Higher Leibniz

by Stephen Compall on Sep 20, 2014


We’ve previously seen the basic implementation and motivation for scalaz.Leibniz. But there’s still quite a bit more to this traditionally esoteric member of the Scalaz collection of well-typed stuff.

Strictly necessarily strict

The word “witness” implies that Leibniz is a passive bystander in your function; sitting back and telling you that some type is equal to another type, otherwise content to let the real code do the real work. The fact that Leibniz lifts into functions (which are a member of the everything set, you’ll agree) might reinforce the notion that Leibniz is spooky action at a distance.

But one of the nice things about Leibniz is that there’s really no cheating: the value with its shiny new type is dependent on the Leibniz actually existing, and its subst, however much a glorified identity function it might be, completing successfully.

To see this in action, let’s check in with the bastion of not evaluating stuff, Haskell.

The Haskell implementation

{-# LANGUAGE RankNTypes, PolyKinds #-}
module Leib
  ( Leib()
  , subst
  , lift
  , symm
  , compose
  ) where

import Data.Functor

data Leib a b = Leib {
  subst :: forall f. f a -> f b

refl :: Leib a a
refl = Leib id

lift :: Leib a b -> Leib (f a) (f b)
lift ab = runOn . subst ab . On $ refl

newtype On c f a b = On {
  runOn :: c (f a) (f b)

symm :: Leib a b -> Leib b a
symm ab = runDual . subst ab . Dual $ refl

newtype Dual c a b = Dual {
  runDual :: c b a

compose :: Leib b c -> Leib a b -> Leib a c
compose = subst

We use newtypes in place of type lambdas, and a value instead of a method, but the implementation is otherwise identical.

It’s really there

OK. Let’s try to make a fake Leib.

badForce :: Leib a b
badForce = Leib $ \_ -> error "sorry for fibbing"

The following code will signal an error only if forcing the head cons of the substed list signals such an error. We never give Haskell the chance to force anything else.

λ> subst (badForce :: Leib Int String) [42] `seq` 33
*** Exception: sorry for fibbing

Oh well, let’s try to bury it behind combinators.

λ> subst (symm . symm $ badForce :: Leib Int String) [42] `seq` 33
*** Exception: sorry for fibbing
λ> subst (compose refl $ badForce :: Leib Int String) [42] `seq` 33
*** Exception: sorry for fibbing

Hmm. We have two properties:

  1. The id from refl? The type-substituted data actually goes through that function. The same goes for the subst method in Scala.
  2. When using Leibniz combinators, the strictness forms a chain to all underlying Leibniz evidence. If there are any missing values, the transform will also fail.

Higher kinded Leibniz

Let’s try a variant on Leib.

sealed abstract class LeibF[G[_], H[_]] {
  def subst[F[_[_]]](fa: F[G]): F[H]

This reads “LeibF[G, H] can replace G with H in any type function”. But, whereas the kind of the types that Leib discusses is *, for LeibF it’s *->*. So, LeibF[List, List] exhibits that the type constructors List and List are equal.

implicit def refl[G[_]]: LeibF[G, G] = new LeibF[G, G] {
  override def subst[F[_[_]]](fa: F[G]): F[G] = fa

Interestingly, except for the kinds of type parameters, these definitions are exactly the same as for Leib. Does that hold for lift?

def lift[F[_[_], _], A[_] , B[_]](ab: LeibF[A, B]): LeibF[F[A, ?], F[B, ?]] =
  ab.subst[Lambda[x[_] => LeibF[F[A, ?], F[x, ?]]]](LeibF.refl[F[A, ?]])

Despite that we are positively buried in type lambdas (yet moderated by Kind Projector) now, absolutely!

As an exercise, adapt your symm and compose methods from the last part for LeibF, by only changing type parameters and switching any refl references.

def symm[A[_], B[_]](ab: LeibF[A, B]): LeibF[B, A]
def compose[A[_], B[_], C[_]](ab: LeibF[A, B], bc: LeibF[B, C]): LeibF[A, C]

You can write a Leibniz and associated combinators for types of any kind; the principles and implementation techniques outlined above for types of kind *->* apply to all kinds.

Whence PolyKinds?

You have to define a new Leib variant and set of combinators for each kind you wish to support. There is no need to do this in Haskell, though.

λ> :k Leib []
Leib [] :: (* -> *) -> *
λ> :t refl :: Leib [] []
refl :: Leib [] [] :: Leib [] []
λ> :t lift (refl :: Leib [] [])
lift (refl :: Leib [] []) :: Leib (f []) (f [])
λ> :t compose (refl :: Leib [] [])
compose (refl :: Leib [] []) :: Leib a [] -> Leib a []

In Haskell, we can take advantage of the fact that the actual implementations are kind-agnostic, by having those definitions be applicable to all kinds via the PolyKinds language extension, mentioned at the top of the Haskell code above. No such luck in Scala.

Better GADTs

In a post from a couple months ago, Kenji Yoshida outlines an interesting way to simulate the missing type-evidence features of Scala’s GADT support with Leibniz. This works in Haskell, too, in case you are comfortable with turning on RankNTypes but not GADTs somehow.

Let’s examine Kenji’s GADT.

sealed abstract class Foo[A, B]
final case class X[A]() extends Foo[A, A]
final case class Y[A, B](a: A, b: B) extends Foo[A, B]

For completeness, let’s also see the Haskell version, including the function that demands so much hoop-jumping in Scala, but just works in Haskell.

module FooXY where

data Foo a b where
  X :: Foo a a
  Y :: a -> b -> Foo a b

hoge :: Foo a b -> f a c -> f b c
hoge X bar = bar

Note that the Haskell type system understands that when hoge’s first argument’s data constructor is X, the type variables a and b must be the same type, and therefore by implication the argument of type f a c must also be of type f b c. This is what we’re trying to get Scala to understand.

def hoge1[F[_, _], A, B, C](foo: Foo[A, B], bar: F[A, C]): F[B, C] =
  foo match {
    case X() => bar

This transliteration of the above Haskell hoge function fails to compile, as Kenji notes, with the following:

…/LeibnizArticle.scala:39: type mismatch;
 found   : bar.type (with underlying type F[A,C])
 required: F[B,C]
      case X() => bar

The overridden cata method

Kenji introduces a cata method on Foo to constrain use of the Leibniz.force hack, while still providing external code with usable Leibniz evidence that can be lifted to implement hoge. However, by implementing the method in a slightly different way, we can use refl instead.

sealed abstract class Foo[A, B] {
  def cata[Z](x: (A Leib B) => Z, y: (A, B) => Z): Z

final case class X[A]() extends Foo[A, A] {
  def cata[Z](x: (A Leib A) => Z, y: (A, A) => Z) =

final case class Y[A, B](a: A, b: B) extends Foo[A, B] {
  def cata[Z](x: (A Leib B) => Z, y: (A, B) => Z) =
    y(a, b)

Now we can replace the pattern match (and all other such pattern matches) with an equivalent cata invocation.

def hoge2[F[_, _], A, B, C](foo: Foo[A, B], bar: F[A, C]): F[B, C] =
  foo.cata(x => x.subst[F[?, C]](bar),
           (_, _) => sys error "nonexhaustive")

So why can we get away with Leib.refl, whereas the function version Kenji presents cannot? Compare the cata signature in Foo versus X:

  def cata[Z](x: (A Leib B) => Z, y: (A, B) => Z): Z
  def cata[Z](x: (A Leib A) => Z, y: (A, A) => Z): Z

We supplied A for both the A and B type parameters in our extends clause, so that substitution also applies in all methods from Foo that we’re implementing, including cata. At that point it’s obvious to the compiler that refl implements the requested Leib.

Incidentally, a similar style of substitution underlies the definition of refl.

The Leib member

What if we don’t want to write or maintain an overriding-style cata? After all, that’s an n² commitment. Instead, we can incorporate a Leib value in the GADT. First, let’s see what the equivalent Haskell is, without the GADTs extension:

data Foo a b = X (Leib a b) | Y a b

hoge :: Foo a b -> f a c -> f b c
hoge (X leib) bar = runDual . subst leib . Dual $ bar

We needed RankNTypes to implement Leib, of course, but perhaps that’s acceptable. It’s useful in Ermine, which supports rank-N types but not GADTs as of this writing.

The above is simple enough to port to Scala, though.

sealed abstract class Foo[A, B]
final case class X[A, B](leib: Leib[A, B]) extends Foo[A, B]
final case class Y[A, B](a: A, b: B) extends Foo[A, B]

def hoge3[F[_, _], A, B, C](foo: Foo[A, B], bar: F[A, C]): F[B, C] =
  foo match {
    case X(leib) => leib.subst[F[?, C]](bar)

It feels a little weird that X now must retain Foo’s type-system-level separation of the two type parameters. But this style may more naturally integrate in your ADTs, and it is much closer to the original non-working hoge1 implementation.

It also feels a little weird that you have to waste a slot carting around this evidence of type equality. As demonstrated in section “It’s really there” above, though, it matters that the instance exists.

You can play games with this definition to make it easier to supply the wholly mechanical leib argument to X, e.g. adding it as an implicit val in the second parameter list so it can be imported and implicitly supplied on X construction. The basic technique is exactly the same as above, though.

Leibniz mastery

This time we talked about

  • Why it matters that subst always executes to use a type equality,
  • the Haskell implementation,
  • higher-kinded type equalities and their Leibnizes,
  • simulating GADTs with Leibniz members of data constructors.

This article was tested with Scala 2.11.2, Kind Projector 0.5.2, and GHC 7.8.3.