我怎么理解 "(.) .(.)"? [英] How can I understand "(.) . (.)"?

问题描述

我相信我理解 fmap .fmap 用于函子，但在函数上它已经让我头疼好几个月了.

I believe I understand fmap . fmap for Functors, but on functions it's hurting my head for months now.

我已经看到您可以将 (.) 的定义应用于 (.) .(.)，但我忘了怎么做.
当我自己尝试时，结果总是错的:

I've seen that you can just apply the definition of (.) to (.) . (.), but I've forgot how to do that.
When I try it myself it always turns out wrong:

(.) f g = x -> f (g x)
(.) (.) (.) = x -> (.) ((.) x)
x f -> (.) ((.) x) f
x f y  -> (((.)(f y)) x)
x f y g-> (((.)(f y) g) x)
x f y g-> ((f (g y)) x)
x f y g-> ((f (g y)) x):: t2 -> (t1 -> t2 -> t) -> t3 -> (t3 -> t1) -> t

如果只是应用定义"是唯一的方法，那么有人是怎么想出 (.) 的.(.)?
我一定缺少一些更深层次的理解或直觉.

If "just applying the definition" is the only way of doing it, how did anybody come up with (.) . (.)?
There must be some deeper understanding or intuition I'm missing.

推荐答案

你也可以使用你对 fmap 的理解.fmap.

如果你有两个 Functors foo 和 bar，那么

If you have two Functors foo and bar, then

fmap . fmap :: (a -> b)  ->  foo (bar a)    ->   foo (bar b)

fmap .fmap 接受一个函数，并为两个 Functor 的组合产生一个诱导函数.

fmap . fmap takes a function and produces an induced function for the composition of the two Functors.

现在，对于任何类型 t，(->) t 是一个 Functor，而 fmap> 对于那个 Functor 是 (.).

Now, for any type t, (->) t is a Functor, and the fmap for that Functor is (.).

所以 (.) .(.) 是 fmap .fmap 对于两个 Functor 是 (->) s 和 (->) t 的情况，以及因此

So (.) . (.) is fmap . fmap for the case where the two Functors are (->) s and (->) t, and thus

(.) . (.) :: (a -> b) -> ((->) s) ((->) t a) -> ((->) s) ((->) t b)
          =  (a -> b) -> (s -> (t -> a))     -> (s -> (t -> b))
          =  (a -> b) -> (s ->  t -> a )     -> (s ->  t -> b )

它组合"了一个函数 f :: a ->b 带有两个参数的函数，g :: s ->t->一个,

it "composes" a function f :: a -> b with a function of two arguments, g :: s -> t -> a,

((.) . (.)) f g = x y -> f (g x y)

该观点还清楚地表明，该模式以及如何扩展到采用更多参数的函数，

That view also makes it clear that, and how, the pattern extends to functions taking more arguments,

(.)             :: (a -> b) -> (s ->           a) -> (s ->           b)
(.) . (.)       :: (a -> b) -> (s -> t ->      a) -> (s -> t ->      b)
(.) . (.) . (.) :: (a -> b) -> (s -> t -> u -> a) -> (s -> t -> u -> b)

等

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