A good first step while building a new application is to define its domain model. TypeScript offers many tools to help you in this task. Algebraic Data Types (ADT for short) are one of these tools.
What is an ADT?
In computer programming, especially functional programming and type theory, an algebraic data type is a kind of composite type, i.e., a type formed by combining other types.
Two common classes of algebraic types are:
- product types
- sum types
Product types
A product type is a collection of types Ti indexed by a set I
.
Two common members of this family are n
-tuples, where I
is a non empty interval of natural numbers...
type Tuple1 = [string] // I = [0]
type Tuple2 = [string, number] // I = [0, 1]
type Tuple3 = [string, number, boolean] // I = [0, 1, 2]
// Accessing by index
type Fst = Tuple2[0] // string
type Snd = Tuple2[1] // number
...and structs, where I
is a set of labels.
// I = {"name", "age"}
interface Person {
name: string
age: number
}
// Accessing by label
type Name = Person['name'] // string
type Age = Person['age'] // number
Why "product" types?
If we write C(A)
for the number of inhabitants of the type A
(aka its cardinality) then the following equality holds
C([A, B]) = C(A) * C(B)
the cardinality of the product is the product of cardinalities
Example
type Hour = 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
type Period = 'AM' | 'PM'
type Clock = [Hour, Period]
The Clock
type has 12 * 2 = 24
inhabitants.
When should I use a product type?
Whenever its components are independent.
type Clock = [Hour, Period]
Here Hour
and Period
are independent, i.e. the value of Hour
doesn't affect the value of Period
and vice versa, all pairs are legal and meaningful.
Sum types
A sum type is a data structure used to hold a value that could take on several different, but fixed, types. Only one of the types can be in use at any one time, and a tag field explicitly indicates which one is in use.
In the TypeScript documentation they are named tagged union types.
Example (redux actions)
type Action =
| {
type: 'ADD_TODO'
text: string
}
| {
type: 'UPDATE_TODO'
id: number
text: string
completed: boolean
}
| {
type: 'DELETE_TODO'
id: number
}
The type
field is the tag and ensures that its members are disjoint.
Constructors
A sum type with n
members requires n
constructors, one for each member
const add = (text: string): Action => ({
type: 'ADD_TODO',
text
})
const update = (id: number, text: string, completed: boolean): Action => ({
type: 'UPDATE_TODO',
id,
text,
completed
})
const del = (id: number): Action => ({
type: 'DELETE_TODO',
id
})
Sum types can be polymorphic and/or recursive.
Example (linked lists)
// ↓ type parameter
type List<A> = { type: 'Nil' } | { type: 'Cons'; head: A; tail: List<A> }
// ↑ recursion
Pattern matching
JavaScript doesn't have pattern matching (and so TypeScript) however we can define a "poor man" pattern matching by defining a fold
function
const fold = <A, R>(
fa: List<A>,
onNil: () => R,
onCons: (head: A, tail: List<A>) => R
): R => (fa.type === 'Nil' ? onNil() : onCons(fa.head, fa.tail))
Example (calculate the length of a List
recursively)
const length = <A>(fa: List<A>): number =>
fold(fa, () => 0, (_, tail) => 1 + length(tail))
Why "sum" types?
The following equality holds
C(A | B) = C(A) + C(B)
the cardinality of the sum is the sum of cardinalities
Example (the Option
type)
type Option<A> =
| { type: 'None' }
| {
type: 'Some'
value: A
}
From the general formula C(Option<A>) = 1 + C(A)
we can derive for example the cardinality of Option<boolean>
: 1 + 2 = 3
inhabitants.
When should I use a sum type?
When its components would be dependent if implemented as a product type.
Example (component props)
interface Props {
editable: boolean
onChange?: (text: string) => void
}
class Textbox extends React.Component<Props> {
render() {
if (this.props.editable) {
// error: Cannot invoke an object which is possibly 'undefined' :(
this.props.onChange(...)
}
}
}
The problem here is that Props
is modelled as a product type but onChange
depends on editable
.
A sum type is a better choice
type Props =
| {
type: 'READONLY'
}
| {
type: 'EDITABLE'
onChange: (text: string) => void
}
class Textbox extends React.Component<Props> {
render() {
switch (this.props.type) {
case 'EDITABLE' :
this.props.onChange(...) // :)
...
}
}
}
Example (node callbacks)
declare function readFile(
path: string,
// ↓ ---------- ↓ CallbackArgs
callback: (err?: Error, data?: string) => void
): void
The result is modelled as a product type
type CallbackArgs = [Error | undefined, string | undefined]
however, its components are dependent: you get either an error or a string
err | data | legal? |
---|---|---|
Error |
undefined |
✓ |
undefined |
string |
✓ |
Error |
string |
✘ |
undefined |
undefined |
✘ |
A sum type would be a better choice, but which one?
Functional error handling
Let's see how to deal with errors in a functional style.
The Option
type
The Option
type represents the effect of a computation that may fail or yield a value of type A
type Option<A> =
| { type: 'None' } // represents a failure
| { type: 'Some'; value: A } // represents a success
Constructors and pattern matching:
// a nullary constructor can be implemented as a constant
const none: Option<never> = { type: 'None' }
const some = <A>(value: A): Option<A> => ({ type: 'Some', value })
const fold = <A, R>(fa: Option<A>, onNone: () => R, onSome: (a: A) => R): R =>
fa.type === 'None' ? onNone() : onSome(fa.value)
The Option
type can be used to avoid throwing exceptions and/or to represent optional values, so we can go from...
// this is a lie ↓
const head = <A>(as: Array<A>): A => {
if (as.length === 0) {
throw new Error('Empty array')
}
return as[0]
}
let s: string
try {
s = String(head([]))
} catch (e) {
s = e.message
}
...where the type system is unaware of possible failures, to...
// ↓ the type system "knows" that this computation may fail
const head = <A>(as: Array<A>): Option<A> => {
return as.length === 0 ? none : some(as[0])
}
const s = fold(head([]), () => 'Empty array', a => String(a))
...where the possibility of failures is lifted to the type system.
The Either
type
A common use of Either
is as an alternative to Option
for dealing with possible missing values. In this usage, None
is replaced with a Left
which can contain useful information. Right
takes the place of Some
. Convention dictates that Left
is used for failure and Right
is used for success.
type Either<L, A> =
| { type: 'Left'; left: L } // represents a failure
| { type: 'Right'; right: A } // represents a success
Constructors and pattern matching:
const left = <L, A>(left: L): Either<L, A> => ({ type: 'Left', left })
const right = <L, A>(right: A): Either<L, A> => ({ type: 'Right', right })
const fold = <L, A, R>(
fa: Either<L, A>,
onLeft: (left: L) => R,
onRight: (right: A) => R
): R => (fa.type === 'Left' ? onLeft(fa.left) : onRight(fa.right))
Getting back to our callback example
declare function readFile(
path: string,
callback: (err?: Error, data?: string) => void
): void
readFile('./myfile', (err, data) => {
let message: string
if (err !== undefined) {
message = `Error: ${err.message}`
} else if (data !== undefined) {
message = `Data: ${data.trim()}`
} else {
// should never happen
message = 'The impossible happened'
}
console.log(message)
})
we can change its signature to
declare function readFile(
path: string,
callback: (result: Either<Error, string>) => void
): void
and then consume the API like this
readFile('./myfile', e => {
const message = fold(e, err => `Error: ${err.message}`, data => `Data: ${data.trim()}`)
console.log(message)
})
Conclusion
In this post we've seen product types and sum types, and how reasoning in terms of the number of states they represent can greatly affect the design of our domain models.
A common pitfall of many real-world APIs is to misuse product types that, in addition to all the legal states, also model many illegal ones.
Sum types are an incredibly useful and fundamental language feature, they are the key to design excellent domain models by allowing to make illegal states irrepresentable.
Top comments (8)
Great article !
What do you think about this approach to pattern matching ?
I think it looks a bit more declarative then checking the type manually.
Check this out github.com/pfgray/ts-adt
There is a small typo and don't know if one can submit a change to a post.
The following code block declare:
Its should have been:
Are you aware of ReasonML, which have real support for ADTs and pattern matching? 🙂
ReasonML is way more readable...
Great article, well written!
Straight and to the point, great writing.
Question, is a 'Maybe' the equivalent to a 'Option' or an 'Either' in your example?
Maybe
is equivalent toOption