Add Combinator Definition

[jethrolarson]

You know what you'd like it to say?

[hemanth]

Are you referring to Fixed-point combinator?

[jethrolarson]

There's a whole bunch of them. S, K, B, C, I, W, and everything in aviary...

On Sun, Dec 11, 2016, 9:06 PM hemanth.hm notifications@github.com wrote:

Are you referring to Fixed-point combinator?

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[hemanth]

And then there is Y 🤗

On Mon, Dec 12, 2016, 4:46 PM Jethro Larson notifications@github.com wrote:

There's a whole bunch of them. S, K, B, C, I, W, and everything in aviary...

On Sun, Dec 11, 2016, 9:06 PM hemanth.hm notifications@github.com wrote:

Are you referring to Fixed-point combinator?

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[stereobooster]

Most known fixed-point combinator is Y combinator, but it will not work in JS, because of order of evaluation. So there is Z combinator to the rescue

// The Y combinator is defined as:
// Y          = λf.(λx.f (x x))(λx.f (x x))
// The Z combinator is defined as:
// Z          = λf.(λx.f (λv.((x x) v))) (λx.f (λv.((x x) v)))

const Z       = f => (x => f (v => ((x (x)) (v)))) (x => f (v => ((x (x)) (v))))

// example of usage: factorial
const factgen = fact => n => n < 2 ? 1 : n * fact(n-1)
const fact    = Z(factgen)

Fixed-point combinator is a function to find fixed point of other function. Simple example of fixed point of function is dotty number e.g. solution of cos(x) = x.

But the most striking thing about fixed-point combinator if it is applied to higher order function e.g. function which returns function, fixed point of higher order function is function too. First time I realized it blowed my mind.

[stereobooster]

Other idea: we can expand Lambda Calculus section.

A lot of simple Lambda Calculus ideas pretty trivial to demonstrate in JS

//True   = λt.λf.t
//False  = λt.λf.f
//Not    = λb.b False True

const True   = t => f => t
const False  = t => f => f
const Not    = b => b (False) (True)

const toBool = (churchBoolean) => churchBoolean (true) (false)

//Zero   = λf.λx.x
//Succ   = λn.λf.λx.f (n f x)

const Zero   = f => z => z
const Succ   = n => f => z => f (n (f) (z))

const toNumber = (churchNumeral) => churchNumeral((x) => x+1) (0)

But I suppose it can be separate page dedicated to Lambda Calculus.

[jethrolarson]

Church encoding is cool but I think it's too far away from being practical to try to teach it here

On Thu, Dec 22, 2016, 4:41 PM stereobooster notifications@github.com wrote:

Other idea: we can expand Lambda Calculus section.

A lot of simple Lambda Calculus ideas pretty trivial to demonstrate in JS

//True = λt.λf.t//False = λt.λf.f//Not = λb.b False True const True = t => f => tconst False = t => f => fconst Not = b => b (False) (True) const toBool = (churchBoolean) => churchBoolean (true) (false) //Zero = λf.λx.x//Succ = λn.λf.λx.f (n f x) const Zero = f => z => zconst Succ = n => f => z => f (n (f) (z))const toNumber = (churchNumeral) => churchNumeral((x) => x+1) (0)

But I suppose it can be separate page dedicated to Lambda Calculus.

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[SeanSilke]

It would be nice to see definition of various combinations. The C combinator is rather practical and can be used in everyday coding.

[jethrolarson]

Would the combinators all be curried?

On Thu, Jan 19, 2017, 1:57 AM Sergey Orlov notifications@github.com wrote:

It would be nice to see definition of various combinations. The C combinator is rather practical and can be used in everyday coding.

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[SeanSilke]

@jethrolarson Don't sure that i get your question right. Yes C combinator is type of curing. My knowledge of combinators is rather limited, actually i get it from this great talk of Reginald Braithwaite https://youtu.be/3t75HPU2c44

[jethrolarson]

I think if we have functional combinators in the doc we should just have a general definition and not try to teach the details of every combinator. We can link out for that.

[stereobooster]

Also worth to elaborate a bit more, about why Y combinator does not work. It is because order of evaluation and add link to Lazy evaluation section

[jethrolarson]

Here's a whack at it:

Functional Combinator

A higher-order function, usually curried, which returns a new function changed in some way. Functional combinators are often used in point-free programming to write especially terse programs.