泛函编程(31)-泛函IO:Free Monad-Running free

来源:转载

    在上节我们介绍了Free Monad的基本情况。可以说Free Monad又是一个以数据结构替换程序堆栈的实例。实际上Free Monad的功能绝对不止如此,以heap换stack必须成为Free Monad的运算模式,这样我们才可以放心的使用Free Monad所产生的Monadic编程语言了。前面我们介绍了Trampoline的运算模式可以有效解决堆栈溢出问题,而上节的Free Monad介绍里还没有把Free Monad与Trampoline运算模式挂上钩。我们先考虑一下如何在Free Monad数据类型里引入Trampoline运算模式。

我们先对比一下Tranpoline和Free这两个数据类型的基本结构:

 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) = 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)( _ flatMap f))

7 case FlatMap(b,g) => FlatMap(b, x => g(x) flatMap f) //FlatMap(b, g andThen (_ flatMap f))

8 }

9 def map[B](f: A => B)(implicit F: Functor[F]): Free[F,B] = flatMap(a => Return(f(a)))

10 }

11 case class Return[F[_],A](a: A) extends Free[F,A]

12 case class Suspend[F[_],A](ffa: F[Free[F,A]]) extends Free[F,A]

13 trait Trampoline[A] {

14 private case class FlatMap[B](a: Trampoline[A], f: A => Trampoline[B]) extends Trampoline[B]

15 final def runT: A = resume match {

16 case Right(a) => a

17 case Left(k) => k().runT

18 }

19 def unit[A](a: A) = Done(a)

20 def flatMap[B](f: A => Trampoline[B]): Trampoline[B] = this match {

21 // case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f)

22 // case FlatMap(b,g) => FlatMap(b, x => FlatMap(g(x),f))

23 case FlatMap(b,g) => FlatMap(b, x => g(x) flatMap f)

24 case x => FlatMap(x,f)

25 }

26 def map[B](f: A => B): Trampoline[B] = flatMap(a => More(() => Done(f(a))))

27 final def resume: Either[() => Trampoline[A],A] = this match {

28 case Done(a) => Right(a)

29 case More(k) => Left(k)

30 case FlatMap(a,f) => a match {

31 case Done(v) => f(v).resume

32 case More(k) => Left(() => FlatMap(k(),f))

33 case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f)).resume

34 }

35 }

36 }

37 case class Done[A](a: A) extends Trampoline[A]

38 case class More[A](k: () => Trampoline[A]) extends Trampoline[A]

这两个数据类型的设计目的都是为了能逐步运行算法:按照算法运算的状态确定下一步该如何运行。这个F[Free[F,A]]就是一个循环递归结构,里面保存了运算当前状态和下一步运算。

我们曾说如果一个数据类型能有个Functor实例,那么我们就可以用它来产生一个Free Monad。这个要求从上面Free[F,A]类型里的map,flatMap可以了解:我们用了implicit F: Functor[F]参数,因为必须有个Functor实例F才能实现map和flatMap。

为了实现Free Monad在运行中采用Trampoline运行机制,我们可以像Trampoline数据类型一样来实现resume,这个确定每一步运算方式的函数:

 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) = 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)( _ flatMap f))

7 case FlatMap(b,g) => FlatMap(b, x => g(x) flatMap f) //FlatMap(b, g andThen (_ flatMap f))

8 }

9 def map[B](f: A => B)(implicit F: Functor[F]): Free[F,B] = flatMap(a => Return(f(a)))

10 final def resume(implicit F: Functor[F]): Either[F[Free[F,A]],A] = this match {

11 case Return(a) => Right(a)

12 case Suspend(k) => Left(k)

13 case FlatMap(a,f) => a match {

14 case Return(v) => f(v).resume

15 case Suspend(k) => Left(F.map(k)(_ flatMap f))

16 case FlatMap(b,g) => FlatMap(b, g andThen (_ flatMap f)).resume

17 }

18 }

19 }

20 case class Return[F[_],A](a: A) extends Free[F,A]

21 case class Suspend[F[_],A](ffa: F[Free[F,A]]) extends Free[F,A]

Free类型的resume函数与Trampoline的基本一致,只有返回类型和增加了参数implicit F: Functor[F],因为Free[F,A]的F必须是个Functor:用Functor F可以产生Free[F,A]。

我们用个实际例子来体验一下用Functor产生Free:

我们可以用上一节的Interact类型:

1 trait Interact[A]

2 case class Ask(prompt: String) extends Interact[String]

3 case class Tell(msg: String) extends Interact[Unit]

 

这个类型太简单了,太单纯了。我还没想到如何得出它的Functor实例。好像没办法实现那个map函数。那么如果修改一下这个Interact类型:

1 trait Interact[A]

2 case class Ask[A](prompt: String, next: A) extends Interact[A]

3 case class Tell[A](msg: String, next: A) extends Interact[A]

这个新类型的两个状态Ask,Tell都增加了个参数next,代表下一步操作。实际上我们是用map来运行next的。这样我们就可以得出Interact的Functor实例。

1 trait Interact[A]

2 case class Ask[A](prompt: String, next: A) extends Interact[A]

3 case class Tell[A](msg: String, next: A) extends Interact[A]

4 implicit val interactFunctor = new Functor[Interact] {

5 def map[A,B](ia: Interact[A])(f: A => B): Interact[B] = ia match {

6 case Ask(x,n) => Ask(x,f(n))

7 case Tell(x,n) => Tell(x,f(n))

8 }

9 } //> interactFunctor : ch13.ex1.Functor[ch13.ex1.Interact] = ch13.ex1$$anonfun$

从上面的Functor实例中我们可以看到如何通过map的f(n)来运行下一步骤next。

接下来我们要把Interact类型升格到Free类型:

 1 def liftF[F[_],A](fa: F[A])(implicit F: Functor[F]): Free[F,A] = {

2 Suspend(F.map(fa)(a => Return(a)))

3 } //> liftF: [F[_], A](fa: F[A])(implicit F: ch13.ex1.Functor[F])ch13.ex1.Free[F,

4 //| A]

5 implicit def LiftInteract[A](ia: Interact[A]): Free[Interact,A] = liftF(ia)

6 //> LiftInteract: [A](ia: ch13.ex1.Interact[A])ch13.ex1.Free[ch13.ex1.Interact,

7 //| A]

8 val prg = for {

9 first <- Ask("What's your first name?",())

10 last <- Ask("What's your last name?",())

11 _ <- Tell(s"Hello $first $last",())

12 } yield () //> prg : ch13.ex1.Free[ch13.ex1.Interact,Unit] = Suspend(Ask(What's your firs

13 //| t name?,Suspend(Ask(What's your last name?,Suspend(Tell(Hello () (),Return(

14 //| ())))))))

看,把Interact升格后就可以使用for-comprehension了。

还是那句话:用一个有Functor实例的类型就可以产生一个Free Monad。然后我们可以用这个产生的Monad来在for-comprehension里面编写一个算法。

解译运算(Interpret)是Free Monad的Interpreter功能。我们说过要把Trampoline运行机制引入Free Monad运算:

1 def foldMap[G[_]](f: F ~> G)(implicit F: Functor[F], G: Monad[G]): G[A] = resume match {

2 case Right(a) => G.unit(a)

3 case Left(k) => G.flatMap(f(k))(_ foldMap f)

4 }

 

foldMap通过调用resume引入了Trampoline运行机制。

前面介绍的Free Monad相对都比较简单。实际上Free Monad的Suspend处理可以是很复杂的,包括返回结果及接受输入等任何组合。下面我们再看一个较复杂的例子:我们可以把State视为一种简单的状态转变编程语言,包括读取及设定状态两种操作指令:

1 trait StateF[S,A]

2 case class Get[S,A](f: S => A) extends StateF[S,A]

3 case class Put[S,A](s: S, a: A) extends StateF[S,A]

我们先看看嫩不能获取StateF的Functor实例:

1 mplicit def stateFFunctor[S] = new Functor[({type l[x] = StateF[S,x]})#l] {

2 def map[A,B](sa: StateF[S,A])(f: A => B): StateF[S,B] = sa match {

3 case Get(g) => Get( s => f(g(s)) )

4 case Put(s,a) => Put(s, f(a))

5 }

6 } //> stateFFunctor: [S]=> ch13.ex1.Functor[[x]ch13.ex1.StateF[S,x]]

既然有了Functor实例,那么我们可以用来产生Free Monad:

1 type FreeState[S,A] = Free[({type l[x] = StateF[S,x]})#l, A]

Free[F,A]里的Functor F只接受一个类型参数。StateF[S,A]有两个类型参数,我们必须用type lambda来解决类型参数匹配问题。

现在我们已经得到了一个FreeState Monad。下面接着实现FreeState的基础组件函数:

1 def unit[S,A](a: A): FreeState[S,A] = Return[({type l[x] = StateF[S,x]})#l, A](a)

2 //> unit: [S, A](a: A)ch13.ex1.FreeState[S,A]

3 def getState[S]: FreeState[S,S] = Suspend[({type l[x] = StateF[S,x]})#l, S](

4 Get(s => Return[({type l[x] = StateF[S,x]})#l, S](s)))

5 //> getState: [S]=> ch13.ex1.FreeState[S,S]

6 def setState[S](s: S): FreeState[S,Unit] = Suspend[({type l[x] = StateF[S,x]})#l, Unit](

7 Put(s, Return[({type l[x] = StateF[S,x]})#l, Unit](())))

8 //> setState: [S](s: S)ch13.ex1.FreeState[S,Unit]

注意类型匹配。我们可以写个函数来运算这个FreeState:

1 def evalS[S,A](s: S, t: FreeState[S,A]): A = t.resume match {

2 case Right(a) => a

3 case Left(Get(f)) => evalS(s, f(s))

4 case Left(Put(n,a)) => evalS(n,a)

5 } //> evalS: [S, A](s: S, t: ch13.ex1.FreeState[S,A])A

这个运算方式还是调用了resume函数。注意:Get(f) 返回 StateF[S,A],StateF是个Functor, F[Free[F,A]]那么A就是Free[F,A]

还是试试运算那个zipIndex函数:

 1 def zipIndex[A](as: List[A]): List[(Int, A)] = {

2 evalS(1, as.foldLeft(unit[Int,List[(Int,A)]](List()))(

3 (acc,a) => for {

4 xs <- acc

5 n <- getState

6 _ <- setState(n+1)

7 } yield (n, a) :: xs)).reverse

8 } //> zipIndex: [A](as: List[A])List[(Int, A)]

9

10 zipIndex((0 to 10000).toList) //> res0: List[(Int, Int)] = List((1,0), (2,1), (3,2), (4,3), (5,4), (6,5), (7,

11 //| 6), (8,7), (9,8), (10,9), (11,10), (12,11), (13,12), (14,13), (15,14), (16,

12 //| 15), (17,16), (18,17), (19,18), (20,19), (21,20), (22,21), (23,22), (24,23)

 

没错,这段程序不但维护了一个状态而且使用了Trampoline运算模式,可以避免StackOverflow问题。

下面我们再用一个例子来示范Free Monad的Monadic Program和Interpreter各自的用途:

我们用一个Stack操作的例子。对Stack中元素的操作包括:Push,Add,Sub,Mul,End。这几项操作也可被视作一种Stack编程语言中的各项操作指令:

1 trait StackOps[A]

2 case class Push[A](value: Int, ops:A) extends StackOps[A]

3 case class Add[A](ops: A) extends StackOps[A]

4 case class Mul[A](ops: A) extends StackOps[A]

5 case class Dup[A](ops: A) extends StackOps[A]

6 case class End[A](ops: A) extends StackOps[A]

我们先推导它的Functor实例:

1 implicit val stackOpsFunctor: Functor[StackOps] = new Functor[StackOps] {

2 def map[A,B](oa: StackOps[A])(f: A => B): StackOps[B] = oa match {

3 case Push(v,a) => Push(v,f(a))

4 case Add(a) => Add(f(a))

5 case Mul(a) => Mul(f(a))

6 case Dup(a) => Dup(f(a))

7 case End(a) => End(f(a))

8 }

9 }

 这里的next看起来是多余的,但它代表的是下一步运算。有了它才可能得到Functor实例,即使目前每一个操作都是完整独立步骤。

有了Functor实例我们就可以实现StackOps的Monadic programming:

 1 def liftF[F[_],A](fa: F[A])(implicit F: Functor[F]): Free[F,A] = {

2 Suspend(F.map(fa)(a => Return(a)))

3 } //> liftF: [F[_], A](fa: F[A])(implicit F: ch13.ex1.Functor[F])ch13.ex1.Free[F,

4 //| A]

5 implicit def liftStackOps[A](sa: StackOps[A]): Free[StackOps,A] = liftF(sa)

6 //> liftStackOps: [A](sa: ch13.ex1.StackOps[A])ch13.ex1.Free[ch13.ex1.StackOps,

7 //| A]

8 val stkprg = for {

9 _ <- Push(1,())

10 _ <- Push(2,())

11 _ <- Add(())

12 } yield x //> stkprg : ch13.ex1.Free[ch13.ex1.StackOps,Unit] = Suspend(Push(1,Suspend(Pu

13 //| sh(2,Suspend(Add(Suspend(Pop(Return(())))))))))

我们用lisftStackOps函数把StackOps升格到Free[StackOps,A]后就可以用for-comprehension进行Monadic programming了。如果不习惯Add(())这样的表达式可以这样:

 1 def push(value: Int) = Push(value,()) //> push: (value: Int)ch13.ex1.Push[Unit]

2 def add = Add(()) //> add: => ch13.ex1.Add[Unit]

3 def sub = Sub(()) //> sub: => ch13.ex1.Sub[Unit]

4 def mul = Mul(()) //> mul: => ch13.ex1.Mul[Unit]

5 def end = End(()) //> end: => ch13.ex1.End[Unit]

6 val stkprg = for {

7 _ <- push(1)

8 _ <- push(2)

9 _ <- add

10 _ <- push(4)

11 _ <- mul

12 } yield () //> stkprg : ch13.ex1.Free[ch13.ex1.StackOps,Unit] = Suspend(Push(1,Suspend(Pu

13 //| sh(2,Suspend(Add(Suspend(Push(4,Suspend(Mul(Return(())))))))))))

这样从文字意思上描述就清楚多了。但是,这个stkprg到底是干什么的?如果不从文字意义上解释我们根本不知道这段程序干了些什么,怎么干的。换句直白的话就是:没有意义。这正是Free Monad功能精妙之处:我们用Monad for-comprehension来编写一段Monadic program,然后在Interpreter中赋予它具体意义:用Interpreter来确定程序具体的意义。

那我们就进入Interpreter来运算这段程序吧。

先申明Stack类型: type Stack = List[Int]

在上面我们有个Interpreter, foldMap:

1 def foldMap[G[_]](f: F ~> G)(implicit F: Functor[F], G: Monad[G]): G[A] = resume match {

2 case Right(a) => G.unit(a)

3 case Left(k) => G.flatMap(f(k))(_ foldMap f)

4 }

但是运行stkprg必须传入Stack起始值,foldMap无法满足。那么我们再写另一个runner吧:

 1 final def foldRun[B](b: B)(f: (B, F[Free[F,A]]) => (B, Free[F,A]))(implicit F: Functor[F]): (B, A) = {

2 @annotation.tailrec

3 def run(t: Free[F,A], z: B): (B, A) = t.resume match {

4 case Right(a) => (z, a)

5 case Left(k) => {

6 val (b1, f1) = f(z, k)

7 run(f1,b1)

8 }

9 }

10 run(this,b)

11 }

foldRun也是个折叠算法:给予一个起始值及一个对数据结构内部元素的处理函数然后可以开始运行。这个函数刚好符合我们的需要。下一步就是给予stkprg意义:确定Push,Add...这些指令具体到底干什么:

 1 type Stack = List[Int]

2 def stackFn(stack: Stack, prg: StackOps[Free[StackOps,Unit]]): (Stack, Free[StackOps,Unit]) = prg match {

3 case Push(v, n) => {

4 (v :: stack, n)

5 }

6 case Add(n) => {

7 val hf :: hs :: t = stack

8 ((hf + hs) :: stack, n)

9 }

10 case Sub(n) => {

11 val hf :: hs :: t = stack

12 ((hs - hf) :: stack, n)

13 }

14 case Mul(n) => {

15 val hf :: hs :: t = stack

16 ((hf * hs) :: stack, n)

17 }

18 } //> stackFn: (stack: ch13.ex1.Stack, prg: ch13.ex1.StackOps[ch13.ex1.Free[ch13.

19 //| ex1.StackOps,Unit]])(ch13.ex1.Stack, ch13.ex1.Free[ch13.ex1.StackOps,Unit])

啊。。。在这里我们才能具体了解每一句StackOps指令的意义。这就是Free Monad Interpreter的作用了。我们试着运算这个stkprg:

1 val stkprg = for {

2 _ <- push(1)

3 _ <- push(2)

4 _ <- add

5 _ <- push(4)

6 _ <- mul

7 } yield () //> stkprg : ch13.ex1.Free[ch13.ex1.StackOps,Unit] = Suspend(Push(1,Suspend(Pu

8 //| sh(2,Suspend(Add(Suspend(Push(4,Suspend(Mul(Return(())))))))))))

9 stkprg.foldRun(List[Int]())(stackFn) //> res0: (List[Int], Unit) = (List(12, 4, 3, 2, 1),())

跟踪一下操作步骤,最终结果是正确的。
我们再试一下用Natural Transformation原理的foldMap函数。我们可以用State的runS来传入Stack初始值:

1 type StackState[A] = State[Stack,A]

2 implicit val stackStateMonad = new Monad[StackState] {

3 def unit[A](a: A) = State(s => (a,s))

4 def flatMap[A,B](sa: StackState[A])(f: A => StackState[B]): StackState[B] = sa flatMap f

5 } //> stackStateMonad : ch13.ex1.Monad[ch13.ex1.StackState] = ch13.ex1$$anonfun$

6 //| [email protected]

这个StackState类型就是一个State类型。我们能够推导它的Monad实例,那我们就可以调用foldMap了。我们先编写Interpreter功能:

 1 object StackOperator extends (StackOps ~> StackState) {

2 def apply[A](sa: StackOps[A]): StackState[A] = sa match {

3 case Push(v,n) => State((s: Stack) => (n, v :: s))

4 case Add(n) => State((s: Stack) => {

5 val hf :: hs :: t = s

6 (n, (hf + hs) :: s)

7 })

8 case Sub(n) => State((s: Stack) => {

9 val hf :: hs :: t = s

10 (n, (hs - hf) :: s)

11 })

12 case Mul(n) => State((s: Stack) => {

13 val hf :: hs :: t = s

14 (n, (hf * hs) :: s)

15 })

16 }

17 }

通过Natural Transformation把StackOps转成StackState状态维护。StackOps具体意义也在这里才能得到体验。我们用foldMap运算stkprg:

1 stkprg.foldMap(StackOperator).runS(List[Int]()) //> res1: (Unit, ch13.ex1.Stack) = ((),List(12, 4, 3, 2, 1))

我们得到了同样的运算结果。

希望通过这些例子能把Free Monad的用途、用法、原理解释清楚了。

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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