Scalaz（38）－ Free ：Coproduct－Monadic语句组合

很多函数式编程爱好者都把FP称为Monadic Programming，意思是用Monad进行编程。我想FP作为一种比较成熟的编程模式，应该有一套比较规范的操作模式吧。因为Free能把任何F[A]升格成Monad，所以Free的算式（AST）、算法（Interpreter）关注分离（separation of concern）模式应该可以成为一种规范的FP编程模式。我们在前面的几篇讨论中都涉及了一些AST的设计和运算，但都是一些功能单一，离散的例子。如果希望通过Free获取一个完整可用的程序，就必须想办法把离散的Free AST组合成一体运算。我们先从单一的Free AST例子开始：

 import scalaz._
 import Scalaz._
 import scala.language.higherKinds
 import scala.language.implicitConversions
 object FreeModules {
   object FreeInteract {
     trait Interact[+A]
     type FreeInteract[A] = Free.FreeC[Interact,A]
     object Interact {
       case class Ask(prompt: String) extends Interact[String]
       case class Tell(msg: String) extends Interact[Unit]
       implicit def interactToFreeC[A](ia: Interact[A]) = Free.liftFC(ia)
       object InteractConsole extends (Interact ~> Id) {
         def apply[A](ia: Interact[A]): Id[A] = ia match {
           case Ask(p) => println(p); readLine
           case Tell(m) => println(m)
         }
       }
     }
     import Interact._
     val interactScript = for {
       first <- Ask("What's your first name?")
       last <- Ask("What's your last name?")
       _ <- Tell(s"Hello ${first} ${last}, nice to meet you!")
     } yield ()
   }
 }

这是一个我们在前面讨论中重复描述几次的简单交互例子，包括了ADT、AST和Interpreter。我们可以直接运行这个程序：

 object freePrgDemo extends App {
   import FreeModules._
   import FreeInteract._
   import Interact._
   Free.runFC(interactScript)(InteractConsole)
 }

运算结果如下：

 What's your first name?
 Tiger
 What's your last name?
 Chan
 Hello Tiger Chan, nice to meet you!

就是简单的两句界面提示和键盘输入，然后提示输入结果，没什么意义。作为测试，我们也可以模拟Console交互：用Map[String,String]来模拟Map[提问，回答]，然后把这个Map提供给Interpreter，返回结果(List[String],A)，其中List[String]是运行跟踪记录，A是模拟的键盘输入：

       type InteractMapTester[A] = Map[String,String] => (List[String], A)
       implicit val mapTesterMonad = new Monad[InteractMapTester] {
          def point[A](a: => A) = _ => (List(), a)
          def bind[A,B](ia: InteractMapTester[A])(f: A => InteractMapTester[B]): InteractMapTester[B] =
            m => {
              val (o1,a1) = ia(m)
              val (o2,a2) = f(a1)(m)
              (o1 ++ o2, a2)
            }
       }
       object InteractTesterMap extends (Interact ~> InteractMapTester) {
         def apply[A](ia: Interact[A]): InteractMapTester[A] = ia match {
           case Ask(p) => { m => (List(), m(p)) } //m(p)返回提问对应的答案作为键盘输入
           case Tell(s) => { m => (List(s), ()) } //List(s)在bind函数中的o1++o2形成跟踪记录
                                                  //在运算AST时就会调用InteractMapTester的bind函数
         }
       }

使用模拟Console的Interpreter来运行：

 object freePrgDemo extends App {
   import FreeModules._
   import FreeInteract._
   import Interact._
   //Free.runFC(interactScript)(InteractConsole)
     val result = Free.runFC(interactScript)(InteractTesterMap).apply(
     Map(
     "What's your first name?" -> "tiger",
     "What's your last name?" -> "chan"
   ))
   println(result)
   }
 //产生以下输出结果
 (List(Hello tiger chan, nice to meet you!),())

从mapTesterMonad定义中的bind看到了这句：o1++o2，是Logger的典型特征。那么用Writer能不能实现同等效果呢？我们先看看WriterT：

final case class WriterT[F[_], W, A](run: F[(W, A)]) { self =>
...

实际上这个W就可以满足Logger的功能，因为在WriterT的flatMap中实现了W|+|W：

  def flatMap[B](f: A => WriterT[F, W, B])(implicit F: Bind[F], s: Semigroup[W]): WriterT[F, W, B] =
    flatMapF(f.andThen(_.run))

  def flatMapF[B](f: A => F[(W, B)])(implicit F: Bind[F], s: Semigroup[W]): WriterT[F, W, B] =
    writerT(F.bind(run){wa =>
      val z = f(wa._2)
      F.map(z)(wb => (s.append(wa._1, wb._1), wb._2))
    })

那么如何把Map[提问,回答]传人呢？我们可以通过WriterT[F[_],W,A]的F[]来实现这一目的：

       type WriterTF[A] = Map[String,String] => A
       type InteractWriterTester[A] = WriterT[WriterTF,List[String],A]

然后我们可以用WriterT的参数run来传人Map[String,String]：run:WriterTF[(W,A)] == Map[String,String]=>(W,A)。

以下是用WriterT实现的Interpreter版本：

       type WriterTF[A] = Map[String,String] => A
       type InteractWriterTester[A] = WriterT[WriterTF,List[String],A]
       def testerToWriter[A](f: Map[String,String] => (List[String], A)) =
         WriterT[WriterTF,List[String],A](f)
       implicit val writerTesterMonad = WriterT.writerTMonad[WriterTF, List[String]]
       object InteractTesterWriter extends (Interact ~> InteractWriterTester) {
         def apply[A](ia: Interact[A]): InteractWriterTester[A] = ia match {
           case Ask(p) => testerToWriter { m => (List(), m(p)) }
           case Tell(s) => testerToWriter { m => (List(s), ())}
         }
       }

我们可以这样运行：

object freePrgDemo extends App {
  import FreeModules._
  import FreeInteract._
  import Interact._
  //Free.runFC(interactScript)(InteractConsole)
  //val result = Free.runFC(interactScript)(InteractTesterMap).apply(
  val result = Free.runFC(interactScript)(InteractTesterWriter).run(
    Map(
    "What's your first name?" -> "tiger",
    "What's your last name?" -> "chan"
  ))
  println(result)

}

我们再设计另一个用户登录Login的例子：

   object FreeUserLogin {
     import Dependencies._
     trait UserLogin[+A]
     type FreeUserLogin[A] = Free.FreeC[UserLogin,A]
     object UserLogin {
       case class Login(user: String, pswd: String) extends UserLogin[Boolean]
       implicit def loginToFree[A](ul: UserLogin[A]) = Free.liftFC(ul)
       type LoginService[A] = Reader[PasswordControl,A]
       object LoginInterpreter extends (UserLogin ~> LoginService) {
         def apply[A](ul: UserLogin[A]): LoginService[A] = ul match {
           case Login(u,p) => Reader( cr => cr.matchPassword(u, p))
         }
       }
     }
     import UserLogin._
     val loginScript = for {
       b <- Login("Tiger","")
     } yield b
   }

这个例子里只有Login一个ADT，它的功能是把输入的User和Password与一个用户登录管理系统内的用户身份信息进行验证。由于如何进行用户密码验证不是这个ADT的功能，它可能涉及另一特殊功能系统的调用，刚好用来做个Reader依赖注入示范。以下是这项依赖定义：

 object Dependencies {
   trait PasswordControl {
     type User = String
     type Password = String
     val pswdMap: Map[User, Password]
     def matchPassword(u: User, p: Password): Boolean
   }
 }

对loginScript进行测试运算时必须先获取PasswordControl实例，然后注入运算：

   import Dependencies._
   import FreeUserLogin._
   import UserLogin._
   object Passwords extends PasswordControl {  //依赖实例
      val pswdMap = Map (
        "Tiger" -> "",
        "John" -> ""
      )
      def matchPassword(u: User, p: Password) = pswdMap.getOrElse(u, p+"!") === p
   }
   val result = Free.runFC(loginScript)(LoginInterpreter).run(Passwords)  //注入依赖
   println(result)

不过即使能够运行，loginScsript的功能明显不完整，还需要像Interact那样的互动部分来获取用户输入信息。那么我们是不是考虑在ADT层次上把Interact和UserLogin合并起来，像这样：

       case class Ask(prompt: String) extends Interact[String]
       case class Tell(msg: String) extends Interact[Unit]
       case class Login(user: String, pswd: String) extends Interact[Boolean]

明显这是可行的。但是，Interact和Login被紧紧捆绑在了一起形成了一个新的ADT。如果我们设计另一个同样需要互动的ADT，我们就需要重复同样的Interact功能设计，显然这样做违背了FP的原则：从功能单一的基本计算开始，按需要对基本函数进行组合实现更复杂的功能。Interact和UserLogin都是基础ADT，从编程语言角度描述Interact和UserLogin属于两种类型的编程语句。我们最终需要的AST是这样的：

   val interLogin: Free[???, A] = for {
     user <- Ask("Enter User ID:")  //Free[Interact,A]
     pswd <- Ask("Enter Password:") //Free[Interact,A]
     ok <- Login(user,pswd) //Free[UserLogin,A]
   } yield ok

不过明显类型对不上，因为Interact和UserLogin是两种语句。scalaz的Coproduct类型可以帮助我们实现两种Monadic语句的语义（sematics）合并。Coproduct是这样定义的：scalaz/Coproduct.scala

/** `F` on the left, and `G` on the right, of [[scalaz.\/]].
  *
  * @param run The underlying [[scalaz.\/]]. */
final case class Coproduct[F[_], G[_], A](run: F[A] \/ G[A]) {
  import Coproduct._

  def map[B](f: A => B)(implicit F: Functor[F], G: Functor[G]): Coproduct[F, G, B] =
    Coproduct(run.bimap(F.map(_)(f), G.map(_)(f)))
...

从run:F[A]\/G[A]可以理解Coproduct是两种语句F,G的联合（union）。在我们上面的例子里我们可以用下面的表达方式代表Interact和UserLogin两种语句的联合（union）：

   type InteractLogin[A] = Coproduct[Interact,UserLogin,A]

这是一个语义更广泛的类型：包含了Interact和UserLogin语义。我们可以用Inject类型来把Interact和UserLogin语句集“注入”到一个更大的句集。Inject是这样定义的：scalaz/Inject.scala

/**
 * Inject type class as described in "Data types a la carte" (Swierstra 2008).
 *
 * @see [[http://www.staff.science.uu.nl/~swier004/Publications/DataTypesALaCarte.pdf]]
 */
sealed abstract class Inject[F[_], G[_]] {
  def inj[A](fa: F[A]): G[A]
  def prj[A](ga: G[A]): Option[F[A]]
}

sealed abstract class InjectInstances {
  implicit def reflexiveInjectInstance[F[_]] =
    new Inject[F, F] {
      def inj[A](fa: F[A]) = fa
      def prj[A](ga: F[A]) = some(ga)
    }

  implicit def leftInjectInstance[F[_], G[_]] =
    new Inject[F, ({type λ[α] = Coproduct[F, G, α]})#λ] {
      def inj[A](fa: F[A]) = Coproduct.leftc(fa)
      def prj[A](ga: Coproduct[F, G, A]) = ga.run.fold(some(_), _ => none)
    }

  implicit def rightInjectInstance[F[_], G[_], H[_]](implicit I: Inject[F, G]) =
      new Inject[F, ({type λ[α] = Coproduct[H, G, α]})#λ] {
        def inj[A](fa: F[A]) = Coproduct.rightc(I.inj(fa))
        def prj[A](ga: Coproduct[H, G, A]) = ga.run.fold(_ => none, I.prj(_))
      }
}
...

实现函数inj(fa:F[A]):G[A]代表把F[A]并入G[A]。这里还提供了三个类型的实例：

1、reflexiceInjectInstance[F[_]]：自我注入

2、leftInjectInstance[F[_],G[_]]：把F[A]注入Coproduct[F,G,A]的left(-\/)

3、rightInjectInstance[F[_],G[_],H[_]]：把F[A]注入Coproduct的right（\/-）。需要先把F注入G（inj(F[A]):G[A]）

我们可以用implicitly来证明Interact和UserLogin的Inject实例存在：

   val selfInj = implicitly[Inject[Interact,Interact]]
   type LeftInterLogin[A] = Coproduct[Interact,UserLogin,A]
   val leftInj = implicitly[Inject[Interact,LeftInterLogin]]
   type RightInterLogin[A] = Coproduct[UserLogin,LeftInterLogin,A]
   val rightInj = implicitly[Inject[Interact,RightInterLogin]]

现在我们需要把Coproduct[F,G,A]的F与G合并然后把F[A]升格成Free[G,A]:

   object coproduct {
     def lift[F[_],G[_],A](fa: F[A])(implicit I: Inject[F,G]): Free.FreeC[G,A] = Free.liftFC(I.inj(fa))
   }

我们可以用这个lift把Interact和UserLogin的ADT统一升格成Free[G,A]：

   object coproduct {
     import FreeInteract._
     import Interact._
     import FreeUserLogin._
     import UserLogin._
     def lift[F[_],G[_],A](fa: F[A])(implicit I: Inject[F,G]): Free.FreeC[G,A] = Free.liftFC(I.inj(fa))
     class Interacts[G[_]](implicit I: Inject[Interact,G]) {
       def ask(prompt: String): Free.FreeC[G,String] = lift(Ask(prompt))
       def tell(msg: String): Free.FreeC[G,Unit] = lift(Tell(msg))
     }
     class Logins[G[_]](implicit I: Inject[UserLogin,G]) {
       def login(u: String, p: String): Free.FreeC[G,Boolean] = lift(Login(u,p))
     }
   }

我们用lift把基础Interact和UserLogin的语句注入了联合的语句集G[A]，然后升格成FreeC[G,A]。现在我们可以把Interact,UserLogin这两种语句用在同一个for-comprehension里了：

   def loginScript[G[_]](implicit I: Interacts[G], L: Logins[G]) ={
     import I._
     import L._
     for {
       uid <- ask("ya id?")
       pwd <- ask("password?")
       login <- login(uid,pwd)
       _ <- if (login) tell("ya lucky bastard!") else tell("geda fk outa here!")
     } yield()
   }

有了Inject和Lift，现在已经成功的用两种ADT集成了一个AST。不过我们还必须提供Interacts[G]和Logins[G]实例：

 object CoproductModules {
   object CoproductFunctions {
     import FreeInteract._
     import Interact._
     import FreeUserLogin._
     import UserLogin._
     def lift[F[_],G[_],A](fa: F[A])(implicit I: Inject[F,G]): Free.FreeC[G,A] = Free.liftFC(I.inj(fa))
     class Interacts[G[_]](implicit I: Inject[Interact,G]) {
       def ask(prompt: String): Free.FreeC[G,String] = lift(Ask(prompt))
       def tell(msg: String): Free.FreeC[G,Unit] = lift(Tell(msg))
     }
     object Interacts {
       implicit def instance[G[_]](implicit I: Inject[Interact,G]) = new Interacts[G]
     }
     class Logins[G[_]](implicit I: Inject[UserLogin,G]) {
       def login(u: String, p: String): Free.FreeC[G,Boolean] = lift(Login(u,p))
     }
     object Logins {
       implicit def instance[G[_]](implicit I: Inject[UserLogin,G]) = new Logins[G]
     }
   }

现在我们的语句集（AST）是一个联合的语句集（Coproduct）。那么，我们应该怎么去运算它呢？我们应该如何实现它的Interpreter？现在我们面对的Monadic程序类型是个Coproduct：

   type InteractLogin[A] = Coproduct[Interact,UserLogin,A]
   val loginPrg = loginScript[InteractLogin]

现在语句集Interact和UserLogin是分别放在Coproduce的左右两边。那么我们可以历遍这个Coproduct来分别运算Interact和UserLogin语句：

   def or[F[_],G[_],H[_]](fg: F ~> G, hg: H ~> G): ({type l[x] = Coproduct[F,H,x]})#l ~> G =
     new (({type l[x] = Coproduct[F,H,x]})#l ~> G) {
     def apply[A](ca: Coproduct[F,H,A]): G[A] = ca.run match {
       case -\/(fa) => fg(fa)
       case \/-(ha) => hg(ha)
     }
   }

值得注意的是如果or函数用在Interact和UserLogin上时它们自然转换（NaturalTransformation）的目标类型必须一致，应该是一个更大的类型，而且必须是Monad，这是NaturalTransformation的要求。所以我们可以把InteractInterpreter的转换目标类型由Id变成Reader，也就是LoginInterpreter的转换目标类型：

   object InteractReader extends (Interact ~> LoginService) {
     def apply[A](ia: Interact[A]): LoginService[A] = ia match {
     case Ask(p) => println(p);  Reader(cr => readLine)
     case Tell(m) => println(m); Reader(cr => ())
    }
   }      

好了，现在我们可以这样来测试运算：

 object freePrgDemo extends App {
   import FreeModules._
   import FreeInteract._
   import Interact._
   //Free.runFC(interactScript)(InteractConsole)
   //val result = Free.runFC(interactScript)(InteractTesterMap).apply(
  /* val result = Free.runFC(interactScript)(InteractTesterWriter).run(
     Map(
     "What's your first name?" -> "tiger",
     "What's your last name?" -> "chan"
   ))
   println(result)
   */
   import Dependencies._
   import FreeUserLogin._
   import UserLogin._

   object Passwords extends PasswordControl {
      val pswdMap = Map (
        "Tiger" -> "",
        "John" -> ""
      )
      def matchPassword(u: User, p: Password) = pswdMap.getOrElse(u, p+"!") === p
   }
   /*
   val result = Free.runFC(loginScript)(LoginInterpreter).run(Passwords)
   println(result)
   */

   import CoproductDemo._
   Free.runFC(loginPrg)(or(InteractReader,LoginInterpreter)).run(Passwords)
 }

我们把密码管理依赖也注入进去了。看看结果：

 ya id?
 Tiger
 password?

 geda fk outa here!

 ya id?
 Tiger
 password?

 ya lucky bastard!

 ya id?
 John
 password?

 ya lucky bastard!

OK, 把这节示范源代码提供在下面：

 package demos
 import scalaz._
 import Scalaz._
 import scala.language.higherKinds
 import scala.language.implicitConversions
 object FreeModules {
   object FreeInteract {
     trait Interact[+A]
     type FreeInteract[A] = Free.FreeC[Interact,A]
     object Interact {
       case class Ask(prompt: String) extends Interact[String]
       case class Tell(msg: String) extends Interact[Unit]
       implicit def interactToFreeC[A](ia: Interact[A]) = Free.liftFC(ia)
       object InteractConsole extends (Interact ~> Id) {
         def apply[A](ia: Interact[A]): Id[A] = ia match {
           case Ask(p) => println(p); readLine
           case Tell(m) => println(m)
         }
       }
       type InteractMapTester[A] = Map[String,String] => (List[String], A)
       implicit val mapTesterMonad = new Monad[InteractMapTester] {
          def point[A](a: => A) = _ => (List(), a)
          def bind[A,B](ia: InteractMapTester[A])(f: A => InteractMapTester[B]): InteractMapTester[B] =
            m => {
              val (o1,a1) = ia(m)
              val (o2,a2) = f(a1)(m)
              (o1 ++ o2, a2)
            }
       }
       object InteractTesterMap extends (Interact ~> InteractMapTester) {
         def apply[A](ia: Interact[A]): InteractMapTester[A] = ia match {
           case Ask(p) => { m => (List(), m(p)) } //m(p)返回提问对应的答案作为键盘输入
           case Tell(s) => { m => (List(s), ()) } //List(s)在bind函数中的o1++o2形成跟踪记录
                                                  //在运算AST时会用到InteractMapTester的bind
         }
       }
       type WriterTF[A] = Map[String,String] => A
       type InteractWriterTester[A] = WriterT[WriterTF,List[String],A]
       def testerToWriter[A](f: Map[String,String] => (List[String], A)) =
         WriterT[WriterTF,List[String],A](f)
       implicit val writerTesterMonad = WriterT.writerTMonad[WriterTF, List[String]]
       object InteractTesterWriter extends (Interact ~> InteractWriterTester) {
         def apply[A](ia: Interact[A]): InteractWriterTester[A] = ia match {
           case Ask(p) => testerToWriter { m => (List(), m(p)) }
           case Tell(s) => testerToWriter { m => (List(s), ())}
         }
       }
     }
     import Interact._
     val interactScript = for {
       first <- Ask("What's your first name?")
       last <- Ask("What's your last name?")
       _ <- Tell(s"Hello ${first} ${last}, nice to meet you!")
     } yield ()
   }
   object FreeUserLogin {
     import Dependencies._
     trait UserLogin[+A]
     type FreeUserLogin[A] = Free.FreeC[UserLogin,A]
     object UserLogin {
       case class Login(user: String, pswd: String) extends UserLogin[Boolean]
       implicit def loginToFree[A](ul: UserLogin[A]) = Free.liftFC(ul)
       type LoginService[A] = Reader[PasswordControl,A]
       object LoginInterpreter extends (UserLogin ~> LoginService) {
         def apply[A](ul: UserLogin[A]): LoginService[A] = ul match {
           case Login(u,p) => Reader( cr => cr.matchPassword(u, p))
         }
       }
     }
     import UserLogin._
     val loginScript = for {
       b <- Login("Tiger","")
     } yield b
   }
 }
 object Dependencies {
   trait PasswordControl {
     type User = String
     type Password = String
     val pswdMap: Map[User, Password]
     def matchPassword(u: User, p: Password): Boolean
   }
 }
 object CoproductDemo {
   import FreeModules._
   import FreeUserLogin._
   import UserLogin._
   import FreeInteract._
   import Interact._
   import Dependencies._
   def lift[F[_],G[_],A](fa: F[A])(implicit I: Inject[F,G]): Free.FreeC[G,A] = Free.liftFC(I.inj(fa))
   class Interacts[G[_]](implicit I: Inject[Interact,G]) {
     def ask(prompt: String) = lift(Ask(prompt))
     def tell(msg: String) = lift(Tell(msg))
   }
   object Interacts {
     implicit def instance[F[_]](implicit I: Inject[Interact,F]) = new Interacts[F]
   }
   class Logins[G[_]](implicit I: Inject[UserLogin,G]) {
     def login(user: String, pswd: String) = lift(Login(user,pswd))
   }
   object Logins {
     implicit def instance[F[_]](implicit I: Inject[UserLogin,F]) = new Logins[F]
   }
   def loginScript[G[_]](implicit I: Interacts[G], L: Logins[G]) ={
     import I._
     import L._
     for {
       uid <- ask("ya id?")
       pwd <- ask("password?")
       login <- login(uid,pwd)
       _ <- if (login) tell("ya lucky bastard!") else tell("geda fk outa here!")
     } yield()
   }

   def or[F[_],G[_],H[_]](fg: F ~> G, hg: H ~> G): ({type l[x] = Coproduct[F,H,x]})#l ~> G =
     new (({type l[x] = Coproduct[F,H,x]})#l ~> G) {
     def apply[A](ca: Coproduct[F,H,A]): G[A] = ca.run match {
       case -\/(fa) => fg(fa)
       case \/-(ha) => hg(ha)
     }
   }

   type InteractLogin[A] = Coproduct[Interact,UserLogin,A]
   val loginPrg = loginScript[InteractLogin]
   object InteractReader extends (Interact ~> LoginService) {
     def apply[A](ia: Interact[A]): LoginService[A] = ia match {
     case Ask(p) => println(p);  Reader(cr => readLine)
     case Tell(m) => println(m); Reader(cr => ())
    }
   }      

 }

 object freePrgDemo extends App {
   import FreeModules._
   import FreeInteract._
   import Interact._
   //Free.runFC(interactScript)(InteractConsole)
   //val result = Free.runFC(interactScript)(InteractTesterMap).apply(
  /* val result = Free.runFC(interactScript)(InteractTesterWriter).run(
     Map(
     "What's your first name?" -> "tiger",
     "What's your last name?" -> "chan"
   ))
   println(result)
   */
   import Dependencies._
   import FreeUserLogin._
   import UserLogin._

   object Passwords extends PasswordControl {
      val pswdMap = Map (
        "Tiger" -> "",
        "John" -> ""
      )
      def matchPassword(u: User, p: Password) = pswdMap.getOrElse(u, p+"!") === p
   }
   /*
   val result = Free.runFC(loginScript)(LoginInterpreter).run(Passwords)
   println(result)
   */

   import CoproductDemo._
   Free.runFC(loginPrg)(or(InteractReader,LoginInterpreter)).run(Passwords)
 }