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([email protected])
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([email protected])
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 //| [email protected]
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 //| [email protected]
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！
原创文章，作者：ItWorker，如若转载，请注明出处：https://blog.ytso.com/12952.html