高阶函数是一类可以接受函数为参数的函数。我们看到过的有 map
，没什么多大的不同。他们提供了一种只有函数式编程语言才有的抽象能力。在Haskell这样的函数式编程语言中，函数和其他成员一样，要理解高阶函数并不难。
本书中有独立的一章是关于高阶函数的，这不是因为他们很难——我们也已经认识它们了——而是因为它们很强大。一旦我们可以像把函数作为参数来调用，函数就可以变得更加强大。一般来说，高阶函数有很高的抽象能力 。 另外,高阶函数使得Haskell 更加有趣。
最快的快排算法编辑
不要太激动，但 快排 quickSort
的确是最快的。假如你已经听过了可以跳过。
= quickSort
的背后编辑
很简单，对于一个表，我们先取其第一个元素，再把这个表分成三部分。
第一部分是那些应该排在这个元素之前的元素，第二部分是那些等于这个元素的元素，第三部分是那些应该排在这个元素之后的元素。然后呢，当然是要把这三部分连接起来。我们就得到了一个更“好”的表，不是吗？
注意，第一部分和第三部分都是没有排好序的；至于第二部分，就根本不需要排序（因为它们都是相等的）；接下来我们应该把第一和第三部分排好序，如何？要这么排呢？只需要把相同的算法重新用一遍就行。但所有都完成后，这个表也排好了。
So Let's Get Down To It!编辑
 空表，什么都不做
 这个是递归的初始条件
quickSort [] = []
 只有一个元素的时候，也不需要排序
 实际上，第三个已经包含这一部分的处理了
 但我们要一步一步来
quickSort [x] = [x]
 整个排序的关键
 拿出一个元素作为 “支点” ，剩下的去排序
 别忘了支点是在表的中间 ！
quickSort (x : xs) = (quickSort less) ++ (x : equal) ++ (quickSort more)
where less = filter (< x) xs
equal = filter (== x) xs
more = filter (> x) xs
完成了！ 我想你已经看过 quickSort
，你也许在想递归是很难实现奇技淫巧而已。
现在，我们要怎么利用它 ？编辑
在我们的 quickSort
中，对任意表的排序都是小菜一碟。假设我们有一个 字符串( String
) 表,例如一堆英语单词，我们也可以用现在的 quickSort
对其排序。 在剩下的章节李，我们会用一个伪字典来做例子。(当然，用 有25000 单词的字典也是可以的）
dictionary = ["I", "have", "a", "thing", "for", "Linux"]
上面的就是给我们用的quickSort
字典
["I", "Linux", "a", "for", "have", "thing"]
假如我们要按降序排列那该怎样做？很容易，把这个表翻转就行了，reverse (quickSort dictionary)
就可以了。
等一下，翻转并不能使得我们把这个表真正的按降序排列，我们只是把它按升序排列然后再翻转而已。虽然这样做结果是一样，但是却不是同一回事。
而且，结果可能并不是你想要的， “a“ 应该在 ”I” 前，“Linux” 应该在 “have” 和 “thing” 之间。出了什么问题呢？
是这样的，字符全 (String
) 使用 ASCII 码来表示的，而在 ASCII码 （和其他几乎所有的编码方法）里，大写字母是小于小写字母的。所以，“Z” 小于 ”a” . 我们得修改一下代码，使得 quickSort
忽略大小写。可能以后有用。
但是，看上去 quickSort
已经加不了什么了。我们得做点额外的东西。
修改我们原来的代码编辑
我们需要从 quickSort
中分离其对比的功能。
What we need to do is to factor out the comparisons quickSort
makes. We need to provide quickSort
with a function that compares two elements, and gives an Ordering
, and as you can imagine, an Ordering
is any of LT, EQ, GT
.
To sort in the descending order, we supply quickSort
with a function that returns the opposite of the usual Ordering
. For the caseinsensitive sort, we may need to define the function ourselves. By all means, we want to make quickSort
applicable to all such functions so that we don't end up writing it over and over again, each time with only minor changes.
quickSort
, Take Two编辑
Our quickSort
will take two things this time: first, the comparison function, and second, the list to sort.
A comparison function will be a function that takes two things, say, x
and y
, and compares them. If x
is less than y
(according to the criteria we want to implement by this function), then the value will be LT
. If they are equal (well, equal with respect to the comparison, we want "Linux" and "linux" to be equal when we are dealing with the insensitive case), we will have EQ
. The remaining case gives us GT
(pronounced: greater than, for obvious reasons).
 no matter how we compare two things
 the first two equations should not change
 they need to accept the comparison function though
 but the result doesn't need it
quickSort _ [] = []
quickSort _ [x] = [x]
 we are in a more general setting now
 but the changes are worth it!
 c is the comparison function
quickSort c (x : xs) = (quickSort c less) ++ (x : equal) ++ (quickSort c more)
where less = filter (\y > y `c` x == LT) xs
equal = filter (\y > y `c` x == EQ) xs
more = filter (\y > y `c` x == GT) xs
Cool!
注解
Almost all the basic data types in Haskell are members of the 
But What Did We Gain?编辑
Reuse. We can reuse quickSort
to serve different purposes.
 the usual ordering
 uses the compare function from the Ord class
usual = compare
 the descending ordering, note we flip the order of the arguments to compare
descending x y = compare y x
 the caseinsensitive version is left as an exercise!
insensitive = ...
 can you think of anything without making a very big list of all possible cases?
And we are done!
quickSort usual dictionary
should, then, give
["I", "Linux", "a", "for", "have", "thing"]
The comparison is just compare
from the Ord
class. This was our quickSort
, before the tweaking.
quickSort descending dictionary
now gives
["thing", "have", "for", "a", "Linux", "I"]
And finally,
quickSort insensitive dictionary
gives
["a", "for", "have", "I", "Linux", "thing"]
Exactly what we wanted!
练习 

Write insensitive , such that quickSort insensitive dictionary gives ["a", "for", "have", "I", "Linux", "thing"] 
HigherOrder Functions and Types编辑
Our quickSort
has type (a > a > Ordering) > [a] > [a]
.
Most of the time, the type of a higherorder function provides a good guideline about how to use it. A straightforward way of reading the type signature would be, "quickSort
takes a function that gives an ordering of a
s, and a list of a
s, to give a list of a
s". It is then natural to guess that the function sorts the list respecting the given ordering function.
Note that the parentheses surrounding a > a > Ordering
is mandatory. It says that a > a > Ordering
altogether form a single argument, an argument that happens to be a function. What happens if we omit the parentheses? We would get a function of type a > a > Ordering > [a] > [a]
, which accepts four arguments instead of the desired two (a > a > Ordering
and [a]
). Furthermore none of the four arguments, neither a
nor Ordering
nor [a]
are functions, so omitting the parentheses would give us something that isn't a higher order function.
Furthermore, it's worth noting that the >
operator is rightassociative, which means that a > a > Ordering > [a] > [a]
means the same thing as a > (a > (Ordering > ([a] > [a])))
. We really must insist that the a > a > Ordering
be clumped together by writing those parentheses... but wait... if >
is rightassociative, wouldn't that mean that the correct signature (a > a > Ordering) > [a] > [a]
actually means... (a > a > Ordering) > ([a] > [a])
?
Is that really what we want?
If you think about it, we're trying to build a function that takes two arguments, a function and a list, returning a list. Instead, what this type signature is telling us is that our function takes one argument (a function) and returns another function. That is profoundly odd... but if you're lucky, it might also strike you as being profoundly beautiful. Functions in multiple arguments are fundamentally the same thing as functions that take one argument and give another function back. It's OK if you're not entirely convinced. We'll go into a little bit more detail below and then show how something like this can be turned to our advantage.
练习 

The following exercise combines what you have learned about higher order functions, recursion and IO. We are going to recreate what programmers from more popular languages call a "for loop". Implement a function for :: a > (a>Bool) > (a>a) > (a> IO ()) > IO () for i p f job =  ???An example of how this function would be used might be for 1 (<10) (+1) (print)which prints the numbers 1 to 9 on the screen. Starting from an initial value
Some more challenging exercises you could try

Currying编辑
I hope you followed the reasoning of the preceding chapter closely enough. If you haven't, you should, so give it another try.
Currying is a technique^{[1]} that lets you partially apply a multiparameter function. When you do that, it remembers those given values, and waits for the remaining parameters.
Our quickSort
takes two parameters, the comparison function, and the list. We can, by currying, construct variations of quickSort
with a given comparison function. The variation just "remembers" the specific comparison, so you can apply it to the list, and it will sort the list using the that comparison function.
descendingSort = quickSort descending
What is the type of descendingSort? quickSort
was (a > a > Ordering) > [a] > [a]
, and the comparison function descending
was a > a > Ordering
. Applying quickSort
to descending
(that is, applying it partially, we haven't specified the list in the definition) we get a function (our descendingSort
) for which the first parameter is already given, so you can scratch that type out from the type of quickSort
, and we are left with a simple [a] > [a]
. So we can apply this one to a list right away!
descendingSort dictionary
gives us
["thing", "have", "for", "a", "Linux", "I"]
It's rather neat. But is it useful? You bet it is. It is particularly useful as sections, you might notice. Currying often makes Haskell programs very concise. More than that, it often makes your intentions a lot clearer. Remember
less = filter (< x) xs
from the first version of quickSort
? You can read it aloud like "keep those in xs
that are less than x
". Although (< x)
is just a shorthand for (\y > y < x)
, try reading that aloud!