prerequisites: The post Product types in Elm, Elixir, and Kotlin

“We are our choices.”
Jean-Paul Sartre

1. Introduction

The goal of this blog post is to define the concept of sum types and compare the implementation of sum types in three different functional programming languages: Kotlin, Elixir, and Elm.

The post is structured as follows. In Section 2, we define the concept of sum types. Then, in Sections 3, 4, and 5 we look at concrete implementations of sum types in Kotlin, Elixir, and Elm, respectively. The post is concluded in Section 6.

2. Sum types

In this section, we define the concept of sum types.

In our post on enum types, we defined an enum type as a “data type consisting of a set of named values which we call the members of the type”, e.g. we defined shape as:

datatype shape
    = Rectangle
    | Circle
    | Triangle

In the case of a sum type (sometimes called a tagged union), we may look at it as a generalization of the enum type, where each member of a sum type may take its own set of arguments. Conversely, we may also look at enum types as the subset of sum types for which each member is a unit type, i.e. each member’s type constructor takes zero arguments.

In our post on product types we implemented three different types of shapes: rectangle, circle, and triangle. Now, with the above definition of sum types in mind, we want to define a shape type that can be either a Rectangle, a Circle, or a Triangle. In our ML-like syntax, we could express our shape type and its members as:

datatype shape
    = Rectangle of float * float (* height and width *)
    | Circle of float (* radius *)
    | Triangle of float * float (* base and height *)

where we declare a datatype with the name shape and three type constructors, Rectangle, Circle, and Triangle, where the first type constructor, Rectangle, takes two float values, the height and width, the second type constructor, Circle, takes a single float value, the radius, and the third type constructor, Triangle, takes two float values, the base and height. This scenario reminds us of the issue with tuple types we discussed in the previous post, i.e. it is not clear which of the arguments has which semantic meaning. Fortunately, we can substitute the float * float arguments to the Rectangle type constructor, with the rectangle product type we defined in the previous post:

datatype rectangle
    = height of float * width of float

and likewise for the Circle type constructor:

datatype circle
    = radius of float

and Triangle type constructor:

datatype triangle
    = base of float * height of float

Combining all this, we get the following definition of the shape sum type:

datatype shape
    = Rectangle of rectangle
    | Circle of circle
    | Triangle of triangle

where the semantic meaning of the arguments to each of the type constructors is now more obvious. Note that being able to create these kinds of types that are compositions of both sum- and product types are what we usually refer to as Algebraic data types.

As in the case of enum types, we can pattern match on sum types. This provides the opportunity for us to merge the three different area functions of the previous post into one area function that takes an instance of the shape sum type as argument, pattern matches on its type constructor and calculates the area of that type of shape. We express this area function in our ML-like syntax as:

fun area shape =
    case shape
    of Rectangle rectangle =>
         rectangle.height * rectangle.width
     | Circle circle =>
         3.14 * circle.radius * circle.radius
     | Triangle triangle =>
         0.5 * triangle.base * triangle.height

We see that the area function pattern matches on the type constructor of the shape argument, i.e. Rectangle, Circle and Triangle, and in doing so also unwraps the arguments of each type constructor and binds these to suitable variable names, thereby allowing us to calculate the area of the shape in the matched clause. As in the previous post, we can also destructure the arguments and directly access their fields in each of the clauses:

fun area shape =
    case shape
    of Rectangle (height, width) =>
         height * width
     | Circle (radius) =>
         3.14 * radius * radius
     | Triangle (base, height) =>
         0.5 * base * height

thus removing the need to qualify the use of the different field values in the body expression of each of the clauses.

In the following three sections, we look at how to express the above shape sum type, along with the area example function, in each of our three programming languages.

3. Sum types in Kotlin

In this section, we implement the shape sum type and area function in Kotlin.

If we look at the definition of the enum type we defined in the previous post:

enum class Shape {
    Rectangle,
    Circle,
    Triangle;
}

we might expect that we could simply add the needed set of arguments to each of the defined members in order to obtain the desired sum type. Unfortunately, while an enum class is actually able to take a set of arguments, these are declared for the whole class and not for the individual member, which is too constrained to fit with our definition above of sum types. Luckily, Kotlin has introduced the concept of a sealed class, which allows us to define a “restricted class hierarchies, when a value can have one of the types from a limited set, but cannot have any other type” which sounds a lot like our definition of a sum type. Thus, in order to define our custom sum type we declare our new type as sealed class Shape followed by a class declaration for each of the members of the sum type, Rectangle, Circle, and Triangle, each of which then has to be declared as a subclass of Shape:

sealed class Shape
data class Rectangle(val height: Float,
                     val width: Float) : Shape()
data class Circle(val radius: Float) : Shape()
data class Triangle(val base: Float,
                    val height: Float) : Shape()

Note that in contrast to our ML-like example, we do not explicitly list each of the members of our Shape sum type when declaring it, but instead do it implicitly as we define each of the actual member types and declare a member type to be a subclass of Shape.

Just as we could pattern match on instances of an enum class using a when (<var>) {...} expression, so is it the case for instances of a sealed class. Thus, we define our area function in Kotlin as:

fun area(shape: Shape) : Number {
    return when (shape) {
        is Rectangle ->
            shape.height * shape.width

        is Circle ->
            Math.PI * shape.radius * shape.radius

        is Triangle ->
            0.5 * triangle.base * triangle.height
    }
}

A few details worth noting:

  • We use the is keyword in each of the matching clauses as we are matching on a subclass type and not a specific value, and
  • Kotlin smart casts the shape variable into its correct member type, e.g. we do not have to cast shape as a Rectangle in order to access shape.height once we are inside the body expression of the is Rectangle clause.

Finally, we can run the above code by implementing the main function, instanting a variable of type Shape and passing it to the area function:

fun main(args: Array<String>) {
    val circle = Circle(4.2F)
    println("The circle has an area of ${area(circle)}!")
    /* ==> The circle has an area of 55.4176893759496! */
}

Having implemented our shape sum type and area function in Kotlin, we move on to repeat the exercise in Elixir.

4. Sum types in Elixir

In this section, we implement the shape sum type and area function in Elixir.

As in the case of the shape enum type, we create a module named Shape and use the @type directive to define a type named t, which is either a Rectangle.t, Circle.t or Triangle.t type:

defmodule Shape do
  alias Rectangle
  alias Circle
  alias Triangle

  @type t :: Rectangle.t | Circle.t | Triangle.t
end

where Rectangle.t, Circle.t and Triangle.t correspond to the product types we defined in our previoust post:

defmodule Rectangle do
  @type t :: %__MODULE__{height: float, width: float}
  defstruct [height: 0.0, width: 0.0]
end

defmodule Circle do
  @type t :: %__MODULE__{radius: float}
  defstruct [radius: 0.0]
end

defmodule Triangle do
  @type t :: %__MODULE__{base: float, height: float}
  defstruct [base: 0.0, height: 0.0]
end

Having defined our Shape.t type and its members, Rectangle.t, Circle.t and Triangle.t, we can now define our area function which takes an argument of type Shape.t and calculates the area of the shape by pattern matching on the concrete member of the shape sum type:

defmodule Example do
  alias Rectangle
  alias Circle
  alias Triangle
  alias Shape

  @spec area(Shape.t) :: float
  def area(shape) do
    case shape do
      %Rectangle{height: height, width: width} ->
        height * width

      %Circle{radius: radius} ->
        :math.pi * radius * radius

      %Triangle{base: base, height: height} ->
        0.5 * base * height
    end
  end
end

While the case <var> do ... expression is similar to the one we used for enum types, we do note that - as in the Kotlin case - we automatically unwrap the arguments of the matching member/type constructor and bind these to suitable variable names.

Finally, we can test the above code by instantiating a value of type Shape.t and pass it to the area function:

circle = %Circle{radius: 4.2}
IO.puts "The circle has an area of #{area(circle)}!"
# ==> The circle has an area of 55.41769440932395!

Having implemented our shape sum type and area function in both Kotlin and Elixir, we move on to our final language example, Elm.

5. Sum types in Elm

In this section, we implement the shape sum type and area function in Elm.

In the case of Elm, we once again return to the ML-like syntax we saw at the beginning of this post, where we define our sum type, Shape, using the type keyword followed by listing each of the members of the type, Rectangle, Circle, and Triangle:

type Shape
    = Rectangle { height: Float, width: Float }
    | Circle { radius: Float }
    | Triangle { base : Float, height: Float }

Here, we simply inline the definition of Rectangle, Circle, Triangle from our previous post into their corresponding clauses in the Shape sum type. Alternatively, we would have to change the names of the clauses or argument types in order to avoid names clashing, e.g.

type Shape
    = RectangleShape Rectangle
    | CircleShape Circle
    | TriangleShape Triangle

which in this case is less aesthetic than the former definition.

It is worth appreciating that in order to go from an enum type to a sum type in Elm, all we had to do was add arguments to the members / type constructors of the type. Unsurprisingly, Elm does not make an actual distinction between enum and sum types, but sees the former as a subset of the later, as we also discussed in Section 2.

The similarity to our ML-like syntax also holds in the case of pattern matching in the area function:

area : Shape -> Float
area shape =
    case shape of
        Rectangle { height, width } ->
            height * width

        Circle { radius } ->
            pi * radius * radius

        Triangle { base, height } ->
            0.5 * base * height

where the difference are minor. Finally, we can run the above code snippets by implementing the main function, where we instantiate a value of type Circle, pass it to the area function and print it as a text DOM element:

main =
    let
        circle =
            Circle 4.2
    in
        text <|
            "The circle has an area of " ++
                (toString <| area <| circle) ++
                "!"
-- ==> "The circle has an area of 55.41769440932395!"

6. Conclusion

In this blog post, we have defined the concept of sum types, and compared the implementation of sum types in the three different programming languages: Kotlin, Elixir, and Elm.

While all three languages supported the concept of sum types, it is noticeable that Elm required the least introduction of new syntax, as it does not really make a distinction between enum types and sum types, as the former can be expressed in terms of the latter.