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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 Thrush 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

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

Authors

Changelog

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

For more detailed changelogs, please check the release changelogs.