摘要：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，并且它是在里首先使用的。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。而推导器是所有类型推导器的基础。
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> m = new HashMap>();
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> 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(": " + 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 "
" $ 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)
