Chuusai

Type-level selection sort in shapeless

Miles Sabin — Fri, 27 Jan 2012 14:07:54 +0000

To celebrate my talk on shapeless being selected for this year’s Northeast Scala Symposium in Boston, I thought I’d share something entertaining (and slightly whimsical) with you — compile time selection sort at the type-level!

Selection sort is just about the simplest possible sorting algorithm — select the least element from your unsorted list as the first element of the sorted list, then recursively sort the remainder — so I’ll take it as a given, and focus on the interesting bit … how can this possibly be done at the type-level?

The first thing we need is a type-level representation of lists of sortable things. For that, we’re going to use HLists of type-level natural numbers.

import shapeless._
import Nat._
import HList._

def typed[T](t : => T) {}

val unsorted = _3 :: _1 :: _4 :: _0 :: _2 :: HNil
typed[_3 :: _1 :: _4 :: _0 :: _2 :: HNil](unsorted)

Here unsorted is a value-level list of Nat‘s, and we can see that its structure is exactly mirrored in its type — in shapeless each Nat value (_0, _1, …) has a corresponding Nat type with the same name (_0, _1, …). We use the typed function here to verify the type of unsorted rather than use a type annotation on its val declaration to ensure that the act of verifying the inferred type doesn’t itself contribute to type inference.

Now we need a way of capturing order over these type-level Nat‘s. We do that using a type class that witnesses the relationship at the type level,

trait LTEq[A <: Nat, B <: Nat]

object LTEq {
  import Nat._0

  type <=[A <: Nat, B <: Nat] = LTEq[A, B]

  implicit def ltEq1 = new <=[_0, _0] {}
  implicit def ltEq2[B <: Nat] = new <=[_0, Succ[B]] {}
  implicit def ltEq3[A <: Nat, B <: Nat](implicit lt : A <= B) =
    new <=[Succ[A], Succ[B]] {}
}

If you're familiar with Peano arithmetic it should be pretty clear how this works — we have two base cases: _0 is less than or equal to _0, and _0 is less than or equal to the successor of any number; and we have one induction case: if A <= B then A+1 <= B+1. With these definitions in place, the magic of Scala's implicit resolution allows us to ask the compiler about relationships between different Nat types,

scala> import shapeless._ ; import Nat._ ; import LTEq._
import shapeless._
import Nat._
import LTEq._

scala> implicitly[_2 <= _5] // OK
res0: shapeless.LTEq.<=[shapeless.Nat._2,shapeless.Nat._5] =
  shapeless.LTEq$$anon$5@4fc8927c

scala> implicitly[_4 <= _2] // Does not compile
:17: error: could not find implicit value for parameter
  e: shapeless.LTEq.<=[shapeless.Nat._4,shapeless.Nat._2]
    implicitly[_4 <= _2]
              ^

Before we go any further, let's define an operation which witnesses whether or not an HList of Nats is in sorted order — we'll want this to verify that our type-level sort is correct,

trait NonDecreasing[L <: HList]
implicit def hnilNonDecreasing = new NonDecreasing[HNil] {}
implicit def hlistNonDecreasing1[H] = new NonDecreasing[H :: HNil] {}
implicit def hlistNonDecreasing2[H1 <: Nat, H2 <: Nat, T <: HList]
  (implicit ltEq : H1 <= H2, ndt : NonDecreasing[H2 :: T]) =
    new NonDecreasing[H1 :: H2 :: T] {}

def acceptNonDecreasing[L <: HList](l : L)(implicit ni : NonDecreasing[L]) = l

// Verify type-level relations
implicitly[NonDecreasing[_1 :: _2 :: _3 :: HNil]]   // OK
implicitly[NonDecreasing[_1 :: _3 :: _2 :: HNil]]   // Does not compile

// Apply at the value-level
acceptNonDecreasing(_1 :: _2 :: _3 :: HNil)         // OK
acceptNonDecreasing(_1 :: _3 :: _2 :: HNil)         // Does not compile

This is a fairly straighforward induction — we have two base cases: an empty list is in sorted order, as is a list of exactly one element; and we have one induction case: a list is in sorted order if its tail is, and if its first element is less than or equal to its second element.

Now we can define an operation to select the least element from an HList of Nats, returning both it and the remainder of the list,

trait SelectLeast[L <: HList, M <: Nat, Rem <: HList] {
  def apply(l : L) : (M, Rem)
}

trait LowPrioritySelectLeast {
  implicit def hlistSelectLeast1[H <: Nat, T <: HList] =
    new SelectLeast[H :: T, H, T] {
      def apply(l : H :: T) : (H, T) = (l.head, l.tail)
    }
}

object SelectLeast extends LowPrioritySelectLeast {
  implicit def hlistSelectLeast3[H <: Nat, T <: HList, TM <: Nat, TRem <: HList]
    (implicit tsl : SelectLeast[T, TM, TRem], ev : TM < H) =
      new SelectLeast[H :: T, TM, H :: TRem] {
        def apply(l : H :: T) : (TM, H :: TRem) = {
          val (tm, rem) = tsl(l.tail)
          (tm, l.head :: rem)
        }
      }
}

val (l1, r1) = selectLeast(_1 :: _2 :: _3 :: HNil)
typed[_1](l1)
typed[_2 :: _3 :: HNil](r1)

val (l2, r2) = selectLeast(_3 :: _1 :: _4 :: _0 :: _2 :: HNil)
typed[_0](l2)
typed[_3 :: _1 :: _4 :: _2 :: HNil](r2)

Here're we're working at both the type-level and at the value-level — we're selecting the least value-level Nat from each of the examples HLists, and these are being assigned the corresponding Nat types. Hopefully the recursive pattern is beginning to look fairly familiar, even in this slightly more complicated case — we have a base case: the least element of a singleton list is its only element; and the least element of a non-singleton list is the least element of its tail unless its head is already the least element of the entire list.

And now we can sort!

trait SelectionSort[L <: HList, S <: HList] {
  def apply(l : L) : S
}

trait LowPrioritySelectionSort {
  implicit def hlistSelectionSort1[S <: HList] = new SelectionSort[S, S] {
    def apply(l : S) : S = l
  }
}

object SelectionSort extends LowPrioritySelectionSort {
  implicit def hlistSelectionSort2[L <: HList, M <: Nat, Rem <: HList, ST <: HList]
    (implicit sl : SelectLeast[L, M, Rem], sr : SelectionSort[Rem, ST]) =
      new SelectionSort[L, M :: ST] {
        def apply(l : L) = {
          val (m, rem) = sl(l)
          m :: sr(rem)
        }
      }
}

def selectionSort[L <: HList, S <: HList](l : L)
  (implicit sort : SelectionSort[L, S]) = sort(l)

val unsorted = _3 :: _1 :: _4 :: _0 :: _2 :: HNil
typed[_3 :: _1 :: _4 :: _0 :: _2 :: HNil](unsorted)
acceptNonDecreasing(unsorted)  // Does not compile!

val sorted = selectionSort(unsorted)
typed[_0 :: _1 :: _2 :: _3 :: _4 :: HNil](sorted)
acceptNonDecreasing(sorted)    // Compiles!

As you can see, the static type of the sorted value reflects the fact that it is in sorted order, and that consequently, unlike the initial unsorted value, it has the correct type to be passed to the acceptNonDecreasing function that we wrote earlier.

Update — Joni Freeman has done a nice job of porting the algorithm used to Prolog here.

If you've made it this far, and want to play around with it yourself, you can find shapeless on github along with the example source this article is based on. You'll also find a mailing list for discussion around shapeless here. Oh, and if you're going to be at Nescala then I'll see you there ...

shapeless at the Northeast Scala Symposium

Miles Sabin — Fri, 27 Jan 2012 10:51:29 +0000

I’ve been selected (against some really excellent competition — it’s a shame more talks couldn’t be squeezed in) to talk at this year’s Northeast Scala Symposium in Boston. Thanks to everyone who voted for my talk about shapeless. It looks like it’s going to be a wonderful event, and I’m really looking forward to meeting up with everyone there.

In advance of that, I’d also be keen to talk to any Scala-using organizations who would be interested in a two/three day Advanced Scala training course on the East coast in the run up to, or just after the event — in depth coverage of the techniques involved in the creation of shapeless, and how they can be applied elsewhere, will be included. Please get in touch if you’re interested.

shapeless — an exploration of generic programming in Scala

Miles Sabin — Mon, 19 Dec 2011 15:25:21 +0000

shapeless is an exploration of generic (aka polytypic) programming in Scala derived from the various talks I’ve given over the course of 2011 on implementing scrap your boilerplate and higher rank polymorphism in Scala. It’s published under the Apache 2.0 licence and hosted on github — fork it and have a play.

In the new year I’ll be posting a (long overdue) series of articles here on the implementation techniques used — heavily type class based, with essential use of dependent types at various junctures, which together enable the relatively smooth encoding of type level functions. In the meantime you’ll find Olivera, Moors and Odersky Type Classes as Objects and Implicits useful background material.

Selected highlights of shapeless include …

A new encoding of polymorphic function values which optionally supports type specific cases, and which is interoperable with Scala’s ordinary monomorphic function values.

// choose is a function from Sets to Options with no type specific cases
object choose extends (Set ~> Option) {
  def default[T](s : Set[T]) = s.headOption
}

// choose is convertible to an ordinary monomorphic function value
val lo = List(Set(1, 3, 5), Set(2, 4, 6)) map choose
lo == List(Option(1), Option(2))

// size is a function from values of arbitrary type to a 'size' which is
// defined via type specific cases
object size extends (Id ~> Const[Int]#λ) {
  def default[T](t : T) = 1
}
implicit def sizeInt = size.λ[Int](x => 1)
implicit def sizeString = size.λ[String](s => s.length)
implicit def sizeList[T] = size.λ[List[T]](l => l.length)
implicit def sizeOption[T](implicit cases : size.λ[T]) =
  size.λ[Option[T]](t => (t map size).getOrElse(0))
implicit def sizeTuple[T, U](implicit st : size.λ[T], su : size.λ[U]) =
  size.λ[(T, U)](t => size(t._1)+size(t._2))

size(23) == 1
size("foo") == 3
size((23, "foo")) == 4

An implementation of Scrap your Boilerplate with Class which provides generic map and fold operations over arbitrarily nested data structures,

val nested = List(Option(List(Option(List(Option(23))))))

val succ = everywhere(inc)(nested)
succ == List(Option(List(Option(List(Option(24))))))

A Typeable type class which provides a type safe cast operation,

val a : Any = List(Vector("foo", "bar", "baz"), Vector("wibble"))

val lvs : Option[List[Vector[String]]] = a.cast[List[Vector[String]]]
lvs.isDefined == true

val lvi : Option[List[Vector[Int]]] = a.cast[List[Vector[Int]]]
lvi.isEmpty == true

The mother of all Scala HList‘s which, amongst other things, is covariant,
```
trait Fruit
case class Apple() extends Fruit
case class Pear() extends Fruit

type FFFF = Fruit :: Fruit :: Fruit :: Fruit :: HNil
type APAP = Apple :: Pear :: Apple :: Pear :: HNil

val a : Apple = Apple()
val p : Pear = Pear()

val apap : APAP = a :: p :: a :: p :: HNil
val ffff : FFFF = apap  // APAP <: FFFF
```
It has a map operation, applying a polymorphic function value (possibly with type specific cases) across its elements. This means that it subsumes both typical HList's and also KList's (HList's whose elements share a common outer type constructor),
```
type SISS = Set[Int] :: Set[String] :: HNil
type OIOS = Option[Int] :: Option[String] :: HNil

val sets : SISS = Set(1) :: Set("foo") :: HNil
val opts : OIOS = sets map choose
opts == Option(1) :: Option("foo") :: HNil
```
It has a zipper for traversal and persistent update,
```
val l = 1 :: "foo" :: 3.0 :: HNil

val l2 = l.toZipper.right.put("wibble", 45).toHList
l2 == 1 :: ("wibble", 45) :: 3.0 :: HNil

val l3 = l.toZipper.right.delete.toHList
l3 == 1 :: 3.0 :: HNil

val l4 = l.toZipper.last.left.insert("bar").toHList
l4 == 1 :: "foo" :: "bar" :: 3.0 :: HNil
```
It has a unify operation which converts it to an HList of elements of the least upper bound of the original types,
```
val ffff = apap.unify // type inferred as FFFF
```
It supports conversion to an ordinary Scala List of elements of the least upper bound of the original types,
```
val lf = apap.toList  // type inferred as List[Fruit]
lf == List(a, p, a, p)
```
And it has a Typeable type class instance, allowing, eg. vanilla List[Any]'s or HList's with elements of type Any to be safely cast to precisely typed HList's,
```
val ffff : FFFF = apap.unify               // discard precise typing
val apap2 : Option[APAP] = ffff.cast[APAP] // reestablish precise typing
apap2.get == apap
```
These last three features make this HList dramatically more practically useful than HList's are typically thought to be — normally the full type information required to work with them is too fragile to cross subtyping or I/O boundaries. This implementation supports the discarding of precise information where necessary (eg. to serialize a precisely typed record after construction), and its later reconstruction (eg. a weakly typed deserialized record with a known schema can have it's precise typing reestabilished).

Conversions between tuples and HList's, and between ordinary Scala functions of arbitrary arity and functions which take a single corresponding HList argument,

// Round trip from tuple to HList and back
val t1 = (23, "foo", 2.0, true)

val l1 = t1.hlisted
h1 == 23 :: "foo" :: 2.0 :: true :: HNil

val t2 = l1.tupled
t1 == t2

One application of this is the liftO function which lifts an ordinary function of arbitrary arity into Option.

// Lift these ordinary Scala function values into Option
val sum : (Int, Int) => Int = _ + _
val prd : (Int, Int, Int) => Int = _ * _ * _

// Nb. liftO abstracts over the arity of its function arguments
val sumO = liftO(sum) // (Option[Int], Option[Int]) => Option[Int]
val prdO = liftO(prd) // (Option[Int], Option[Int], Option[Int]) => Option[Int]

val s1 = sumO(Some(1), Some(2))
s1 == Option(3)

val s2 = sumO(Some(1), None)
s2 == None

val p1 = prdO(Some(2), Some(3), Some(4))
p1 == Option(24)

val p2 = prdO(Some(2), None, Some(4))
p2 == None

The library is targetted at Scala 2.10-SNAPSHOT by default, but currently should build against Scala 2.9.1.final with the -Ydependent-method-types switch enabled. I make no promises that it'll continue to build with 2.9.x, or even vanilla Scala 2.10-SNAPSHOT — in 2012 I plan to investigate, amongst other things, what can be done with singleton types for literal values — at a minimum they would make the clunky encoding of type level natural numbers redundant, and might enable a whole lot more.

Fork it, kick the tires over the holiday and let me know how you get on!

Functional Dependencies in Scala

Miles Sabin — Sat, 16 Jul 2011 17:02:24 +0000

Functional dependencies are a near-standard extension to Haskell (present in GHC and elsewhere) which allow constraints on the type parameters of type classes to be expressed and then enforced by the compiler. They allow the programmer to state that some of the type parameters of a multi-parameter type class are completely determined by the others. One particularly common application of this is allowing the result type of a function to be a parameter but nevertheless determined by the type(s) of it’s argument(s) — the examples on the Haskell wiki page that I linked to above illustrate that extremely well.

Does this have an equivalent in Scala? We certainly have the equivalent problem — whenever we have multiple type parameters (of a type or method) we might want to be able to express functional dependency-like constraints between those type parameters and have those constraints checked by the compiler. And ideally we want that to get along well with type-inference — we would like the compiler to be able to infer any types which are determined by types which it is already able to infer.

It turns out that there’s a very simple translation of Haskell-style functional dependencies to Scala, but for some reason this doesn’t seem to have been commented on. Quite the opposite in fact — I’ve seen it reported that encoding fundeps in Scala is either impossible, or possible but not useful because of issues with type inference. What makes this even more surprising is that a crucial feature of Scala’s (now not so) new collections framework (CanBuildFrom) appears to be a perfect example of this encoding in practice … strangely this isn’t advertised in Type Classes as Objects and Implicits (Olivera, Odersky and Moors, 2010) either.

Maybe this has all been noticed before and assumed to be too obvious to mention, nevertheless, I think it’s useful to be able to make the connection. I’ll illustrate the translation using the examples from the Haskell wiki. By the time I’m done translating them into Scala the fact that CanBuildFrom is an instance of a fundep should be quite clear.

Let’s start with the matrix/vector/scalar multiplication example. Here we want to express the fact that the result type of the multiplication is fully determined by it’s arguments,

Matrix	*	Matrix	-> Matrix
Matrix	*	Vector	-> Vector
Matrix	*	Int	-> Matrix
Int	*	Matrix	-> Matrix

Here we have a triple of types where the first two (the argument types) determine the third (the result type). Lets express this relationship directly via a trait with three type parameters accompanied by some implicit definitions which capture the relationship we want to enforce,

trait Matrix // Dummy definitions for expository purposes
trait Vector

trait MultDep[A, B, C]
// trait MultDep[-A, -B, +C]    // for Scala <= 2.9.0-1 these variance
//                              // annotations are required

implicit object mmm extends MultDep[Matrix, Matrix, Matrix]
implicit object mvv extends MultDep[Matrix, Vector, Vector]
implicit object mim extends MultDep[Matrix, Int, Matrix]
implicit object imm extends MultDep[Int, Matrix, Matrix]

Given these definitions we can ask the compiler what is and what isn't allowed,

scala> implicitly[MultDep[Matrix, Matrix, Matrix]] // OK: Matrix * Matrix -> Matrix
res0: MultDep[Matrix,Matrix,Matrix] = mmm$@1ddeda2

scala> implicitly[MultDep[Matrix, Vector, Vector]] // OK: Matrix * Vector -> Vector
res1: MultDep[Matrix,Vector,Vector] = mvv$@1a8fb1b

scala> implicitly[MultDep[Matrix, Vector, Matrix]] // Error
:15: error: could not find implicit value for
  parameter e: MultDep[Matrix,Vector,Matrix]
       implicitly[MultDep[Matrix, Vector, Matrix]]

The first two cases are fine, because they both correspond to one of the implicit instances we provided earlier. The third fails, quite rightly, because there is no instance corresponding to Matrix * Vector -> Matrix.

So, we've been able to capture the desired relationship between the type variables. Here's how we put that to use,

def mult[A, B, C](a : A, b : B)(implicit instance : MultDep[A, B, C]) : C =
  error("TODO")

// Type annotations soley to verify that the correct result type has been inferred
val r1 : Matrix = mult(new Matrix {}, new Matrix{}) // Compiles
val r2 : Vector = mult(new Matrix {}, new Vector{}) // Compiles
val r3 : Matrix = mult(new Matrix {}, 2)            // Compiles
val r4 : Matrix = mult(2, new Matrix {})            // Compiles

// This next one doesn't compile ...
val r5 : Matrix = mult(new Matrix {}, new Vector{})

Notice how the third type parameter, C, isn't used in the first (explicit) parameter list. If it weren't for its appearance in the second (implicit) parameter list the compiler would normally infer it as Nothing — not what we want at all. But the combination of type inference and implicit search saves the day. The first two type parameters A and B are inferred from the explicit arguments to mult(), then implicit search kicks in, attempting to locate an implicit definition of MultDep[_, _, _] consistent with those two types. By construction, it will only ever find one, and that is sufficient to uniquely determine C for use as the result type of the function.

We have the signature of our mult() function sorted out, now for the implementation. The (hopefully fairly obvious) trick here is to make our MultDep instances provide the implementations of the different cases as well as the type constraints,

implicit object mmm extends MultDep[Matrix, Matrix, Matrix] {
  def apply(m1 : Matrix, m2 : Matrix) : Matrix = error("TODO")
}

implicit object mvv extends MultDep[Matrix, Vector, Vector] {
  def apply(m1 : Matrix, v2 : Vector) : Vector = error("TODO")
}

implicit object mim extends MultDep[Matrix, Int, Matrix] {
  def apply(m1 : Matrix, i2 : Int) : Matrix = error("TODO")
}

implicit object imm extends MultDep[Int, Matrix, Matrix] {
  def apply(i1 : Int, m2 : Matrix) : Matrix = error("TODO")
}

(filling out the bodies of the dummy apply() methods is an exercise left for the reader). We can now complete the definition of mult(),

def mult[A, B, C](a : A, b : B)(implicit instance : MultDep[A, B, C]) : C =
  instance(a, b)

And that's all there is to it. The implicit objects correspond exactly to the type class instances in the Haskell original, and the combination of type inference and implicit search captures the functional dependency directly.

Here's the second example from Haskell wiki page, translated in the same way,

trait ExtractDep[A, B] {
  def apply(a : A) : B
}

implicit def ep[A, B] = new ExtractDep[(A, B), A] {
  def apply(p : (A, B)) : A = p._1
}

def extract[A, B](a : A)(implicit instance : ExtractDep[A, B]) : B = instance(a)

// Type annotation soley to verify that the correct result type has been inferred
val c : Char = extract(('x', 3))

Of course, neither of these two examples are particularly exciting in Scala as they stand — Scala has Java-style overloading, which makes the matrix/vector/scalar example mostly redundant; and the second example is a little too artificial to be compelling. Nevertheless, there are many situations where being able to express similar constraints amongst type parameters can be extremely useful — Scala embedded DSLs often have to deal with situations which are equivalent to wanting a function result type to be determined by its argument types, and I know of a number of DSL designers who've struggled with the problem of unhelpful Nothings being inferred for unconstrained result types.

But the best widely known example of a fundep is right under our noses in the Scala standard library itself: CanBuildFrom. Let's take a look at (a slight simplification of) of the part of its definition which expresses type constraints,

trait CanBuildFrom[From, Elem, To] {
  def apply ...
}

Instances of this trait capture collection-type-specific constraints on whether or not a collection of type To with elements of type Elem can be created from a collection of type From. For "regular" collection types (collections which can hold elements of any type) the corresponding instance is equivalent to,

implicit def regularCanBuildFrom[CC[_], A] = new CanBuildFrom[CC[_], A, CC[A]] {
  def apply ...
}

In effect this asserts that for a regular collection type it's possible to create a new collection of the same shape, but with an arbitrarily substituted element type. We can ask the compiler about these relationships like so,

scala> import scala.collection.generic._
import scala.collection.generic._

scala> implicitly[CanBuildFrom[List[Int], String, List[String]]]
res0: scala.collection.generic.CanBuildFrom[List[Int],String,List[String]] = ...

scala> implicitly[CanBuildFrom[Set[String], Double, Set[Double]]]
res1: scala.collection.generic.CanBuildFrom[Set[String],Double,Set[Double]] = ...

scala> implicitly[CanBuildFrom[List[String], Int, List[Double]]]
:11: error: Cannot construct a collection of type List[Double] with
  elements of type Int based on a collection of type List[String].
       implicitly[CanBuildFrom[List[String], Int, List[Double]]]
                 ^

scala> implicitly[CanBuildFrom[List[String], Int, Set[Int]]]
:11: error: Cannot construct a collection of type Set[Int] with
  elements of type Int based on a collection of type List[String].
       implicitly[CanBuildFrom[List[String], Int, Set[Int]]]
                 ^

As you can see from these examples, the third type argument in CanBuildFrom is determined by its first two — it takes its type constructor from the first type argument and its element type from the second type argument. Any attempt to deviate from that pattern results in a static compile time error. This should be sounding familiar ...

Now lets look at where this abstraction is used: the definition of TraversibleLike.map(),

def map[B, That](f: A => B)(implicit bf: CanBuildFrom[Repr, B, That]): That = {
  val b = bf(repr)
  b.sizeHint(this)
  for (x <- this) b += f(x)
  b.result
}

It should be apparent that here, like the matrix/vector/scalar example, we have a method with a parametrized result type which is not used in any of its explicit argument types, and hence which can't be inferred from its explicit arguments. Just as in the earlier case, it's the implicit argument which is determining the otherwise unconstrained result type via implicit search for a CanBuildFrom instance with type parameters B (determined by type inference from the explicit parameter list) and Repr (effectively a constant). It should also be apparent that the implicit argument is providing a significant component of the implementation of this method via the call bf(repr) — this is exactly analogous to the calls of instance(a, b) in the earlier example.

Hopefully the discussion above will have persuaded you that these are the hallmarks of a Haskell-like functional dependency, right at the heart of the Scala collections framework. And in my book, that makes fundeps in Scala neither impossible nor useless.

Unboxed union types in Scala via the Curry-Howard isomorphism

Miles Sabin — Thu, 09 Jun 2011 15:07:18 +0000

Update — I’ll be giving a talk on this material on the 16th of August in London. Please come along.

Scala has a highly expressive type system, but it doesn’t include everything you might find yourself hankering after — at least, not as primitives. There are a few genuinely useful things which fall under this heading — higher-rank polymorphic function types and recursive structural types are two I’ll talk about more in later posts. Today I’m going to show how we can encode union types in Scala, in the course of which I’ll have an opportunity to shed a little light on the Curry-Howard isomorphism and show how it can be put to work for us.

So, first up, what is a union type? A union type is pretty much what you’d expect: it’s the union of two (or more, but I’ll limit this discussion to just two) types. The values of that type are all of the values of each of the types that it’s the union of. An example will help to make this clear, but first a little notation — for reasons which will become apparent later I’ll write the union of types T and U as T ∨ U (ie. the two types flanking the logical ‘or’ operator), and so we write the union of the types Int and String as Int ∨ String. The values of this union type are all the Ints and all the Strings.

What does this mean more concretely? It means that if we could express this type directly in Scala we would be able to write,

def size(x : Int ∨ String) = x match {
  case i : Int => i
  case s : String => s.length
}

size(23) == 23   // OK
size("foo") == 3 // OK
size(1.0)        // Not OK, compile time error

In other words, the size method would accept arguments of either type Int or type String (and their subtypes, Null and Nothing) and nothing else.

It’s important to recognize the difference between this use of a union type and the similar use of Scala’s standard Either. Either is what’s known as a sum type, the analog of union types in languages which don’t support subtyping. Recasting our example in terms of Either we get,

def size(x : Either[Int, String]) = x match {
  case Left(i) => i
  case Right(s) => s.length
}

size(Left(23)) == 23    // OK
size(Right("foo")) == 3 // OK

Either[Int, String] can model the union type Int ∨ String because there is an isomorphism between the two types and their values. But equally clearly the Either type manages this by way of a layer of boxed representation, rather then by being an unboxed primitive feature of the type system. Can we do better than Either? Can we find a way of representing union types in Scala which doesn’t require boxing, and which provides all of the static guarantees we would expect?

It turns out that we can, but to get there we have take a detour through first-order logic via the Curry-Howard isomorphism. Curry-Howard tells us that the relationships between types in a type system can be viewed as an image of the relationships between propositions in a logical system (and vice versa). There are various ways that we can fill that claim out, depending on the type system we’re talking about and the logical system we’re working with, but for the purposes of this discussion I’m going to ignore most of the details and focus on simple examples.

To illustrate Curry-Howard (in the context of a type system with subtyping like Scala’s), we can see that there is a correspondence between intersection types (A with B in Scala) and logical conjunction (A ∧ B); between my hypothetical union types (A ∨ B) and logical disjunction (also A ∨ B, as hinted earlier); and between subtyping (A <: B in Scala) and logical implication (A ⇒ B). On the left hand side of each row in the table below we have a subtype relationship which is valid in Scala (although, in the case of the union types at the bottom, not directly expressible), and on the right hand side we have a logical formula which is obtained from the type relationship on the left by simply rewriting ("with" to "∧" and "<:" to "⇒") — in each case the result of the rewriting is a logically valid.

(A with B) <: A	(A ∧ B) ⇒ A
(A with B) <: B	(A ∧ B) ⇒ B
A <: (A ∨ B)	A ⇒ (A ∨ B)
B <: (A ∨ B)	B ⇒ (A ∨ B)

The essence of Curry-Howard is that this mechanical rewriting process (whichever direction you go in) will always preserve validity — valid type formulae will always rewrite to valid logical formulae, and vice versa. This isn’t only true for conjunction, disjunction and implication. We can also generalize the correspondence to logical formulae which include negation (the key one for us here) and universal and existential quantification.

So what would it mean to add negation to the mix? The conjunction of two types (ie. A with B) has values which are instances of both A and B, so similarly we should expect the negation of a type A (I’ll write it as ¬[A]) to have as it’s values everything which isn’t an instance of A. This is also something which can’t be directly expressed in Scala, but suppose it was?

If it was, then we would be able to crank on the Curry-Howard isomorphism and De Morgan’s laws to give us a definition of union types in terms of intersection types (A with B) and type negation. Here’s how that might go …

First recall the De Morgan equivalence,

(A ∨ B) ⇔ ¬(¬A ∧ ¬B)

Now apply Curry-Howard (using Scala’s =:= type equality operator),

(A ∨ B) =:= ¬[¬[A] with ¬[B]]

If we could work out a way of expressing this in Scala, we’d be home and dry and have our union types. So can we express type negation?

Unfortunately we can’t. But what we can do is transform all of our types in a way which allows negation to be expressed in the transformed context. We’ll then need to work out how make that work for us back in the original untransformed context.

Some readers might have been a little surprised earlier when I illustrated Curry-Howard using intersection types as the correlate of conjunction, union types as the correlate of disjunction and the subtype relation as the correlate of implication. That’s not how it’s normally done — usually product types (ie. (A, B)) model conjunction, sum types (ie. Either[A, B]) model disjunction and function types model implication. If we recast our earlier table in terms of products, sums and functions we end up with this,

(A, B) => A	(A ∧ B) ⇒ A
(A, B) => B	(A ∧ B) ⇒ B
A => Either[A, B]	A ⇒ (A ∨ B)
B => Either[A, B]	B ⇒ (A ∨ B)

On the left hand side we’re no longer looking for validity with respect to the subtype relation, instead we’re looking for evidence of the principle of parametricity, which allows us to determine if a function type is implementable just by reading it’s signature. It’s clear that all the function signatures on the left in the table above can be implemented — for the first two we have an (A, B) pair as our function argument, so we can easily evalutate to either an A or a B, using _1 or _2,

val conj1 : ((A, B)) => A = p => p._1
val conj2 : ((A, B)) => B = p => p._2

and for the last two we have either an A or a B as our function argument, so we can evalute to Either[A, B] (as Left[A] or Right[B] respectively).

val disj1 : A => Either[A, B] = a => Left(a)
val disj2 : B => Either[A, B] = b => Right(b)

This is the form in which the Curry-Howard isomorphism is typically expressed for languages without subtyping. Because this mapping doesn’t reflect subtype relations it isn’t going to be much direct use to us for expressing union types which, like intersection types, are inherently characterized in terms of subtyping. But it can help us out with negation, which is the missing piece that we need.

Either with or without subtyping, the bottom type (Scala’s Nothing type) maps to logical falsehood, so for example, the following equivalences all hold,

A => Either[A, Nothing]	A ⇒ (A ∨ false)
B => Either[Nothing, B]	B ⇒ (false ∨ B)

because the function signatures on the left are once again all implementable, and the logical formulae on the right are again all valid (see this post from James Iry for an explanation of why I haven’t shown the corresponding cases for products/conjunctions). Now we need to think about what a function signature like,

A => Nothing

corresponds to. On the logical side of Curry-Howard this maps to A ⇒ false, which is equivalent to ¬A. This seems fairly intuitively reasonable — there are no values of type Nothing, so the signature A => Nothing can’t be implemented (other than by throwing an exception, which isn’t allowed).

Let’s see what happens if we take this as our representation of the negation of a type,

type ¬[A] = A => Nothing

and apply it back in the subtyping context that we started with to see if we can now use De Morgan’s laws to get the union types we’re after,

type ∨[T, U] = ¬[¬[T] with ¬[U]]

We can test this using the Scala REPL, which will very quickly show us that we’re not quite there yet,

scala> type ¬[A] = A => Nothing
defined type alias $u00AC

scala> type ∨[T, U] = ¬[¬[T] with ¬[U]]
defined type alias $u2228

scala> implicitly[Int <:< (Int ∨ String)]
:11: error: Cannot prove that Int <:<
  ((Int) => Nothing with (String) => Nothing) => Nothing.
       implicitly[Int <:< (Int ∨ String)]

The expression “implicitly[Int <:< (Int ∨ String)]” is asking the compiler if it can prove that Int is a subtype of Int ∨ String, which it would be if we had succeeded in coming up with an encoding of union types.

So what’s gone wrong? The problem is that we have transformed the types on the right hand side of the <:< operator into function types so that we can make use of the encoding of type negation as A => Nothing. This means that the union type is itself a function type. That’s clearly not consistent with Int being a subtype of it — as the error message from the REPL shows. To make this work, then, we also need to transform the left hand side of the <:< operator into a type which could possibly be a subtype of the type on the right. What could that transformation be? How about double negation?

type ¬¬[A] = ¬[¬[A]]

Lets see what the compiler says now,

scala> type ¬¬[A] = ¬[¬[A]]
defined type alias $u00AC$u00AC

scala> implicitly[¬¬[Int] <:< (Int ∨ String)]
res5: <:<[((Int) => Nothing) => Nothing,
  ((Int) => Nothing with (String) => Nothing) => Nothing] = 

scala> implicitly[¬¬[String] <:< (Int ∨ String)]
res6: <:<[((String) => Nothing) => Nothing,
  ((Int) => Nothing with (String) => Nothing) => Nothing] =

Bingo! ¬¬[Int] and ¬¬[String] are both now subtypes of Int ∨ String!

Let's just check that this isn't succeeding vacuously,

scala> implicitly[¬¬[Double] <:< (Int ∨ String)]
:12: error: Cannot prove that ((Double) => Nothing) => Nothing <:<
  ((Int) => Nothing with (String) => Nothing) => Nothing.
       implicitly[¬¬[Double] <:< (Int ∨ String)]

We're almost there, but there's one remaining loose end to tie up — we have subtype relationships which are isomorphic to the ones we want (because ¬¬[T] is isomorphic to T), but we don't yet have a way to express those relationships with respect to the the untransformed types that we really want to work with.

We can do that by treating our ¬[T], ¬¬[T] and (T ∨ U) as phantom types, using them only to represent the subtype relationships on the underlying type rather that working directly with their values. Here's how that goes for our motivating example,

def size[T](t : T)(implicit ev : (¬¬[T] <:< (Int ∨ String))) = t match {
  case i : Int => i
  case s : String => s.length
}

This is using a generalized type constraint to require the compiler to be able to prove that any T inferred as the argument type of the size method must be such that it's double negation is a subtype of (Int ∨ String). That's only ever true when T is Int or T is String, as this REPL session shows,

scala> def size[T](t : T)(implicit ev : (¬¬[T] <:< (Int ∨ String))) = t match {
     |     case i : Int => i
     |     case s : String => s.length
     |   }
size: [T](t: T)(implicit ev: <:<[((T) => Nothing) => Nothing,
  ((Int) => Nothing with (String) => Nothing) => Nothing])Int

scala> size(23)
res8: Int = 23

scala> size("foo")
res9: Int = 3

scala> size(1.0)
:13: error: Cannot prove that ((Double) => Nothing) => Nothing <:<
  ((Int) => Nothing with (String) => Nothing) => Nothing.
       size(1.0)

One last little trick to finesse this slightly. The implicit evidence parameter is syntactically a bit ugly and heavyweight, and we can improve things a little by converting it to a context bound on the type parameter T like so,

type |∨|[T, U] = { type λ[X] = ¬¬[X] <:< (T ∨ U) }

def size[T : (Int |∨| String)#λ](t : T) = t match {
    case i : Int => i
    case s : String => s.length
}

And there you have it — an unboxed, statically type safe encoding of union types in unmodified Scala!

Obviously it would be nicer if Scala supported union types as primitives, but at least this construction shows that the Scala compiler has all the information it needs to be able to do that. Now we just need to pester Martin and Adriaan to make it directly accessible.

Chuusai becomes the first Scala Fly Space Partner

Miles Sabin — Fri, 03 Jun 2011 12:40:25 +0000

Today Chuusai and Zink Digital announced an agreement for Chuusai to become a Fly Space consulting and training partner. The agreement means that Chuusai will offer both integration assistance for Fly customers using the Scala interface, and provide training and support to ensure that projects using Fly and Scala get the most from this powerful new combination of technologies.

Nigel Warren, designer of Fly, said “I’m delighted that Miles and his team at Chuusai will be bringing all of their deep understanding of Scala in the High Performance, Financial and Scientific Computing domains to bear on projects using Fly to cluster their distributed Scala apps. The Fly team has worked very hard to make the Scala interface natural for Scala developers by, for example, including direct support for Actors, whilst maximising performance by writing native Scala objects directly to the Fly server.”

Miles Sabin of Chuusai said “We’re delighted to be working with the Fly team to bring the combination of tuple spaces and Scala to a wider audience. Tuple spaces are coming into their own in our new environment of ubiquitous parallelism and distribution, and Fly, being the foremost implementation available on the Java Virtual Machine, and the only one with a Scala interface, deserves serious attention from everyone working in demanding computational domains.”

Contact Miles Sabin, miles.sabin@chuusai.com, +44 1273 319 080, for Chuusai

Contact Nigel Warren, nige@zink-digital.com, for Zink Digital