A gentle introduction to dependent types
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Most of you are certainly familiar with this piece of code:
This is the syntax for generic (parameterized) types, which has been available in Java since 2004. And of course, you know what you’re supposed to put between the brackets: a type (class or interface). But have you ever wondered what else we could use there, besides types?
For example, how about:
List<Integer><2> // not legal Java
Instead of a type, I’m now using a value of a certain type (an integer) as a type parameter.
What could this possibly mean? Well, since I’m making up the syntax (and I’ll use that liberty a bit more in the rest of this post), I might as well make up what it means as well. How about we say that the integer here represents the size of the list?
In other words:
List<Integer><2> l = List(1, 2); // compiles List<Integer><2> l = List(1); // does not compile
Or we could decide that if the type that we are indexing (Integer) supports the + operator, then that value is the sum of all its elements:
List<Integer><2> l = List(2); // compiles List<Integer><2> l = List(0, 1, 1); // compiles List<Integer><2> l = List(0, 1); // does not compile
You probably guessed it by now: List<Integer><2> is a dependent type. A type parameterized not just by other types but by values of other types. The correct terminology here is that the type is “indexed” by the value (not to be confused with the meaning of “index” in the context of a list).
Different values specify different types, and of course, we don’t have to stop there: the value could be of a different type than Integer, e.g String (List<Integer><"abc">>, etc… Obviously, you’re not limited to List either, you can imagine having maps, trees or any other parameterized type.
Unfortunately, the fun stops there. While the appeal of dependent types looks strong, it’s actually very hard to cross over in the realm of concrete code.
Let’s get back to our list: how do we manipulate a dependent list with a fixed size, so that the compiler will flag any attempt to violate this constraint by refusing to type check our program?
I used the easy situation in the code snippets above by initializing my lists with constants, but obviously, this is not very useful: you pretty much never use lists that are initialized with constants at declaration time, except maybe in tests. Very often, we create a list, we populate it at runtime with some algorithm and if it’s mutable, we alter it in various ways.
First, we need to create our list without putting any elements in it, but doesn’t this violate the constraint?
List<Integer><2> l = new ArrayList<Integer><2>(); // should not compile
If we can’t even compile this, then we’ll never be able to use anything but lists initialized with constants, so let’s imagine that the compiler will let us get away with it. How do we populate our list now? As it turns out, we haven’t made much progress:
List<Integer><2> l = new ArrayList<Integer><2>(); // assume this compiles l.add(1); // will not compile l.add(2); // would compile if the previous linecompiled
Having a list with one element would be a type error, so the second line in the snippet above should not compile. You could work around this limitation by having add() return a different list of a different type. It will still be a List, but indexed by the value +1 of the original one. And now you have to determine how a List<Integer><1> and a List<Integer><2> can interact and how compatible they are..
We are at an impasse, and if we want to make progress, we have to configure the compiler to increasingly release the dependent typing condition. For example, maybe it would only apply the type checker on operations that access the content of the list (read) but it wouldn’t perform any checking for operations that alter it (write). Kind of a builder pattern waiver, where the object is allowed to be in an illegal state within well delineated boundaries.
Even so, the burden of type checking will become quite big and will require the compiler to not just generate code but actually perform branch analysis to make sure that the number of elements of the list always satisfies the dependency at certain given points.
Note also that I glossed over another non-trivial aspect: how can we teach the compiler what the syntax above means? Should it be configurable or part of the language definition? What types are allowed to be indexed? What types are allowed as indices? How do types of the same family but with different indices interoperate?
By now, it should be increasingly clear that mutable types can’t realistically be dependent, which confines dependent types to immutable ones. How useful is this, really? Still working on that.
If the subject interests you, the best paper I have found so far is Dependent types at work (PDF), which is a short introduction of dependent types through Agda. Agda is probably the first language you should look at since it’s much more recent than Coq (which I was already studying as part of my PhD fifteen years ago, since I was working at the same research institute that created Coq). This paper (and most of the literature on the subject) doesn’t touch any of the topics I described in this short introduction, they are more focused on describing how you can use Agda to formalize the proofs of your type system, which in turn will give you guidance on how your type checker should be implemented.
And if you have some interesting practical examples of dependent types, comment away!
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