Fantas, Eel, and Specification 3: Setoid

Congratulations! You’ve mastered the fundamentals of daggy, nailed the intro to type signatures, and are ready to begin your journey through Fantasy Land. First stop: the setoid.

A setoid is any type with a notion of equivalence. You already use plenty of setoids (integers, booleans, strings) almost every time you use the == operator, so this shouldn’t be too tricky. You also use things that aren’t setoids, like functions.

This may seem weird, but how could we reliably know whether two functions were equivalent? While our compiler will confidently tell us that 100 * 10 is equivalent to 1000, it won’t be brave enough to say x => x * x is equivalent to x => Math.pow(x, 2); it’s really not a trivial thing to work out!*

Now, for a type to be a Fantasy Land-compliant setoid, it must have a prototype method called equals with the following signature:

equals :: Setoid a => a ~> a -> Boolean

Nothing too scary, I hope? Just a way of finding out whether one thing equals another. Let’s write some Setoid instances for types from our first article:

// Check that each point matches
// equals :: Coord ~> Coord -> Bool
Coord.prototype.equals = function (that) {
  return this.x === that.x
      && this.y === that.y
      && this.z === that.z

// Check each Coord with Coord.equals
// equals :: Line ~> Line -> Bool
Line.prototype.equals = function (that) {
  return this.from.equals(that.from)

// The this' "true-ness" must match that's!
// equals :: Bool ~> Bool -> Bool
Bool.prototype.equals = function (that) {
  return this instanceof Bool.True
    === that instanceof Bool.True

// Check the lists' heads, then their tails
// equals :: Setoid a => [a] ~> [a] -> Bool
List.prototype.equals = function (that) {
  return this.cata({
    // Note the two different Setoid uses:
    Cons: (head, tail) =>
      head.equals(that.head) // a
        && tail.equals(that.tail), // [a]

    Nil: () => that instanceof List.Nil

You get the idea, right? If we have multiple constructors, we check for the same constructor. If the constructor/s take/s arguments, then we probably check those as well. Of course, if we do, the arguments must be setoids; how else could we check that they’re equivalent?

In fact, this requirement is exactly why we have the type constraint on List’s equals implementation: we need to be able to compare the whole structure, innards and all!

Sadly, the ugly side-effect of using JavaScript is that we’re going to have to mix === and .equals depending on whether we’re working with primitive types or not. It’s a shame; in other languages, we could override the behaviour of === for custom types, but not JavaScript. You could add .equals to the primitive values’ prototypes, but this is generally considered a bad idea. Best not fiddle with standard prototypes.

Still, these .equals implementations are quite pretty, right?

All the Fantasy Land structures come with laws that must be obeyed for the instance to be valid, and Setoid is no exception. In order to make sure your type behaves itself when used with other libraries and algorithms, there are just three things we have to remember. In all cases:

  • a.equals(a) === true, which we call reflexivity.

  • a.equals(b) === b.equals(a). This is symmetry or commutativity - you can give the values either way round. Remember that operations like subtraction aren’t commutative, and there are other non-commutative examples that may surprise you!

  • If a.equals(b) and b.equals(c), then it’s always true that a.equals(c): the law of transitivity.

We can see without too much trouble that all these would hold for the .equals implementations above, as long as we respect the type signatures!

If none of these laws are particularly surprising to you, that’s a great thing! This means that you have a good intuition for what a Setoid is. Later in the series, we’ll get to more complex structures, and finding an intuition will be incredibly valuable for working out how to use them.

If you’re now desperate for an exercise, why not write an .equals implementation for the built-in Array type to make it a Setoid? Add it to Array.prototype - I won’t tell - and be sure that your implementation obeys the laws above.

If you wanted to, you could also derive a function called notEquals using Setoid’s shiny new .equals method:

// notEquals :: Setoid a => a -> a -> Bool
const notEquals = x => y => ...

If you’re not desperate for exercises, (or you’ve managed to sate your burning desire at long last), shall we move onto what the point of all this fuss is? If we have formal definitions of things like Setoid (however straightforward it may be), we can define sensible interfaces for working with all sorts of data. Consider this function:

nub :: Setoid a => [a] -> [a]

I think nub might be my favourite name of any function. In practice, nub returns a copy of the given array with the duplicates removed. That’s it! You might also have heard it called uniq. At first glance, this is easy to write in JavaScript:

const nub = xs => xs.filter(
  (x, i) => xs.indexOf(x) === i)

This is okay, but we run into a problem: for non-primitive structures, this only works if equivalent values always inhabit the same space in memory. This, however, is not usually the case: if we try [[]].indexOf([]), we get back -1, even though we can clearly see [] in that array! How could we fix this? Setoid to the rescue!

// indexOf :: Setoid a => [a] -> a -> Int
const indexOf = xs => x => {
  for (let i = 0; i < xs.length; i++)
    if (xs[i].equals(x)) return i

  return -1

// nub_ :: Setoid a => [a] -> [a]
const nub_ = xs => xs.filter(
  (x, i) => indexOf(xs)(x) === i

Now, we have a function that will work for any array of a Setoid type. If we know our function will be used responsibly (that is, only ever with arrays of a Setoid type), we could even add an exception to make it work for primitives - exactly how Ramda’s equality works! Goodness, would you look at all this Polymorphism.

I think I most often see mention of Setoid (and Eq, as they call it in the Haskell world) among List and Array functions, which give plenty of opportunities for exercises to cement your understanding:

  • Write a function to determine whether a given list’s values form a palindrome (e.g. whether a list is equivalent to itself reversed). We’ll need a Setoid instance for the inner type to make sure it’s nice and general. As a small hint, you could write a naïve solution with that zipWith function we mentioned earlier…

  • Use daggy to build a Set type that stores a unique set of values; you can even reuse nub_! You’ll need methods for adding and removing elements, and the former will need a check to see whether the element already exists in the internal store (probably an array).

Setoid is, without a doubt, the simplest structure within the Fantasy Land spec, but that makes it a really good one to start with. For most, the intuition required to understand this one will be perfectly natural, and none of the laws should come as a shock.

Don’t get too cosy, though! Next time, we’ll move onto a far more weird and wonderful structure: the semigroup. Ooer.

Until then, I hope you’ve enjoyed this post. Please get in touch with any feedback and suggestions - I really want to make this series as useful as possible! - and don’t hesitate to ask for more examples, exercises, or explanations. Oh, and, as always:

Take care ♥

* The important point here is that equivalence is much deeper than pointer equality. Just try typing (x => x) === (x => x) into your Node REPL.