Scalaz(6)- typeclass:Functor-just map

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  Functor是范畴学(Category theory)里的概念。不过无须担心,我们在scala FP编程里并不需要先掌握范畴学知识的。在scalaz里,Functor就是一个普通的typeclass,具备map over特性。我的理解中,Functor的主要用途是在FP过程中更新包嵌在容器(高阶类)F[T]中元素T值。典型例子如:List[String], Option[Int]等。我们曾经介绍过FP与OOP的其中一项典型区别在于FP会尽量避免中间变量(temp variables)。FP的变量V是以F[V]这种形式存在的,如:List[Int]里一个Int变量是包嵌在容器List里的。所以FP需要特殊的方式来更新变量V,这就是Functor map over的意思。scalaz提供了Functor typeclass不但使用户能map over自定义的高阶类型F[T],并且用户通过提供自定义类型的Functor实例就可以免费使用scalaz Functor typeclass提供的一系列组件函数(combinator functions)。

  scalaz中Functor的trait是这样定义的:scalaz/Functor.scala

1 trait Functor[F[_]] extends InvariantFunctor[F] { self =>

2 ////

3 import Liskov.<~<

4

5 /** Lift `f` into `F` and apply to `F[A]`. */

6 def map[A, B](fa: F[A])(f: A => B): F[B]

7

8 ...

任何类型的实例只需要实现这个抽象函数map就可以使用scalaz Functor typeclass的这些注入方法了:scalaz/syntax/FunctorSyntax.scala

 1 final class FunctorOps[F[_],A] private[syntax](val self: F[A])(implicit val F: Functor[F]) extends Ops[F[A]] {

2 ////

3 import Leibniz.===

4 import Liskov.<~<

5

6 final def map[B](f: A => B): F[B] = F.map(self)(f)

7 final def distribute[G[_], B](f: A => G[B])(implicit D: Distributive[G]): G[F[B]] = D.distribute(self)(f)

8 final def cosequence[G[_], B](implicit ev: A === G[B], D: Distributive[G]): G[F[B]] = D.distribute(self)(ev(_))

9 final def cotraverse[G[_], B, C](f: F[B] => C)(implicit ev: A === G[B], D: Distributive[G]): G[C] = D.map(cosequence)(f)

10 final def ∘[B](f: A => B): F[B] = F.map(self)(f)

11 final def strengthL[B](b: B): F[(B, A)] = F.strengthL(b, self)

12 final def strengthR[B](b: B): F[(A, B)] = F.strengthR(self, b)

13 final def fpair: F[(A, A)] = F.fpair(self)

14 final def fproduct[B](f: A => B): F[(A, B)] = F.fproduct(self)(f)

15 final def void: F[Unit] = F.void(self)

16 final def fpoint[G[_]: Applicative]: F[G[A]] = F.map(self)(a => Applicative[G].point(a))

17 final def >|[B](b: => B): F[B] = F.map(self)(_ => b)

18 final def as[B](b: => B): F[B] = F.map(self)(_ => b)

19 final def widen[B](implicit ev: A <~< B): F[B] = F.widen(self)

20 ////

21 }

以上的注入方法中除了map外其它方法的应用场景我还没有确切的想法,不过这不会妨碍我们示范它们的用法。Functor必须遵循一些定律:

1、map(fa)(x => x) === fa

2、map(map(fa)(f1))(f2) === map(fa)(f2 compose f1)

scalaz/Functor.scala

 1  trait FunctorLaw extends InvariantFunctorLaw {

2 /** The identity function, lifted, is a no-op. map(fa)(x => x*/

3 def identity[A](fa: F[A])(implicit FA: Equal[F[A]]): Boolean = FA.equal(map(fa)(x => x), fa)

4

5 /**

6 * A series of maps may be freely rewritten as a single map on a

7 * composed function.

8 */

9 def composite[A, B, C](fa: F[A], f1: A => B, f2: B => C)(implicit FC: Equal[F[C]]): Boolean = FC.equal(map(map(fa)(f1))(f2), map(fa)(f2 compose f1))

10 }

我们可以用List来证明:map(fa)(x => x) === fa

1 scala> List(1,2,3).map(x => x) assert_=== List(1,2,3)

2

3 scala> List(1,2,3).map(identity) assert_=== List(1,2)

4 java.lang.RuntimeException: [1,2,3] ≠ [1,2]

5 at scala.sys.package$.error(package.scala:27)

6 at scalaz.syntax.EqualOps.assert_$eq$eq$eq(EqualSyntax.scala:16)

7 ... 43 elided

map(map(fa)(f1))(f2) === map(fa)(f2 compose f1)

1 scala> Functor[List].map(List(1,2,3).map(i => i + 1))(i2 => i2 * 3) assert_=== List(1,2,3).map(((i2:Int) => i2 * 3) compose ((i:Int) => i + 1))

2

3 scala> Functor[List].map(List(1,2,3).map(i => i + 1))(i2 => i2 * 3) assert_=== List(1,2,3).map(((i:Int) => i + 1) compose ((i2:Int) => i2 * 3))

4 java.lang.RuntimeException: [6,9,12] ≠ [4,7,10]

5 at scala.sys.package$.error(package.scala:27)

6 at scalaz.syntax.EqualOps.assert_$eq$eq$eq(EqualSyntax.scala:16)

7 ... 43 elided

注意:compose对f1,f2的施用是互换的。

针对我们自定义的类型,我们只要实现map函数就可以得到这个类型的Functor实例。一旦实现了这个类型的Functor实例,我们就可以使用以上scalaz提供的所有Functor组件函数了。

我们先试着创建一个类型然后推算它的Functor实例:

1 case class Item3[A](i1: A, i2: A, i3: A)

2 val item3Functor = new Functor[Item3] {

3 def map[A,B](ia: Item3[A])(f: A => B): Item3[B] = Item3(f(ia.i1),f(ia.i2),f(ia.i3))

4 } //> item3Functor : scalaz.Functor[scalaz.functor.Item3] = scalaz.functor$$anonf

5 //| [email protected]

scalaz同时在scalaz-tests下提供了一套scalacheck测试库。我们可以对Item3的Functor实例进行测试:

1 scala> functor.laws[Item3].check

2 <console>:27: error: could not find implicit value for parameter af: org.scalacheck.Arbitrary[Item3[Int]]

3 functor.laws[Item3].check

4 ^

看来我们需要提供自定义类型Item3的随意产生器(Generator):

 1 scala> implicit def item3Arbi[A](implicit a: Arbitrary[A]): Arbitrary[Item3[A]] = Arbitrary {

2 | def genItem3: Gen[Item3[A]] = for {

3 | b <- Arbitrary.arbitrary[A]

4 | c <- Arbitrary.arbitrary[A]

5 | d <- Arbitrary.arbitrary[A]

6 | } yield Item3(b,c,d)

7 | genItem3

8 | }

9 item3Arbi: [A](implicit a: org.scalacheck.Arbitrary[A])org.scalacheck.Arbitrary[Item3[A]]

10

11 scala> functor.laws[Item3].check

12 + functor.invariantFunctor.identity: OK, passed 100 tests.

13 + functor.invariantFunctor.composite: OK, passed 100 tests.

14 + functor.identity: OK, passed 100 tests.

15 + functor.composite: OK, passed 100 tests.

Item3的Functor实例是合理的。

实际上map就是(A => B) => (F[A] => F[B]),就是把(A => B)升格(lift)成(F[A] => F[B]):

 1 case class Item3[A](i1: A, i2: A, i3: A)

2 implicit val item3Functor = new Functor[Item3] {

3 def map[A,B](ia: Item3[A])(f: A => B): Item3[B] = Item3(f(ia.i1),f(ia.i2),f(ia.i3))

4 } //> item3Functor : scalaz.Functor[scalaz.functor.Item3] = scalaz.functor$$anonf

5 //| [email protected]

6 val F = Functor[Item3] //> F : scalaz.Functor[scalaz.functor.Item3] = scalaz.functor$$anonfun$main$1$$

7 //| [email protected]

8 F.map(Item3("Morning","Noon","Night"))(_.length) //> res0: scalaz.functor.Item3[Int] = Item3(7,4,5)

9 F.apply(Item3("Morning","Noon","Night"))(_.length)//> res1: scalaz.functor.Item3[Int] = Item3(7,4,5)

10 F(Item3("Morning","Noon","Night"))(_.length) //> res2: scalaz.functor.Item3[Int] = Item3(7,4,5)

11 F.lift((s: String) => s.length)(Item3("Morning","Noon","Night"))

12 //> res3: scalaz.functor.Item3[Int] = Item3(7,4,5)

虽然函数升格(function lifting (A => B) => (F[A] => F[B])是Functor的主要功能,但我们说过:一旦能够获取Item3类型的Functor实例我们就能免费使用所有的注入方法:

scalaz提供了Function1的Functor实例。Function1 Functor的map就是 andThen 也就是操作方调换的compose:

 1 scala> (((_: Int) + 1) map((k: Int) => k * 3))(2)

2 res20: Int = 9

3

4 scala> (((_: Int) + 1) map((_: Int) * 3))(2)

5 res21: Int = 9

6

7 scala> (((_: Int) + 1) andThen ((_: Int) * 3))(2)

8 res22: Int = 9

9

10 scala> (((_: Int) * 3) compose ((_: Int) + 1))(2)

11 res23: Int = 9

我们也可以对Functor进行compose:

1 scala> val f = Functor[List] compose Functor[Item3]

2 f: scalaz.Functor[[α]List[Item3[α]]] = scalaz.Functor$$anon$1@647ce8fd

3

4 scala> val item3 = Item3("Morning","Noon","Night")

5 item3: Item3[String] = Item3(Morning,Noon,Night)

6

7 scala> f.map(List(item3,item3))(_.length)

8 res25: List[Item3[Int]] = List(Item3(7,4,5), Item3(7,4,5))

反过来操作:

1 scala> val f1 = Functor[Item3] compose Functor[List]

2 f1: scalaz.Functor[[α]Item3[List[α]]] = scalaz.Functor$$anon$1@5b6a0166

3

4 scala> f1.map(Item3(List("1"),List("22"),List("333")))(_.length)

5 res26: Item3[List[Int]] = Item3(List(1),List(2),List(3))

我们再试着在Item3类型上调用那些免费的注入方法:

 1 scala> item3.fpair

2 res28: Item3[(String, String)] = Item3((Morning,Morning),(Noon,Noon),(Night,Night))

3

4 scala> item3.strengthL(3)

5 res29: Item3[(Int, String)] = Item3((3,Morning),(3,Noon),(3,Night))

6

7 scala> item3.strengthR(3)

8 res30: Item3[(String, Int)] = Item3((Morning,3),(Noon,3),(Night,3))

9

10 scala> item3.fproduct(_.length)

11 res31: Item3[(String, Int)] = Item3((Morning,7),(Noon,4),(Night,5))

12

13 scala> item3 as "Day"

14 res32: Item3[String] = Item3(Day,Day,Day)

15

16 scala> item3 >| "Day"

17 res33: Item3[String] = Item3(Day,Day,Day)

18

19 scala> item3.void

20 res34: Item3[Unit] = Item3((),(),())

我现在还没有想到这些函数的具体用处。不过从运算结果来看,用这些函数来产生一些数据模型用在游戏或者测试的模拟(simulation)倒是可能的。

scalaz提供了许多现成的Functor实例。我们先看看一些简单直接的实例:

 1 scala> Functor[List].map(List(1,2,3))(_ + 3)

2 res35: List[Int] = List(4, 5, 6)

3

4 scala> Functor[Option].map(Some(3))(_ + 3)

5 res36: Option[Int] = Some(6)

6

7 scala> Functor[java.util.concurrent.Callable]

8 res37: scalaz.Functor[java.util.concurrent.Callable] = scalaz.std.java.util.concurrent.CallableInstances$$anon$1@4176ab89

9

10 scala> Functor[Stream]

11 res38: scalaz.Functor[Stream] = scalaz.std.StreamInstances$$anon$1@4f5374b9

12

13 scala> Functor[Vector]

14 res39: scalaz.Functor[Vector] = scalaz.std.IndexedSeqSubInstances$$anon$1@4367920a

对那些多个类型变量的类型我们可以采用部分施用方式:即type lambda来表示。一个典型的类型:Either[E,A],我们可以把Left[E]固定下来: Either[String, A],我们可以用type lambda来这样表述:

1 scala> Functor[({type l[x] = Either[String,x]})#l].map(Right(3))(_ + 3)

2 res41: scala.util.Either[String,Int] = Right(6)

如此这般我可以对Either类型进行map操作了。

函数类型的Functor是针对返回类型的:

1 scala> Functor[({type l[x] = String => x})#l].map((s: String) => s + "!")(_.length)("Hello")

2 res53: Int = 6

3

4 scala> Functor[({type l[x] = (String,Int) => x})#l].map((s: String, i: Int) => s.length + i)(_ * 10)("Hello",5)

5 res54: Int = 100

6

7 scala> Functor[({type l[x] = (String,Int,Boolean) => x})#l].map((s: String,i: Int, b: Boolean)=> s + i.toString + b.toString)(_.toUpperCase)("Hello",3,true)

8 res56: String = HELLO3TRUE

tuple类型的Functor是针对最后一个元素类型的: 

 

1 cala> Functor[({type l[x] = (String,x)})#l].map(("a",1))(_ + 2)

2 res57: (String, Int) = (a,3)

3

4 scala> Functor[({type l[x] = (String,Int,x)})#l].map(("a",1,"b"))(_.toUpperCase)

5 res58: (String, Int, String) = (a,1,B)

6

7 scala> Functor[({type l[x] = (String,Int,Boolean,x)})#l].map(("a",1,true,Item3("a","b","c")))(i => i.map(_.toUpperCase))

8 res62: (String, Int, Boolean, Item3[String]) = (a,1,true,Item3(A,B,C))

 

 

 

 

 

 

 

 

 

 

 

 


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