Functional programming features have been making their way into mainstream languages lately, such as with new programming languages like Scala and Rust, or new type systems for existing languages like TypeScript or Flow.

One popular feature is null-tracking: the compiler statically ensures that null values are handled explicitly, instead of allowing failure at runtime with the equivalent of a null-pointer exception. This can make it easier to reason about the correctness of code.

Null-tracking is typically implemented in one of two ways:

• with a union type, requiring type refinement to extract the non-null value;
• or with a sum type, requiring pattern matching to extract the non-null value.

Union types and sum types are different beasts! But for some reason, there doesn’t seem to be any treatment of the subject available with a quick Google search.

In this article, I explain the difference between union and sum types and how they differ in practice. I discuss both of these approaches for null-tracking, but also cover how they differ for other purposes.

## Union types

### In TypeScript

TypeScript is a new statically-typed language by Microsoft that compiles down to Javascript. It uses union types for null-tracking.

The syntax is mostly a superset of Javascript’s, but you can also annotate your variables with types:

function greet(name: string): string {
return "Hello, " + name + "!";
}


We can denote a union type using |. A type of A | B means “any value which is either an A or a B (or both)”. We can express a nullable type by writing null | A for any type A:

function filterString(str: string): null | string {
if (str === "filtered") {
return null;
} else {
return str;
}
}


Then the following code is rejected by the typechecker because of the possibility that name is null:

const name: string = readNameFromUser();
greet(filterString(name));
// error: filterString returns a null | string,
//        but greet takes a string.


We could fix this by rewriting greet to take a nullable string and handle it explicitly:

function greet(name: null | string): string {
if (name === null) {
// The type of name has been refined to null in this branch.
return "Hello, no-name!";
} else {
// The type of name has been refined to string in this branch.
return "Hello, " + name + "!";
}
}


When seeing a special construct like === null in an if-condition, the typechecker knows to refine the type of name within the branch. Within that branch, the type of name is known to be a more specific subset of its type outside that branch.

### Theory

Let’s think of a type as a set of values. For example, the integer type could be defined as the infinite set

$\text{int} = \left\{\cdots, -2, -1, 0, 1, 2, \cdots\right\}$

A union type between two types $A$ and $B$ is simply the union of the two sets, denoted as $A \cup B$ So

$\text{bool} \cup \text{int} = \left\{\text{true}, \text{false}, \cdots, -2, -1, 0, 1, 2, \cdots \right\}$

The null type is the set containing one value, the null value. For clarity, we’ll denote the null value as $\emptyset$, and the null type as $\text{null}$:

$\text{null} = \left\{\emptyset\right\}$

So the type of null | int is

$\text{null} \cup \text{int} = \left\{\emptyset, \cdots, -2, -1, 0, 1, 2, \cdots\right\}$

## Sum types

### In OCaml

OCaml is a mature functional programming language that uses the Hindley-Milner type system. In OCaml, null-checking uses sum types.

ML-family languages represent the absence of a value using an algebraic data type. The following sum type is built into OCaml:

type 'a option =
| Some of 'a
| None


The option type takes a generic type parameter 'a, and can take on either the value Some containing that value, or None, containing no value.

Note that although both union types and sum types are denoted with | in their respective languages, these do not represent the same thing. This can be a source of confusion for someone acquainted with languages in both families.

We can use this algebraic data type to force the programmer to check for null when unpacking an option. For example:

let filter_string (str : string) : string option =
if str = "filtered"
then None
else Some str

let greet (name : string option) : unit =
(* It wouldn't compile to just write

Printf.printf "Hello, %s!" name

here, because printf is expecting a string but will have received a
string option. *)
match name with
| None -> Printf.printf "Hello, no-name!"
| Some name_ ->
(* We've pattern-matched a string called name_ out of the
string option called name. Note that we could also call this
variable name and shadow the outer name, but I don't here for
clarity. *)
Printf.printf "Hello, %s!" name_


Instead of type refinement, where the compiler recognizes a special syntactic pattern in order to do type inference, the language uses pattern-matching to extract a value with the appropriate type out of an algebraic data type.

### Theory

A sum type is like a union type, but every type in the sum type is accompanied by a label, which is a unique identifier for this element of the sum type. If we have types $A$ and $B$, and labels $l_A$ and $l_B$, then we can define the sum type $A + B$ as

$A + B = \left(\left\{l_A\right\} \times A\right) \cup \left(\left\{l_B\right\} \times B\right)$

where $\times$ denotes Cartesian product.

This overall operation is called disjoint union. Basically, it can be summarized as “a union, but each element remembers what set it came from”. Sum types are also called “tagged unions” or “discriminated unions”.

As a concrete example, consider this data type:

type identifier =
| Name of string
| IdNumber of int


This represents an identifier which is either a name or an ID number. Here are the corresponding variables:

\begin{aligned} A &= \text{string} \\ B &= \text{int} \\ l_A &= \text{"name"} \\ l_B &= \text{"id number"} \end{aligned}

Then the sum type $A + B$ using these labels is the the set of all values depicted below:

\begin{aligned} (\text{"name"},& &\text{""}) \\ (\text{"name"},& &\text{"a"}) \\ (\text{"name"},& &\text{"aa"}) \\ (\text{"name"},& &\text{"ab"}) \\ & \vdots & \end{aligned}\quad\cup\quad\begin{aligned} (\text{"id number"},& & 0) \\ (\text{"id number"},& & 1) \\ (\text{"id number"},& & 2) \\ (\text{"id number"},& & 3) \\ & \vdots & \end{aligned}

In the option type, the labels are Some and None. Note that labels must be unique across all sum types (or at least they must reside in different namespaces). That is, this wouldn’t compile in OCaml:

type foo =
| Foo

type foo_or_bar =
| Foo
| Bar


## Variant types

Sometimes you may hear about “variant types”. Please don’t call either sum or union types “variant types”. This terminology is confusing and can mean either union or sum types, depending on the language and environment. For example, in C++, std::variant describes a union type, but in OCaml, “variant types” are sum types.

## Differences

### Singleton null versus a null for every type

In the union type scheme, there is one null value that is shared across every type. This can have some practical disadvantages. For example, let’s say that you have a map that, when looking up a value, returns null if the key is not found. Then there may be some ambiguity in a case like this:

const key: string = getKey();
const map: Map<string, null | int> = getMap();

// Returns null | null | int, which is the same as null | int.
const value: null | int = map.get(key);

if (value === null) {
}


This can be circumvented by having an additional null-like sentinel values, like undefined in Javascript, but it can get confusing to remember the exact semantics of them all, and still doesn’t let you store undefined in an unambiguous way.

Sum types wrap their values in such a way that there is no ambiguity. For example:

let key : string = get_key () in
let map : string (int option) map = get_map () in

(* Note the double option -- the map wraps the result in another option. *)
let value : int option option = Map.get key map in
match value with
| None -> Printf.printf "Map did not contain key"
| Some x ->
match x with
| Some x -> Printf.printf "Map had value %d" x
| None -> Printf.printf "Map had None"


### Distinguishing identical types via labels

With a union type between $A$ and $B$, if $A$ and $B$ are the same, there is no way to distinguish between elements from $A$ versus elements from $B$.

For example, we can express this in OCaml:

type file =
| Filename of string
| FileContents of string


This gives us some type-safety because then we can never mix the two up:

let print_file_contents file =
let contents =
match file with
| Filename filename -> read_file_from_disk filename
| FileContents file_contents -> file_contents
in
Printf.printf "File contents: %s" contents


However, sum types can be implemented in terms of union types, as described later.

Union types can be used to implement a form of function overloading:

function printAnything(value: int | string | bool): void {
if (typeof value === "int") {
console.log("The int was ", value);
} else if (typeof value === "string") {
console.log("The string was ", value);
} else {
console.log("The bool was ", value);
}
}


OCaml does not support ad-hoc polymorphism like this. Indeed, one of the warts of the language that it needs two different operators + and +. for addition between integers and floats, respectively.

### Mixing labels of different types

In a sum type, the labels of different sum types are not interchangeable. You cannot write this in OCaml:

type foo =
| Foo of string

type bar =
| Bar of int

let takes_a_foo_or_bar foo_or_bar =
match foo_or_bar with
| Foo foo -> Printf.printf "Got string %s" foo
| Bar bar -> Printf.printf "Got int %d" bar


This can be worked around with so-called polymorphic variants in OCaml, but they are not a general staple of ML-family languages. (SML doesn’t have them.)

### Adding fields to an existing record type

Union types are usually accompanied by intersection types. We can use an intersection type to add fields to an existing record type:

type User = { name: string } | { id: number }

// Same as the User type, but every record also has an email field.
type UserWithEmail = User & { email: string }


This is especially useful for processing data in stages and attaching additional fields to records as we determine their values. For example, in a compiler, we might start with a container of all the variables in the programs, and then attach the type to each variable as we determine it.

With sum types (even polymorphic variants, as in OCaml), we can’t add new fields to existing record types. We can only create new sum types with the added fields, and convert between the two. This is more difficult to work with in some situations, since one has to implement boilerplate code to convert from one to the other.

## Implementing sum types in terms of union types

The definition of sum type involves a union. So if we can add a label to our union types, we can emulate a sum type. The general technique is as follows:

interface Filename {
label: "filename";
filename: string;
}

interface FileContents {
label: "file_contents";
contents: string;
}

type File = Filename | FileContents

function printFileContents(file: File): void {
switch (file.label) {
case "filename":