Combinator

Description

This package provides a list of well known Combinators.

A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments. It was introduced in 1920 by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. Combinators which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions - and to remove any mention of variables - particularly in predicate logic.

Requirements

• PHP >= 8

Installation

composer require loophp/combinator

Available combinators

Name	Alias	Composition	Composition using S and K	Haskell	Lambda calculus	Term definition (JS like)	Type	# Arguments
A	Apply	SK(SK)	(S(K))(S(K))	$	λab.ab	a => b => a(b)	(a -> b) -> a -> b	2
B	Bluebird	S(KS)K	S(KS)K	.	λabc.a(bc)	a => b => c => a(b(c))	(a -> b) -> (c -> a) -> c -> b	3
Blackbird	Blackbird	BBB	(S(K(S(KS)K)))(S(KS)K)	...	λabcd.a(bcd)	a => b => c => => d => a(b(c)(d))	(c -> d) -> (a -> b -> c) -> a -> b -> d	4
C	Cardinal	S(BBS)(KK)	((S((S(K(S(KS)K)))S))(KK))	flip	λabc.acb	a => b => c => a(c)(b)	(a -> b -> c) -> b -> a -> c	3
D	Dove	BB	(S(K(S(KS)K)))		λabcd.ab(cd)	a => b => c => d => a(b)(c(d))	(a -> c -> d) -> a -> (b -> c) -> b -> d	4
E	Eagle	B(BBB)	(S(K((S(K(S(KS)K)))((S(KS))K))))		λabcde.ab(cde)	a => b => c => d => e => a(b)(c(d)(e))	(a -> d -> e) -> a -> (b -> c -> d) -> b -> c -> e	5
F	Finch	ETTET	((S(K((S((SK)K))(K((S(K(S((SK)K))))K)))))((S(K((S(K(S(KS)K)))((S(KS))K))))((S(K(S((SK)K))))K)))		λabc.cba	a => b => c => c(b)(a)	a -> b -> (b -> a -> c) -> c	3
G	Goldfinch	BBC	((S(K(S(KS)K)))((S((S(K(S(KS)K)))S))(KK)))		λabcd.ad(bc)	a => b => c => d => a(d)(b(c))	(a -> b -> c) -> (d -> b) -> d -> a -> c	4
H	Hummingbird	BW(BC)	((S(K((S(K(S((S(K((S((SK)K))((SK)K))))((S(K(S(KS)K)))((S(K(S((SK)K))))K))))))K)))(S(K((S((S(K(S(KS)K)))S))(KK)))))		λabc.abcb	a => b => c => a(b)(c)(b)	(a -> b -> a -> c) -> a -> b -> c	3
I	Idiot	SKK	((SK)K)	id	λa.a	a => a	a -> a	1
J	Jay	B(BC)(W(BC(E)))	((S(K(S(K((S((S(K(S(KS)K)))S))(KK))))))((S((S(K((S((SK)K))((SK)K))))((S(K(S(KS)K)))((S(K(S((SK)K))))K))))(K((S(K((S((S(K(S(KS)K)))S))(KK))))(S(K((S(K(S(KS)K)))((S(KS))K))))))))		λabcd.ab(adc)	a => b => c => d => a(b)(a(d)(c))	(a -> b -> b) -> a -> b -> a -> b	4
K	Kestrel	K	K	const	λab.a	a => b => a	a -> b -> a	2
Ki	Kite	KI	(K((SK)K))		λab.b	a => b => b	a -> b -> b	2
L	Lark	CBM	((S((S(KS))K))(K((S((SK)K))((SK)K))))		λab.a(bb)	a => b => a(b(b))		2
M	Mockingbird	SII	((S((SK)K))((SK)K))		λa.aa	a => a(a)		1
O	Owl	SI	(S((SK)K))		λab.b(ab)	a => b => b(a(b))	((a -> b) -> a) -> (a -> b) -> b	2
Omega	Ω	MM	(((S((SK)K))((SK)K))((S((SK)K))((SK)K)))		λa.(aa)(aa)	a => (a(a))(a(a))		1
Phoenix					λabcd.a(bd)(cd)	a => b => c => d => a(b(d))(c(d))	(a -> b -> c) -> (d -> a) -> (d -> b) -> d -> c	4
Psi				on	λabcd.a(bc)(bd)	a => b => c => d => a(b(c))(b(d))	(a -> a -> b) -> (c -> a) -> c -> c -> b	4
Q	Queer	CB	((S(K(S((S(KS))K))))K)	(##)	λabc.b(ac)	a => b => c => b(a(c))	(a -> b) -> (b -> c) -> a -> c	3
R	Robin	BBT	((S(K(S(KS)K)))((S(K(S((SK)K))))K))		λabc.bca	a => b => c => b(c)(a)	a -> (b -> a -> c) -> b -> c	3
S	Starling	S	S	<*>	λabc.ac(bc)	a => b => c => a(c)(b(c))	(a -> b -> c) -> (a -> b) -> a -> c	3
S_				<*>	λabc.a(bc)c	a => b => c => a(b(c))(c)	(a -> b -> c) -> (b -> a) -> b -> c	3
S2				<*>	λabcd.a((bd)(cd))	a => b => c => d => a(b(d))(c(d))	(b -> c -> d) -> (a -> b) -> (a -> c) -> a -> d	4
T	Trush	CI	((S(K(S((SK)K))))K)	(#)	λab.ba	a => b => b(a)	a -> (a -> b) -> b	2
U	Turing	LO	((S(K(S((SK)K))))((S((SK)K))((SK)K)))		λab.b(aab)	a => b => b(a(a)(b))		2
V	Vireo	BCT	((S(K((S((S(K(S(KS)K)))S))(KK))))((S(K(S((SK)K))))K))		λabc.cab	a => b => c => c(a)(b)	a -> b -> (a -> b -> c) -> c	3
W	Warbler	C(BMR)	((S(K(S((S(K((S((SK)K))((SK)K))))((S(K(S(KS)K)))((S(K(S((SK)K))))K))))))K)		λab.abb	a => b => a(b)(b)	(a -> a -> b) -> a -> b	2
Y	Y-Fixed point				λa.(λb(a(bb))(λb(a(bb))))	a => (b => b(b))(b => a(c => b(b)(c)))		1
Z	Z-Fixed point				λa.M(λb(a(Mb)))			1

Usage

Simple combinators

<?php

declare(strict_types=1);

include 'vendor/autoload.php';

use loophp\combinator\Combinators;

// Lambda calculus: I combinator correspond to λa.a
Combinators::I()('a'); // a

// Lambda calculus: K combinator correspond to λa.λb.a (or λab.a)
Combinators::K()('a')('b'); // a

// Lambda calculus: C combinator correspond to λf(λa(λb(fba)))
// and thus: C K a b = b
Combinators::C()(Combinators::K())('a')('b'); // b

// Lambda calculus: Ki combinator correspond to λa.λb.b (or λab.b)
Combinators::Ki()('a')('b'); // b

Y combinator

<?php

declare(strict_types=1);

namespace Test;

include __DIR__ . '/vendor/autoload.php';

use Closure;
use loophp\combinator\Combinators;

// Example 1
$factorialGenerator = static fn (Closure $fact): Closure =>
static fn (int $n): int  => (0 === $n) ? 1 : ($n * $fact($n - 1));

$factorial = Combinators::Y()($factorialGenerator);

var_dump($factorial(6)); // 720

// Example 2
$fibonacciGenerator = static fn (Closure $fibo): Closure =>
static fn (int $number): int => (1 >= $number) ? $number : $fibo($number - 1) + $fibo($number - 2);

$fibonacci = Combinators::Y()($fibonacciGenerator);

var_dump($fibonacci(10)); // 55

More on the wikipedia page.

Suggested reading and resources

• To Mock a Mockingbird
• http://dkeenan.com/Lambda/
• https://gist.github.com/Avaq/1f0636ec5c8d6aed2e45
• https://en.wikipedia.org/wiki/Combinatory_logic
• https://github.com/sanctuary-js/sanctuary
• https://en.wikipedia.org/wiki/Lambda_calculus
• https://hackage.haskell.org/package/data-aviary-0.4.0/docs/src/Data-Aviary-BirdsInter.html
• https://github.com/fantasyland/fantasy-birds/blob/master/README.md
• https://www.cis.upenn.edu/~bcpierce/tapl/
• https://plato.stanford.edu/entries/lambda-calculus/
• https://github.com/glebec/lambda-talk
• https://www.youtube.com/watch?v=seVSlKazsNk

Contributing

Feel free to contribute by sending pull requests. We are a usually very responsive team and we will help you going through your pull request from the beginning to the end.

For some reasons, if you can't contribute to the code and willing to help, sponsoring is a good, sound and safe way to show us some gratitude for the hours we invested in this package.

Sponsor me on Github and/or any of the contributors.

Thanks

• Marco Pivetta
• Danny Willems

Authors

• Pol Dellaiera

Changelog

See CHANGELOG.md for a changelog based on git commits.

For more detailed changelogs, please check the release changelogs.