Common combinators in JavaScript

combinators.js

// list of combinators http://www.angelfire.com/tx4/cus/combinator/birds.html
const I = x => x;
const K = x => y => x;  // T
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 P = f => g => x => y => f(g(x))(g(y));
const Y = f => (g => g(g))(g => f(x => g(g)(x)));

// Boolean
const T = a => b => a // K
const F = a => b => b // KI
const not = p => p(F)(T)
// usage
not(T)
not(F)

// Numbers (Church encoding)
const zero = f => a => a // KI
const once = f => a => f(a) // I
const twice = f => a => f(f(a))
const thrice = f => a => f(f(f(a)))
succ = n => f => a => f(n(f)(a))
// usage
twice(not)(F)
thrice(not)(T)

// birds
const B = f => g => a => f(g(a)) // blue bird (compose function)
// usage
B(not)(not)(T) // from right to left

combinators.md

Name	#	Haskell	Ramda	Sanctuary	Signature
identity	I	id	identity	I	a → a
constant	K	const	always	K	a → b → a
apply¹	A	($)	call		(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	ap²	ap²	ap²	(a → b → c) → (a → b) → a → c
psi	P	on		on	(b → b → c) → (a → b) → a → a → c
fix-point³	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) 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.

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)