Cats（1）－从Free开始，Free cats

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cats是scala的一个新的函数式编程工具库，其设计原理基本继承了scalaz：大家都是haskell typeclass的scala版实现。当然，cats在scalaz的基础上从实现细节、库组织结构和调用方式上进行了一些优化，所以对用户来说：cats的基础数据类型、数据结构在功能上与scalaz是大致相同的，可能有一些语法上的变化。与scalaz著名抽象、复杂的语法表现形式相比，cats的语法可能更形象、简单直白。在scalaz的学习过程中，我们了解到所谓函数式编程就是monadic Programming：即用monad这样的数据类型来构建程序。而实际可行的monadic programming就是用Free-Monad编程了。因为Free-Monad程序是真正可运行的，或者说是可以实现安全运行的，因为它可以保证在固定的堆栈内实现无限运算。我们知道：函数式编程模式的运行方式以递归算法为主，flatMap函数本身就是一种递归算法。这就预示着monadic programming很容易造成堆栈溢出问题（StackOverflowError）。当我们把普通的泛函类型F[A]升格成Free-Monad后就能充分利用Free-Monad安全运算能力来构建实际可运行的程序了。由于我们在前面已经详细的了解了scalaz的大部分typeclass，包括Free，对cats的讨论就从Free开始，聚焦在cats.Free编程模式方面。同时，我们可以在使用cats.Free的过程中对cats的其它数据类型进行补充了解。

cats.Free的类型款式如下：

sealed abstract class Free[S[_], A] extends Product with Serializable {...}

S是个高阶类，就是一种函数式运算。值得注意的是：现在S不需要是个Functor了。因为Free的一个实例Suspend类型是这样的：

/** Suspend the computation with the given suspension. */  private final case class Suspend[S[_], A](a: S[A]) extends Free[S, A]

我们不需要map就可以把F[A]升格成Free：

/**   * Suspend a value within a functor lifting it to a Free.   */  def liftF[F[_], A](value: F[A]): Free[F, A] = Suspend(value)

我们在scalaz.Free的讨论中并没能详尽地分析在什么情况下S[_]必须是个Functor。下面我们需要用一些篇幅来解析。

Free程序的特点是算式（description）／算法（implementation）关注分离（separation of concern）：我们用一组数据类型来模拟一种编程语句ADT（algebraic data type），这一组ADT就形成了一种定制的编程语言DSL（domain specific language）。Free的编程部分就是用DSL来描述程序功能（description of purpose），即算式了。算法即用DSL描述的功能的具体实现，可以有多种的功能实现方式。我们先看个简单的DSL：

import cats.free._import cats.Functorobject catsFree {  object ADTs {    sealed trait Interact[+A]    object Interact {      case class Ask(prompt: String) extends Interact[String]      case class Tell(msg: String) extends Interact[Unit]            def ask(prompt: String): Free[Interact,String] = Free.liftF(Ask(prompt))      def tell(msg: String): Free[Interact,Unit] = Free.liftF(Tell(msg))      implicit object interactFunctor extends Functor[Interact]  {        def map[A,B](ia: Interact[A])(f: A => B): Interact[B] = ???      /*   ia match {           case Ask(p) => ???           case Tell(m) => ???        } */      }      }  }  object DSLs {    import ADTs._    import Interact._    val prg: Free[Interact,Unit] = for {      first <- ask("What's your first name?")      last <- ask("What's your last name?")      _ <- tell(s"Hello $first $last")    } yield()  }

在这个例子里Interact并不是一个Functor，因为我们无法获取Interact Functor实例的map函数。先让我们分析一下Functor的map：

     implicit object interactFunctor extends Functor[Interact]  {        def map[A,B](ia: Interact[A])(f: A => B): Interact[B] = ia match {           case Ask(p) => ???           case Tell(m) => ???        }      }

map的作用是用一个函数A => B把F[A]转成F[B]。也就是把语句状态从F[A]转成F[B]，但在Interact的情况里F[B]已经是明确的Interact[Unit]和Interact[String]两种状态，而map的f是A => B，在上面的示范里我们该如何施用f来获取这个Interact[B]呢？从上面的示范里我们观察可以得出Ask和Tell这两个ADT纯粹是为了模拟ask和tell这两个函数。ask和tell分别返回Free版本的String,Unit结果。可以说：Interact并没有转换到下一个状态的要求。那么假如我们把ADT调整成下面这样呢：

      sealed trait FunInteract[NS]      object FunInteract {        case class FunAsk[NS](prompt: String, onInput: String =>  NS) extends FunInteract[NS]        case class FunTell[NS](msg: String, ns: NS) extends FunInteract[NS]                def funAsk(prompt: String): Free[FunInteract,String] = Free.liftF(FunAsk(prompt,identity))        def funAskInt(prompt: String): Free[FunInteract,Int] = Free.liftF(FunAsk(prompt,_.toInt))        def funTell(msg: String): Free[FunInteract,Unit] = Free.liftF(FunTell(msg,()))                implicit object funInteract extends Functor[FunInteract] {           def map[A,NS](fa: FunInteract[A])(f: A => NS) = fa match {              case FunAsk(prompt,input) => FunAsk(prompt,input andThen f)              case FunTell(msg,ns) => FunTell(msg,f(ns))           }        }      }

现在这两个ADT是有类型参数NS的了：FunAsk[NS],FunTell[NS]。NS代表了ADT当前类型，如FunAsk[Int]、FunTell[String]...，现在这两个ADT都通过类型参数NS变成了可map的对象了，如FunAsk[String] >>> FunAsk[String], FunAsk[String] >>> FunAsk[Int]...。所以我们可以很顺利的实现object funInteract的map函数。但是，一个有趣的现象是：为了实现这种状态转换，如果ADT需要返回操作结果，就必须具备一个引领状态转换的机制，如FunAsk类型里的onInput: String => NS：它代表funAsk函数返回的结果可以指向下一个状态。新增函数funAskInt是个很好的示范：通过返回的String结果将状态转换到FunAsk[Int]状态。函数funTell不返回结果，所以FunTell没有状态转换机制。scalaz旧版本Free.Suspend的类型款式是：Suspend[F[Free,A]]，这是一个递归类型，内部的Free代表下一个状态。由于我们必须用F.map才能取出下一个状态，所以F必须是个Functor。我们应该注意到如果ADT是Functor的话会造成Free程序的冗余代码。既然cats.Free对F[A]没有设置Functor门槛，那么我们应该尽量避免使用Functor。

得出对ADT类型要求结论后，我们接着示范cats的Free编程。下面是Free程序的功能实现interpret部分（implementation）：

    import ADTs._    object iconsole extends (Interact ~> Id) {      def apply[A](ia: Interact[A]): Id[A] = ia match {         case Ask(p) => {println(p); readLine}         case Tell(m) => println(m)      }    }  }

DSL程序的功能实现就是把ADT F[A]对应到实际的指令集G[A]，在Free编程里用NaturalTransformation ～>来实现。注意G[A]必须是个Monad。在上面的例子里对应关系是：Interact～>Id，代表直接对应到运算指令println和readLine。我们也可以实现另一个版本：

    type Prompt = String    type Reply = String    type Message = String    type Tester[A] = Map[Prompt,Reply] => (List[Message],A)    object tester extends (Interact ~> Tester) {      def apply[A](ia: Interact[A]): Tester[A] = ia match {        case Ask(p) => { m => (List(), m(p)) }        case Tell(m) => { _ => (List(m), ()) }      }    }    import cats.Monad    implicit val testerMonad = new Monad[Tester] {      override def pure[A](a: A): Tester[A] = _ => (List(),a)      override def flatMap[A,B](ta: Tester[A])(f: A => Tester[B]): Tester[B] = m => {        val (o1,a1) = ta(m)        val (o2,a2) = f(a1)(m)        (o1 ++ o2, a2)      }      override def tailRecM[A,B](a: A)(f: A => Tester[Either[A,B]]): Tester[B] =         defaultTailRecM(a)(f)    }  }

上面是个模拟测试：我们用个Map[K,V]来模拟互动，K模拟问prompt，V模拟获取回答Input。测试方式是个Function1，输入测试数据Map，在List[Message]里返回所有Tell产生的信息。上面提到过Tester[A]必须是个Monad，所以我们实现了Tester的Monad实例testMonad。实际上 m=>(List,a)就是个writer函数。所谓的Writer就是包嵌一个对值pair(L,V)的Monad，L代表Log，V代表运算值。Writer的特性就是log所有V的运算过程。我们又可以用Writer来实现这个tester：

   import cats.data.WriterT    type WF[A] = Map[Prompt,Reply] => A    type WriterTester[A] = WriterT[WF,List[Message],A]    def testerToWriter[A](f: Map[Prompt,Reply] => (List[Message],A)) =    WriterT[WF,List[Message],A](f)    object testWriter extends (Interact ~> WriterTester) {      import Interact._      def apply[A](ia: Interact[A]): WriterTester[A] = ia match {        case Ask(p) => testerToWriter(m => (List(),m(p)))        case Tell(m) => testerToWriter(_ => (List(m),()))      }    }

如果我们用Writer来实现Interact，实际上就是把Ask和Tell都升格成Writer类型。

我们再来看看在cats里是如何运算Free DSL程序的。相对scalaz而言，cats的运算函数简单的多，就一个foldMap，我们来看看它的定义：

/**   * Catamorphism for `Free`.   *   * Run to completion, mapping the suspension with the given   * transformation at each step and accumulating into the monad `M`.   *   * This method uses `tailRecM` to provide stack-safety.   */  final def foldMap[M[_]](f: FunctionK[S, M])(implicit M: Monad[M], r: RecursiveTailRecM[M]): M[A] =    r.sameType(M).tailRecM(this)(_.step match {      case Pure(a) => M.pure(Right(a))      case Suspend(sa) => M.map(f(sa))(Right(_))      case FlatMapped(c, g) => M.map(c.foldMap(f))(cc => Left(g(cc)))    })

除了要求M是个Monad之外，cats还要求M的RecursiveTailRecM隐式实例。那么什么是RecursiveTailRecM呢：

/** * This is a marker type that promises that the method * .tailRecM for this type is stack-safe for arbitrary recursion. */trait RecursiveTailRecM[F[_]] extends Serializable {  /*   * you can call RecursiveTailRecM[F].sameType(Monad[F]).tailRec   * to have a static check that the types agree   * for safer usage of tailRecM   */  final def sameType[M[_[_]]](m: M[F]): M[F] = m}

我们用RecursiveTailRecM来保证这个Monad类型与tailRecM是匹配的，这是一种运算安全措施，所以在foldMap函数里r.sameType(M).tailRecM保证了tailRecM不会造成StackOverflowError。cats.Free里还有一种不需要类型安全检验的函数foldMapUnsafe：

/**   * Same as foldMap but without a guarantee of stack safety. If the recursion is shallow   * enough, this will work   */  final def foldMapUnsafe[M[_]](f: FunctionK[S, M])(implicit M: Monad[M]): M[A] =    foldMap[M](f)(M, RecursiveTailRecM.create)

这个函数不需要RecursiveTailRecM。下面我们选择能保证运算安全的方法来运算tester：首先我们需要Tester类型的Monad和RecursiveTailRecM实例：

    import cats.Monad    implicit val testerMonad = new Monad[Tester] with RecursiveTailRecM[Tester]{      override def pure[A](a: A): Tester[A] = _ => (List(),a)      override def flatMap[A,B](ta: Tester[A])(f: A => Tester[B]): Tester[B] = m => {        val (o1,a1) = ta(m)        val (o2,a2) = f(a1)(m)        (o1 ++ o2, a2)      }      override def tailRecM[A,B](a: A)(f: A => Tester[Either[A,B]]): Tester[B] =        defaultTailRecM(a)(f)    }

然后我们制造一些测试数据：

  val testData = Map("What's your first name?" -> "Tiger",  "What's your last name?" -> "Chan")             //> testData  : scala.collection.immutable.Map[String,String] = Map(What's your first name? -> Tiger, What's your last name? -> Chan)

测试运算：

import ADTs._,DSLs._,IMPLs._   val testData = Map("What's your first name?" -> "Tiger",  "What's your last name?" -> "Chan")    //> testData  : scala.collection.immutable.Map[String,String] = Map(What's your first name? -> Tiger, What's your last name? -> Chan)  val prgRunner = prg.foldMap(tester)    //> prgRunner  : demo.ws.catsFree.IMPLs.Tester[Unit] = <function1>  prgRunner(testData)                    //> res0: (List[demo.ws.catsFree.IMPLs.Message], Unit) = (List(Hello Tiger Chan),())

那么如果运算testWriter呢？我们先取得WriterT的Monad实例：

   implicit val testWriterMonad =  WriterT.catsDataMonadWriterForWriterT[WF,List[Message]]

然后构建一个RecursiveTailRecM实例后再用同样的测试数据来运算：

 implicit val testWriterRecT = new RecursiveTailRecM[WriterTester]{}           //> testWriterRecT  : cats.RecursiveTailRecM[demo.ws.catsFree.IMPLs.WriterTester] = demo.ws.catsFree$$anonfun$main$1$$anon$2@6093dd95  val prgRunner = prg.foldMap(testWriter)         //> prgRunner  : demo.ws.catsFree.IMPLs.WriterTester[Unit] = WriterT(<function1>)  prgRunner.run(testData)._1.map(println)         //> Hello Tiger Chan                                                  //| res0: List[Unit] = List(())

运算结果一致。

我们再示范一下cats官方文件里关于free monad例子：模拟一个KVStore的put,get,delete功能。ADT设计如下：

  object ADTs {    sealed trait KVStoreA[+A]    case class Put[T](key: String, value: T) extends KVStoreA[Unit]    case class Get[T](key: String) extends KVStoreA[Option[T]]    case class Del(key: String) extends KVStoreA[Unit]  }

对应的模拟功能函数设计如下：

    type KVStore[A] = Free[KVStoreA,A]    object KVStoreA {      def put[T](key: String, value: T): KVStore[Unit] =        Free.liftF[KVStoreA,Unit](Put[T](key,value))      def get[T](key: String): KVStore[Option[T]] =        Free.liftF[KVStoreA,Option[T]](Get[T](key))      def del(key: String): KVStore[Unit] =        Free.liftF[KVStoreA,Unit](Del(key))      def mod[T](key: String, f: T => T): KVStore[Unit] =        for {          opt <- get[T](key)          _ <- opt.map {t => put[T](key,f(t))}.getOrElse(Free.pure(()))        } yield()    }

注意一下mod函数：它是由基础函数get和put组合而成。我们要求所有在for内的类型为Free[KVStoreA,?]，所以当f函数施用后如果opt变成None时就返回结果Free.pure(())，它的类型是：Free[Nothing,Unit]，Nothing是KVStoreA的子类。

现在我们可以用这个DSL来编制KVS程序了：

 object DSLs {    import ADTs._    import KVStoreA._    def prg: KVStore[Option[Int]] =    for {      _ <- put[Int]("wild-cats", 2)      _ <- mod[Int]("wild-cats", (_ + 12))      _ <- put[Int]("tame-cats", 5)      n <- get[Int]("wild-cats")      _ <- del("tame-cats")    } yield n  }

我们可以通过State数据结纯代码（pure code）方式来实现用immutable map的KVStore：

 object IMPLs {    import ADTs._    import cats.{~>}    import cats.data.State       type KVStoreState[A] = State[Map[String, Any], A]    val kvsToState: KVStoreA ~> KVStoreState = new (KVStoreA ~> KVStoreState) {      def apply[A](fa: KVStoreA[A]): KVStoreState[A] =        fa match {          case Put(key, value) => State { (s:Map[String, Any]) =>             (s.updated(key, value),()) }          case Get(key) => State { (s:Map[String, Any]) =>            (s,s.get(key).asInstanceOf[A]) }          case Del(key) => State { (s:Map[String, Any]) =>              (s - key, (())) }        }    }  }

我们把KVStoreA ADT模拟成对State结构的S转换（mutation），返回State{S=>(S,A)}。KVStoreState[A]类型的S参数为immutable.Map[String, Any]，所以我们在S转换操作时用immutable map的操作函数来构建新的map返回，典型的pure code。我们来运算一下KVStoreA程序：

  import ADTs._,DSLs._,IMPLs._  val prgRunner = prg.foldMap(kvsToState)    //> prgRunner  : demo.ws.catsFreeKVS.IMPLs.KVStoreState[Option[Int]] = cats.data.StateT@2cfb4a64  prgRunner.run(Map.empty).value       //> res0: (Map[String,Any], Option[Int]) = (Map(wild-cats -> 14),Some(14))

但是难道不需要Monad、RecursiveTailRecM实例了吗？实际上cats已经提供了State的Monad和RecursiveTailRecM实例：

  import cats.{Monad,RecursiveTailRecM}  implicitly[Monad[KVStoreState]]      //> res1: cats.Monad[demo.ws.catsFreeKVS.IMPLs.KVStoreState] = cats.data.StateT Instances$$anon$2@71bbf57e  implicitly[RecursiveTailRecM[KVStoreState]]     //> res2: cats.RecursiveTailRecM[demo.ws.catsFreeKVS.IMPLs.KVStoreState] = cats.RecursiveTailRecM$$anon$1@7f13d6e

在cats的StateT.scala里可以找到这段代码：

private[data] sealed trait StateTInstances2 {  implicit def catsDataMonadForStateT[F[_], S](implicit F0: Monad[F]): Monad[StateT[F, S, ?]] =    new StateTMonad[F, S] { implicit def F = F0 }  implicit def catsDataRecursiveTailRecMForStateT[F[_]: RecursiveTailRecM, S]: RecursiveTailRecM[StateT[F, S, ?]] = RecursiveTailRecM.create[StateT[F, S, ?]]  implicit def catsDataSemigroupKForStateT[F[_], S](implicit F0: Monad[F], G0: SemigroupK[F]): SemigroupK[StateT[F, S, ?]] =    new StateTSemigroupK[F, S] { implicit def F = F0; implicit def G = G0 }}

我把上面两个示范的源代码提供在下面：

Interact:

import cats.free._import cats.{Functor, RecursiveTailRecM}object catsFree {  object ADTs {    sealed trait Interact[+A]    object Interact {      case class Ask(prompt: String) extends Interact[String]      case class Tell(msg: String) extends Interact[Unit]      def ask(prompt: String): Free[Interact,String] = Free.liftF(Ask(prompt))      def tell(msg: String): Free[Interact,Unit] = Free.liftF(Tell(msg))      implicit object interactFunctor extends Functor[Interact]  {        def map[A,B](ia: Interact[A])(f: A => B): Interact[B] = ???        /*   ia match {             case Ask(p) => ???             case Tell(m) => ???          } */      }      sealed trait FunInteract[NS]      object FunInteract {        case class FunAsk[NS](prompt: String, onInput: String =>  NS) extends FunInteract[NS]        case class FunTell[NS](msg: String, ns: NS) extends FunInteract[NS]        def funAsk(prompt: String): Free[FunInteract,String] = Free.liftF(FunAsk(prompt,identity))        def funAskInt(prompt: String): Free[FunInteract,Int] = Free.liftF(FunAsk(prompt,_.toInt))        def funTell(msg: String): Free[FunInteract,Unit] = Free.liftF(FunTell(msg,()))        implicit object funInteract extends Functor[FunInteract] {          def map[A,NS](fa: FunInteract[A])(f: A => NS) = fa match {            case FunAsk(prompt,input) => FunAsk(prompt,input andThen f)            case FunTell(msg,ns) => FunTell(msg,f(ns))          }        }      }    }  }  object DSLs {    import ADTs._    import Interact._    val prg: Free[Interact,Unit] = for {      first <- ask("What's your first name?")      last <- ask("What's your last name?")      _ <- tell(s"Hello $first $last")    } yield()  }  object IMPLs {    import cats.{Id,~>}    import ADTs._    import Interact._    object iconsole 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 Prompt = String    type Reply = String    type Message = String    type Tester[A] = Map[Prompt,Reply] => (List[Message],A)    object tester extends (Interact ~> Tester) {      def apply[A](ia: Interact[A]): Tester[A] = ia match {        case Ask(p) => { m => (List(), m(p)) }        case Tell(m) => { _ => (List(m), ()) }      }    }    import cats.Monad    implicit val testerMonad = new Monad[Tester] with RecursiveTailRecM[Tester]{      override def pure[A](a: A): Tester[A] = _ => (List(),a)      override def flatMap[A,B](ta: Tester[A])(f: A => Tester[B]): Tester[B] = m => {        val (o1,a1) = ta(m)        val (o2,a2) = f(a1)(m)        (o1 ++ o2, a2)      }      override def tailRecM[A,B](a: A)(f: A => Tester[Either[A,B]]): Tester[B] =        defaultTailRecM(a)(f)    }    import cats.data.WriterT    import cats.instances.all._    type WF[A] = Map[Prompt,Reply] => A    type WriterTester[A] = WriterT[WF,List[Message],A]    def testerToWriter[A](f: Map[Prompt,Reply] => (List[Message],A)) =      WriterT[WF,List[Message],A](f)    implicit val testWriterMonad =  WriterT.catsDataMonadWriterForWriterT[WF,List[Message]]    object testWriter extends (Interact ~> WriterTester) {      import Interact._      def apply[A](ia: Interact[A]): WriterTester[A] = ia match {        case Ask(p) => testerToWriter(m => (List(),m(p)))        case Tell(m) => testerToWriter(_ => (List(m),()))      }    }  }  import ADTs._,DSLs._,IMPLs._   val testData = Map("What's your first name?" -> "Tiger",  "What's your last name?" -> "Chan")  //val prgRunner = prg.foldMap(tester)  //prgRunner(testData)  implicit val testWriterRecT = new RecursiveTailRecM[WriterTester]{}  val prgRunner = prg.foldMap(testWriter)  prgRunner.run(testData)._1.map(println)}

KVStore：

import cats.free._import cats.instances.all._object catsFreeKVS {  object ADTs {    sealed trait KVStoreA[+A]    case class Put[T](key: String, value: T) extends KVStoreA[Unit]    case class Get[T](key: String) extends KVStoreA[Option[T]]    case class Del(key: String) extends KVStoreA[Unit]    type KVStore[A] = Free[KVStoreA,A]    object KVStoreA {      def put[T](key: String, value: T): KVStore[Unit] =        Free.liftF[KVStoreA,Unit](Put[T](key,value))      def get[T](key: String): KVStore[Option[T]] =        Free.liftF[KVStoreA,Option[T]](Get[T](key))      def del(key: String): KVStore[Unit] =        Free.liftF[KVStoreA,Unit](Del(key))      def mod[T](key: String, f: T => T): KVStore[Unit] =        for {          opt <- get[T](key)          _ <- opt.map {t => put[T](key,f(t))}.getOrElse(Free.pure(()))        } yield()    }  }  object DSLs {    import ADTs._    import KVStoreA._    def prg: KVStore[Option[Int]] =    for {      _ <- put[Int]("wild-cats", 2)      _ <- mod[Int]("wild-cats", (_ + 12))      _ <- put[Int]("tame-cats", 5)      n <- get[Int]("wild-cats")      _ <- del("tame-cats")    } yield n  }  object IMPLs {    import ADTs._    import cats.{~>}    import cats.data.State       type KVStoreState[A] = State[Map[String, Any], A]    val kvsToState: KVStoreA ~> KVStoreState = new (KVStoreA ~> KVStoreState) {      def apply[A](fa: KVStoreA[A]): KVStoreState[A] =        fa match {          case Put(key, value) => State { (s:Map[String, Any]) =>             (s.updated(key, value),()) }          case Get(key) => State { (s:Map[String, Any]) =>            (s,s.get(key).asInstanceOf[A]) }          case Del(key) => State { (s:Map[String, Any]) =>              (s - key, (())) }        }    }  }  import ADTs._,DSLs._,IMPLs._  val prgRunner = prg.foldMap(kvsToState)  prgRunner.run(Map.empty).value    import cats.{Monad,RecursiveTailRecM}  implicitly[Monad[KVStoreState]]  implicitly[RecursiveTailRecM[KVStoreState]]}

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Cats（1）－ 从Free开始，Free cats

Cats（1）－从Free开始，Free cats