泛函编程（33）－泛函IO：Free Functor - Coyoneda

用户1150956

发布于 2018-01-05 09:46:24

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发布于 2018-01-05 09:46:24

在前几期讨论中我们终于推导出了Free Monad。这是一个Monad工厂，它可以把任何F[A]变成Monad。可惜的是它对F[A]是有所要求的：F必须是个Functor。Free Monad由此被称为由Functor F 产生的Monad。F必须是Functor,这个门槛使我们在使用Free Monad时很不方便。举个前面讨论过的例子：

1 trait Console[A]
2 case object GetLine extends Console[String]
3 case class PutLine(line: String) extends Console[Unit]

我们想用Free Monad把Console[A]变成Monad: Free[Console,A]，但我们必须先得到Console的Functor实例：

1 implicit val consoleFunctor = new Functor[Console] {
2     def map[A,B](ca: Console[A])(f: A => B): Console[B] = ca match {
3         case GetLine => ?????
4         case PutLine(l) => ????
5     }
6 }

讲老实话，我到现在还没能想出如何实现这个map函数。除非把Console类型修改一下，这个可以参考前面讨论中的代码。

现在的问题是如果能有个什么方法把F[A]变成Functor，就像Free Monad那样有个Free Functor就好了。范畴学中Yoneda lemma结论中的Coyoneda就是一个Free Functor。

Yoneda lemma是这样推论的：如果我们有个这样的函数定义：def map[B](f: A => B): F[B]，那我们就肯定能得出F[A]值，因为我们只需要把一个恒等函数当作f就能得到F[A]。反过来推论：如果我们有个F[A]，F是任何Functor，A是任何类型，我们同样可以得出以上的map函数。我们可以用个类型来表示：

1 trait Yoneda[F[_],A] {
2    def map[B](f: A => B): F[B]
3 }

当然，这也意味着如果：有个类型B，一个函数(B => A)，A是任意类型，一个F[B]，F是任意Functor，我们肯定能得出F[A]：因为我们只要把(B => A)和F[B]传入map：

map(fb: F[B])(f: B => A): F[A]。

我们同样可以用一个类型来表示：

1 trait Coyoneda[F[_],A] { coyo =>
2  type I
3  def fi: F[I]
4  def k(i: I): A
5 }

在下面我们可以证明F[A]同等Coyoneda[F,A]，而Coyoneda是个Functor。我们只需将F[A]升格（lift）到Coyoneda就能得到一个Free Functor了。

 1 trait Functor[F[_]] {
 2     def map[A,B](fa: F[A])(f: A => B): F[B]
 3 }
 4 object Functor {
 5     def apply[F[_]: Functor]: Functor[F] = implicitly[Functor[F]]
 6 }
 7 trait Monad[M[_]] {
 8     def unit[A](a: A): M[A]
 9     def flatMap[A,B](ma: M[A])(f: A => M[B]): M[B]
10     def map[A,B](ma: M[A])(f: A => B) = flatMap(ma)(a => unit(f(a)))
11 }
12 object Monad {
13     def apply[M[_]: Monad]: Monad[M] = implicitly[Monad[M]]
14 }
15 trait Yoneda[F[_],A] { yo =>
16     def apply[B](f: A => B): F[B]
17     def run: F[A] = apply(a => a)  //无需Functor实例就可以将Yoneda转变成F[A]
18     def toCoyoneda: Coyoneda[F,A] = new Coyoneda[F,A] { //转Coyoneda无需Functor
19         type I = A
20         def fi = yo.run
21         def k(i: A) = i
22     }
23     def map[B](f: A => B): Yoneda[F,B] = new Yoneda[F,B] { //纯粹的函数组合 map fusion
24         def apply[C](g: B => C): F[C] = yo( f andThen g)
25     }
26 }
27 trait Coyoneda[F[_],A] { coyo =>
28  type I
29  def fi: F[I]
30  def k(i: I): A
31  def run(implicit F: Functor[F]): F[A] =  //Coyoneda转F需要F Functor实例
32    F.map(fi)(k)
33  def toYoneda(implicit F: Functor[F]): Yoneda[F,A] = new Yoneda[F,A] { //转Yoneda需要Functor
34      def apply[B](f: A => B): F[B] = F.map(fi)(k _ andThen f)
35  }
36  def map[B](f: A => B): Coyoneda[F,B] = new Coyoneda[F,B] {
37      type I = coyo.I
38      def fi = coyo.fi
39      def k(i: I) = f(coyo k i)
40  }
41 }
42 object Yoneda {
43     def apply[F[_]: Functor,A](fa: F[A]) = new Yoneda[F,A] { //F转Yoneda需要Functor
44         def apply[B](f: A => B): F[B] = Functor[F].map(fa)(f)
45     }
46     implicit def yonedaFunctor[F[_]] = new Functor[({type l[x] = Yoneda[F,x]})#l] {
47         def map[A,B](ya: Yoneda[F,A])(f: A => B) = ya map f
48         
49     }
50 }
51 object Coyoneda {
52     def apply[F[_],A](fa: F[A]): Coyoneda[F,A] = new Coyoneda[F,A] {
53         type I = A          //把F[A]升格成Coyoneda, F无须为Functor
54         def fi = fa
55         def k(a: A) = a
56     }
57     implicit def coyonedaFunctor[F[_]] = new Functor[({type l[x] = Coyoneda[F,x]})#l] {
58         def map[A,B](ca: Coyoneda[F,A])(f: A => B) = ca map f   //Coyoneda本身就是Functor
59     }
60 }

以上值得注意的是：F[A]可以直接升格等于Coyoneda，而Coyoneda是个Functor。换句话说我们把F[A]升格到Coyoneda就可以当Functor来用了。

我们的目的是把任何F[A]变成Free Monad，那么我们就需要有一个用Coyoneda产生的Free：

 1 trait Free[F[_],A] {
 2  private case class FlatMap[B](a: Free[F,A], f: A => Free[F,B]) extends Free[F,B]
 3  def unit(a: A): Free[F,A] = Return(a)
 4  def flatMap[B](f: A => Free[F,B])(implicit F: Functor[F]): Free[F,B] = this match {
 5      case Return(a) => f(a)
 6      case Suspend(k) => Suspend(F.map(k)(a => a flatMap f))
 7   case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f))
 8  }
 9  
10  def map[B](f: A => B)(implicit F: Functor[F]): Free[F,B] = flatMap(a => Return(f(a)))
11  def resume(implicit F: Functor[F]): Either[F[Free[F,A]],A] = this match {
12      case Return(a) => Right(a)
13      case Suspend(k) => Left(k)
14      case FlatMap(a,f) => a match {
15          case Return(b) => f(b).resume
16          case Suspend(k) => Left(F.map(k)(_ flatMap f))
17          case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f)).resume
18      }
19  }
20  def foldMap[G[_]](f: (F ~> G))(implicit F: Functor[F], G: Monad[G]): G[A] = resume match {
21        case Right(a) => G.unit(a)
22        case Left(k) => G.flatMap(f(k))(_ foldMap f)
23  }
24 }
25 case class Return[F[_],A](a: A) extends Free[F,A]
26 case class Suspend[F[_],A](ffa: F[Free[F,A]]) extends Free[F,A]
27 object Free {
28 import scalaz.Unapply
29   /** A free monad over the free functor generated by `S` */
30   type FreeC[S[_], A] = Free[({type f[x] = Coyoneda[S, x]})#f, A]
31 
32   /** Suspends a value within a functor in a single step. Monadic unit for a higher-order monad. */
33   def liftF[S[_], A](value: => S[A])(implicit S: Functor[S]): Free[S, A] =
34     Suspend(S.map(value)(Return[S, A]))
35 
36   /** A version of `liftF` that infers the nested type constructor. */
37   def liftFU[MA](value: => MA)(implicit MA: Unapply[Functor, MA]): Free[MA.M, MA.A] =
38     liftF(MA(value))(MA.TC)
39 
40   /** A free monad over a free functor of `S`. */
41   def liftFC[S[_], A](s: S[A]): FreeC[S, A] =
42     liftFU(Coyoneda(s))
43     
44   /** Interpret a free monad over a free functor of `S` via natural transformation to monad `M`. */
45   def runFC[S[_], M[_], A](sa: FreeC[S, A])(interp: S ~> M)(implicit M: Monad[M]): M[A] =
46     sa.foldMap[M](new (({type λ[α] = Coyoneda[S, α]})#λ ~> M) {
47       def apply[A](cy: Coyoneda[S, A]): M[A] =
48         M.map(interp(cy.fi))(cy.k)
49       })
50 }

我们把前面推导出来的Free搬过来。然后在Free companion object里增加了FreeC类型：

type FreeC[S[_],A] = Free[({type f[x] = Coyoneda[F,x]})#f, A]

这个可以说是一个由Coyoneda产生的Free。

现在我们要想办法把S[A]升格成FreeC：liftFC[S[_],A](s: S[A]): FreeC[S,A]，这里需要先把S[A]升格成Coyoneda：Coyoneda(s)。

由于Coyoneda[S,A]是个多层嵌入类型。我们在liftFU函数中需要借用scalaz的Unapply类型来分解出Coyoneda, S[A]然后施用在liftF；

def liftF[S[_],A](sa: S[A])(implicit S: Functor[S])，这里的S就是Coyoneda。

Interpreter沿用了foldMap但是调整了转换源目标类型 Functor >>> Coyoneda。其它如Trampoline机制维持不变。

现在我们可以直接用任何F[A]来产生Free了。先试试上面的那个Console。这个Console不是个Functor：

 1 trait Console[A]
 2 case object GetLine extends Console[String]
 3 case class PutLine(line: String) extends Console[Unit]
 4 import Free._
 5 implicit def liftConsole[A](ca: Console[A]): FreeC[Console,A] = liftFC(ca)
 6                                                   //> liftConsole: [A](ca: ch13.ex11.Console[A])ch13.ex11.Free.FreeC[ch13.ex11.Co
 7                                                   //| nsole,A]
 8 for {
 9     _ <- PutLine("What is your first name ?")
10     first <- GetLine
11     _ <- PutLine("What is your last name ?")
12     last <- GetLine
13     _ <- PutLine(s"Hello, $first $last !")
14 } yield ()                                        //> res0: ch13.ex11.Free[[x]ch13.ex11.Coyoneda[ch13.ex11.Console,x],Unit] = Sus
15                                                   //| pend(ch13.ex11$Coyoneda$$anon$4@50f8360d)

可以使用Free的Monadic语言了。下面再试试Interpreter部分：

 1 val ioprg = for {
 2     _ <- PutLine("What is your first name ?")
 3     first <- GetLine
 4     _ <- PutLine("What is your last name ?")
 5     last <- GetLine
 6     _ <- PutLine(s"Hello, $first $last !")
 7 } yield ()                                        //> ioprg  : ch13.ex11.Free[[x]ch13.ex11.Coyoneda[ch13.ex11.Console,x],Unit] = 
 8                                                   //| Suspend(ch13.ex11$Coyoneda$$anon$4@13c78c0b)
 9 
10 type Id[A] = A
11 implicit val idMonad = new Monad[Id] {
12     def unit[A](a: A) = a
13     def flatMap[A,B](fa: A)(f: A => B): B = f(fa)
14 }                                                 //> idMonad  : ch13.ex11.Monad[ch13.ex11.Id] = ch13.ex11$$anonfun$main$1$$anon$
15                                                   //| 10@12843fce
16 
17 object RealConsole extends (Console ~> Id) {
18     def apply[A](ca: Console[A]): A = ca match {
19         case GetLine => readLine
20         case PutLine(l) => println(l)
21     }
22 }
23 Free.runFC(ioprg)(RealConsole)                    //> What is your first name ?/

也很顺利呢。再试试加了State维护的IO程序：

 1 case class State[S,A](runState: S => (A,S)) {
 2     def map[B](f: A => B) = State[S,B](s => {
 3         val (a1,s1) = runState(s)
 4         (f(a1),s1)
 5     })
 6     def flatMap[B](f: A => State[S,B]) = State[S,B](s => {
 7         val (a1,s1) = runState(s)
 8         f(a1).runState(s1)
 9     })
10 }
11 case class InOutLog(inLog: List[String], outLog: List[String])
12 type LogState[A] = State[InOutLog, A]
13 implicit val logStateMonad = new Monad[LogState] {
14     def unit[A](a: A) = State(s => (a, s))
15     def flatMap[A,B](sa: LogState[A])(f: A => LogState[B]) = sa flatMap f
16 }                                                 //> logStateMonad  : ch13.ex11.Monad[ch13.ex11.LogState] = ch13.ex11$$anonfun$m
17                                                   //| ain$1$$anon$11@3dd3bcd
18 object MockConsole extends(Console ~> LogState) {
19     def apply[A](c: Console[A]): LogState[A] = State(
20         s => (c,s) match {
21             case (GetLine, InOutLog(in,out)) => (in.head, InOutLog(in.tail, out))
22           case (PutLine(l), InOutLog(in,out)) => ((),InOutLog(in, l :: out))
23         })
24 }
25 val s = Free.runFC(ioprg)(MockConsole)            //> s  : ch13.ex11.LogState[Unit] = State(<function1>)
26 val ls = s.runState(InOutLog(List("Tiger","Chan"),List()))
27                                                   //> ls  : (Unit, ch13.ex11.InOutLog) = ((),InOutLog(List(),List(Hello, Tiger Ch
28                                                   //| an !, What is your last name ?, What is your first name ?)))

也能正确地维护状态。

现在我们可以把任何F[A]类型变成Free Monad并用它实现Monadic programming及副作用解译运算！

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泛函编程（33）－泛函IO：Free Functor - Coyoneda

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