Scalaz（8）－ typeclass：Monoid and Foldable

Monoid是种最简单的typeclass类型。我们先看看scalaz的Monoid typeclass定义：scalaz/Monoid.scala

 trait Monoid[F] extends Semigroup[F] { self =>
   ////
   /** The identity element for `append`. */
   def zero: F
 ...

Monoid trait又继承了Semigroup：scalaz/Semigroup.scala

 trait Semigroup[F]  { self =>
   ////
   /**
    * The binary operation to combine `f1` and `f2`.
    *
    * Implementations should not evaluate the by-name parameter `f2` if result
    * can be determined by `f1`.
    */
   def append(f1: F, f2: => F): F
 ...

所以获取一个类型的Monoid实例需要实现zero和append这两个抽象函数。实际上Monoid typeclass也就是支持了append（|+|）这么一个简单的操作。scalaz为一些标准类型定义了Monoid实例：

  |+|                                          //> res0: Int = 50
 .some |+| .some                               //> res1: Option[Int] = Some(50)
 List(,,) |+| List(,,)                       //> res2: List[Int] = List(1, 2, 3, 4, 5, 6)
 Tags.Multiplication() |+| Monoid[Int @@ Tags.Multiplication].zero
                                                   //> res3: scalaz.@@[Int,scalaz.Tags.Multiplication] = 3
 Tags.Conjunction(true) |+| Tags.Conjunction(false)//> res4: scalaz.@@[Boolean,scalaz.Tags.Conjunction] = false
 Tags.Disjunction(true) |+| Tags.Disjunction(false)//> res5: scalaz.@@[Boolean,scalaz.Tags.Disjunction] = true
 Monoid[Boolean @@ Tags.Conjunction].zero          //> res6: scalaz.@@[Boolean,scalaz.Tags.Conjunction] = true
 Monoid[Boolean @@ Tags.Disjunction].zero          //> res7: scalaz.@@[Boolean,scalaz.Tags.Disjunction] = false

就这么来看好像没什么值得提的。不过Ordering的Monoid倒是值得研究一下。我们先看看Ordering trait：scalaz/Ordering.scala

  implicit val orderingInstance: Enum[Ordering] with Show[Ordering] with Monoid[Ordering] = new Enum[Ordering] with Show[Ordering] with Monoid[Ordering] {
     def order(a1: Ordering, a2: Ordering): Ordering = (a1, a2) match {
       case (LT, LT)      => EQ
       case (LT, EQ | GT) => LT
       case (EQ, LT)      => GT
       case (EQ, EQ)      => EQ
       case (EQ, GT)      => LT
       case (GT, LT | EQ) => GT
       case (GT, GT)      => EQ
     }

     override def shows(f: Ordering) = f.name

     def append(f1: Ordering, f2: => Ordering): Ordering = f1 match {
       case Ordering.EQ => f2
       case o           => o
     }
 ...

这里定义了Ordering的Monoid实例。它的append函数意思是：两个Ordering类型值f1,f2的append操作结果：假如f1是EQ就是f2，否则是f1：

 (Ordering.EQ: Ordering) |+| (Ordering.GT: Ordering)
                                                   //> res8: scalaz.Ordering = GT
 (Ordering.EQ: Ordering) |+| (Ordering.LT: Ordering)
                                                   //> res9: scalaz.Ordering = LT
 (Ordering.GT: Ordering) |+| (Ordering.EQ: Ordering)
                                                   //> res10: scalaz.Ordering = GT
 (Ordering.LT: Ordering) |+| (Ordering.EQ: Ordering)
                                                   //> res11: scalaz.Ordering = LT
 (Ordering.LT: Ordering) |+| (Ordering.GT: Ordering)
                                                   //> res12: scalaz.Ordering = LT
 (Ordering.GT: Ordering) |+| (Ordering.LT: Ordering)
                                                   //> res13: scalaz.Ordering = GT

如果我用以上的特性来比较两个String的长度：如果长度相等则再比较两个String的字符顺序。这个要求刚好符合了Ordering Monoid实例的append操作：

  ?|?                                            //> res14: scalaz.Ordering = LT
 "abc" ?|? "bac"                                   //> res15: scalaz.Ordering = LT
 def strlenCompare(lhs: String, rhs: String): Ordering =
  (lhs.length ?|? rhs.length) |+| (lhs ?|? rhs)    //> strlenCompare: (lhs: String, rhs: String)scalaz.Ordering

 strlenCompare("abc","aabc")                       //> res16: scalaz.Ordering = LT
 strlenCompare("abd","abc")                        //> res17: scalaz.Ordering = GT

这个示范倒是挺新鲜的。

好了，单看Monoid操作会觉着没什么特别，好像不值得研究。实际上Monoid的主要用途是在配合可折叠数据结构（Foldable）对结构内部元素进行操作时使用的。我们再看看这个Foldable typeclass：scalaz/Foldable.scala

 trait Foldable[F[_]]  { self =>
   ////
   import collection.generic.CanBuildFrom
   import collection.immutable.IndexedSeq

   /** Map each element of the structure to a [[scalaz.Monoid]], and combine the results. */
   def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B
   /** As `foldMap` but returning `None` if the foldable is empty and `Some` otherwise */
   def foldMap1Opt[A,B](fa: F[A])(f: A => B)(implicit F: Semigroup[B]): Option[B] = {
     import std.option._
     foldMap(fa)(x => some(f(x)))
   }

   /**Right-associative fold of a structure. */
   def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B): B
 ...

Foldable typeclass提供了许多注入方法支持折叠操作： scalaz/syntax/FoldableSyntax.scala

 final class FoldableOps[F[_],A] private[syntax](val self: F[A])(implicit val F: Foldable[F]) extends Ops[F[A]] {
   ////
   import collection.generic.CanBuildFrom
   import Leibniz.===
   import Liskov.<~<

   final def foldMap[B: Monoid](f: A => B = (a: A) => a): B = F.foldMap(self)(f)
   final def foldMap1Opt[B: Semigroup](f: A => B = (a: A) => a): Option[B] = F.foldMap1Opt(self)(f)
   final def foldRight[B](z: => B)(f: (A, => B) => B): B = F.foldRight(self, z)(f)
   final def foldMapRight1Opt[B](z: A => B)(f: (A, => B) => B): Option[B] = F.foldMapRight1Opt(self)(z)(f)
   final def foldRight1Opt(f: (A, => A) => A): Option[A] = F.foldRight1Opt(self)(f)
   final def foldLeft[B](z: B)(f: (B, A) => B): B = F.foldLeft(self, z)(f)
   final def foldMapLeft1Opt[B](z: A => B)(f: (B, A) => B): Option[B] = F.foldMapLeft1Opt(self)(z)(f)
   final def foldLeft1Opt(f: (A, A) => A): Option[A] = F.foldLeft1Opt(self)(f)
   final def foldRightM[G[_], B](z: => B)(f: (A, => B) => G[B])(implicit M: Monad[G]): G[B] = F.foldRightM(self, z)(f)
   final def foldLeftM[G[_], B](z: B)(f: (B, A) => G[B])(implicit M: Monad[G]): G[B] = F.foldLeftM(self, z)(f)
   final def foldMapM[G[_] : Monad, B : Monoid](f: A => G[B]): G[B] = F.foldMapM(self)(f)
   final def fold(implicit A: Monoid[A]): A = F.fold(self)(A)
   final def foldr[B](z: => B)(f: A => (=> B) => B): B = F.foldr(self, z)(f)
   final def foldr1Opt(f: A => (=> A) => A): Option[A] = F.foldr1Opt(self)(f)
   final def foldl[B](z: B)(f: B => A => B): B = F.foldl(self, z)(f)
   final def foldl1Opt(f: A => A => A): Option[A] = F.foldl1Opt(self)(f)
   final def foldrM[G[_], B](z: => B)(f: A => ( => B) => G[B])(implicit M: Monad[G]): G[B] = F.foldrM(self, z)(f)
   final def foldlM[G[_], B](z: B)(f: B => A => G[B])(implicit M: Monad[G]): G[B] = F.foldlM(self, z)(f)
   final def length: Int = F.length(self)
   final def index(n: Int): Option[A] = F.index(self, n)
   final def indexOr(default: => A, n: Int): A = F.indexOr(self, default, n)
   final def sumr(implicit A: Monoid[A]): A = F.foldRight(self, A.zero)(A.append)
   final def suml(implicit A: Monoid[A]): A = F.foldLeft(self, A.zero)(A.append(_, _))
   final def toList: List[A] = F.toList(self)
   final def toVector: Vector[A] = F.toVector(self)
   final def toSet: Set[A] = F.toSet(self)
   final def toStream: Stream[A] = F.toStream(self)
   final def toIList: IList[A] = F.toIList(self)
   final def toEphemeralStream: EphemeralStream[A] = F.toEphemeralStream(self)
   final def to[G[_]](implicit c: CanBuildFrom[Nothing, A, G[A]]) = F.to[A, G](self)
   final def all(p: A => Boolean): Boolean = F.all(self)(p)
   final def ∀(p: A => Boolean): Boolean = F.all(self)(p)
   final def allM[G[_]: Monad](p: A => G[Boolean]): G[Boolean] = F.allM(self)(p)
   final def anyM[G[_]: Monad](p: A => G[Boolean]): G[Boolean] = F.anyM(self)(p)
   final def any(p: A => Boolean): Boolean = F.any(self)(p)
   final def ∃(p: A => Boolean): Boolean = F.any(self)(p)
   final def count: Int = F.count(self)
   final def maximum(implicit A: Order[A]): Option[A] = F.maximum(self)
   final def maximumOf[B: Order](f: A => B): Option[B] = F.maximumOf(self)(f)
   final def maximumBy[B: Order](f: A => B): Option[A] = F.maximumBy(self)(f)
   final def minimum(implicit A: Order[A]): Option[A] = F.minimum(self)
   final def minimumOf[B: Order](f: A => B): Option[B] = F.minimumOf(self)(f)
   final def minimumBy[B: Order](f: A => B): Option[A] = F.minimumBy(self)(f)
   final def longDigits(implicit d: A <:< Digit): Long = F.longDigits(self)
   final def empty: Boolean = F.empty(self)
   final def element(a: A)(implicit A: Equal[A]): Boolean = F.element(self, a)
   final def splitWith(p: A => Boolean): List[NonEmptyList[A]] = F.splitWith(self)(p)
   final def selectSplit(p: A => Boolean): List[NonEmptyList[A]] = F.selectSplit(self)(p)
   final def collapse[X[_]](implicit A: ApplicativePlus[X]): X[A] = F.collapse(self)
   final def concatenate(implicit A: Monoid[A]): A = F.fold(self)
   final def intercalate(a: A)(implicit A: Monoid[A]): A = F.intercalate(self, a)
   final def traverse_[M[_]:Applicative](f: A => M[Unit]): M[Unit] = F.traverse_(self)(f)
   final def traverseU_[GB](f: A => GB)(implicit G: Unapply[Applicative, GB]): G.M[Unit] =
     F.traverseU_[A, GB](self)(f)(G)
   final def traverseS_[S, B](f: A => State[S, B]): State[S, Unit] = F.traverseS_(self)(f)
   final def sequence_[G[_], B](implicit ev: A === G[B], G: Applicative[G]): G[Unit] = F.sequence_(ev.subst[F](self))(G)
   final def sequenceS_[S, B](implicit ev: A === State[S,B]): State[S,Unit] = F.sequenceS_(ev.subst[F](self))
   def sequenceF_[M[_],B](implicit ev: F[A] <~< F[Free[M,B]]): Free[M, Unit] = F.sequenceF_(ev(self))
   final def msuml[G[_], B](implicit ev: A === G[B], G: PlusEmpty[G]): G[B] = F.foldLeft(ev.subst[F](self), G.empty[B])(G.plus[B](_, _))
   ////
 }

这简直就是一个完整的函数库嘛。scalaz为大多数标准库中的集合类型提供了Foldable实例，也就是说大多数scala集合类型都支持这么一堆折叠操作函数。我还看不到任何需要去自定义集合类型，标准库的集合类型加上Foldable typeclass应该足够用了。

在Foldable typeclass中比较重要的函数就是foldMap了：

 trait Foldable[F[_]]  { self =>
   ////
   import collection.generic.CanBuildFrom
   import collection.immutable.IndexedSeq

   /** Map each element of the structure to a [[scalaz.Monoid]], and combine the results. */
   def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B

首先，foldMap需要Monoid[B]实例来实现。用List来举例：List trait 继承了Traverse：scalaz/std/List.scala

 trait ListInstances extends ListInstances0 {
   implicit val listInstance = new Traverse[List] with MonadPlus[List] with Zip[List] with Unzip[List] with Align[List] with IsEmpty[List] with Cobind[List] {
 ...

在Traverse typeclass里定义了Foldable实例：scalaz/Traverse.scala

  def foldLShape[A,B](fa: F[A], z: B)(f: (B,A) => B): (B, F[Unit]) =
     runTraverseS(fa, z)(a => State.modify(f(_, a)))

   override def foldLeft[A,B](fa: F[A], z: B)(f: (B,A) => B): B = foldLShape(fa, z)(f)._1

   def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B = foldLShape(fa, F.zero)((b, a) => F.append(b, f(a)))._1

   override def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B) =
     foldMap(fa)((a: A) => (Endo.endo(f(a, _: B)))) apply z
 ...

这个foldMap就是一个游览可折叠结构的函数。在游览过程中用Monoid append对结构中元素进行操作。值得注意的是这个f: A => B参数：这个函数是用来在append操作之前先对内部元素进行一次转变（transform）：

 List(,,) foldMap {x => x}                      //> res18: Int = 6
 List(,,) foldMap {x => (x + ).toString}       //> res19: String = 456 变成String操作

我们试着用一些实际的例子来示范Monoid的用法。上面提到Monoid在可折叠数据结构里的元素连续处理有着很好的应用，我们先试一个例子：确定一个可折叠数据结构F[A]中的元素A是否排序的：

def ordered(xs: List[Int]): Boolean  //判断xs是否按序排列

由于我们必须游览List xs，所以用Monoid对元素Int进行判断操作是可行的方法。我们先设计一个对比数据结构：

Option[(min: Int, max: Int. ordered: Boolean)], 它记录了当前元素的状态，包括最小，最大，是否排序的：

 /判断xs是否是排序的
 def ordered(xs: List[Int]): Boolean = {
     val monoid = new Monoid[Option[(Int,Int,Boolean)]] {  //对类型Option[(Int,Int,Boolean)]定义一个Monoid实例
         def zero = None
         def append(a1: Option[(Int,Int,Boolean)], a2: => Option[(Int,Int,Boolean)]) =  //对连续两个元素进行对比操作
           (a1,a2) match {
             case (x,None) => x
             case (None,x) => x    //保留不为None的状态
             case (Some((min1,max1,ord1)),Some((min2,max2,ord2))) =>  //如果max1 <= min2状态即为true
                  Some((min1 min min2, max1 max max2, ord1 && ord2 && (max1 <= min2)))  //更新min,max和ord
           }
     }  //我们需要把元素转换成Option((Int,Int,Boolean))
     (xs.foldMap(i => Option((i, i, true)))(monoid)).map(_._3) getOrElse(true)
 }                                                 //> ordered: (xs: List[Int])Boolean

 ordered(List(,,,))                          //> res21: Boolean = true
 ordered(List(,,,))                          //> res22: Boolean = false

注意这个i => Option((i,i,true)) 转换（transform）。

由于Monoid是种极简单的类型，所以很容易对Monoid进行组合。Monoid组合产生的结果还是Monoid，并且用起来可以更方便：

 def productMonoid[A,B](ma: Monoid[A], mb: Monoid[B]): Monoid[(A,B)] = new Monoid[(A,B)] {
     def zero = (ma.zero, mb.zero)
     def append(x: (A,B), y: => (A,B)): (A,B) = (ma.append(x._1, y._1), mb.append(x._2, y._2))
 }                                                 //> productMonoid: [A, B](ma: scalaz.Monoid[A], mb: scalaz.Monoid[B])scalaz.Mon
                                                   //| oid[(A, B)]
 val pm = productMonoid(Monoid[Int],Monoid[List[Int]])
                                                   //> pm  : scalaz.Monoid[(Int, List[Int])] = Exercises.monoid$$anonfun$main$1$$a
                                                   //| non$3@72d1ad2e

以上的pm就是两个Monoid的组合，结果是一个tuple2Monoid。我们可以使用这个tuple2Monoid对可折叠数据结构中元素进行并行操作。比如我们可以在游览一个List[Int]时同时统计长度（list length）及乘积（product）:

 val intMultMonoid = new Monoid[Int] {
     def zero =
     def append(a1: Int, a2: => Int): Int = a1 * a2
 }                                                 //> intMultMonoid  : scalaz.Monoid[Int] = Exercises.monoid$$anonfun$main$1$$ano
                                                   //| n$1@6c64cb25
 def productMonoid[A,B](ma: Monoid[A], mb: Monoid[B]): Monoid[(A,B)] = new Monoid[(A,B)] {
     def zero = (ma.zero, mb.zero)
     def append(x: (A,B), y: => (A,B)): (A,B) = (ma.append(x._1, y._1), mb.append(x._2, y._2))
 }                                                 //> productMonoid: [A, B](ma: scalaz.Monoid[A], mb: scalaz.Monoid[B])scalaz.Mon
                                                   //| oid[(A, B)]
 val pm = productMonoid(Monoid[Int @@ Tags.Multiplication],Monoid[Int])
                                                   //> pm  : scalaz.Monoid[(scalaz.@@[Int,scalaz.Tags.Multiplication], Int)] = Exe
                                                   //| rcises.monoid$$anonfun$main$1$$anon$3@72d1ad2e
 List(,,,,).foldMap(i => (i, ))(productMonoid(intMultMonoid,Monoid[Int]))
                                                   //> res23: (Int, Int) = (144,5)

我们再来一个合并多层map的Monoid:

 def mapMergeMonoid[K,V](V: Monoid[V]): Monoid[Map[K, V]] =
   new Monoid[Map[K, V]] {
     def zero = Map[K,V]()
     def append(a: Map[K, V], b: => Map[K, V]) =
       (a.keySet ++ b.keySet).foldLeft(zero) { (acc,k) =>
         acc.updated(k, V.append(a.getOrElse(k, V.zero),
                             b.getOrElse(k, V.zero)))
       }
   }                                               //> mapMergeMonoid: [K, V](V: scalaz.Monoid[V])scalaz.Monoid[Map[K,V]]

  val M: Monoid[Map[String, Map[String, Int]]] = mapMergeMonoid(mapMergeMonoid(Monoid[Int]))
                                                   //> M  : scalaz.Monoid[Map[String,Map[String,Int]]] = Exercises.monoid$$anonfun
                                                   //| $main$1$$anon$4@79e2c065
  val m1 = Map("o1" -> Map("i1" -> , "i2" -> ))  //> m1  : scala.collection.immutable.Map[String,scala.collection.immutable.Map[
                                                   //| String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 2))
  val m2 = Map("o1" -> Map("i2" -> ))             //> m2  : scala.collection.immutable.Map[String,scala.collection.immutable.Map[
                                                   //| String,Int]] = Map(o1 -> Map(i2 -> 3))
  val m3 = M.append(m1, m2)                        //> m3  : Map[String,Map[String,Int]] = Map(o1 -> Map(i1 -> 1, i2 -> 5))

我们可以用这个组合成的M的append操作进行map的深度合并。m1,m2合并后：Map(o1->Map("i1"->1,"i2" -> 5))。

我们还可以用这个Monoid来统计一段字串内字符发生的频率：

 def frequencyMap[A](as: List[A]): Map[A, Int] =
     as.foldMap((a: A) => Map(a -> ))(mapMergeMonoid[A, Int](Monoid[Int]))
                                                   //> frequencyMap: [A](as: List[A])Map[A,Int]
 frequencyMap("the brown quik fox is running quikly".toList)
   //> res24: Map[Char,Int] = Map(e -> 1, s -> 1, x -> 1, n -> 4, y -> 1, t -> 1,
   //| u -> 3, f -> 1, i -> 4,   -> 6, q -> 2, b -> 1, g -> 1, l -> 1, h -> 1, r -
   //| > 2, w -> 1, k -> 2, o -> 2)

我们现在可以体会到Monoid必须在可折叠数据结构（Foldable）内才能正真发挥作用。