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Who implements the typeclass instance?

The typeclass pattern in Scala invites you to place implementation-specific knowledge directly in the typeclass instances, with the interface defined as the typeclass’s abstract interface.

However, GADTs permit a different organization of code. It is even possible to define a typeclass that seems to do nothing at all, yet still permits full type-safe typeclass usage.

The possibilities between these two extremes form a design space. If you wish to practice ad-hoc polymorphism in Scala, this space is well worth exploring.

A glorified overloader

Refactoring a set of overloads into a typeclass is a fine way to get some free flexibility and dynamism, because expressing overloads as a typeclass gives you free fixes for common overload problems.

  1. Methods calling the overloaded method do not themselves need to be overloaded just to avoid suppressing the flexibility of the overload beneath. (See addThree and zipAdd below for examples.)
  2. Return-type overloading works, even in Scala, where it does not when attempting to write overloads in the Java style, i.e. multiple methods with the same name.
  3. Overloads may be defined as recursive type rules, admitting a combinatorial explosion or even infinite “effective overloads”.

Let’s make a quick example of something like a typical overload.

object OverAdd {
  def add(x: Int, y: Int): Int = x + y
  
  def add(x: String, y: String): String = s"$x$y"
  
  def add[A](l: Vector[A], r: Vector[A]): Vector[A] =
    l ++ r
}

This mechanically translates to a newly introduced type, some implicit instances of that type, and a function to let us call add the same way we used to.

// typeclasses are often defined with trait, but this is not required
final case class Adder[A](addImpl: (A, A) => A)

// easier if all implicits are in this block
object Adder {
  implicit val addInts: Adder[Int] = Adder((x, y) => x + y)
  
  implicit val addStrings: Adder[String] =
    Adder((x, y) => s"$x$y")
    
  implicit def addVects[A]: Adder[Vector[A]] =
    Adder((l, r) => l ++ r)
}

// and to tie it back together
def add[A](x: A, y: A)(implicit adder: Adder[A]): A =
  adder.addImpl(x, y)

Overloaded wrapping without overloading

While a bit more ceremonious, this allows us to write some nice functions more easily. Here’s a function to add three values.

def addThree[A: Adder](l: A, m: A, r: A): A =
  add(l, add(m, r))

addThree supports all three “overloads” of add.

scala> addThree(1, 2, 3)
res0: Int = 6

scala> addThree("a", "ba", "cus")
res1: String = abacus

scala> addThree(Vector(1, 1), Vector(0, 0), Vector(1, 0, 0, 1))
res2: scala.collection.immutable.Vector[Int] =
  Vector(1, 1, 0, 0, 1, 0, 0, 1)

With the overload style, we need three variants of this function, too, each with the exact same body. The typeclass version need only be written once, and automatically supports new overloads, that is, new instances of Adder.

Same with this function.

def zipAdd[A: Adder](l: List[A], r: List[A]): List[A] =
  l zip r map {case (x, y) => add(x, y)}

Functions like addThree and zipAdd are called derived combinators. The more that you can do in derived combinators, the more abstract and orthogonal your program will be.

   +=============+            |   +=============+
   |   derived   |  (open     |   |  primitive  |  (closed
   | combinators |    set)    |   | combinators |      set)
   +=============+            |   +=============+
                              |
   +----------+               |    +----------+
   | addThree |---→---→---→---→---→| Adder    |
   +----------+       calls   |    | -addImpl |    +===========+
    ↑                       |→---→ +----------+    | Instances |
    | +--------+            | |                    +===========+
    | | zipAdd |---→---→---→- |
    | +--------+    calls     |  +------+ +---------+ +-------+
    ↑        ↑                |  | Ints | | Strings | | Vects |
    |   calls|                |  +------+ +---------+ +-------+
    |      +-----+            |      |
    |      | ??? |            |      |
    ↑      +-----+            |      |
    |  (derived combinators          ↓
    |   can derive from each other)  ---→---→---→
    |                                           |
    ↑          -------------------------------  |
    |          To evaluate `addThree(1, 2, 3)`  |
    |          -------------------------------  ↓
    |          1. Fetch `Adder` implicitly      |
    |-←---←---←---←---←---←---←---←---←---←---←-|
               2. Pass to `addThree`
               3. `addThree` uses the abstract interface to
                  invoke the primitive `add` combinator on what,
                  to it, is an abstract type, `A`.

Infinite overloads via recursion

Making derived combinators easier to write is very useful, but typeclasses go further by letting you describe overloading rules that would be impossible with normal overloading.

Given that I can add Int and Int together, I should be able to add (Int, Int) and (Int, Int) to get (Int, Int).

implicit val addIntPairs: Adder[(Int, Int)] =
  Adder{case ((x1, x2), (y1, y2)) =>
    (x1 + y1, x2 + y2)}
    
scala> add((2, 7), (3, 8))
res3: (Int, Int) = (5,15)

But I should also be able to add pairs of String. And (Int, String) pairs. And (String, Vector[Boolean]) pairs. And pairs of pairs of pairs.

Typeclasses let you declare newly supported types recursively, with an implicit argument list to the implicit def.

implicit def addPairs[A: Adder, B: Adder]
    : Adder[(A, B)] =
  Adder{case ((a1, b1), (a2, b2)) =>
    (add(a1, a2), add(b1, b2))
  }

Surely this must be going somewhere new

If you’re familiar with type classes, all this must be old hat. But this time, we’re going to expand the boundaries of the typeclass design space, by exploiting GADT pattern matching.

We could have designed the Adder type class to include addThree as a primitive combinator, and implemented it afresh for each of the four instances we’ve defined so far, as well as any future instances someone might define. Thinking orthogonally, however, shows us that there’s a more primitive concept which strictly generalizes it: if we primitively define a two-value adder, we can use it to add three items, simply by using it twice.

This has a direct impact on how we structure the functions related to Adder. The primitives must be split up, their separate implementations appearing directly in the implicit instances. Derived combinators may occur anywhere that is convenient to us: outside the typeclass for full flexibility of location, or within the typeclass for possible overrides for performance.

But how much of the primitive implementations must occur in the instances, really?

Empty tags as instances

There is a progression of design refinements here.

  1. Ad hoc overloads, Java-style, impossible to abstract over.
  2. Flip into a typeclass.
  3. Refine the primitive/derived distinction to minimize code in instances.

For some typeclasses, no code needs to be put in the instances. For example, if we want to support only Int, String, and Vector, here is a perfectly sensible typeclass definition.

sealed trait ISAdder[A]

object ISAdder {
  implicit object AddInts extends ISAdder[Int]
  implicit object AddStrs extends ISAdder[String]
  
  final case class AddVects[E]() extends ISAdder[Vector[E]]
  
  implicit def addVects[E]: ISAdder[Vector[E]] =
    AddVects()
}

If the instances cannot add values of the types indicated by the type parameters, surely that code must exist somewhere! And it has a place, in the definition of add.

If you recall, this method merely called addImpl on the typeclass instance before. Now there is no such thing; the instances are empty.

Well, they are not quite empty; they contain a type. So we can define add, with complete type safety, as follows.

def isadd[A](x: A, y: A)(implicit adder: ISAdder[A]): A =
  adder match {
    case ISAdder.AddInts => x + y
    case ISAdder.AddStrings => s"$x$y"
    case ISAdder.AddVects() =>
      x ++ y
  }

More specifically, they contain a runtime tag, which allows information about the type of A to be extracted with a pattern match. For example, determining that adder is AddInts reveals that A = Int, because that’s what the extends clause says. This is GADT pattern matching.

The Vector case is a little tricky here, because we can only determine that A is Vector[e] for some unknown e, but that’s enough information to invoke ++ and get a result also of Vector[e] for the same e.

You can see this in action by using a variable type pattern to assign the name e (a lowercase type parameter is required for this usage), so you can refer to it in types.

    case _: ISAdder.AddVects[e] =>
      (x: Vector[e]) ++ y

The lowercase e names a GADT skolem

In the AddVects[e] pattern immediately above, e is a variable type pattern. This is a type that exists only in the scope of the case.

It’s existential because we don’t know what it is, only that it is some type and we don’t get to pick here what that is. In this way, it is no different from a type parameter’s treatment by the implementation, which is existential on the inside.

It’s a GADT skolem because it was bound by the pattern matching mechanism to a “fresh” type, unequal to any other. Recall the way AddVects was defined:

AddVects[E] extends ISAdder[Vector[E]]

Matching ISAdder with AddVects doesn’t tell us anything about bounds on the type passed to AddVects at construction time. This isn’t true of all GADT skolems, but is only natural for this one.

scalac will create this GADT skolem regardless of whether we give it a name. In the pattern case AddVects(), it’s still known that A = Vector[e] for some e; the only difference is that you haven’t bound the e name, so you can’t actually refer to this unspeakable type.

Usually, you do not need to assign names such as e to such types; _ is sufficient. However, if you have problems getting scalac to apply all the type equalities it ought to know about, a good first step is to assign names to any skolems and try type ascriptions. You’ll need a variable type pattern in other situations that don’t infer, too. By contrast, with the e name bound, we can confirm that x: Vector[e] in the above example, and y is sufficiently well-typed for the whole expression to type-check.

Porting addPairs and other recursive cases

Suppose we add support for pairs to ISAdder.

final case class AddPairs[A, B](
    val fst: ISAdder[A],
    val snd: ISAdder[B]
  ) extends ISAdder[(A, B)]

This should permit us to pattern-match in isadd to make complex determinations about the A type given to isadd. This ought to be a big win for GADT-style typeclasses, allowing “short-circuiting” patterns that work in an obvious way.

// this pattern means A=(Int, String)
case AddPairs(AddInts, AddStrs) =>

// this pattern means A=(ea, Vector[eb])
// where ea and eb are GADT skolems
case AddPairs(fst, _: AddVects[eb]) =>

// here, A=(ea, eb) (again, GADT skolems)
// calling `isadd` recursively is the most
// straightforward implementation
case AddPairs(fst, snd) =>
  val (f1, s1) = x
  val (f2, s2) = y
  (isadd(f1, f2)(fst), isadd(s1, s2)(snd))

The final case’s body is fine. scalac effectively introduces skolems ea and eb so that A = (ea, eb), fst: Adder[ea], and so on, and everything lines up nicely. We are not so lucky with the other cases.

....scala:76: pattern type is incompatible with expected type;
 found   : ISAdder.AddInts.type
 required: ISAdder[Any]
      case AddPairs(AddInts, AddStrs) =>
                    ^
....scala:76: pattern type is incompatible with expected type;

 found   : ISAdder.AddStrs.type
 required: ISAdder[Any]
      case AddPairs(AddInts, AddStrs) =>
                             ^
....scala:79: pattern type is incompatible with expected type;
 found   : ISAdder.AddVects[eb]
 required: ISAdder[Any]
      case AddPairs(fst, _: AddVects[eb]) =>
                            ^

This is nonsensical; the underlying code is sound, we just have to go the long way around so that scalac doesn’t get confused. Instead of the above form, you must assign names to the AddPairs skolems as we described above, and do a sub-pattern-match.

case p: AddPairs[ea, eb] =>
  val (f1, s1) = x
  val (f2, s2) = y
  (p.fst, p.snd) match {
    case (AddInts, AddStrs) =>
    case (fst, _: AddVects[eb]) =>
    case (fst, snd) =>

Note that we had to give up on the AddPairs pattern entirely, because

  1. More complex situations require type ascription.
  2. You cannot ascribe with skolems unless you’ve bound the skolems to names with variable type patterns.
  3. You can’t use variable type patterns with the structural “ADT-style” patterns; you must instead use inelegant and inconvenient [non-variable] type patterns. (This may be improved in Typelevel Scala 4.)

Yet this remains entirely up to shortcomings in the current pattern matcher implementation. An improved pattern matcher could make the nice version work, safely and soundly.

As such, I don’t want these shortcomings to discourage you from trying out the pure type-tagging, “GADT-style” typeclasses. It is simply nicer for many applications, and you aren’t going to code yourself into a hole with them, because should you wind up in the buggy territory we’ve been exploring, there’s still a way out.

Same typeclass, new “primitive” combinators

“Empty” typeclasses like ISAdder contain no implementations of primitive combinators, only “tags”. As such, they are in a sense the purest form of “typeclass”; to classify types is the beginning and end of what they do!

Every type that is a member of the “class of types” ISAdder is either

  1. the type Int,
  2. the type String,
  3. a type Vector[e], where e is any type, or
  4. a type (x, y) where x and y are types that are also in the ISAdder class.

This is the end of ISAdder’s definition; in particular, there is nothing here about “adding two values to get a value”. All that is said is what types are in the class!

Given this ‘undefinedness’, if we have another function we want to write over the exact same class-of-types, we can just write it without making any changes to ISAdder.

def backwards[X](x: X)(implicit adder: ISAdder[X]): X = adder match {
  case AddInts => -x
  case AddStrs => x.reverse
  case _: AddVects[e] => x.reverse
  case p: AddPairs[ea, eb] =>
    val (a, b): (ea, eb) = x
    (backwards(a)(p.fst), backwards(b)(p.snd))
}

Set aside the question of whether the class of “backwards-able” types ought to remain in lockstep with the class of “addable” types. Supposing that it should, the class need be defined only once.

More practically speaking, if you expose the subclasses of a typeclass to users of your library, they can define primitives “in lockstep”, too. The line between primitive and derived combinators is also blurred: a would-be derived combinator can pattern-match on the typeclass to supply special cases for improved performance, becoming “semi-primitive” in the process. You decide whether these are good things or not.

Hybrid “clopen” typeclasses

Pattern-matching typeclass GADTs is subject to the same exhaustiveness concerns and compiler warnings as pattern-matching ordinary ADTs. If you eliminate a case from def isadd, you’ll see something like

....scala:57: match may not be exhaustive.
It would fail on the following input: AddInts
    adder match {
    ^

We could unseal ISAdder, which would eliminate the warning, but wouldn’t really solve anything. The function would still crash upon encountering the missing case.

Pattern matches of unsealed hierarchies typically include a “fallback” case, code used when none of the “special” cases match. However, for pure typeclasses like ISAdder, this strategy is a dead end too. Consider a hypothetical fallback case.

  case _ => ???

Each of the other patterns in isadd, by their success, taught us something useful about the A type parameter. For example, case AddInts tells us that A = Int, and accordingly x: Int and y: Int. It also meant that the expected result type of that block is also Int. That’s plenty of information to actually implement “adding”.

By contrast, case _ tells us nothing about the A type. We don’t know anything new about x, y, or the type of value we ought to return. All we can do is return either x or y without further combination; while this is a sort of “adding” in abstract algebra, there’s a good chance it’s not really what the caller was expecting.

Instead, we can reformulate a closed typeclass like ISAdder with one extension point, where the typeclass is specially encoded in the usual “embedded implementation” style. It’s closed and open, so “clopen”.

sealed doesn’t seal subclasses

Our GADT typeclass instances work by embedding type information within the instances, to be rediscovered at runtime. To support open extension, we need a data case that contains functions instead of types. We know how to encode that, because that is how standard, non-GADT typeclasses work.

sealed trait ISOAdder[A]

trait ExtISOAdder[A] extends ISOAdder[A] {
  val addImpl: (A, A) => A
}

object ISOAdder {
  implicit object AddInts extends ISOAdder[Int]
  implicit object AddStrs extends ISOAdder[String]
  
  final class AddVects[A] extends ISOAdder[Vector[A]]
  
  implicit def addVects[A]: ISOAdder[Vector[A]] =
    new AddVects
    
  def isoadd[A](x: A, y: A)(implicit adder: ISOAdder[A]): A =
    adder match {
      case AddInts => x + y
      case AddStrs => s"$x$y"
      case _: AddVects[e] =>
        (x: Vector[e]) ++ y
      // NB: no unchecked warning here, which makes sense
      case e: ExtISOAdder[A] =>
        e.addImpl(x, y)
    }
}

By sealing ISOAdder, we ensure that the pattern match in isoadd remains exhaustive. However, one of those cases, ExtISOAdder, admits new subclasses, itself! This is fine because no matter how many subclasses of ExtISOAdder we make, they’ll still match the last pattern of isoadd.

We could also define ExtISOAdder as a final case class. The point is that you can make this “extension point” in your otherwise-closed typeclass using whatever style you like.

One caveat, though: “clopen” typeclasses cannot have arbitrary new primitive combinators added to them. They are like ordinary open typeclasses in that regard. Consider a version of backwards for ISOAdder: what you could do in the ExtISOAdder case?

Whoever you like

With type parameters vs. members, you can get pretty far with the “rule of thumb”. Beyond that, even bugs in scalac typechecking can guide you to the “right” choice.

There is no similar rule for this design space. It might seem that typeclass newcomers might have an easier time with the OO-style “unimplemented method” signposts in the open style, but I have also seen them lament the loss of flexibility that would be provided by the GADT style.

Likewise, as an advanced practitioner, your heart will be rent by the tug-of-war between the boilerplate of the open style and the pattern-matcher’s finickiness with the GADT style. You may then be tempted to adopt the hybrid ‘clopen’ style, but this, too, is too often a form of design excess.

Given all that, the only help I can offer, aside from describing the design space above, is “pick whichever you like”. You know your program; if you are not sure which will be nicer, try both!

This article was tested with Scala 2.12.4.

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