Haskell 中的求幂

发布于 2024-11-16 06:33:09 字数 166 浏览 3 评论 0原文

有人可以告诉我为什么 Haskell Prelude 定义了两个单独的求幂函数（即 ^ 和 **）？我认为类型系统应该消除这种重复。

Prelude> 2^2
4
Prelude> 4**0.5
2.0

原文

Can someone tell me why the Haskell Prelude defines two separate functions for exponentiation (i.e. ^ and **)? I thought the type system was supposed to eliminate this kind of duplication.

Prelude> 2^2
4
Prelude> 4**0.5
2.0

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月亮邮递员 2024-11-23 06:33:09

实际上存在三个求幂运算符：(^)、(^^) 和 (**)。 ^ 是非负整数幂，^^ 是整数幂，** 是浮点幂：

(^) :: (Num a, Integral b) => a -> b -> a
(^^) :: (Fractional a, Integral b) => a -> b -> a
(**) :: Floating a => a -> a -> a

原因是类型安全：数值运算的结果通常与输入参数具有相同的类型。但您无法将 Int 进行浮点幂并获得 Int 类型的结果。因此类型系统会阻止您执行此操作：(1::Int) ** 0.5 会产生类型错误。 (1::Int) ^^ (-1) 也是如此。

另一种说法是：Num 类型在 ^ 下封闭（它们不需要具有乘法逆元），Fractional 类型在 ^ 下封闭^^、Floating 类型在 ** 下封闭。由于 Int 没有 Fractional 实例，因此您无法将其计算为负幂。

理想情况下，^ 的第二个参数静态地约束为非负数（目前，1 ^ (-2) 会引发运行时异常）。但 Prelude 中没有自然数的类型。

There are actually three exponentiation operators: (^), (^^) and (**). ^ is non-negative integral exponentiation, ^^ is integer exponentiation, and ** is floating-point exponentiation:

(^) :: (Num a, Integral b) => a -> b -> a
(^^) :: (Fractional a, Integral b) => a -> b -> a
(**) :: Floating a => a -> a -> a

The reason is type safety: results of numerical operations generally have the same type as the input argument(s). But you can't raise an Int to a floating-point power and get a result of type Int. And so the type system prevents you from doing this: (1::Int) ** 0.5 produces a type error. The same goes for (1::Int) ^^ (-1).

Another way to put this: Num types are closed under ^ (they are not required to have a multiplicative inverse), Fractional types are closed under ^^, Floating types are closed under **. Since there is no Fractional instance for Int, you can't raise it to a negative power.

Ideally, the second argument of ^ would be statically constrained to be non-negative (currently, 1 ^ (-2) throws a run-time exception). But there is no type for natural numbers in the Prelude.

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命硬 2024-11-23 06:33:09

Haskell 的类型系统不够强大，无法将三个求幂运算符表示为一个。你真正想要的是这样的：

class Exp a b where (^) :: a -> b -> a
instance (Num a,        Integral b) => Exp a b where ... -- current ^
instance (Fractional a, Integral b) => Exp a b where ... -- current ^^
instance (Floating a,   Floating b) => Exp a b where ... -- current **

即使你打开多参数类型类扩展，这也不会真正起作用，因为实例选择需要比 Haskell 目前允许的更聪明。

Haskell's type system isn't powerful enough to express the three exponentiation operators as one. What you'd really want is something like this:

class Exp a b where (^) :: a -> b -> a
instance (Num a,        Integral b) => Exp a b where ... -- current ^
instance (Fractional a, Integral b) => Exp a b where ... -- current ^^
instance (Floating a,   Floating b) => Exp a b where ... -- current **

This doesn't really work even if you turn on the multi-parameter type class extension, because the instance selection needs to be more clever than Haskell currently allows.

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