Haskell:关于部分应用的问题

Question

我正在读《为大善而学 Haskell!》一书作者:Miran Lipovaca，并在第 5 章中学习高阶函数。

其中一个示例涉及以下函数:

applyTwice :: (a -> a) -> a -> a
applyTwice f x = f (f x)

以下是函数输出的示例:

ghci> applyTwice (++ " HAHA") "HEY"
"HEY HAHA HAHA"

ghci> applyTwice ("HAHA " ++) "HEY"
"HAHA HAHA HEY"

对于第一个示例，我知道字符串是通过以下方式使用连接运算符生成的:

"HEY" ++ " HAHA"
"HEY HAHA" ++ " HAHA"
"HEY HAHA HAHA"

但是，我不明白第二个示例中的串联运算符是如何工作的。输出字符串“HAHA HAHA HEY”是如何产生的？任何见解表示赞赏。

Answer 1

For the first example, I understand that the string was produced by using the concatenation operator in the following manner:

"HEY" ++ " HAHA"
"HEY HAHA" ++ " HAHA"
"HEY HAHA HAHA"

如果您从函数的角度思考，它会有所帮助，而不是直接跳转到中缀表达式(即 ++ 位于中间)。

(++ " HAHA") :: [Char] -> [Char]   -- #1 this is a function (++ is partially applied)     
"HEY" :: [Char]

(++ " HAHA") "HEY"                 -- apply "HEY" as an argument to #1
-- same as "HEY" ++ " HAHA"

(+) :: (Num a) => a -> a -> a      -- #2 a binary function
(+) 1 2                            -- #3 apply 1 and 2 as arguments to #2
-- same as 1 + 2

-- technically, #3 is curried as
--    ((+) 1) 2                    -- i.e. (+) 1 is a partially applied function, which is then applied to 2     

如果将 (++ "HAHA") 代入 applyTwice 的定义中，您将得到

applyTwice f x = f (f x)
applyTwice (++ " HAHA") "HEY" = (++ " HAHA") ((++ " HAHA") "HEY")

                              = (++ " HAHA") ("HEY" ++ " HAHA")
                              = (++ " HAHA") ("HEY HAHA")
                              = "HEY HAHA" ++ " HAHA"
                              = "HEY HAHA HAHA"

现在对 applyTwice ("HAHA "++) "HEY"执行相同的操作。

applyTwice f x = f (f x)
applyTwice ("HAHA " ++) "HEY" = ("HAHA " ++) (("HAHA " ++) "HEY")

                              = ("HAHA " ++) ("HAHA " ++ "HEY")
                              = ("HAHA " ++) ("HAHA HEY")
                              = "HAHA " ++ "HAHA HEY"
                              = "HAHA HAHA HEY"