如果你有一个类型构造函数和返回该类型族值的函数，你就需要单子。最终，你会想把这些函数组合在一起。这是回答为什么的三个关键要素。

让我详细说明一下。你有Int, String和Real和函数类型Int -> String, String -> Real等等。你可以很容易地组合这些函数，以Int -> Real结尾。生活是美好的。

然后，有一天，您需要创建一个新的类型家族。这可能是因为您需要考虑不返回值(Maybe)、返回错误(Either)、多个结果(List)等的可能性。

注意Maybe是一个类型构造函数。它接受一个类型，比如Int，然后返回一个新的类型，可能是Int。首先要记住，没有类型构造函数，没有单子。

当然，你想在你的代码中使用你的类型构造函数，很快你就会以像Int -> Maybe String和String -> Maybe Float这样的函数结束。现在，你不能轻易地组合你的功能。生活不再美好。

这时单子就来拯救我们了。它们允许你再次组合这类功能。你只需要改变成分。> = =。

为什么我们需要单一类型?

由于I/O的困境和它在非严格语言(如Haskell)中可观察到的影响，使得单元接口如此突出:

[...] monads are used to address the more general problem of computations (involving state, input/output, backtracking, ...) returning values: they do not solve any input/output-problems directly but rather provide an elegant and flexible abstraction of many solutions to related problems. [...] For instance, no less than three different input/output-schemes are used to solve these basic problems in Imperative functional programming, the paper which originally proposed `a new model, based on monads, for performing input/output in a non-strict, purely functional language'. [...] [Such] input/output-schemes merely provide frameworks in which side-effecting operations can safely be used with a guaranteed order of execution and without affecting the properties of the purely functional parts of the language. Claus Reinke (pages 96-97 of 210). (emphasis by me.) [...] When we write effectful code – monads or no monads – we have to constantly keep in mind the context of expressions we pass around. The fact that monadic code ‘desugars’ (is implementable in terms of) side-effect-free code is irrelevant. When we use monadic notation, we program within that notation – without considering what this notation desugars into. Thinking of the desugared code breaks the monadic abstraction. A side-effect-free, applicative code is normally compiled to (that is, desugars into) C or machine code. If the desugaring argument has any force, it may be applied just as well to the applicative code, leading to the conclusion that it all boils down to the machine code and hence all programming is imperative. [...] From the personal experience, I have noticed that the mistakes I make when writing monadic code are exactly the mistakes I made when programming in C. Actually, monadic mistakes tend to be worse, because monadic notation (compared to that of a typical imperative language) is ungainly and obscuring. Oleg Kiselyov (page 21 of 26). The most difficult construct for students to understand is the monad. I introduce IO without mentioning monads. Olaf Chitil.

更普遍的:

Still, today, over 25 years after the introduction of the concept of monads to the world of functional programming, beginning functional programmers struggle to grasp the concept of monads. This struggle is exemplified by the numerous blog posts about the effort of trying to learn about monads. From our own experience we notice that even at university level, bachelor level students often struggle to comprehend monads and consistently score poorly on monad-related exam questions. Considering that the concept of monads is not likely to disappear from the functional programming landscape any time soon, it is vital that we, as the functional programming community, somehow overcome the problems novices encounter when first studying monads. Tim Steenvoorden, Jurriën Stutterheim, Erik Barendsen and Rinus Plasmeijer.

如果有另一种方法可以在Haskell中指定“有保证的执行顺序”，同时保持将常规Haskell定义与那些涉及I/O(及其可观察的效果)的定义分开的能力-翻译Philip Wadler的这种变化:

val echoML    : unit -> unit
fun echoML () = let val c = getcML () in
                if c = #"\n" then
                  ()
                else
                  let val _ = putcML c in
                  echoML ()
                end

fun putcML c  = TextIO.output1(TextIO.stdOut,c);
fun getcML () = valOf(TextIO.input1(TextIO.stdIn));

.．.然后可以像这样简单:

echo :: OI -> ()                         
echo u = let !(u1:u2:u3:_) = partsOI u in
         let !c = getChar u1 in          
         if c == '\n' then               
           ()                            
         else                            
           let !_ = putChar c u2 in      
           echo u3                       

地点:

data OI  -- abstract

foreign import ccall "primPartOI" partOI :: OI -> (OI, OI)
                      ⋮

foreign import ccall "primGetCharOI" getChar :: OI -> Char
foreign import ccall "primPutCharOI" putChar :: Char -> OI -> ()
                      ⋮

and:

partsOI         :: OI -> [OI]
partsOI u       =  let !(u1, u2) = partOI u in u1 : partsOI u2 

这是如何运作的呢?在运行时，Main。main接收一个初始OI伪数据值作为参数:

module Main(main) where

main            :: OI -> ()
          ⋮

使用parttoi或partsOI从其中产生其他OI值。您所要做的就是确保每个新的OI值在每次调用基于OI的定义(外部或其他)时最多使用一次。作为回报，你会得到一个普通的结果——它并没有与一些奇怪的抽象状态配对，或者需要使用回调延续等等。

使用OI，而不是像标准ML那样使用单元类型()，意味着我们可以避免总是必须使用单一的接口:

一旦你进入IO单子，你就永远被困在那里，并被简化为algolstyle命令式编程。 罗伯特·哈珀。

但如果你真的需要它:

type IO a       =  OI -> a

unitIO          :: a -> IO a
unitIO x        =  \ u -> let !_ = partOI u in x

bindIO          :: IO a -> (a -> IO b) -> IO b
bindIO m k      =  \ u -> let !(u1, u2) = partOI u in
                          let !x        = m u1 in
                          let !y        = k x u2 in
                          y

                      ⋮

所以，单体类型并不总是需要的-有其他的接口:

LML早在1989年就有了一个完整的oracle多处理器(sequence Symmetry)实现。Fudgets论文中的描述引用了这个实现。和它一起工作很愉快，也很实用。 […] 现在所有的事情都是用单子完成的，所以其他的解决方案有时会被遗忘。 Lennart Augustsson(2006)。

等一下:既然它与标准ML直接使用效果非常相似，那么这种方法及其使用的伪数据引用是透明的吗?

当然，只要找到一个合适的“参考透明度”的定义;有很多选择…

Monads are just a convenient framework for solving a class of recurring problems. First, monads must be functors (i.e. must support mapping without looking at the elements (or their type)), they must also bring a binding (or chaining) operation and a way to create a monadic value from an element type (return). Finally, bind and return must satisfy two equations (left and right identities), also called the monad laws. (Alternatively one could define monads to have a flattening operation instead of binding.)

列表单子通常用于处理不确定性。绑定操作选择列表中的一个元素(直观地说，它们都在并行世界中)，让程序员对它们进行一些计算，然后将所有世界中的结果组合到一个列表中(通过连接或平铺嵌套列表)。下面是如何在Haskell的单元框架中定义一个排列函数:

perm [e] = [[e]]
perm l = do (leader, index) <- zip l [0 :: Int ..]
            let shortened = take index l ++ drop (index + 1) l
            trailer <- perm shortened
            return (leader : trailer)

下面是一个示例repl会话:

*Main> perm "a"
["a"]
*Main> perm "ab"
["ab","ba"]
*Main> perm ""
[]
*Main> perm "abc"
["abc","acb","bac","bca","cab","cba"]

需要注意的是，列表单子绝不是计算的副作用。一个数学结构是一个单子(即符合上面提到的接口和规律)并不意味着副作用，尽管副作用现象通常很好地适合单子框架。

本杰明·皮尔斯在TAPL中说

一个类型系统可以看作是计算一种静态 近似于程序中项的运行时行为。

这就是为什么配备了强大类型系统的语言严格来说比类型差的语言更具表现力。你可以用同样的方式来思考单子。

正如@Carl和sigfpe所指出的那样，你可以为一个数据类型配备你想要的所有操作，而无需求助于单子、类型类或任何其他抽象的东西。然而，单子不仅允许你编写可重用的代码，还可以抽象出所有冗余的细节。

举个例子，假设我们想过滤一个列表。最简单的方法是使用filter函数:filter (> 3) [1..]10]，等于[4,5,6,7,8,9,10]。

filter的一个稍微复杂一点的版本也是从左向右传递累加器

swap (x, y) = (y, x)
(.*) = (.) . (.)

filterAccum :: (a -> b -> (Bool, a)) -> a -> [b] -> [b]
filterAccum f a xs = [x | (x, True) <- zip xs $ snd $ mapAccumL (swap .* f) a xs]

获取所有i，使i <= 10, sum [1..]I] > 4, sum [1..I] < 25，我们可以写

filterAccum (\a x -> let a' = a + x in (a' > 4 && a' < 25, a')) 0 [1..10]

等于[3,4,5,6]。

或者我们可以重新定义nub函数，它从列表中删除重复的元素，使用filterAccum:

nub' = filterAccum (\a x -> (x `notElem` a, x:a)) []

nub' [1,2,4,5,4,3,1,8,9,4] equals [1,2,4,5,3,8,9]. A list is passed as an accumulator here. The code works, because it's possible to leave the list monad, so the whole computation stays pure (notElem doesn't use >>= actually, but it could). However it's not possible to safely leave the IO monad (i.e. you cannot execute an IO action and return a pure value — the value always will be wrapped in the IO monad). Another example is mutable arrays: after you have leaved the ST monad, where a mutable array live, you cannot update the array in constant time anymore. So we need a monadic filtering from the Control.Monad module:

filterM          :: (Monad m) => (a -> m Bool) -> [a] -> m [a]
filterM _ []     =  return []
filterM p (x:xs) =  do
   flg <- p x
   ys  <- filterM p xs
   return (if flg then x:ys else ys)

filterM对列表中的所有元素执行一个单体操作，生成元素，单体操作返回True。

一个带有数组的过滤示例:

nub' xs = runST $ do
        arr <- newArray (1, 9) True :: ST s (STUArray s Int Bool)
        let p i = readArray arr i <* writeArray arr i False
        filterM p xs

main = print $ nub' [1,2,4,5,4,3,1,8,9,4]

按预期打印[1,2,4,5,3,8,9]。

还有一个带有IO单子的版本，它询问要返回哪些元素:

main = filterM p [1,2,4,5] >>= print where
    p i = putStrLn ("return " ++ show i ++ "?") *> readLn

E.g.

return 1? -- output
True      -- input
return 2?
False
return 4?
False
return 5?
True
[1,5]     -- output

最后一个例子，filterAccum可以用filterM来定义:

filterAccum f a xs = evalState (filterM (state . flip f) xs) a

StateT单子，它在底层使用，只是一个普通的数据类型。

这个例子说明，单子不仅允许您抽象计算上下文和编写干净的可重用代码(由于单子的可组合性，正如@Carl解释的那样)，而且还可以统一对待用户定义的数据类型和内置原语。

为什么我们需要单子?

We want to program only using functions. ("functional programming (FP)" after all). Then, we have a first big problem. This is a program: f(x) = 2 * x g(x,y) = x / y How can we say what is to be executed first? How can we form an ordered sequence of functions (i.e. a program) using no more than functions? Solution: compose functions. If you want first g and then f, just write f(g(x,y)). This way, "the program" is a function as well: main = f(g(x,y)). OK, but ... More problems: some functions might fail (i.e. g(2,0), divide by 0). We have no "exceptions" in FP (an exception is not a function). How do we solve it? Solution: Let's allow functions to return two kind of things: instead of having g : Real,Real -> Real (function from two reals into a real), let's allow g : Real,Real -> Real | Nothing (function from two reals into (real or nothing)). But functions should (to be simpler) return only one thing. Solution: let's create a new type of data to be returned, a "boxing type" that encloses maybe a real or be simply nothing. Hence, we can have g : Real,Real -> Maybe Real. OK, but ... What happens now to f(g(x,y))? f is not ready to consume a Maybe Real. And, we don't want to change every function we could connect with g to consume a Maybe Real. Solution: let's have a special function to "connect"/"compose"/"link" functions. That way, we can, behind the scenes, adapt the output of one function to feed the following one. In our case: g >>= f (connect/compose g to f). We want >>= to get g's output, inspect it and, in case it is Nothing just don't call f and return Nothing; or on the contrary, extract the boxed Real and feed f with it. (This algorithm is just the implementation of >>= for the Maybe type). Also note that >>= must be written only once per "boxing type" (different box, different adapting algorithm). Many other problems arise which can be solved using this same pattern: 1. Use a "box" to codify/store different meanings/values, and have functions like g that return those "boxed values". 2. Have a composer/linker g >>= f to help connecting g's output to f's input, so we don't have to change any f at all. Remarkable problems that can be solved using this technique are: having a global state that every function in the sequence of functions ("the program") can share: solution StateMonad. We don't like "impure functions": functions that yield different output for same input. Therefore, let's mark those functions, making them to return a tagged/boxed value: IO monad.

总幸福!

如果你有一个类型构造函数和返回该类型族值的函数，你就需要单子。最终，你会想把这些函数组合在一起。这是回答为什么的三个关键要素。

让我详细说明一下。你有Int, String和Real和函数类型Int -> String, String -> Real等等。你可以很容易地组合这些函数，以Int -> Real结尾。生活是美好的。

然后，有一天，您需要创建一个新的类型家族。这可能是因为您需要考虑不返回值(Maybe)、返回错误(Either)、多个结果(List)等的可能性。

注意Maybe是一个类型构造函数。它接受一个类型，比如Int，然后返回一个新的类型，可能是Int。首先要记住，没有类型构造函数，没有单子。

当然，你想在你的代码中使用你的类型构造函数，很快你就会以像Int -> Maybe String和String -> Maybe Float这样的函数结束。现在，你不能轻易地组合你的功能。生活不再美好。

这时单子就来拯救我们了。它们允许你再次组合这类功能。你只需要改变成分。> = =。