我想为一个函数定义一个类型,它做某事然后返回另一个相同类型的函数[可以是它本身]。显而易见的想法行不通(“非法循环类型引用”错误):
type Behavior[S] = S => Behavior[S]
我在这里明显遗漏了什么吗?我也不明白如何表达“函数返回自身”的想法。
最佳答案
case class Behavior[S](step: S => Behavior[S])
Terminal F-Coalgebras很酷。
警告:很多带刺铁丝网和香蕉,或其他东西......
好吧,假设你有一个仿函数 F
的概念,它捕捉了你的行为“做某事”的含义。在大多数图书馆中是这样的:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
一个 F
-coalgebra A
本质上只是一个从 A
到 F[A]
的函数:
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
现在,一个 terminal F
-coalgebra T
是一个 F
-coalgebra,它还有一个属性从所有其他 F
-coalgebra A
有一个中介态射 A => T
(这样一切都通勤,等等):
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
我们可以为任意的 F
实现它吗?事实证明我们可以:
case class TerminalFCoalgCarrier[F[_]: Functor](
step: () => F[TerminalFCoalgCarrier[F]]
)
given tfcImpl[F[_]: Functor]: TerminalFCoalg[F, TerminalFCoalgCarrier[F]] with
def apply(a: TerminalFCoalgCarrier[F]): F[TerminalFCoalgCarrier[F]] = a.step()
def mediate[A](coalg: FCoalg[F, A]): A => TerminalFCoalgCarrier[F] = a =>
TerminalFCoalgCarrier(() => summon[Functor[F]].map(coalg(a))(mediate(coalg)))
为了一个具体的例子,让我们看看这个装置对最简单的仿函数 Option
做了什么:
given Functor[Option] with
def map[A, B](fa: Option[A])(f: A => B): Option[B] = fa.map(f)
type ConaturalNumber = TerminalFCoalgCarrier[Option]
事实证明,Option
的终结符 F
-coalgebra 是所谓的自然数。这些基本上是自然数,加上可数无穷大。这些东西非常适合表示潜在无限“点击”过程的长度。
让我们在有限行为上尝试一下:
enum WelshCounting:
case Eeny
case Meeny
case Miny
case Moe
object WelshCountingOptionCoalg extends FCoalg[Option, WelshCounting]:
def apply(w: WelshCounting): Option[WelshCounting] =
import WelshCounting._
w match
case Eeny => None
case Meeny => Some(Eeny)
case Miny => Some(Meeny)
case Moe => Some(Miny)
val welshMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(WelshCountingOptionCoalg)
现在,上述机器自动为我们提供了一种通用方法,可以将那些计数词翻译成自然数。让我们添加一个辅助方法来描述自然数(大约):
def describe(c: ConaturalNumber): String =
var counter = 0
var curr = c
while true do
curr.step() match
case None => return s"${counter}"
case Some(next) =>
if counter > 42 then
return "probably infinite"
else {
counter += 1
curr = next
}
throw new Error("We have counted to infinity, yay! :D")
威尔士语数词说明了什么?
@main def demo(): Unit =
for w <- WelshCounting.values do
val conat = welshMediatingMorphism(w)
println(s"${w} -> ${describe(conat)}")
// Eeny -> 0
// Meeny -> 1
// Miny -> 2
// Moe -> 3
好的,这很好。让我们尝试一个无限点击过程,只有一个状态是其自身的后继状态:
object LoopForever extends FCoalg[Option, Unit]:
def apply(u: Unit) = Some(())
val loopForeverMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(LoopForever)
现在如何描述单一状态 ()
?
println(s"${()} -> ${describe(loopForeverMediatingMorphism(()))}")
// () -> probably infinite
似乎有效。
完整代码:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
case class TerminalFCoalgCarrier[F[_]: Functor](
step: () => F[TerminalFCoalgCarrier[F]]
)
given tfcImpl[F[_]: Functor]: TerminalFCoalg[F, TerminalFCoalgCarrier[F]] with
def apply(a: TerminalFCoalgCarrier[F]): F[TerminalFCoalgCarrier[F]] = a.step()
def mediate[A](coalg: FCoalg[F, A]): A => TerminalFCoalgCarrier[F] = a =>
TerminalFCoalgCarrier(() => summon[Functor[F]].map(coalg(a))(mediate(coalg)))
given Functor[Option] with
def map[A, B](fa: Option[A])(f: A => B): Option[B] = fa.map(f)
type ConaturalNumber = TerminalFCoalgCarrier[Option]
def describe(c: ConaturalNumber): String =
var counter = 0
var curr = c
while true do
curr.step() match
case None => return s"${counter}"
case Some(next) =>
if counter > 42 then
return "probably infinite"
else {
counter += 1
curr = next
}
throw new Error("We cannot count to infinity :(")
enum WelshCounting:
case Eeny
case Meeny
case Miny
case Moe
object WelshCountingOptionCoalg extends FCoalg[Option, WelshCounting]:
def apply(w: WelshCounting): Option[WelshCounting] =
import WelshCounting._
w match
case Eeny => None
case Meeny => Some(Eeny)
case Miny => Some(Meeny)
case Moe => Some(Miny)
val welshMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(WelshCountingOptionCoalg)
object LoopForever extends FCoalg[Option, Unit]:
def apply(u: Unit) = Some(())
val loopForeverMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(LoopForever)
@main def demo(): Unit =
for w <- WelshCounting.values do
val conat = welshMediatingMorphism(w)
println(s"${w} -> ${describe(conat)}")
println(s"${()} -> ${describe(loopForeverMediatingMorphism(()))}")
https://stackoverflow.com/questions/70086713/
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