Java中的Hindley-Milner算法

Type = {TypeVariable,ConcreteType}
TypeRelation = {LeftType,RightType}
Substitution = {OldType,NewType}
TypeRelations = set of TypeRelation
Substitutions = set of Substitution

1) Initialize TypeRelations to the constraints,Initialize Substitutions to empty
2) Take a TypeRelation
3) If LeftType and RightType are both TypeVariables or are concrete 
      types with LeftType <: RightType Then do nothing
4) If only LeftType is a TypeVariable Then
    replace all occurrences of RightType in TypeRelations and Substitutions
    put LeftType,RightType into Substitutions
5) If only RightType is a TypeVariable then
    replace all occurrences of LeftType in TypeRelations and Substitutions
    put RightType,LeftType into Substitutions
6) Else fail

我阅读了至少20篇文章,发现了一个(Francois Pottier：在介绍子类型：从理论到实践的类型推理),我可以使用：

输入：

Type = { TypeVariable,ConcreteType }
TypeRelation = { Left: Type,Right: Type }
TypeRelations = Deque<TypeRelation>

助手功能：

ExtendsOrEquals = #(ConcreteType,ConcreteType) => Boolean
Union = #(ConcreteType,ConcreteType) => ConcreteType | fail
Intersection = #(ConcreteType,ConcreteType) => ConcreteType
SubC = #(Type,Type) => List<TypeRelation>

如果第一个扩展或等于第二个扩展名,则ExtendsOrEquals可以告诉两个具体类型,例如(String,Object)== true,(Object,String)== false.

如果可能,联盟计算两个具体类型的公共子类型,例如(Object,Serializable)== Object& Serializable,(Integer,String)== null.

交点计算两个具体类型的最接近的超类型,例如(List,Set)== Collection,String)== Object.

SubC是结构分解函数,在这种简单的情况下,只返回一个包含其参数的新TypeRelation的单例列表.

跟踪结构：

UpperBounds = Map<TypeVariable,Set<Type>>
LowerBounds = Map<TypeVariable,Set<Type>>
Reflexives = List<TypeRelation>

UpperBounds跟踪可能是类型变量的超类型的类型,LowerBounds跟踪可能是类型变量的子类型的类型.反射子跟踪对类型变量之间的关系,以帮助算法的有界重写.

算法如下：

While TypeRelations is not empty,take a relation rel

  [Case 1] If rel is (left: TypeVariable,right: TypeVariable) and 
           Reflexives does not have an entry with (left,right) {

    found1 = false;
    found2 = false
    for each ab in Reflexives
      // apply a >= b,b >= c then a >= c rule
      if (ab.right == rel.left)
        found1 = true
        add (ab.left,rel.right) to Reflexives
        union and set upper bounds of ab.left 
          with upper bounds of rel.right

      if (ab.left == rel.right)
        found2 = true
        add (rel.left,ab.right) to Reflexives
        intersect and set lower bounds of ab.right 
          with lower bounds of rel.left

    if !found1
        union and set upper bounds of rel.left
          with upper bounds of rel.right
    if !found2
        intersect and set lower bounds of rel.right
          with lower bounds of rel.left

    add TypeRelation(rel.left,rel.right) to Reflexives

    for each lb in LowerBounds of rel.left
      for each ub in UpperBounds of rel.right
        add all SubC(lb,ub) to TypeRelations
  }
  [Case 2] If rel is (left: TypeVariable,right: ConcreteType) and 
      UpperBound of rel.left does not contain rel.right {
    found = false
    for each ab in Reflexives
      if (ab.right == rel.left)
        found = true
        union and set upper bounds of ab.left with rel.right
    if !found 
        union the upper bounds of rel.left with rel.right
    for each lb in LowerBounds of rel.left
      add all SubC(lb,rel.right) to TypeRelations
  }
  [Case 3] If rel is (left: ConcreteType,right: TypeVariable) and
      LowerBound of rel.right does not contain rel.left {
    found = false;
    for each ab in Reflexives
      if (ab.left == rel.right)
         found = true;
         intersect and set lower bounds of ab.right with rel.left
    if !found
       intersect and set lower bounds of rel.right with rel.left
    for each ub in UpperBounds of rel.right
       add each SubC(rel.left,ub) to TypeRelations
  }
  [Case 4] if rel is (left: ConcreteType,Right: ConcreteType) and 
      !ExtendsOrEquals(rel.left,rel.right)
    fail
  }

一个基本的例子：

Merge = (T,T) => T
Sink = U => Void

Sink(Merge("String",1))

这个表达的关系：

String >= T
Integer >= T
T >= U

1.)rel是(String,T);情况3被激活.因为反身是空的,所以T的LowerBounds设置为String.不存在T的UpperBounds,因此TypeRelations保持不变.

2.)rel是(整数,T);情况3再次被激活.反身仍然为空,T的下限设置为String和Integer的交集,产生Object,仍然没有T的上限,TypeRelations中没有更改

3.)rel是T> = U.情况1被激活.因为反身是空的,所以T的上限与U的上限相结合,这个上限保持为空.然后下限U被设置为下限t T,产生Object> = U.TypeRelation(T,U)被加到反射器.

4.)算法终止.从边界Object> = T和Object> = U

在另一个示例中,示出了类型冲突：

Merge = (T,T) => T
Sink = Integer => Void

Sink(Merge("String",1))

关系：

String >= T
Integer >= T
T >= Integer

步骤1.)和2.)与上述相同.

3.)rel是T> = U.情况2被激活.该情况尝试将T的上限(此时为Object)与Integer并联,失败并且算法失败.

类型系统的扩展

向类型系统添加通用类型在主要情况和SubC功能中需要扩展.

Type = { TypeVariable,ConcreteType,ParametricType<Type,...>)

一些想法：

>如果ConcreteType和ParametricType符合,那就是一个错误.>如果TypeVariable和ParametricType满足,例如T = C(U1,…,Un),则创建新的类型变量和关系,如T1> = U1,Tn> = Un并与它们一起工作.>如果两个ParametricType满足(D和C),则检查D> = C和类型参数的数量是否相同,然后提取每一对作为关系.