泛函编程（33）－泛函IO：Free Functor - Coyoneda

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：

trait Console[A]
case object GetLine extends Console[String]
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：

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

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：

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

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：

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

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。

trait Functor[F[_]] {
def map[A,B](fa: F[A])(f: A => B): F[B]
}
object Functor {
def apply[F[_]: Functor]: Functor[F] = implicitly[Functor[F]]
}
trait Monad[M[_]] {
def unit[A](a: A): M[A]
def flatMap[A,B](ma: M[A])(f: A => M[B]): M[B]
def map[A,B](ma: M[A])(f: A => B) = flatMap(ma)(a => unit(f(a)))
}
object Monad {
def apply[M[_]: Monad]: Monad[M] = implicitly[Monad[M]]
}
trait Yoneda[F[_],A] { yo =>
def apply[B](f: A => B): F[B]
def run: F[A] = apply(a => a)  //无需Functor实例就可以将Yoneda转变成F[A]
def toCoyoneda: Coyoneda[F,A] = new Coyoneda[F,A] { //转Coyoneda无需Functor
type I = A
def fi = yo.run
def k(i: A) = i
}
def map[B](f: A => B): Yoneda[F,B] = new Yoneda[F,B] { //纯粹的函数组合 map fusion
def apply[C](g: B => C): F[C] = yo( f andThen g)
}
}
trait Coyoneda[F[_],A] { coyo =>
type I
def fi: F[I]
def k(i: I): A
def run(implicit F: Functor[F]): F[A] =  //Coyoneda转F需要F Functor实例
F.map(fi)(k)
def toYoneda(implicit F: Functor[F]): Yoneda[F,A] = new Yoneda[F,A] { //转Yoneda需要Functor
def apply[B](f: A => B): F[B] = F.map(fi)(k _ andThen f)
}
def map[B](f: A => B): Coyoneda[F,B] = new Coyoneda[F,B] {
type I = coyo.I
def fi = coyo.fi
def k(i: I) = f(coyo k i)
}
}
object Yoneda {
def apply[F[_]: Functor,A](fa: F[A]) = new Yoneda[F,A] { //F转Yoneda需要Functor
def apply[B](f: A => B): F[B] = Functor[F].map(fa)(f)
}
implicit def yonedaFunctor[F[_]] = new Functor[({type l[x] = Yoneda[F,x]})#l] {
def map[A,B](ya: Yoneda[F,A])(f: A => B) = ya map f

}
}
object Coyoneda {
def apply[F[_],A](fa: F[A]): Coyoneda[F,A] = new Coyoneda[F,A] {
type I = A          //把F[A]升格成Coyoneda, F无须为Functor
def fi = fa
def k(a: A) = a
}
implicit def coyonedaFunctor[F[_]] = new Functor[({type l[x] = Coyoneda[F,x]})#l] {
def map[A,B](ca: Coyoneda[F,A])(f: A => B) = ca map f   //Coyoneda本身就是Functor
}
}

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：

trait Free[F[_],A] {
private case class FlatMap[B](a: Free[F,A], f: A => Free[F,B]) extends Free[F,B]
def unit(a: A): Free[F,A] = Return(a)
def flatMap[B](f: A => Free[F,B])(implicit F: Functor[F]): Free[F,B] = this match {
case Return(a) => f(a)
case Suspend(k) => Suspend(F.map(k)(a => a flatMap f))
case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f))
}

def map[B](f: A => B)(implicit F: Functor[F]): Free[F,B] = flatMap(a => Return(f(a)))
def resume(implicit F: Functor[F]): Either[F[Free[F,A]],A] = this match {
case Return(a) => Right(a)
case Suspend(k) => Left(k)
case FlatMap(a,f) => a match {
case Return(b) => f(b).resume
case Suspend(k) => Left(F.map(k)(_ flatMap f))
case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f)).resume
}
}
def foldMap[G[_]](f: (F ~> G))(implicit F: Functor[F], G: Monad[G]): G[A] = resume match {
case Right(a) => G.unit(a)
case Left(k) => G.flatMap(f(k))(_ foldMap f)
}
}
case class Return[F[_],A](a: A) extends Free[F,A]
case class Suspend[F[_],A](ffa: F[Free[F,A]]) extends Free[F,A]
object Free {
import scalaz.Unapply
/** A free monad over the free functor generated by `S` */
type FreeC[S[_], A] = Free[({type f[x] = Coyoneda[S, x]})#f, A]

/** Suspends a value within a functor in a single step. Monadic unit for a higher-order monad. */
def liftF[S[_], A](value: => S[A])(implicit S: Functor[S]): Free[S, A] =
Suspend(S.map(value)(Return[S, A]))

/** A version of `liftF` that infers the nested type constructor. */
def liftFU[MA](value: => MA)(implicit MA: Unapply[Functor, MA]): Free[MA.M, MA.A] =
liftF(MA(value))(MA.TC)

/** A free monad over a free functor of `S`. */
def liftFC[S[_], A](s: S[A]): FreeC[S, A] =
liftFU(Coyoneda(s))

/** Interpret a free monad over a free functor of `S` via natural transformation to monad `M`. */
def runFC[S[_], M[_], A](sa: FreeC[S, A])(interp: S ~> M)(implicit M: Monad[M]): M[A] =
sa.foldMap[M](new (({type λ[α] = Coyoneda[S, α]})#λ ~> M) {
def apply[A](cy: Coyoneda[S, A]): M[A] =
M.map(interp(cy.fi))(cy.k)
})
}

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：

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

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

val ioprg = for {
_ <- PutLine("What is your first name ?")
first <- GetLine
_ <- PutLine("What is your last name ?")
last <- GetLine
_ <- PutLine(s"Hello, $first $last !")
} yield ()                                        //> ioprg  : ch13.ex11.Free[[x]ch13.ex11.Coyoneda[ch13.ex11.Console,x],Unit] =
//| Suspend(ch13.ex11$Coyoneda$$anon$4@13c78c0b)

type Id[A] = A
implicit val idMonad = new Monad[Id] {
def unit[A](a: A) = a
def flatMap[A,B](fa: A)(f: A => B): B = f(fa)
}                                                 //> idMonad  : ch13.ex11.Monad[ch13.ex11.Id] = ch13.ex11$$anonfun$main$1$$anon$
//| 10@12843fce

object RealConsole extends (Console ~> Id) {
def apply[A](ca: Console[A]): A = ca match {
case GetLine => readLine
case PutLine(l) => println(l)
}
}
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：

case class State[S,A](runState: S => (A,S)) {
def map[B](f: A => B) = State[S,B](s => {
val (a1,s1) = runState(s)
(f(a1),s1)
})
def flatMap[B](f: A => State[S,B]) = State[S,B](s => {
val (a1,s1) = runState(s)
f(a1).runState(s1)
})
}
case class InOutLog(inLog: List[String], outLog: List[String])
type LogState[A] = State[InOutLog, A]
implicit val logStateMonad = new Monad[LogState] {
def unit[A](a: A) = State(s => (a, s))
def flatMap[A,B](sa: LogState[A])(f: A => LogState[B]) = sa flatMap f
}                                                 //> logStateMonad  : ch13.ex11.Monad[ch13.ex11.LogState] = ch13.ex11$$anonfun$m
//| ain$1$$anon$11@3dd3bcd
object MockConsole extends(Console ~> LogState) {
def apply[A](c: Console[A]): LogState[A] = State(
s => (c,s) match {
case (GetLine, InOutLog(in,out)) => (in.head, InOutLog(in.tail, out))
case (PutLine(l), InOutLog(in,out)) => ((),InOutLog(in, l :: out))
})
}
val s = Free.runFC(ioprg)(MockConsole)            //> s  : ch13.ex11.LogState[Unit] = State(<function1>)
val ls = s.runState(InOutLog(List("Tiger","Chan"),List()))
//> ls  : (Unit, ch13.ex11.InOutLog) = ((),InOutLog(List(),List(Hello, Tiger Ch
//| 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！