loophp / combinator Goto Github PK

Hello,

not considering myself an expert I wonder whether the definition of the Omega combinator is correct.
Shouldn't it be like this?

Ω := MM = (λa.aa)λa.aa

That way Ω does not consume further arguments. The loop is entered when M takes M as an argument, right?

Ωx = MMx = ((λa.aa)λa.aa)x

Whereas

(λa.(aa)(aa))x = (xx)(xx)

takes one more argument and would terminate when x was I := λa.a for instance.

Accordingly, the JS expression for omega would be a function without arguments.

() => (a => a(a))(a => a(a))

I might be wrong.

Anyway, I was happy to come across your project.
Thanks for sharing it!

Cheers!

TAPL (Types And Programming Languages) by B. Pierce is considered as « The Bible » for the beginners in type systems.

Not sure if you are interested in this, but it is an interesting exercise, and good added value for downstream consumers.

Given (for example) combinator A:

$next = function (string $input) : int {
    return strlen($input); // simplistic example, ignore the fact that it is redundant
};

$a = new A($next, 'foo');

$a(); // int - can be inferred, in theory

This can be achieved via something like:

/**
 * @psalm-template XType
 * @psalm-template ResultType
 */
final class A extends Combinator
{
    /**
     * @psalm-param callable(XType) : ResultType $f
     * @psalm-param XType $x
     */
    public function __construct(callable $f, $x)
    {
        // ...
    }

    /** @psalm-return ResultType */
    public function __invoke()
    {
        // ...
    }

    /** @psalm-return callable() : ResultType */
    public static function closure(): callable
    {
        return static function (callable $f, $x) {
            return (new self($f, $x))();
        };
    }
}

This is currently tested in (for example) phpunit via some static analysis scripts: https://github.com/sebastianbergmann/phpunit/tree/619868463e06e5072c6e42dbae174d99cdd10fc2/tests/static-analysis

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Thanks for the great list!

I have a question regarding the S2 combinator. Is it the same as Curry's formalizing combinator Φ?
Because if it is, it should be λabcd.a(bd)(cd) instead of λabcd.a((bd)(cd)), like in the JS. I am asking because I have never seen the combinator λabcd.a((bd)(cd)) and would like to add it to my own list if it's a widely-used combinator.

Moony