怎样理解Functor与Monad

1. 复合函数操作符

Prelude> :t (.)
(.) :: (b -> c) -> (a -> b) -> a -> c
Prelude> (.) ((+) 5) ((*) 2) 4
13

　　

所以(.)操作符的作用，是将4作为参数传递给((*) 2)函数，再将结果传递给((+) 5)函数。

这就是数学里面的复合函数：

f(x) = 2x

g(x) = x + 5

g(f(x)) = g(2x) = (2x) + 5 = 2x + 5

g(f(4)) = 2*4 + 5 = 13

2. Functor

194 {- | The 'Functor' class is used for types that can be mapped over.
195 Instances of 'Functor' should satisfy the following laws:
196
197 > fmap id  ==  id
198 > fmap (f . g)  ==  fmap f . fmap g
199
200 The instances of 'Functor' for lists, 'Data.Maybe.Maybe' and 'System.IO.IO'
201 satisfy these laws.
202 -}
203
204 class  Functor f  where
205     fmap        :: (a -> b) -> f a -> f b
206
207     -- | Replace all locations in the input with the same value.
208     -- The default definition is @'fmap' . 'const'@, but this may be
209     -- overridden with a more efficient version.
210     (<$)        :: a -> f b -> f a
211     (<$)        =  fmap . const

　　id是一个函数

Prelude> :t id
id :: a -> a
Prelude> id "Daniel"
"Daniel"
Prelude> id 1
1
Prelude> id True
True
Prelude> id Just "Happy"
Just "Happy"
Prelude> id Nothing
Nothing

　　Functor是一种typeclass，用来定义一种允许的操作集合，这里是fmap，并且对于fmap提出了需要满足的条件：

• fmap id == id
• fmap (f . g) == fmap f . fmap g

可以视为"交换律"和“分配律”

Prelude Data.Char> fmap isDigit ((++) ['0'..'9'] ['a'..'f'])
[True,True,True,True,True,True,True,True,True,True,False,False,False,False,False,False]

　　

Prelude Data.Char> :t isDigit
isDigit :: Char -> Bool

　　

fmap不仅可以操作List（此时与List的函数map作用相同），还可以操作比如Maybe

Prelude Data.Char> fmap isDigit (Just 'a')
Just False
Prelude Data.Char> fmap isDigit (Nothing)
Nothing
Prelude Data.Char> fmap isDigit (Just '1')
Just True

　　

3. Manod

Monad与Functor一样，也是用来定义类型可以进行的操作集合。

Prelude Data.Char> :i Monad
class Monad m where
  (>>=) :: m a -> (a -> m b) -> m b
  (>>) :: m a -> m b -> m b
  return :: a -> m a
  fail :: String -> m a
  	-- Defined in `GHC.Base'
instance Monad Maybe -- Defined in `Data.Maybe'
instance Monad (Either e) -- Defined in `Data.Either'
instance Monad [] -- Defined in `GHC.Base'
instance Monad IO -- Defined in `GHC.Base'
instance Monad ((->) r) -- Defined in `GHC.Base'

　　

可以看到，这些操作包括(>>=) (>>) return fail

（>>）执行两个操作，但是放弃前一个操作的结果

Prelude Data.Char> :t (>>)
(>>) :: Monad m => m a -> m b -> m b
Prelude Data.Char> (>>) "Daniel" "King"
"KingKingKingKingKingKing"

　　

212
213 {- | The 'Monad' class defines the basic operations over a /monad/,
214 a concept from a branch of mathematics known as /category theory/.
215 From the perspective of a Haskell programmer, however, it is best to
216 think of a monad as an /abstract datatype/ of actions.
217 Haskell's @do@ expressions provide a convenient syntax for writing
218 monadic expressions.
219
220 Minimal complete definition: '>>=' and 'return'.
221
222 Instances of 'Monad' should satisfy the following laws:
223
224 > return a >>= k  ==  k a
225 > m >>= return  ==  m
226 > m >>= (\x -> k x >>= h)  ==  (m >>= k) >>= h
227
228 Instances of both 'Monad' and 'Functor' should additionally satisfy the law:
229
230 > fmap f xs  ==  xs >>= return . f
231
232 The instances of 'Monad' for lists, 'Data.Maybe.Maybe' and 'System.IO.IO'
233 defined in the "Prelude" satisfy these laws.
234 -}
235
236 class  Monad m  where
237     -- | Sequentially compose two actions, passing any value produced
238     -- by the first as an argument to the second.
239     (>>=)       :: forall a b. m a -> (a -> m b) -> m b
240     -- | Sequentially compose two actions, discarding any value produced
241     -- by the first, like sequencing operators (such as the semicolon)
242     -- in imperative languages.
243     (>>)        :: forall a b. m a -> m b -> m b
244         -- Explicit for-alls so that we know what order to
245         -- give type arguments when desugaring
246
247     -- | Inject a value into the monadic type.
248     return      :: a -> m a
249     -- | Fail with a message.  This operation is not part of the
250     -- mathematical definition of a monad, but is invoked on pattern-match
251     -- failure in a @do@ expression.
252     fail        :: String -> m a
253
254     {-# INLINE (>>) #-}
255     m >> k      = m >>= \_ -> k
256     fail s      = error s

　　

(>>=)是将前一个操作的结果作为参数传递给后一个操作，但是注意需要使用return将从a到b的正常转换(a -> b)变成(a -> mb)，即(a -> ma)(a - b) = (a -> mb)

Prelude Data.Char> :t return
return :: Monad m => a -> m a
Prelude Data.Char> :t (>>=)
(>>=) :: Monad m => m a -> (a -> m b) -> m b
Prelude Data.Char> (>>=) (return ((++) "Daniel" "King")) ((:) 'X')
"XDanielKing"

　　

Prelude Data.Char> (>>=) (return ((++) "Daniel" "King")) ((++) "Hello ")
"Hello DanielKing"

　　

所以(>>=)和return是配合使用的，效果很类似于Unix下的管道操作。

Monad的设计想法是允许pure的Haskell可以产生一些side effect，或者说除了自身的值以外，可以保存下一些状态信息。

比如在这里， ((++) "Daniel" "King")的结果就传递给了后面的action，这样就可以用来更新状态信息。

比较明显的应用是在IO以及Exception Handling上面。

---

如果参考数学中的概念，Monad可以被看作是虚数体系，或者是另外一个维度的类型；

普通的类型与Monad类型之间需要显式地进行转换，

return ： 普通类型转换成Monad类型

<-：Monad类型转换成普通类型

IO action都是操作的Monad类型。

*RecursiveContents Control.Monad Data.Char System.FilePath> :t forM
forM :: Monad m => [a] -> (a -> m b) -> m [b]
*RecursiveContents Control.Monad Data.Char System.FilePath> forM "Daniel King" $ \ch -> do return (toUpper ch)
"DANIEL KING"