Row Polymorphism without the Jargon

April 15, 2020tutorialstype theory

As we delve deeper into Type Theory, we tend to discover concepts that don’t have many details written out. Or, if there is writing about them, it is filled with technical jargon and advanced diagrams. The Wikipedia page for Row Polymorphism is a great example of this. The first line states:

In programming language type theory, row polymorphism is a kind of polymorphism that allows one to write programs that are polymorphic on record field types.

This description isn’t helpful, and the rest of the article fails to explain what it can be used for. Today, I’ll be doing my best to explain Row Polymorphism without the jargon.

Subtyping

To explain something new, it can be nice to put it in terms of what you know. Most developers have seen subtyping in the wild. It’s the core of OOP, the first thing you learned in your Java course: types can inherit each other. In Java, every class has a superclass that it inherits fields and methods from.

public class Human {
    Color hairColor;
    Color eyeColor;
    int height;
    int weight;
}

public class Employee extends Human {
    Job job;
    int salary;
}

Human here contains a number a fields relating to what a human might have. When we say Employee extends Human, we say that the fields and methods in Human are also in Employee.

Here’s where the polymorphism comes in: we can use Employee anywhere a Human is expected.

public feed(Human human) {
    human.weight += 16;
}

...

Employee employee = new Employee();
feed(employee); // works correctly!

This function expects a Human, but we can give it an Employee or any other type that inherits Human.

What if my rows are equal?

A row in this context is the list of fields in our structure. Human is a row comprised of weight and height. To be polymorphic over a row means that we can provide any row that contains the fields we expect. Let’s look at an example.

public class Fish {
    int height;
    int weight;
}

...

Fish fish = new Fish();
feed(fish); // doesn't compile!

Oh no! Our fish won’t compile! Even though Fish has the weight field that feed expects, this won’t compile because Fish doesn’t extend Human. How do we fix this? We’ll, we’d need to make Fish and Human subtypes of the same class, something like Entity.

public class Entity { /* ... */ }
public class Fish extends Entity { /* ... */ }
public class Human extends Entity { /* ... */ }

// Now this works with Fish and Humans!
public feed(Entity entity) {
    entity.weight += 16;
}

Let’s sum up what’s happening here: to add a Fish to our code, we had to:

Make a new class that Fish and Human both extend
Cut any fields from the subclasses and paste them in the new class
Do the same for any methods that might be applicable
Refactor any methods using the existing classes to use Entity

This is a lot of work to add a fish!

What if we could, instead, write our functions in a way that checked if our row was equivalent? Our feed function only cares about a subset of the fields we’re providing it: weight. In Java like languages, we’re required to use subtypes or make an interface that declares our field, but what if we could write a generic function that could accept a subset of the row we provide?

Polymorphism over Rows

Let’s look at an example in a language that does have Row Polymorphism: PureScript.

area thing = thing.width * thing.height

This defines a function called area that will calculate the area for a thing. The type of this function reads:

area :: forall r. { width :: Int, height :: Int | p } -> Int

which says the first parameter is a record ({}) that contains two ints, width of type Int and height of type Int. This record is also allowed to have other fields in it (| p). It returns the area as an Int.

This function allows us to supply any type that contains the width and height fields. We are allowed to have whatever other fields we want, because the function specifies a rho (| p) at the end of the record’s field list. This is Row Polymorphism!

Let’s make our fish again:

newtype Fish = Fish { width :: Int, height :: Int, length :: Int }

This time our fish has information about its dimensions. Can we pass in Fish to area? Yes! We are free to use any type that contains the expected fields! This changes things dramatically, and can greatly reduce refactoring. We can create functions that apply to many different kinds of structures without needing to specify every structure it’s applicable to.

volume :: forall r. { width :: Int, height :: Int, length :: Int | r } -> Int
volume thing = thing.width * thing.height * thing.length

volume automatically accepts Fish and any other types that contain the correct fields.

Building off of existing types

Row Polymorphism not only allows us to not only accept types with a variable amount of fields, but it also allows us to extend existing types with new fields. Here’s an example in a little language called Ordo:

(* we define a function that takes in a record and
   returns the record with a new field added to it *)
let f r = { y = 0 | r }

(* we can then call this function with a record,
   { x = 0 }, and the function returns a new record
   with the old fields and the new 'y' field *)
f { x = 0 } = { x = 0, y = 0 }

I like Ordo’s syntax here because it’s very clear what is happening. Functions can arbitrarily extend records with additional fields. This is another great property of Row Polymorphism!

Outro

Using Row Polymorphism as a replacement for subtyping hasn’t been explored in many languages. I think this is because most programmers are used to subtyping so many newer languages try to include it. Row Polymorphism can also lead to confusion where calling a function works but semantically it doesn’t make sense, like finding the area of a Fish. A Fish can have a surface area, but that is a different formula that expects a length. In a language with subtyping, we might restrict area to only work on an Entity2D and volume to only work on an Entity3D, which would make sure our functions are only called with the types we expect. I think both solutions bring something good to the table, and we should definitely explore both options more.

Here are some languages that have Row Polymorphism:

Errata

Edit 4/21/20: I misinterpreted the term row as one field instead of a set of fields! This has been fixed.

Jadon.