Haskell语言学习笔记（8）Monoid

Monoid

class Monoid a where
  mempty :: a
  mappend :: a -> a -> a
  mconcat :: [a] -> a

Monoid（幺半群）是个类型类。存在单位元mempty，二元结合操作mappend，以及列表折叠操作mconcat。

注：幺半群是群论中的概念。所谓半群是指一个集合，其中存在一种满足结合律的结合运算。所谓幺，是指集合之内存在一个单位元，它与集合中任何元素e结合（包括左结合和右结合）的结果都是e。比如实数的求和以及求积都是幺半群。显然求和运算以及求积运算都满足结合律。求和的单位元是0，求积的单位元是1，因为0+e=e+0=e，而1*e=e*1=e。再比如时钟也是幺半群。

(<>)

(<>) :: Monoid m => m -> m -> m
(<>) = mappend

Monoid的法则

mempty <> x = x
x <> mempty = x
(x <> y) <> z = x <> (y <> z)

幺半群满足结合律（半群），存在单位元（幺）。

[a] 是 Monoid

instance Monoid [a] where
        mempty  = []
        mappend = (++)
        mconcat xss = [x | xs <- xss, x <- xs]

列表是个幺半群。二元结合操作(++)满足结合律。单位元为空列表[]。

Prelude> [1,2,3] <> [4,5,6][1,2,3,4,5,6]Prelude> "pang" <> mempty"pang"Prelude> mconcat [[1,2],[3,6],[9]][1,2,3,6,9]

Ordering 是 Monoid

data Ordering = LT | EQ | GT
instance Monoid Ordering where
        mempty         = EQ
        LT `mappend` _ = LT
        EQ `mappend` y = y
        GT `mappend` _ = GT

排序这个幺半群用于实现按字典排序。单位元为相等即EQ。

Prelude> LT <> GTLTPrelude> GT <> LTGTPrelude> mempty <> LTLTPrelude> mempty <> GTGT

Sum 和 Product 都是 Monoid

newtype Sum a = Sum { getSum :: a }
newtype Product a = Product {getProduct :: a}
instance Num a => Monoid (Sum a) where
    mempty = Sum 0
    Sum x `mappend` Sum y = Sum (x + y)
instance Num a => Monoid (Product a) where
    mempty = Product 1
    Product x `mappend` Product y = Product (x * y)

求和以及求积都是幺半群。显然都满足结合律。求和的单位元为0，求积的单位元为1。

Prelude Data.Monoid> Sum 5 <> Sum 6 <> Sum 10Sum {getSum = 21}Prelude Data.Monoid> getSum . mconcat . fmap Sum $ [5, 6, 10]21Prelude Data.Monoid> Product 5 <> Product 6 <> Product 10Product {getProduct = 300}Prelude Data.Monoid> getProduct . mconcat . fmap Product $ [5, 6, 10]300

Any 和 All 都是 Monoid

newtype Any = Any { getAny :: Bool }
newtype All = All { getAll :: Bool }
instance Monoid Any where
        mempty = Any False
        Any x `mappend` Any y = Any (x || y)
instance Monoid All where
        mempty = All True
        All x `mappend` All y = All (x && y)

求与以及求或都是幺半群。显然都满足结合律。求与的单位元为True，求或的单位元为False。

Prelude Data.Monoid> Any True <> Any FalseAny {getAny = True}Prelude Data.Monoid> All True <> All FalseAll {getAll = False}Prelude Data.Monoid> getAny . mconcat . map Any $ [False, False, False, True]TruePrelude Data.Monoid> getAll . mconcat . map All $ [False, False, False, True]False

如果 a 是 Monoid，那么 Maybe a 也是 Monoid

instance Monoid a => Monoid (Maybe a) where
  mempty = Nothing
  Nothing `mappend` m = m
  m `mappend` Nothing = m
  Just m1 `mappend` Just m2 = Just (m1 `mappend` m2)

Maybe a 是幺半群（前提是 a 是幺半群）。单位元是 Nothing。

Prelude Data.Monoid> Nothing <> Just "andy"Just "andy"Prelude Data.Monoid> Just LT <> NothingJust LTPrelude Data.Monoid> Just (Sum 3) <> Just (Sum 4) Just (Sum {getSum = 7})

First 和 Last 都是 Monoid

newtype First a = First { getFirst :: Maybe a }
newtype Last a = Last { getLast :: Maybe a }
instance Monoid (First a) where
        mempty = First Nothing
        First Nothing `mappend` r = r
        l `mappend` _             = l
instance Monoid (Last a) where
        mempty = Last Nothing
        l `mappend` Last Nothing = l
        _ `mappend` r            = r

Prelude Data.Monoid> First (Just 'a') <> First (Just 'b')First {getFirst = Just 'a'}Prelude Data.Monoid> Last (Just 'a') <> Last (Just 'b')Last {getLast = Just 'b'}Prelude Data.Monoid> getFirst . mconcat . map First $ [Nothing, Just 9, Just 10]  Just 9Prelude Data.Monoid> getLast . mconcat . map Last $ [Nothing, Just 9, Just 10]  Just 10

Min 和 Max 都是 Monoid

newtype Min a = Min { getMin :: a }
newtype Max a = Max { getMax :: a }
instance (Ord a, Bounded a) => Monoid (Min a) where
  mempty = maxBound
  mappend = coerce (min :: a -> a -> a)
instance (Ord a, Bounded a) => Monoid (Max a) where
  mempty = minBound
  mappend = coerce (max :: a -> a -> a)

Prelude Data.Semigroup> Min 3 <> Min 5Min {getMin = 3}Prelude Data.Semigroup> Max 3 <> Max 5Max {getMax = 5}Prelude Data.Semigroup> getMin . mconcat . map Min $ [1,2,3] :: Int1Prelude Data.Semigroup> getMax . mconcat . map Max $ [1,2,3] :: Int3

元组是 Monoid

instance (Monoid a, Monoid b) => Monoid (a,b) where
        mempty = (mempty, mempty)
        (a1,b1) `mappend` (a2,b2) =
                (a1 `mappend` a2, b1 `mappend` b2)

元组是幺半群，前提是其成员都是幺半群。

Prelude Data.Semigroup> mconcat $ map (\x -> (Min x, Max x)) [1..10] :: (Min Int, Max Int)(Min {getMin = 1},Max {getMax = 10})

秒客网