Common combinators in JavaScript

combinators.js

const I  = x => x
const K  = x => y => x
const A  = f => x => f (x)
const T  = x => f => f (x)
const W  = f => x => f (x) (x)
const C  = f => y => x => f (x) (y)
const B  = f => g => x => f (g (x))
const S  = f => g => x => f (x) (g (x))
const S_ = f => g => x => f (g (x)) (x)
const S2 = f => g => h => x => f (g (x)) (h (x))
const P  = f => g => x => y => f (g (x)) (g (y))
const Y  = f => (g => g (g)) (g => f (x => g (g) (x)))

combinators.md

Name	#	Haskell	Ramda	Sanctuary	Signature
identity	I	id	identity	I	a → a
constant	K	const	always	K	a → b → a
apply	A	($)	call	I¹	(a → b) → a → b
thrush	T	(&)	applyTo	T	a → (a → b) → b
duplication	W	join²	unnest²	join²	(a → a → b) → a → b
flip	C	flip	flip	flip	(a → b → c) → b → a → c
compose	B	(.), fmap²	map²	compose, map²	(b → c) → (a → b) → a → c
substitution	S	(<*>)²	ap²	ap²	(a → b → c) → (a → b) → a → c
chain	S_³	(=<<)²	chain²	chain²	(a → b → c) → (b → a) → b → c
converge	S2³	apply2way, liftA2², liftM2²		lift2²	(b → c → d) → (a → b) → (a → c) → a → d
psi	P	on	on	on	(b → b → c) → (a → b) → a → a → c
fix-point4	Y	fix			(a → a) → a

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¹) The A-combinator can be implemented as an alias of the I-combinator. Its implementation in Haskell exists because the infix nature gives it some utility. Its implementation in Ramda exists because it is overloaded with additional functionality.

²) Algebras like ap have different implementations for different types. They work like Function combinators only for Function inputs.

³) I could not find a consistent name for these combinators, but they are common enough in the JavaScript ecosystem to justify their inclusion. I named them myself in order to refer to their implementation.

⁴) In JavaScript and other non-lazy languages, it is impossible to implement the Y-combinator. Instead a variant known as the applicative or strict fix-point combinator is implemented. This variant is sometimes rererred to as the Z-combinator. The implementation found in combinators.js is the strictly evaluated "Z" combinator, which needs the extra wrapper around g (g) on the right hand side.

references.md

Note that when I use the word "combinator" in this context, it implies "function combinator in the untyped lambda calculus".

• To Mock a Mocking Bird (book of combinatory puzzles and examples)
• Combinator Birds (summary of all combinators named by the book)
• CombinatorsJS (implementation of all the same combinators in JavaScripts)
• Fantasy Combinators (implementation of common combinators in JavaScript)
• Collected Lambdas (collection of noteworthy combinators)
• Programming With Nothing (talk)
• Church Encoding (wikipedia)
• Fixed-point combinator (wikipedia)
• Lambda calculus (wikipedia)
• SKI combinator calculus (wikipedia)
• BCKW combinator calculus (wikipedia)

a → b → b is also useful:

const right = a => b => b;
const log = a => right(console.log(a))(a);

This behaves like identity function: a => a which does not affect to the original code but console.log(a) in the process.

It's possible to rewrite with a → b → a

const left = a => b => a;
const log = a => left(a)(console.log(a));

but, It's not intuitive in terms of the evaluation order.

@stken2050 Fortunately its trivial

const CK = C(K)

I don't see why console.log is a good case though. Maybe you meant 'tap':

const tap = (f) => (x) => {
  f(x);
  return x;
};

tap(console.log)(2)

// Or more usefully: 
doThing().then(tap(console.log))

It is impure though

@jethrolarson
Thanks.
Sure, log is IO and impure, and actually, I use the right for any IO operations.

Hello! Ramda v0.28.0 has shipped with some new functions, including on: https://ramdajs.com/docs/#on — Do you think we could update the table above?

It looks like that new on is actually the Psi combinator.

Thanks for the heads-up. Ramda on added to the table! 🎉

@JAForbes:

I have this as a pinned tab and I just stare at it sometimes like the obelisk in 2001.

Six years later, and I'm still doing that!

@Avaq: Thank you for a wonderful resource!

Thank you @CrossEye ❤️
I also still commonly refer to this resource, myself. :)

Same as gazing avianly at the Smullyan book, which I got in ebook after buying it physically.