Reading Ur/Web signatures, part 1

So you stumbled upon Ur/Web and you rather like what it’s about, but after trudging through the tutorials and the examples in the few blog posts you’ve seen around you can’t seem to find your own footing. The compiler errors are long and life is short and you’re about to throw your computer out the window.

I understand.

In this post, I’ll walk you through the signatures of a few functions from the standard library, hopefully providing you with enough context to make it through the rest on your own. You might want to grab a copy or search for the one on your hard drive so you can follow along.

I’m going to be frank: given the current state of the ecosystem and of the documentation, you have close to no chance of learning Ur/Web if you don’t already know some OCaml/ReasonML or another language in the ML family, so if you don’t you might want to get well acquainted with one first. Elm is a good starting point.

Open up string.urs and have a look around. I’m going to assume you can read these signatures:

type t = string

val length : t -> int

val append : t -> t -> t

val all : (char -> bool) -> string -> bool

Still here? Good, let’s introduce some new syntax.

val index : string -> char -> option int

If you come from OCaml or SML you might notice that the argument of the type constructor option is to the right of the constructor, as in normal function application. This unification of function and type constructor application is not just a matter of syntax like in ReasonML; put this in the back of your mind for the moment, we’ll come back to it later.

val substring : string -> {Start : int, Len : int} -> string

Much like SML, Ur/Web makes up for the lack of labeled arguments by using anonymous records as function arguments. Also, record field names must start with a capital letter like the members of a variant.

Generics

Let’s kick things up a notch. Open up list.urs and you’ll be greeted by something like this:

val rev : a ::: Type -> list a -> list a

As you might have guessed, List.rev is a function that takes a list of elements and returns another list with the elements of the first, in reverse order. rev can reverse lists that contain any element, so we say that it is polymorphic.

a is the type of the values contained in the input and output list. The argument a ::: Type is just a way of saying that we don’t know what type a will be when we declare the function; it’s up to the caller to bind it to a valid type. The triple colon (:::) means that this type parameter is implicit, so the compiler will take care of inserting the correct type when calling it.

OCaml and most other languages don’t require you to explicitly declare these type parameter, but sometimes it is useful to ensure the well-typedness of a polymorphic function.

Quoting from the tutorial:

Unlike in ML and Haskell, polymorphic functions in Ur/Web often require full type annotations. That is because more advanced features make Ur type inference undecidable.

Let’s pull up the implementation for a moment (found in list.ur):

fun rev [a] (ls : list a) = ...

The a in square brackets here corresponds to a ::: Type in the signature above. We could also write it like [a ::: Type] if we wanted to be more explicit.

Now let’s look at List.mp. (Which is just the List.map function, but it can’t be called map because map is a keyword in Ur/Web. More on that later.)

val mp : a ::: Type -> b ::: Type -> (a -> b) -> list a -> list b

mp has two polymorphic type parameters, so they are both made explicit in the signature.

Interestingly, we can write a function so that the type parameter has to be passed explicitly by replacing ::: with a double colon (::):

(* id.urs *)
val id : a :: Type -> a -> a

(* id.ur *)
fun id [a :: Type] (x : a) = x

val x = id [int] 451

::: indicates a type parameter that may be inferred by the compiler, while :: indicates one that has to be passed explicitly. The compiler will be able to infer a type parameter by itself most of the time, but in some cases which we’ll see later you’ll have to be explicit and use the double colon.

Basics of type constructors

At this point I should introduce Ur/Web’s type constructors, because they’re a lot more powerful than those in most other languages. Open up json.ur (not json.urs) and the first thing you’ll see will be this:

con json a = {ToJson : a -> string,
              FromJson : string -> a * string}

While the con keyword might throw you off, you might recognize this as a simple type declaration. Ur/Web actually makes a distinction between simple aliases, like the one we encountered at the top of string.urs, and type constructors, which take one (or more!) arguments, and have to be declared with the keyword con.

The json type constructor is simply a record with an encoder function which takes an a and returns a JSON string, and a decoder function which takes a JSON string and returns an a and the remaining JSON string.

Remember that thing earlier about unifying function application and type constructor application syntax? The two are actually very closely related: just as normal functions are functions from values to values, type constructors can be thought of as type-level functions from types to types, and the purpose of this unification is just to make the similarity more apparent.

This insight might not net you much in OCaml or SML because type constructors have a lot of limitations compared to functions: they can’t be curried, and you can perform very few operations inside them.

Ur/Web’s type constructors are much more interesting. The json declaration above is actually syntactic sugar for a type-level function:

con json = fn (a :: Type) =>
  {ToJson : a -> string,
   FromJson : string -> a * string}

We can define a curried constructor that takes two types and returns the type of a 2-tuple:

con pair a b = a * b

con intAnd :: Type -> Type = pair int

val p : intAnd string = (451, "what a shame")

We can also perform various useful operations on record types, as we’ll see later.

Type classes

Ur/Web’s = (equals) operator works just like you’d expect it to for types provived by the standard library: 1 = 1, "line" = "line", and Some "just" = Some "just". So is it implemented like in OCaml, using a “magic” internal function that structurally compares record fields and variant members? Not quite.

If we try to compare two records, we’ll get a surprisingly helpful error message:

(* test.ur *)
val ok = { A = 1 } = { A = 1 }

(* test.ur:1:5: (to 1:26) Can't resolve type class instance
   Class constraint: eq {A : int} *)

Let’s take a look into basis.urs. At line 26, you’ll see these declarations:

class eq
val eq : t ::: Type -> eq t -> t -> t -> bool

The signature of the eq function looks familiar enough. We still don’t know what class eq means, but by the way it’s used in the function we can infer that it’s a constructor that takes one argument.

In fact, eq is just an abstract type, i.e. a type whose implementation isn’t specified in its signature so that only the underlying module can access it. If you don’t know what that is, you can think of it as an opaque pointer in C.

In this case we can’t look at its actual implementation because Basis is implemented directly in C, but it would look somewhat like this:

con eq t = t -> t -> bool

So now we should have all the pieces to understand the eq function above. Or do we?

If you were to define your own eq constructor and your own eq function, you’d always have to pass a function of type eq t as first argument. (This kind of function can also be called witness.)

con eq' t = t -> t -> bool

fun eq' [t] (cmp : eq' t) (a : t) (b : t) =
  cmp a b

fun eq'_bool (a : bool) (b : bool) =
  case (a, b) of
  | (True, True) => True
  | (False, False) => True
  | _ => False

val test = eq' eq'_bool True False

But if we were to do the same with eq, we would get a compiler error.

Turns out that the eq function is just the desugared name of the = operator, and as we’ve seen above, we can use it transparently without having to worry about the witness function. This is where the class keyword comes into play.

When we mark eq with the class keyword in a signature file, the compiler will automatically search for a fitting implementation of eq t every time we call = with a given t.

The option constructor also defines an eq witness in the Option module. This is its signature:

val eq : a ::: Type -> eq a -> eq (option a)

This should be straightforward by now. Option.eq implicitly takes a witness eq a and maps it to the value stored inside the option, if any. Let’s take a look at its implementation.

fun eq [a] (_ : eq a) =
    mkEq (fn x y =>
             case (x, y) of
                 (None, None) => True
               | (Some x, Some y) => x = y
               | _ => False)

The wildcard corresponds to the witness argument, even though the function doesn’t use it directly. In a way, the witness argument is just there to constrain the types we can call Option.eq with to those for which there exists an implementation of eq.

Now the error message should make sense: the compiler is telling us that this invocation of = has a constraint of type eq {A : int} on its arguments, so we need to implement a witness of eq for {A : int}. We’ll have to use the mkEq function to do this.

val eq_a_int = mkEq
  (fn (a : {A : int}) (b : {B : int}) =>
    a.A = b.A)

(* this will compile now *)
val ok = { A = 1 } = { A = 1 }
To be continued…

This post is getting pretty long, so I’ll wrap it up here for this week. If you already knew most of the things I covered here, don’t worry, the next one is gonna cover some of the most foreign parts of the type system.

Watch this space for part 2! (I promise I’ll implement an RSS feed soon.) In the meantime, you might want to brush up on monads, or take a look at the more dense official tutorial.