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Scala类型推导

Scala类型推导

之剑

2016.5.1 00:38:12

类型系统

什么是静态类型?为什么它们很有用?

根据Picrce的说法:“类型系统是一个可以根据代码段计算出来的值对它们进行分类,然后通过语法的手段来自动检测程序错误的系统。”

类型可以让你表示函数的域和值域。例如,在数学里,我们经常看到下面的函数:

 f: R -> N 

这个定义告诉我们函数”f”的作用是把实数集里的数映射到自然集里。

抽象地说,这才是具体意义上的类型。类型系统给了我们一些表示这些集合的更强大的方式。

有了这些类型标识,编译器现在可以 静态地(在编译期)判断这个程序是正确的。

换句话说,如果一个值(在运行期)不能够满足程序里的限制条件的话,那么在编译期就会出错。

通常来说,类型检测器(typechecker)只能够保证不正确的的程序不能通过编译。但是,它不能够保证所有正确的程序都能通过编译。

由于类型系统的表达能力不断增加,使得我们能够生成出更加可靠的代码,因为它使得我们能够控制程序上的不可变,即是是程序还没有运行的情况下(在类型上限制bug的出现)。学术界一直在努力突破类型系统的表达能力的极限,包含值相关的类型。

注意,所有的类型信息都会在编译期擦除。后面不再需要。这个被称为类型擦除。比如,Java里面的泛型的实现.

Scala中的类型

Scala强大的类型系统让我们可以使用更具有表现力的表达式。一些主要的特点如下:

  • 支持参数多态,泛型编程

  • 支持(局部)类型推导,这就是你为什么不需要写val i: Int = 12: Int

  • 支持存在向量(existential quantification),给一些没有名称的类型定义一些操作

  • 支持视图。 给定的值从一个类型到其他类型的“可转换性”

参数多态

多态可以用来编写泛型代码(用于处理不同类型的值),并且不会减少静态类型的表达能力。

例如,没有参数多态的话,一个泛型的列表数据结构通常会是下面这样的写法(在Java还没有泛型的时候,确实是这样的):

 scala> 2 :: 1 :: "bar" :: "foo" :: Nil  res5: List[Any] = List(2, 1, bar, foo)   

这样的话,我们就不能够恢复每个元素的类型信息。

 scala> res5.head  res6: Any = 2 

这样一来,我们的应用里会包含一系列的类型转换(“asInstanceOf[]“),代码会缺少类型安全(因为他们都是动态类型的)。

多态是通过指定类型变量来达到的。

 scala> def drop1[A](l: List[A]) = l.tail  drop1: [A](l: List[A])List[A]     scala> drop1(List(1,2,3))  res1: List[Int] = List(2, 3) 

多态是scala里的一等公民

简单来说,这意味着有一些你想在Scala里表达的类型概念会显得“太过于泛型”,从而导致编译器无法理解。所有的类型变量在运行期必须是确定的。

对于静态类型的一个比较常见的缺陷就是有太多的类型语法。Scala提供了类型推导来解决这个问题。

函数式语言里比较经典的类型推导的方法是 Hindlry-Milner,并且它是在ML里首先使用的。

Scala的类型推导有一点点不同,不过思想上是一致的:推导所有的约束条件,然后统一到一个类型上。

在Scala里,例如,你不能这样写:

 scala> { x => x }  :7: error: missing parameter type         { x => x }  

但是在OCaml里,你可以:

 # fun x -> x;;  - : 'a -> 'a =   

在Scala里,所有的类型推导都是局部的。Scala一次只考虑一个表达式。例如:

 scala> def id[T](x: T) = x  id: [T](x: T)T     scala> val x = id(322)  x: Int = 322     scala> val x = id("hey")  x: java.lang.String = hey     scala> val x = id(Array(1,2,3,4))  x: Array[Int] = Array(1, 2, 3, 4)   

在这里,类型都被隐藏了。Scala编译器自动推导参数的类型。注意我们也没有必要显示指定返回值的类型了。

型变

Scala的类型系统需要把类的继承关系和多态结合起来。类的继承使得类之间存在父子的关系。当把面向对象和多态结合在一起时,一个核心的问题就出来了:如果T’是T的子类,那么Container[T’]是不是Container[T]的子类呢?Variance注释允许你在类继承和多态类型之间表达下面的这些关系:

含义 Scala中的标记
covariant(协变) C[T’]是C[T]的子类 [+T]
contravariant(逆变) C[T]是C[T’]子类 [-T]
invariant(不变) C[T]和C[T’]不相关 [T]

子类关系的真正意思是:对于一个给定的类型T,如果T’是它的子类,那么T’可以代替T吗?

 scala> class Contravariant[-A]  defined class Contravariant     scala> val cv: Contravariant[String] = new Contravariant[AnyRef]  cv: Contravariant[AnyRef] = Contravariant@49fa7ba     scala> val fail: Contravariant[AnyRef] = new Contravariant[String]  :6: error: type mismatch;   found   : Contravariant[String]   required: Contravariant[AnyRef]         val fail: Contravariant[AnyRef] = new Contravariant[String]        

量化(Quantification)

有时候你不需要给一个类型变量以名称,例如

 scala> def count[A](l: List[A]) = l.size  count: [A](List[A])Int  

你可以用“通配符”来替代:

 scala> def count(l: List[_]) = l.size  count: (List[_])Int 

什么是类型推导

先看个代码:

Map<Integer, Map<String, String>> m = new HashMap<Integer, Map<String, String>>();  

是啊, 这简直太长了,我们不禁感叹,这编译器也太愚蠢了.几乎一半字符都是重复的!

针对泛型定义和实例太过繁琐的问题,在java 7 中引入了钻石运算符. 神奇的Coin项目,满足了你的心愿.

于是,你在java 7之后可以这样写了:

 Map<Integer, Map<String, String>> m = new HashMap();  

钻石运算符通常用于简化创建带有泛型对象的代码,可以避免运行时 的异常,并且它不再要求程序员在编码时显示书写冗余的类型参数。实际上,编译器在进行词法解析时会自动推导类型,自动为代码进行补全,并且编译的字节码与 以前无异。

当时在提案中,这个问题叫"Improved Type Inference for Generic Instance Creation",缩写ITIGIX听起来怪怪的,但是为啥叫钻石算法? 世界上, 哪有那么多为什么.

Scala正是因为做了类型推导, 让Coders感觉仿佛在写动态语言的代码.

在Scala中,高阶函数经常传递匿名函数.举个栗子:

一段定义泛型函数的代码

 def dropWhile[A](list: List[A], f: A => Boolean): List[A] 

当我们传入一个匿名函数f来调用它,

 val mylist: List[Int] = List(1,2,3,4,5)   val listDropped = dropWhile( mylist, (x: Int) => x < 4 ) 

listDropped的值是List(4,5)

我们用大脑可以轻易判断, 当list: List[A] 中的类型A在mylist声明的时候已经指定了Int, 那么很明显, 在第二个参数中,我们的x也必是Int.

很幸运Scala设计者们早已考虑到这一点,Scala编译器可以推导这种情况.但是你得按照Scala的规范限制来写你的dropWhile函数的签名(柯里化的): dropWhile( mylist )( f )

 def dropWhile[A] ( list: List[A] ) ( f: A => Boolean ) : List[A] = list match {  case Cons(h,t) if f(h) => dropWhile(t)(f)  case _ => list  } 

如此而来,我们就可以直接像下面这样使用这个函数了:

 val mylist: List[Int] = List(1,2,3,4,5)  val droppedList = dropWhile( mylist ) ( x => x < 4 ) 

注意, x参数没有指定Int类型, 因为编译器直接通过mylist的泛型信息Int推导出x的类型也是Int.

类型推导是一门博大的学问,背后有繁冗的理论, 这在编译器设计开发的时候需要解决的问题.

Scala Haskell,ML
局部的(local)、基于流的(flow-based)类型推断 全局化的Hindley-Milner类型推断

在《Programming in Scala》一书中提到基于流的类型推断有它的局限性,但是对于面向对象的分支类型处理比Hindley-Mlner更加优雅。

基于流的类型推导在偏应用函数场景下,不能对参数类型省略

类型推导算法

类型推导(Type Inference)是现代高级语言中一个越来越常见的特性。其实,这个特性在函数式语言

中早有了广泛应用。而HindleyMilner推导器是所有类型推导器的基础。

Scala实现的一个简单的HindleyMilner推导器:

     /*       * http://dysphoria.net/code/hindley-milner/HindleyMilner.scala       * Andrew Forrest       *       * Implementation of basic polymorphic type-checking for a simple language.       * Based heavily on Nikita Borisov’s Perl implementation at       * http://web.archive.org/web/20050420002559/www.cs.berkeley.edu/~nikitab/courses/cs263/hm.html       * which in turn is based on the paper by Luca Cardelli at       * http://lucacardelli.name/Papers/BasicTypechecking.pdf       *       * If you run it with "scala HindleyMilner.scala" it will attempt to report the types       * for a few example expressions. (It uses UTF-8 for output, so you may need to set your       * terminal accordingly.)       *       * Changes       * June 30, 2011 by Liang Kun(liangkun(AT)baidu.com)       * 1. Modify to enhance readability       * 2. Extend to Support if expression in syntax       *       *       *       * Do with it what you will. :)       */        /** Syntax definition. This is a simple lambda calculous syntax.       * Expression ::= Identifier       * | Constant       * | "if" Expression "then" Expression "else" Expression       * | "lambda(" Identifier ") " Expression       * | Expression "(" Expression ")"       * | "let" Identifier "=" Expression "in" Expression       * | "letrec" Identifier "=" Expression "in" Expression       * | "(" Expression ")"       * See the examples below in main function.       */      sealed abstract class Expression        case class Identifier(name: String) extends Expression {          override def toString = name      }        case class Constant(value: String) extends Expression {          override def toString = value      }        case class If(condition: Expression, then: Expression, other: Expression) extends Expression {          override def toString = "(if " + condition + " then " + then + " else " + other + ")"      }        case class Lambda(argument: Identifier, body: Expression) extends Expression {          override def toString = "(lambda " + argument + " → " + body + ")"      }        case class Apply(function: Expression, argument: Expression) extends Expression {          override def toString = "(" + function + " " + argument + ")"      }        case class Let(binding: Identifier, definition: Expression, body: Expression) extends Expression {          override def toString = "(let " + binding + " = " + definition + " in " + body + ")"      }        case class Letrec(binding: Identifier, definition: Expression, body: Expression) extends Expression {          override def toString = "(letrec " + binding + " = " + definition + " in " + body + ")"      }          /** Exceptions may happened */      class TypeError(msg: String) extends Exception(msg)      class ParseError(msg: String) extends Exception(msg)          /** Type inference system */      object TypeSystem {          type Env = Map[Identifier, Type]          val EmptyEnv: Map[Identifier, Type] = Map.empty            // type variable and type operator          sealed abstract class Type          case class Variable(id: Int) extends Type {              var instance: Option[Type] = None              lazy val name = nextUniqueName()                override def toString = instance match {                  case Some(t) => t.toString                  case None => name              }          }            case class Operator(name: String, args: Seq[Type]) extends Type {              override def toString = {                  if (args.length == 0)                      name                  else if (args.length == 2)                      "[" + args(0) + " " + name + " " + args(1) + "]"                  else                      args.mkString(name + "[", ", ", "]")              }          }            // builtin types, types can be extended by environment          def Function(from: Type, to: Type) = Operator("→", Array(from, to))          val Integer = Operator("Integer", Array[Type]())          val Boolean = Operator("Boolean", Array[Type]())              protected var _nextVariableName = 'α';          protected def nextUniqueName() = {              val result = _nextVariableName              _nextVariableName = (_nextVariableName.toInt + 1).toChar              result.toString          }          protected var _nextVariableId = 0          def newVariable(): Variable = {              val result = _nextVariableId              _nextVariableId += 1              Variable(result)          }              // main entry point          def analyze(expr: Expression, env: Env): Type = analyze(expr, env, Set.empty)          def analyze(expr: Expression, env: Env, nongeneric: Set[Variable]): Type = expr match {              case i: Identifier => getIdentifierType(i, env, nongeneric)                case Constant(value) => getConstantType(value)                case If(cond, then, other) => {                  val condType = analyze(cond, env, nongeneric)                  val thenType = analyze(then, env, nongeneric)                  val otherType = analyze(other, env, nongeneric)                  unify(condType, Boolean)                  unify(thenType, otherType)                  thenType              }                case Apply(func, arg) => {                  val funcType = analyze(func, env, nongeneric)                  val argType = analyze(arg, env, nongeneric)                  val resultType = newVariable()                  unify(Function(argType, resultType), funcType)                  resultType              }                case Lambda(arg, body) => {                  val argType = newVariable()                  val resultType = analyze(body,                                           env + (arg -> argType),                                           nongeneric + argType)                  Function(argType, resultType)              }                case Let(binding, definition, body) => {                  val definitionType = analyze(definition, env, nongeneric)                  val newEnv = env + (binding -> definitionType)                  analyze(body, newEnv, nongeneric)              }                case Letrec(binding, definition, body) => {                  val newType = newVariable()                  val newEnv = env + (binding -> newType)                  val definitionType = analyze(definition, newEnv, nongeneric + newType)                  unify(newType, definitionType)                  analyze(body, newEnv, nongeneric)              }          }            protected def getIdentifierType(id: Identifier, env: Env, nongeneric: Set[Variable]): Type = {              if (env.contains(id))                  fresh(env(id), nongeneric)              else                  throw new ParseError("Undefined symbol: " + id)          }            protected def getConstantType(value: String): Type = {              if(isIntegerLiteral(value))                  Integer              else                  throw new ParseError("Undefined symbol: " + value)          }            protected def fresh(t: Type, nongeneric: Set[Variable]) = {              import scala.collection.mutable              val mappings = new mutable.HashMap[Variable, Variable]              def freshrec(tp: Type): Type = {                  prune(tp) match {                      case v: Variable =>                          if (isgeneric(v, nongeneric))                              mappings.getOrElseUpdate(v, newVariable())                          else                              v                        case Operator(name, args) =>                          Operator(name, args.map(freshrec(_)))                  }              }                freshrec(t)          }            protected def unify(t1: Type, t2: Type) {              val type1 = prune(t1)              val type2 = prune(t2)              (type1, type2) match {                  case (a: Variable, b) => if (a != b) {                      if (occursintype(a, b))                          throw new TypeError("Recursive unification")                      a.instance = Some(b)                  }                  case (a: Operator, b: Variable) => unify(b, a)                  case (a: Operator, b: Operator) => {                      if (a.name != b.name ||                          a.args.length != b.args.length) throw new TypeError("Type mismatch: " + a + " ≠ " + b)                                            for(i <- 0 until a.args.length)                          unify(a.args(i), b.args(i))                  }              }          }            // Returns the currently defining instance of t.          // As a side effect, collapses the list of type instances.          protected def prune(t: Type): Type = t match {              case v: Variable if v.instance.isDefined => {                  val inst = prune(v.instance.get)                  v.instance = Some(inst)                  inst              }              case _ => t          }            // Note: must be called with v 'pre-pruned'          protected def isgeneric(v: Variable, nongeneric: Set[Variable]) = !(occursin(v, nongeneric))            // Note: must be called with v 'pre-pruned'          protected def occursintype(v: Variable, type2: Type): Boolean = {              prune(type2) match {                  case `v` => true                  case Operator(name, args) => occursin(v, args)                  case _ => false              }          }            protected def occursin(t: Variable, list: Iterable[Type]) =              list exists (t2 => occursintype(t, t2))            protected val checkDigits = "^(//d+)$".r          protected def isIntegerLiteral(name: String) = checkDigits.findFirstIn(name).isDefined      }          /** Demo program */      object HindleyMilner {          def main(args: Array[String]){              Console.setOut(new java.io.PrintStream(Console.out, true, "utf-8"))                // extends the system with a new type[pair] and some builtin functions              val left = TypeSystem.newVariable()              val right = TypeSystem.newVariable()              val pairType = TypeSystem.Operator("×", Array(left, right))                val myenv: TypeSystem.Env = TypeSystem.EmptyEnv ++ Array(                  Identifier("pair") -> TypeSystem.Function(left, TypeSystem.Function(right, pairType)),                  Identifier("true") -> TypeSystem.Boolean,                  Identifier("false")-> TypeSystem.Boolean,                  Identifier("zero") -> TypeSystem.Function(TypeSystem.Integer, TypeSystem.Boolean),                  Identifier("pred") -> TypeSystem.Function(TypeSystem.Integer, TypeSystem.Integer),                  Identifier("times")-> TypeSystem.Function(TypeSystem.Integer,                          TypeSystem.Function(TypeSystem.Integer, TypeSystem.Integer))              )                // example expressions              val pair = Apply(                  Apply(                      Identifier("pair"), Apply(Identifier("f"), Constant("4"))                  ),                  Apply(Identifier("f"), Identifier("true"))              )              val examples = Array[Expression](                  // factorial                  Letrec(Identifier("factorial"), // letrec factorial =                      Lambda(Identifier("n"), // lambda n =>                          If(                              Apply(Identifier("zero"), Identifier("n")),                                Constant("1"),                                Apply(                                  Apply(Identifier("times"), Identifier("n")),                                  Apply(                                      Identifier("factorial"),                                      Apply(Identifier("pred"), Identifier("n"))                                  )                              )                          )                      ), // in                      Apply(Identifier("factorial"), Constant("5"))                  ),                    // Should fail:                  // fn x => (pair(x(3) (x(true))))                  Lambda(Identifier("x"),                      Apply(                          Apply(Identifier("pair"),                              Apply(Identifier("x"), Constant("3"))                          ),                          Apply(Identifier("x"), Identifier("true"))                      )                  ),                    // pair(f(3), f(true))                  Apply(                      Apply(Identifier("pair"), Apply(Identifier("f"), Constant("4"))),                      Apply(Identifier("f"), Identifier("true"))                  ),                      // letrec f = (fn x => x) in ((pair (f 4)) (f true))                  Let(Identifier("f"), Lambda(Identifier("x"), Identifier("x")), pair),                    // Should fail:                  // fn f => f f                  Lambda(Identifier("f"), Apply(Identifier("f"), Identifier("f"))),                    // let g = fn f => 5 in g g                  Let(                      Identifier("g"),                      Lambda(Identifier("f"), Constant("5")),                      Apply(Identifier("g"), Identifier("g"))                  ),                    // example that demonstrates generic and non-generic variables:                  // fn g => let f = fn x => g in pair (f 3, f true)                  Lambda(Identifier("g"),                      Let(Identifier("f"),                          Lambda(Identifier("x"), Identifier("g")),                          Apply(                              Apply(Identifier("pair"),                                    Apply(Identifier("f"), Constant("3"))                              ),                              Apply(Identifier("f"), Identifier("true"))                          )                      )                  ),                    // Function composition                  // fn f (fn g (fn arg (f g arg)))                  Lambda( Identifier("f"),                      Lambda( Identifier("g"),                          Lambda( Identifier("arg"),                              Apply(Identifier("g"), Apply(Identifier("f"), Identifier("arg")))                          )                      )                  )              )                for(eg <- examples){                  tryexp(myenv, eg)              }          }            def tryexp(env: TypeSystem.Env, expr: Expression) {              try {                  val t = TypeSystem.analyze(expr, env)                  print(t)                }catch{                  case t: ParseError => print(t.getMessage)                  case t: TypeError => print(t.getMessage)              }              println(":/t" + expr)          }      }        HindleyMilner.main(argv) 

Haskell写的一个 合一算法的简单实现:

https://github.com/yihuang/haskell-snippets/blob/master/Unif.hs

 module Main where        import Data.List (intersperse)      import Control.Monad        -- utils --        mapFst :: (a -> b) -> (a, c) -> (b,   c)      mapFst    f           (a, c) =  (f a, c)        -- types --        type Name = String        data Term = Var Name               | App Name [Term]        -- 表示一个替换关系      type Sub = (Term, Name)        -- implementation --        -- 检查变量 Name 是否出现在 Term 中      occurs :: Name -> Term -> Bool      occurs x t = case t of       (Var y)    -> x==y       (App _ ts) -> and . map (occurs x) $ ts        -- 使用 Sub 对 Term 进行替换      sub :: Sub -> Term -> Term      sub (t1, y) t@(Var a)       | a==y      = t1       | otherwise = t      sub s (App f ts) = App f $ map (sub s) ts        -- 使用 Sub 列表对 Term 进行替换      subs :: [Sub] -> Term -> Term      subs ss t = foldl (flip sub) t ss        -- 把两个替换列表组合起来,同时用新加入的替换对其中所有 Term 进行替换      compose :: [Sub] -> [Sub] -> [Sub]      compose []     s1 = s1      compose (s:ss) s1 = compose ss $ s : iter s s1       where         iter :: Sub -> [Sub] -> [Sub]         iter s ss = map (mapFst (sub s)) ss        -- 合一函数      unify :: Term -> Term -> Maybe [Sub]      unify t1 t2 = case (t1, t2) of       (Var x,   Var y)   -> if x==y        then Just [] else Just [(t1, y)]       (Var x,   App _ _) -> if occurs x t2 then Nothing else Just [(t2, x)]       (App _ _, Var x)   -> if occurs x t1 then Nothing else Just [(t1, x)]       (App n1 ts1, App n2 ts2)                          -> if n1/=n2      then Nothing else unify_args ts1 ts2       where         unify_args [] [] = Just []         unify_args _  [] = Nothing         unify_args [] _  = Nothing         unify_args (t1:ts1) (t2:ts2) = do           u <- unify t1 t2           let update = map (subs u)           u1 <- unify_args (update ts1) (update ts2)           return (u1 `compose` u)        -- display --        instance Show Term where         show (Var s) = s         show (App name ts) = name++"("++(concat . intersperse "," $ (map show ts))++")"        showSub (t, s) = s ++ " -> " ++ show t        -- test cases --        a = Var "a"      b = Var "b"      c = Var "c"      d = Var "d"      x = Var "x"      y = Var "y"      z = Var "z"      f = App "f"      g = App "g"      j = App "j"        test t1 t2 = do         putStrLn $ show t1 ++ "  <==>  " ++ show t2         case unify t1 t2 of           Nothing -> putStrLn "unify fail"           Just u  -> putStrLn $ concat . intersperse "/n" $ map showSub u        testcases = [(j [x,y,z],                   j [f [y,y], f [z,z], f [a,a]])                 ,(x,                   f [x])                 ,(f [x],                   y)                 ,(f [a, f [b, c], g [b, a, c]],                   f [a, a, x])                 ,(f [d, d, x],                   f [a, f [b, c], f [b, a, c]])                 ]        main = forM testcases (uncurry test)     

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