具有上下文的 Haskell 数据类型

Question

我想为 Vertex 编写基本实现。

data Point a = Point a a

class XY c where

    x :: c a -> a

    y :: c a -> a

class XY c => Vertex c where

    translate :: c a -> c a -> c a

    scale :: a -> c a -> c a

    rotate :: a -> c a -> c a

instance XY Point where

    x (Point first second) = first

    y (Point first second) = second

instance Vertex Point where

    translate xy1 xy2 = Point (x xy1 + x xy2) (y xy1 + y xy2)

    scale a xy = Point ((*a) $ x xy) ((*a) $ y xy)

    rotate a xy = Point (x * cosA - y * sinA) (x * sinA + y * cosA) where
                    cosA = cos a
                    sinA = sin a

我必须创建 Vertex 类型类的实例，并在 Point 类型参数中实现 Floating 类型类。如果我像 instance (Floating a) => Vertex Point a where 那样实现它，我得到:

 Expected kind ‘* -> Constraint’,
        but ‘Vertex Point’ has kind ‘Constraint’

用 Haskell 编写它的正确方法是什么？

Answer 1

哇哦。这个众所周知的问题是我最讨厌的问题。正确的解决方案是使 XY 和 Point 类不用于参数类型。标量参数成为关联的类型同义词，一切都很容易进行:

{-# LANGUAGE TypeFamilies #-}

class XY p where
  type Component p :: *
  x :: p -> Component p
  y :: p -> Component p

class XY p => Vertex p where
  translate :: p -> p -> p
  scale :: Component p -> p -> p
  rotate :: Component p -> p -> p

N.B. In fact you could even consider simplifying this to always use the same component type, since you'll likely never need anything else:

class XY p where
  x :: p -> Double
  y :: p -> Double
class XY p => Vertex p where
  translate :: p -> p -> p
  scale :: Double -> p -> p
  rotate :: Double -> p -> p

通过非参数形式，您现在可以轻松地在需要的地方添加数字类型约束，即在 instance Vertex Point 实例中:

instance XY (Point a) where
  type Component (Point a) = a
  x (Point xc _) = xc
  y (Point _ yc) = yc

instance Floating a => Vertex (Point a) where
  translate xy1 xy2 = Point (x xy1 + x xy2) (y xy1 + y xy2)
  scale a xy = Point ((*a) $ x xy) ((*a) $ y xy)
  rotate a xy = Point (x * cosA - y * sinA) (x * sinA + y * cosA)
   where cosA = cos a
         sinA = sin a

出于某种原因^†，大多数人更喜欢为几何实体创建参数化类而不是标量类型，这不仅完全没有必要，而且非几何，因为正确的几何形状显然不依赖于实际的基础分解。

<小时/> ^† _{实际上我相当确定原因是什么:Edward Kmett 决定在 the linear library 中使用参数化类型。他应该更清楚，尤其是自从 Conal Elliott 的 vector-space library ，它以正确的方式做到这一点，已经存在了更长时间。}