Scalaz（7）－ typeclass：Applicative－idomatic function application

Applicative，正如它的名称所示，就是FP模式的函数施用（function application）。我们在前面的讨论中不断提到FP模式的操作一般都在管道里进行的，因为FP的变量表达形式是这样的：F[A]，即变量A是包嵌在F结构里的。Scalaz的Applicative typeclass提供了各种类型的函数施用(function application)和升格（lifting）方法。与其它scalaz typeclass使用方式一样，我们只需要实现了针对自定义类型的Applicative实例就可以使用这些方法了。以下是Applicative trait的部分定义：scalaz/Applicative.scala

 trait Applicative[F[_]] extends Apply[F] { self =>
   ////
   def point[A](a: => A): F[A]

   // alias for point
   final def pure[A](a: => A): F[A] = point(a)
 。。。

我们首先需要实现抽象函数point，然后由于Applicative继承了Apply，我们看看Apply trait有什么抽象函数需要实现的；scalaz/Apply.scala

 trait Apply[F[_]] extends Functor[F] { self =>
   ////
   def ap[A,B](fa: => F[A])(f: => F[A => B]): F[B]
 。。。

我们还需要实现抽象函数ap。注意Apply又继承了Functor，所以我们还需要实现map，一旦实现了Applicative实例就能同时获取了Functor实例。

现在我们先设计一个自定义类型作为下面的范例：

 trait Configure[+A] {
     def get: A
 }
 object Configure {
     def apply[A](data: => A) = new Configure[A] { def get = data }
 }
 Configure("env string")                           //> res0: Exercises.ex4.Configure[String] = Exercises.ex4$Configure$$anon$1@6bf2
                                                   //| d08e

Configure[+A]是个典型的FP类型。通过实现特殊命名apply的函数作为类型构建器，我们可以这样构建实例：Configure("some string")。现在我们按照scalaz隐式解析（implicit resolution）惯例在伴生对象（companion object）里定义隐式Applicative实例：

 import scalaz._
 import Scalaz._
 object ex4 {
 trait Configure[+A] {
     def get: A
 }
 object Configure {
     def apply[A](data: => A) = new Configure[A] { def get = data }
     implicit val configFunctor = new Functor[Configure] {
         def map[A,B](ca: Configure[A])(f: A => B): Configure[B] = Configure(f(ca.get))
     }
     implicit val configApplicative = new Applicative[Configure] {
         def point[A](a: => A) = Configure(a)
         def ap[A,B](ca: => Configure[A])(cfab: => Configure[A => B]): Configure[B] = cfab map {fab => fab(ca.get)}
     }
 }

由于Apply继承了Functor，我们必须先获取Configure的Functor实例。现在我们可以针对Configure类型使用Applicative typeclass的功能函数了。Applicative typeclass的组件函数可以分为几种主要类型：

1、Applicative实例构建函数，point：

 "abc".point[Configure]                            //> res1: Exercises.ex4.Configure[String] = Exercises.ex4$Configure$$anon$3@3c41
                                                   //| 9631
 .point[Configure]                               //> res2: Exercises.ex4.Configure[Int] = Exercises.ex4$Configure$$anon$3@6302168
                                                   //| 9
 .point[Option]                                   //> res3: Option[Int] = Some(5)

看款式应该是通过隐式转换实现的：scalaz/syntax/ApplicativeSyntax.scala

 trait ToApplicativeOps extends ToApplicativeOps0 with ToApplyOps {
   implicit def ToApplicativeOps[F[_],A](v: F[A])(implicit F0: Applicative[F]) =
     new ApplicativeOps[F,A](v)

   ////
   implicit def ApplicativeIdV[A](v: => A) = new ApplicativeIdV[A] {
     lazy val self = v
   }

   trait ApplicativeIdV[A] extends Ops[A] {
     def point[F[_] : Applicative]: F[A] = Applicative[F].point(self)
     def pure[F[_] : Applicative]: F[A] = Applicative[F].point(self)
     def η[F[_] : Applicative]: F[A] = Applicative[F].point(self)
   }  ////
 }

是通过implicit def ApplicativeIDV[A](v: => A)实现的。

2、对F[T}类型进行F[A =>B]式的函数施用（从管道里提供作用函数）。施用函数款式是这样的：

   def ap[A,B](fa: => F[A])(f: => F[A => B]): F[B]

对比Functor函数map：map[A,B](fa: F[A])(f: A => B]): F[B], 分别只在提供操作函数A=>B的方式：ap在F结构内部提供，又或者换句话说ap提供的是高阶函数F[A=>B]。从函数款式看来，ap要比map功能更加强大。因为我们可以用ap实现map, 反之不可：

         def map[A,B](fa: Configure[A])(f: A => B) = ap(fa)(point(f))

通过ap2,ap3,ap4 ...款式的函数我们可以把 F[A],F[B],F[C],F[D]...多个值连接起来：scalaz/Apply.scala

  def ap2[A,B,C](fa: => F[A], fb: => F[B])(f: F[(A,B) => C]): F[C] =
     ap(fb)(ap(fa)(map(f)(_.curried)))
   def ap3[A,B,C,D](fa: => F[A], fb: => F[B], fc: => F[C])(f: F[(A,B,C) => D]): F[D] =
     ap(fc)(ap2(fa,fb)(map(f)(f => ((a:A,b:B) => (c:C) => f(a,b,c)))))
   def ap4[A,B,C,D,E](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D])(f: F[(A,B,C,D) => E]): F[E] =
     ap2(fc, fd)(ap2(fa,fb)(map(f)(f => ((a:A,b:B) => (c:C, d:D) => f(a,b,c,d)))))
   def ap5[A,B,C,D,E,R](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D], fe: => F[E])(f: F[(A,B,C,D,E) => R]): F[R] =
     ap2(fd, fe)(ap3(fa,fb,fc)(map(f)(f => ((a:A,b:B,c:C) => (d:D, e:E) => f(a,b,c,d,e)))))
 ...

试着在Configure类型上使用ap:

 Apply[Configure].ap2(Configure(),Configure())(((_: Int) + (_: Int)).point[Configure])
                                                   //> res4: Exercises.ex4.Configure[Int] = Exercises.ex4$Configure$$anon$3@64cd705
                                                   //| f或者用注入方法（injected method）<*>：scalaz/Syntax/ApplySyntax.scala

或者用注入方法（injected method）<*>：scalaz/Syntax/ApplySyntax.scala

 (Configure() <*> {Configure() <*> {Configure() <*> {(((_:Int)+(_:Int)+(_:Int)).curried).point[Configure]}}}).get
                                                   //> res5: Int = 6

以上的Apply[Configure]是通过Apply typeclass的构建函数apply实现的：scalaz/Apply.scala

 object Apply {
   @inline def apply[F[_]](implicit F: Apply[F]): Apply[F] = F

3、简化一下ap的写法，只用提供f:(A,B) => C这样的基本操作函数：scalaz/Apply.scala

  def apply2[A, B, C](fa: => F[A], fb: => F[B])(f: (A, B) => C): F[C] =
     ap(fb)(map(fa)(f.curried))
   def apply3[A, B, C, D](fa: => F[A], fb: => F[B], fc: => F[C])(f: (A, B, C) => D): F[D] =
     apply2(apply2(fa, fb)((_, _)), fc)((ab, c) => f(ab._1, ab._2, c))
   def apply4[A, B, C, D, E](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D])(f: (A, B, C, D) => E): F[E] =
     apply2(apply2(fa, fb)((_, _)), apply2(fc, fd)((_, _)))((t, d) => f(t._1, t._2, d._1, d._2))
   def apply5[A, B, C, D, E, R](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D], fe: => F[E])(f: (A, B, C, D, E) => R): F[R] =
     apply2(apply3(fa, fb, fc)((_, _, _)), apply2(fd, fe)((_, _)))((t, t2) => f(t._1, t._2, t._3, t2._1, t2._2))
 ...

用在Configure类型上：

 (Apply[Configure].apply2(Configure(),Configure())(((_: Int) + (_: Int)))).get
                                                   //> res6: Int = 3
 (^(Configure(),Configure())((_:Int)+(_:Int))).get
                                                   //> res7: Int = 3
 (^^(Configure(),Configure(),Configure())((_:Int)+(_:Int)+(_:Int))).get
                                                   //> res8: Int = 6

这个^,^^是apply2,apply3的注入方法：scalaz/syntax/ApplySyntax.scala

   def ^[A,B,C](fa: => F[A], fb: => F[B])(
                f: (A, B) => C): F[C] =
     F.apply2(fa, fb)(f)

   def ^^[A,B,C,D](fa: => F[A], fb: => F[B], fc: => F[C])(
                  f: (A, B, C) => D): F[D] =
     F.apply3(fa, fb, fc)(f)

   def ^^^[A,B,C,D,E](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D])(
                    f: (A,B,C,D) => E): F[E] =
     F.apply4(fa, fb, fc, fd)(f)

   def ^^^^[A,B,C,D,E,I](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D], fe: => F[E])(
                      f: (A,B,C,D,E) => I): F[I] =
     F.apply5(fa, fb, fc, fd, fe)(f)
 ...

另一种表达方式是通过ApplicativeBuilder typeclass实现的注入方法|@|：

 ((Configure() |@| Configure() |@| Configure())((_:Int)+(_:Int)+(_:Int))).get
                                                   //> res9: Int = 6

效果是一样的。我们用一个实际的简单例子来示范一下Applicative的具体函数施用：

 def configName(name: String): Configure[String] = Configure(name)
                                                   //> configName: (name: String)Exercises.ex4.Configure[String]
 def configID(userid: String): Configure[String] = Configure(userid)
                                                   //> configID: (userid: String)Exercises.ex4.Configure[String]
 def configPwd(pwd: String): Configure[String] = Configure(pwd)
                                                   //> configPwd: (pwd: String)Exercises.ex4.Configure[String]
 case class WebLogForm(name:String, id: String, pwd: String)

 def logOnWeb(name: String, userid: String, pwd: String) =
   ^^(configName(name),configID(userid), configPwd(pwd))(WebLogForm(_,_,_))
                                                   //> logOnWeb: (name: String, userid: String, pwd: String)Exercises.ex4.Configur
                                                   //| e[Exercises.ex4.WebLogForm]
 def logOnWeb1(name: String, userid: String, pwd: String) =
   (configName(name) |@| configID(userid) |@| configPwd(pwd))(WebLogForm(_,_,_))
                                                   //> logOnWeb1: (name: String, userid: String, pwd: String)Exercises.ex4.Configu
                                                   //| re[Exercises.ex4.WebLogForm]

值得注意的是：用Applicative施用configName,configID,configPwd时，这三个函数之间没有依赖关系。特别适合并行运算或fail-fast，因为无论如何这三个函数都一定会运行。这种Applicative的函数施用体现了它在并行运算中的优势。

4、Applicative style 函数施用。上面提到的|@|操作并不是一种操作函数而是一种层级式持续函数施用模式。具体实现在ApplicativeBuilder typeclass里：scalaz/ApplicativeBuilder.scala

 private[scalaz] trait ApplicativeBuilder[M[_], A, B] {
   val a: M[A]
   val b: M[B]

   def apply[C](f: (A, B) => C)(implicit ap: Apply[M]): M[C] = ap.apply2(a, b)(f)

   def tupled(implicit ap: Apply[M]): M[(A, B)] = apply(Tuple2.apply)

   def ⊛[C](cc: M[C]) = new ApplicativeBuilder3[C] {
     val c = cc
   }

   def |@|[C](cc: M[C]) = ⊛(cc)

   sealed trait ApplicativeBuilder3[C] {
     val c: M[C]

     def apply[D](f: (A, B, C) => D)(implicit ap: Apply[M]): M[D] = ap.apply3(a, b, c)(f)

     def tupled(implicit ap: Apply[M]): M[(A, B, C)] = apply(Tuple3.apply)

     def ⊛[D](dd: M[D]) = new ApplicativeBuilder4[D] {
       val d = dd
     }

     def |@|[D](dd: M[D]) = ⊛(dd)

     sealed trait ApplicativeBuilder4[D] {
       val d: M[D]

       def apply[E](f: (A, B, C, D) => E)(implicit ap: Apply[M]): M[E] = ap.apply4(a, b, c, d)(f)

       def tupled(implicit ap: Apply[M]): M[(A, B, C, D)] = apply(Tuple4.apply)

       def ⊛[E](ee: M[E]) = new ApplicativeBuilder5[E] {
         val e = ee
       }

       def |@|[E](ee: M[E]) = ⊛(ee)
 ...

可以看得出(F[A] |@| F[B] |@| F[C])((A,B,C) => D)这个表达式中的两个|@|符号分别代表ApplicativeBuilder2(F[B])及ApplicativeBuilder3(F[C])。

这是另一种通过函数施用实现连接Applicative类型值的方式。

5、产生tuple：(F[A],F[B])合并成F[(A,B)]：scalaz/Apply.scala

  def tuple2[A,B](fa: => F[A], fb: => F[B]): F[(A,B)] =
     apply2(fa, fb)((_,_))
   def tuple3[A,B,C](fa: => F[A], fb: => F[B], fc: => F[C]): F[(A,B,C)] =
     apply3(fa, fb, fc)((_,_,_))
   def tuple4[A,B,C,D](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D]): F[(A,B,C,D)] =
     apply4(fa, fb, fc, fd)((_,_,_,_))
   def tuple5[A,B,C,D,E](fa: => F[A], fb: => F[B], fc: => F[C], fd: => F[D], fe: => F[E]): F[(A,B,C,D,E)] =
     apply5(fa, fb, fc, fd, fe)((_,_,_,_,_))
 ...

比如：

 Apply[Configure].tuple2(Configure("abc"),Configure())
                                                   //> res10: Exercises.ex4.Configure[(String, Int)] = Exercises.ex4$Configure$$an
                                                   //| on$3@5ffead27
 Apply[Configure].tuple3(Configure("abc"),Configure(),Configure(true))
                                                   //> res11: Exercises.ex4.Configure[(String, Int, Boolean)] = Ex

具体用来干什么，我现在还说不上来。

6、把一个普通函数升格（lift）成高阶函数，如：(A,B) => C 升格成 (F[A],F[B]) => F[C]: scalaz/Apply.scala

  def lift2[A, B, C](f: (A, B) => C): (F[A], F[B]) => F[C] =
     apply2(_, _)(f)
   def lift3[A, B, C, D](f: (A, B, C) => D): (F[A], F[B], F[C]) => F[D] =
     apply3(_, _, _)(f)
   def lift4[A, B, C, D, E](f: (A, B, C, D) => E): (F[A], F[B], F[C], F[D]) => F[E] =
     apply4(_, _, _, _)(f)
   def lift5[A, B, C, D, E, R](f: (A, B, C, D, E) => R): (F[A], F[B], F[C], F[D], F[E]) => F[R] =
     apply5(_, _, _, _, _)(f)
 ...

这种函数升格方式在用FP方式使用OOP库函数时更加方便。最典型的例子是Option类型在FP中结合OOP函数库的使用。如果我们希望在使用OOP库函数时使用Option类型的输入参数和返回值，那我们就可以通过函数升格（function lifting）来实现这样的功能。

 val of2 = Apply[Option].lift2((_: Int) + (_: Int))//> of2  : (Option[Int], Option[Int]) => Option[Int] = <function2>
 of2(Some(),Some())                              //> res12: Option[Int] = Some(3)
 val of3 = Apply[List].lift3((s1: String, s2: String, s3: String) => s1 + " "+s2+" "+s3)
                                                   //> of3  : (List[String], List[String], List[String]) => List[String] = <functi
                                                   //| on3>
 of3(List("How"),List("are"),List("you?"))         //> res13: List[String] = List(How are you?)

我们分别用lift2,lift3把普通函数升格成Option和List高阶函数。

再来个更实际一点的例子：在java.sql.DriverManager库里有个getConnection函数。它的函数款式是：getConnection(p1:String,p2:String,p3:String): java.sql.Connection

虽然我没有它的源代码，但我还是想使用我自定义的类型Configure作为参数，我可以这样：

 import java.sql.DriverManager

 val sqlConnect = Apply[Configure] lift3 java.sql.DriverManager.getConnection
                                                   //> sqlConnect  : (Exercises.ex4.Configure[String], Exercises.ex4.Configure[Str
                                                   //| ing], Exercises.ex4.Configure[String]) => Exercises.ex4.Configure[java.sql.
                                                   //| Connection] = <function3>
 sqlConnect(Configure("Source"),Configure("User"),Configure("Password"))
                                                   //> res12: Exercises.ex4.Configure[java.sql.Connection] = Exercises.ex4$Configu
                                                   //| re$$anon$3@79924b

的确这样可以使我继续在FP模式中工作。

总结来说：Applicative typeclass提供了一套函数施用方式。它是通过一个包嵌在容器结构的高阶函数实现管道内的施用。Applicative typeclass还提供了方法将普通函数升格到高阶函数使FP和OOP混合模式的函数施用更安全方便。