djspiewak / skolems

A microlibrary for Scala encodings of higher-rank quantifiers

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Skolems Build Status

This library simply contains a few different encodings of universal and existential quantification of types. Whereas in Haskell one might write something like this (with appropriate extensions):

type Uni f g = forall a . f a -> g a
type Exi a = exists b . (b, b -> a)

Unfortunately, Scala does not directly provide support for universally quantified higher-rank types, and its support for existential quantification of higher-rank types is spotty and extremely buggy. This project provides better, more uniform alternatives with sane type inference and sound behavior.

As a note, in most of the documentation I will be using the symbols ∀ ("for all") and ∃ ("exists"). Skolems allows you to use these symbols exactly as written, or if you prefer Latin characters, you can use Forall and Exists. Similarly, kind-projector allows you to use Lambda rather than λ if you find that easier to read (or type). I also have a habit of using α (lowercase "alpha") as a variable in my type lambdas, but this is not in any way required.


libraryDependencies += "com.codecommit" %% "skolems" % "<version>"

Published for Scala 2.13, 2.12, and 2.11. This library does not have any upstream dependencies (though I would dearly love to add a Cats module at some point, since Forall and Exists both form useful classes).


Just to conceptualize some of this, it's worth taking a moment to digress on the functionality available in this department with vanilla Scala 2.13 (note that Scala 3 improves on this in some important areas, and actually makes it worse in others). Also note that when I say "vanilla Scala", I actually mean "vanilla Scala with kind-projector", since it's almost impossible to talk about any of this stuff without kind-projector.

In Scala today, what follows is the only way to express universal quantification:

def foo[A](a: A): List[A] = List(a)

This function is universally quantified over A. We often refer to this special case of quantification as polymorphism, or sometimes as "parametricity" or "generics". Whatever word you choose to use, it all comes down to the same thing. Unfortunately, this functionality in the language has a very important limitation: you cannot write it as a value. As an example:

def foo[A](a: A): List[A] = List(a)
val otherFoo = foo(_)    // ?????

That doesn't work. Or rather, it works in the sense that it compiles, but the resulting otherFoo will have the very unhelpful Nothing => List[Nothing] type signature, which is not at all the type that foo has. This is because the quantification of foo is lost when it is expressed as a free value, which is why this form of quantification (as present in Scala) is sometimes referred to as let-bound polymorphism, since the polymorphism is lost if you detach the value (which is to say, the function) from its declaration.

Now, let-bound polymorphism is quite useful, but it is simply insufficient for many cases. As a trivial example, imagine we wanted to write a function that takes two parameters, one String and one Int, and also takes a function with the same theoretical type signature as foo above, and then applies it to both. So in other words, callers of the function would pass their String and Int values, as well as foo (or something with the same type as foo), and this function would apply the foo-like thing to both types.

We can't write this today in Scala. Writing it in Haskell would look something like this:

bippy :: String -> Int -> (forall a . a -> [a]) -> ([String], [Int])
bippy s i f = (f s, f i)

That's actually... really straightforward, but you notice the trick: Haskell is allowing us to write the type of a value which itself is universally quantified. In other words, in Haskell, foo isn't stuck being a def, it is actually a value just like anything else.

Skolems allows you to write this function in Scala. It looks like this:

import skolems.

def bippy(s: String, i: Int, f: ∀[λ[α => α => List[α]]]): (List[String], List[Int]) =
  (f[String](s), f[Int](i))

Or, if you prefer the Latin character version:

import skolems.Forall

def bippy(
    s: String,
    i: Int,
    f: Forall[Lambda[a => a => List[a]]])
    : (List[String], List[Int]) =
  (f[String](s), f[Int](i))

Calling this function is relatively straightforward as well:

def foo[A](a: A): List[A] = List(a)

bippy("hi", 42, ∀[λ[α => α => List[α]]](foo))   // => (List("hi"), List(42))

In some cases, it may be cleaner to use a named type alias rather than the type lambda:

def foo[A](a: A): List[A] = List(a)

type ListConstr[A] = A => List[A]
bippy("hi", 42, ∀[ListConstr](foo))   // => (List("hi"), List(42))

So far, we've only looked at universal quantification, but what about existential? Here, Scala has, on paper, slightly better support in the form of the forSome type operator:

def foo(unapplied: (A, A => String) forSome { type A }): String = {
  val (v, f) = unapplied

foo((42, (i: Int) => i.toString))
foo(("hi", (s: String) => s))

Or, more interestingly:

def foo(unapplied: List[(A, A => String) forSome { type A }]): List[String] =
  unapplied map {
    case (v, f) => f(v)

    (42, (i: Int) => i.toString),
    ("hi", (s: String) => s)))

Unfortunately, forSome is extremely brittle. The compiler will often "lose" the quantifier as it passes through various transformations in more complex types. Additionally, there are certain cases where the compiler will do straight-up unsound things with the type. For example, I've seen scalac convert the following types into one another (or rather, more complex variants of these types):

type One = (F[A] forSome { type A }) => B
type Two = (F[A] => B) forSome { type A }

These are not the same type! Not even close. In fact, One and Two could actually be rewritten as the following:

type Neo = ∀[λ[α => F[α] => B]]
type Wot = ∀[F] => B

In other words, pretending that One and Two are the same type (which, as I said, I've seen the compiler do at times when using forSome) is not just limiting, it's actually wrong and can generate type unsoundness. tldr, forSome is buggy in Scala, and you should try not to use it.

But of course, if you don't use forSome, then your only option for encoding existential quantification is the ad-hoc approach using type members. For example:

trait Unapplied {
  type A
  val a: A
  def apply(v: A): String

def foo(unapplied: List[Unapplied]): List[String] =
  unapplied map { u =>

    new Unapplied {
      type A = Int
      val a = 42
      def apply(i: Int): String = i.toString
    new Unapplied {
      type A = String
      val a = "hi"
      def apply(s: String): String = s

This works a lot better than the forSome approach, and the Scala compiler is much better about not losing its mind when using this encoding, but it's a little awkward to create a new wrapper type every time you need an existential. Also it's unbelievably verbose.

For this reason, Skolems introduces the Exists () type, which is analogous to Forall () except for existential rather than universal types.

import skolems._

def foo(unapplied: List[∃[λ[α => (α, α => String)]]]): List[String] =
  unapplied map { u =>
    val (v, f) = u.value

    ∃[λ[α => (α, α => String)]]((42, (i: Int) => i.toString)),
    ∃[λ[α => (α, α => String)]](("hi", (s: String) => s))))

Basically, think of Exists as a "better forSome". It will type infer sanely and it's almost exactly as easy to use.

Implicit Evidence

Another interesting use of higher-rank universal types is in defining polymorphic implicit values. As an example, Cats defines an instance of Monad for Either[A, ?], for any type A. This is defined in the following way:

implicit def eitherMonad[A]: Monad[Either[A, ?]] = ???

Here again we see let-bound polymorphism rearing its ugly head. While it is possible to define such a value (a monad for Either for all type instantiations), it isn't possible to take it as a parameter. For example:

// we can write this function!
def foo[F[_], A](fa: F[A])(implicit F: Monad[F]) = ???

// but writing this function is awkward
def bar[F[_, _], A, B, C](
    one: F[A, C],
    two: F[B, C])(
    implicit F1: Monad[F[A, ?]],
    F2: Monad[F[B, ?]]) = ???

Notice how we had to take two Monad instances (F1 and F2) for the same type constructor, which is more than a little weird when the instances are actually the same instance: eitherMonad. There are other cases where this problem can get even worse, or even intractable.

What we really want to write is this:

def bar[F[_, _], A, B, C](
    one: F[A, C],
    two: F[B, C])(
    implicit F: ∀[λ[α => Monad[F[α, ?]]]]) = ???

In other words, a universally quantified version of the Monad which has the same expressivity as the original definition, eitherMonad.

With Skolems, you can do exactly this. In fact, the above works exactly as written, and you can even call it in exactly the way you expect!

bar[Either, String, Boolean, Int](Right(42), Left(false))

And that's it. It just works. In fact, you can even omit the type parameters:

bar(Right(42), Left(false))

That works too.

The reason this works is Skolems is able to materialize an implicit Forall type given an implicit definition that uses let-bound polymorphism (such as eitherMonad). Unfortunately, it cannot currently go in the other direction. In other words:

def foo[F[_, _]](implicit F: ∀[λ[α => Monad[F[α, ?]]]]) = Monad[F[String, ?]]  // nope!

This doesn't work. Yet. The problem is that scalac lacks a particular form of symbolic unification in its type checker. This isn't really a theoretical problem, scalac's implementation just can't handle the kind of equation that is necessary to typecheck this. Skolems will add support for this using a macro in an upcoming release.

But until then... you can materialize the instance manually:

def foo[F[_, _]](implicit F: ∀[λ[α => Monad[F[α, ?]]]]) = {
  implicit val fs = F[String]
  implicit val fi = F[Int]

  Monad[F[String, ?]]  // no problem!
  Monad[F[Int, ?]]     // also cool

That works fine for a lot of cases, and we'll be making things even better soon!

As an aside, the same implicit materialization should work for existentially-quantified implicit declarations just the same as it works for universally-quantified ones, but I haven't been able to come up with a good motivating example for this.


There are a number of identities which hold for rank-n quantification in first-order logic (which is to say, Scala's type system). These identities are very difficult to access and highly opaque when using let-bound polymorphism and forSome, but Skolems can make them very easy and direct. These identities are specifically as follows (with their corresponding implementation in the API):

    • Left-to-right: Forall.raise
    • Right-to-left: Forall.lower
      • Also Forall.lowerA if the let-bound polymorphic quantifier encoding is more useful than the Forall-based version
    • Left-to-right: Exists.raise
    • Right-to-left: Exists.lower
      • Also Exists.lowerE if the forSome quantifier encoding is more useful than the Exists-based version

As a note on the naming convention, you should think of raise as "raise the rank". In other words, you're going from a rank-1 type (with the quantifier on the outside) to a rank-2 type (with the quantifier on the inside). Obviously, lower is the inverse. These functions are relatively trivial to define (except for Exists.lower, which requires an asInstanceOf due to the fact that Scala only has local type inference), but it's nice to have them already available.

Related Work

The encodings in this project occurred in their (to my knowledge) original forms in Scalaz. The use of the inexpressible sentinel type as a mechanism for inferring a rank-2 universal from a Scala expression has varied origins. Miles Sabin mentioned the trick to me a few years ago and had some (now discarded) prototype constructs in Shapeless which took advantage of it. Ed Kmett has also discussed it as an encoding which infers better, though I cannot currently find a link to his use of it, and based on my memory of what he was doing, it may have been unsound all along. Scalaz's Forall has a peculiar apply function which uses a doubly-negated existential type (expressed using forSome and A => Nothing) to infer a rank-2 universal. This trick was also devised originally by Miles Sabin (waaaaaay back in the mists of time), and if you look very very closely, it's really just yet another way of expressing the "inexpressible sentinel type" idea. As for the idea of using an abstract type member within an object to encode the sentinel, I first learned of this from Kai, though Alex Konovalov also uses this extensively in several of his projects and I honestly don't know where it originates.

Speaking of Alex, he has a really interesting library called polymorphic which may be a better use of your time than my failings here! Some differences between polymorphic and skolems:

  • His Forall and Exists are entirely unboxed
  • He provides a very clever Instance constructor for the common implicit case of existential pairs
  • He also provides Pi and Sigma for dependent type quantification
  • Polymorphic depends on cats-core, while Skolems has no dependencies (for better or worse)
  • Skolems defines a mechanism for materializing rank-n implicit values, while this is not provided by polymorphic in any form except Instance
  • The type inference seems to be nicer with polymorphic than with skolems