# Functional design: combinators

A style of organizing libraries centered around the idea of combining things. Usually there is some type T, some "primitive" values of type T, and some "combinators" which can combine values of type T in various ways to build up more complex values of type T

So the general shape of a combinator is
combinator: Thing -> Thing

The goal of a combinator is to create new "things" from previously defined "things".
Since the result can be passed back as input, you get a combinatorial explosion of possibilities, which makes this pattern very powerful.
If you mix and match several combinators together, you get an even larger combinatorial explosion.
So a design that you may often find in a functional module is

a small set of very simple "primitives"
a set of "combinators" for combining them into more complicated structures

Let’s see some examples.

Example 1: Setoid

The getArraySetoid combinator: given an instance of Setoid for A, we can derive an instance of Setoid for Array
import { Setoid } from ‘fp-ts/lib/Setoid’

export function getArraySetoid<A>(S: Setoid<A>): Setoid<Array<A>> {
return {
equals: (xs, ys) => xs.length === ys.length && xs.every((x, i) => S.equals(x, ys[i]))
}
}

Usage
/** a primitive `Setoid` instance for `number` */
export const setoidNumber: Setoid<number> = {
equals: (x, y) => x === y
}

// derived
export const setoidArrayOfNumber: Setoid<Array<number>> = getArraySetoid(setoidNumber)

// derived
export const setoidArrayOfArrayOfNumber: Setoid<Array<Array<number>>> = getArraySetoid(setoidArrayOfNumber)

// derived
export const setoidArrayOfArrayOfArrayOfNumber: Setoid<Array<Array<Array<number>>>> = getArraySetoid(setoidArrayOfArrayOfNumber)

// etc…

Another combinator, contramap: given an instance of Setoid for A and a function from B to A, we can derive an instance of Setoid for B
export const contramap = <A, B>(f: (b: B) => A, S: Setoid<A>): Setoid<B> => {
return {
equals: (x, y) => S.equals(f(x), f(y))
}
}

Usage
export interface User {
id: number,
name: string
}

export const setoidUser: Setoid<User> = contramap((user: User) => user.id, setoidNumber)

// mix with `getArraySetoid`
export const setoidArrayOfUser: Setoid<Array<User>> = getArraySetoid(setoidUser)

Example 2: Monoid

The getIOMonoid combinator: given an instance of Monoid for A, we can derive an instance of Monoid for IO<A>
import { IO } from ‘fp-ts/lib/IO’
import { Monoid } from ‘fp-ts/lib/Monoid’

export function getIOMonoid<A>(M: Monoid<A>): Monoid<IO<A>> {
return {
concat: (x, y) => new IO(() => M.concat(x.run(), y.run())),
empty: new IO(() => M.empty)
}
}

We can use getIOMonoid to derive another combinator, replicateIO: given a number n and an action ma of type IO<void>, we can derive an action that performs n times ma
import { fold } from ‘fp-ts/lib/Monoid’
import { replicate } from ‘fp-ts/lib/Array’

/** a primitive `Monoid` instance for `void` */
export const monoidVoid: Monoid<void> = {
concat: () => undefined,
empty: undefined
}

export function replicateIO(n: number, mv: IO<void>): IO<void> {
return fold(getIOMonoid(monoidVoid))(replicate(n, mv))
}

Usage
//
// helpers
//

/** logs to the console */
export function log(message: unknown): IO<void> {
return new IO(() => console.log(message))
}

/** returns a random integer between `low` and `high` */
export const randomInt = (low: number, high: number): IO<number> => {
return new IO(() => Math.floor((high – low + 1) * Math.random() + low))
}

//
// program
//

function fib(n: number): number {
return n <= 1 ? 1 : fib(n – 1) + fib(n – 2)
}

/** calculates a random fibonacci and prints the result to the console */
const printFib: IO<void> = randomInt(30, 35).chain(n => log(fib(n)))

replicateIO(3, printFib).run()
/*
1346269
9227465
3524578
*/

Example 3: IO

We can build many other combinators for IO, for example the time combinator mimics the analogous Unix command: given an action IO<A>, we can derive an action IO<A> that prints to the console the elapsed time
import { IO } from ‘fp-ts/lib/IO’

/** returns the current time in millis */
export const now = new IO(() => new Date().getTime())

export function time<A>(ma: IO<A>): IO<A> {
return now.chain(start =>
ma.chain(a => now.chain(end => log(`Elapsed: \${end – start}`).map(() => a)))
)
}

Usage
time(replicateIO(3, printFib)).run()
/*
5702887
1346269
14930352
Elapsed: 193
*/

With partials…
time(replicateIO(3, time(printFib))).run()
/*
3524578
Elapsed: 32
14930352
Elapsed: 125
3524578
Elapsed: 32
Elapsed: 189
*/ 