我想为一个函数定义一个类型,该函数执行某些操作,然后返回另一个相同类型的函数[可以是它自己]。显而易见的想法不起作用(“非法循环类型引用”错误):
type Behavior[S] = S => Behavior[S]
我在这里有什么明显的遗漏吗?我也不明白如何表达“函数返回自身”的想法。
我想为一个函数定义一个类型,该函数执行某些操作,然后返回另一个相同类型的函数[可以是它自己]。显而易见的想法不起作用(“非法循环类型引用”错误):
type Behavior[S] = S => Behavior[S]
我在这里有什么明显的遗漏吗?我也不明白如何表达“函数返回自身”的想法。
case class Behavior[S](step: S => Behavior[S])
警告:很多铁丝网和香蕉,或其他东西......
好的,所以,假设你有一个仿函数的概念,F
它捕捉到你的行为“做某事”的含义。在大多数库中是这样的:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
-coalgebraF
本质A
上只是一个从A
到的函数F[A]
:
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
现在,一个终端 F
-coalgebraT
是一个F
-coalgebra 它还具有一个属性,即从所有其他F
-coalgebraA
那里有一个中介态射A => T
(这样一切都可以交换,等等):
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
我们可以任意实现它F
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)))
为了举一个具体的例子,让我们看看这个装置对最简单的可想象函子做了什么Option
:
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,终末F
代数Option
是所谓的余数。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可能无限“点击”过程的长度。
让我们尝试一个有限的行为:
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,可以将这些计数词转换为自然数。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"${counter}"
case Some(next) =>
if counter > 42 then
return "probably infinite"
else {
counter += 1
curr = next
}
throw new Error("We have counted to infinity, yay! :D")
威尔士语数词显示了什么?
@main def demo(): Unit =
for w <- WelshCounting.values do
val conat = welshMediatingMorphism(w)
println(s"${w} -> ${describe(conat)}")
// Eeny -> 0
// Meeny -> 1
// Miny -> 2
// Moe -> 3
好的,这很整洁。让我们尝试一个无限点击的过程,其中只有一个状态是自身的后继状态:
object LoopForever extends FCoalg[Option, Unit]:
def apply(u: Unit) = Some(())
val loopForeverMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(LoopForever)
现在它将如何描述单一状态()
?
println(s"${()} -> ${describe(loopForeverMediatingMorphism(()))}")
// () -> probably infinite
似乎工作。
完整代码:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
case class TerminalFCoalgCarrier[F[_]: Functor](
step: () => F[TerminalFCoalgCarrier[F]]
)
given tfcImpl[F[_]: Functor]: TerminalFCoalg[F, TerminalFCoalgCarrier[F]] with
def apply(a: TerminalFCoalgCarrier[F]): F[TerminalFCoalgCarrier[F]] = a.step()
def mediate[A](coalg: FCoalg[F, A]): A => TerminalFCoalgCarrier[F] = a =>
TerminalFCoalgCarrier(() => summon[Functor[F]].map(coalg(a))(mediate(coalg)))
given Functor[Option] with
def map[A, B](fa: Option[A])(f: A => B): Option[B] = fa.map(f)
type ConaturalNumber = TerminalFCoalgCarrier[Option]
def describe(c: ConaturalNumber): String =
var counter = 0
var curr = c
while true do
curr.step() match
case None => return s"${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(()))}")