.

II

I do most of my blogging while in coffee shops, mainly because I need a constant stream of caffeine to stay productive. Unfortunately, that means there is a tangible cost to me writing. Let’s say I want to figure out how much I have to pay to get a blog post done, where do I begin?

Well a good first strategy is to write down any variables I have:

\[ \begin{alignedat}{2} p &= 2.5 \text{Є / cup}\qquad& \text{(price per coffee)} \\\\ b &= 3 \text{days / post}\qquad& \text{(duration of post)} \\\\ n &= 400 \text{mg / day}\qquad& \text{(necessaary caffeine)} \\\\ c &= 100 \text{mg / cup}\qquad& \text{(caffeine per coffee)} \\\\ v &= 250 \text{ml / cup}\qquad& \text{(volume of cup)} \end{alignedat} \]

And then, work backwards. Given that I want my result to be in terms of \(\text{Є / post}\), I start by looking at \(p\) and \(b\), which are \(\text{unit}(p\cdot b) = \text{Є} \cdot \text{days / cup / post}\). Good start, now it’s just a matter of multiplying and dividing out the other variables I know in an attempt to get those extra units to cancel.

In the end, it works out that:

\[ p\cdot b\cdot n \mathbin{/} c = 30 \text{Є / post} \]

which has the right units. And is actually a lot of money to be paying for coffee! Notice that I didn’t end up using \(v\) whatsoever – just because you have some piece of information doesn’t always mean it will be useful!

You’ll also notice that I found this answer through purely mechanical means, there was no insight necessary. Just multiply terms through and pay attention to what units the final result has. When the units become right, there’s a pretty good chance that my answer is correct.

What’s going on here is we’re using the units as a type system to ensure our answer is coherent. The units are doing work for us by ensuring we’re not doing anything stupid, and additionally, they help guide us to the right answer, even if we don’t know what we’re doing.

If you’re coming from an imperative background, this might strike you as being a contrived metaphor, but I don’t think it is. As we’ll see later, when you’re dealing with highly-parametric types, lots of times getting the right implementation is just slapping things together until you get the right type.

But let’s not get ahead of ourselves.

III

Before we dive too deeply into Haskell’s type system, we need to first take a moment to discuss Haskell’s type notation. Concrete types, like in Java, always begin with a capital letter (eg. Int is the type of the integers).

Functions of one argument are typed as a -> b (in this case, taking a a and returning a b). So far, so good, but functions of higher arity might take a little while to get used to:

A function taking two as and returning a b is typed as a -> a -> b, not (a, a) -> b, as you might expect. Turns out, there is a really good reason for this weird syntax, which bears diving into. Take for example, the Haskell function which performs addition:

add :: Int -> Int -> Int
add x y = x + y

You might be tempted to write this in C++11 as:

int add(int x, int y) {
    return x + y;
}

// usage: add(3, 4) == 7

but you’d be wrong. In fact it is closer to:

std::function<int(int)> add(int x) {
    // return an anonymous function of type `int(int)`
    return [x](int y) { return x + y; };
}

// usage: add(3)(4) == 7

Which is to say, add is a function which takes an integer, and returns a function which takes the other integer. This becomes more evident when we add braces to our original type signature: Int -> (Int -> Int). Our function, add, is a function which takes an Int and returns another function of type Int -> Int.

There are two implications to this: we can partially apply functions (this is known as currying in the literature), and functions are first-class values in Haskell.

Now, that is not to say that all types must be definite. Take for example, the polymorphic function head, which returns the first element of a list, and is typed as [a] -> a. Here, a is a type-variable, which can refer to any other type it please. The equivalent C++ of course, is something like:

template<class T>
T head(const std::list<T>& list) {
    return list.front();
}

So far, nothing special, type-variables seem to correspond directly to template parameters, but this is not in fact the case. You may have noticed that our earlier add function was too specific; addition is defined over all numeric types (not just Int), so instead let’s rewrite it as:

add :: (Num a) => a -> a -> a
add x y = x + y

Here, (Num a) acts as a constraint over a, informing you and the compiler (but mostly you – the compiler is really good at type inference), that this function is defined over any numeric a types. You can do something similar in C++ with SFINAE or maybe with traits, but it’s not nearly as elegant.

In effect, what we are doing here with Haskell’s constraints is informing the type system of the semantics of add, while a SFINAE solution is dependent on syntax: it will work for any case in which operator+ is in scope, regardless of whether this is actually meaningful.

IV

Because Haskell encourages you to write functions as general as possible (in C++ this reads as “write lots and lots of templates”), we get lots of abstract functions. It might surprise you to learn that abstract functions are usually more informative than their specialized counterparts, and it is herein that I think Haskell’s type system really shines through. As was the case with dimensional analysis, looking at generalized types gives us a lot of analytic power.

Consider a function of the type [Int] -> Int (a function from a list of Ints to a single Int). It’s not very clear what this function is doing; there are at least four non-trivial implementations for it (length, sum, head, last, and probably others).

Instead, if we change the type of our hypothetical function to [a] -> Int (from a homogeneous list of any type to an Int), there becomes only one non-trivial implementation: the length function. It has to be, since there’s no way to get an Int out of the list any other way.

Let’s consider another hypothetical function, one of type (a -> Bool) -> [a] -> [a] (a function which takes a predicate on the type variable a, a list of as, and returns a list of as). The only reasonable interpretation of this function is that the predicate is being used to determine whether or not to keep each element in the list. There are a few possible implementations, but you will already have a good idea of what would be necessary: iterating through the list and applying the predicate as you go.

To really drive this point home, let’s look at the function of type a -> a. I say “the function” because there is exactly one possible implementation of this function. Why? Well, what do we know about the parameters to this function? Absolutely nothing. While in C++ we could assume that a has a default value, or equality, or something, we can make no such assumptions here. In fact, a could have zero possible values (ie. be isomorphic to type Void), and thus be non-instantiable. Since we have no means of creating an a value, the only possible implementation of this function is to return our parameter as-is:

id :: a -> a
id x = x

So what does this tell us? Lots of times, if you know the type of a function, you get the implementation for free; just like with dimensional analysis, we can get answers through a purely mechanical means of combining the types we have until we get the type we need. Cool!

V

“But wait,” says the skeptic! “The <algorithm> header already gives us a lot of these reusable, abstract functions. Don’t all your arguments apply to C++ as well?”

Of course, this is not to say that C++ doesn’t have similar things. The <algorithm> header is indeed pretty comparable for most of the use-cases we’ve looked at so far. However, this does not hold in general, as C++ is much less specific about its types.

Let’s take transform from <algorithm> for example, which comes with the ultra crusty type signature:

template<
    class InputIterator,
    class OutputIterator,
    class UnaryOperation>
OutputIterator transform (
    InputIterator,
    InputIterator,
    OutputIterator,
    UnaryOperation);

There are a few issues with this function, but the most egregious is that its type doesn’t convey any information (in Haskell it would be a -> a -> b -> c -> b, with some implicit constraints based on what template expansion allows).

Compare the equivalent function in Haskell: (a -> b) -> [a] -> [b] (a function for turning as into bs, and a list of as). Much more concise, and significantly more informative.

It’s easy to see that just because you can get the C++ version to compile, you have no guarantees it is going to actually work. As a matter of fact, the documentation explicitly mentions this:

Exceptions:
Throws if any of the function calls, the assignments or the operations on iterators throws.
Note that invalid arguments cause undefined behavior.

The C++ version is unsafe, unintuitive, and frankly, not very helpful.

There are two major (if related) things to take away from this: types strongly inform implementation and, because of that, a type signature usually implies its use-cases. These happen to be exceptionally useful properties.

VI

The point I’ve been trying to make is twinfold: many polymorphic functions have very few reasonable implementations, and that Haskell’s types convey a lot of information to the programmer. These properties lend themselves to a big, searchable standard library.

For example, the other week I wanted the functional equivalent to a traditional while loop. The defining characteristics of a while loop are: a predicate for stopping, an operation to transform states from one iteration to the next, and a starting state. while loops seemed like something that ought to already exist in Haskell, so I decided to look around a bit before implementing it myself.

I went to Hoogle (“Google” for Haskell), and searched for the function signature I was looking for, namely (a -> Bool) -> (a -> a) -> a -> a. Sure enough, at the top of the list, there it was: until – the function that did exactly what I wanted.

And I find this is usually the case when I’m writing Haskell. Almost all of the effort comes from figuring out exactly how I’m trying to transform my types. Once that’s sorted, it’s just a matter of hoogling for the right functions and composing them together. There’s a surprisingly little amount of actual code being written.

What I take away from this is that Haskell allows you to focus much more on what you’re trying to do, rather than stringing plumbing around to get everything working. The abstractions exist at a high-enough level that you never actually deal with storage or iteration or any of the nitty gritty details. It’s the complete opposite of my historical control-freak attitude towards C++, and it’s a spectacular feeling.


Thanks for Josh Otto for reading drafts of this post.