I want to define a type for a function that does something and then returns another function of the same type [can be itself]. The obvious idea didn't work ("Illegal cyclic type reference" error):
type Behavior[S] = S => Behavior[S]
Is there something obvious that I am missing here? Also I do not understand how to express an idea of "function returning itself".
CodePudding user response：
Short answer
case class Behavior[S](step: S => Behavior[S])
Long answer (short version)
Terminal F-Coalgebras are pretty cool.
Long answer
Warning: lots of barbed wire & co-bananas, or something...
Ok, so, suppose that you have the concept of a functor F that captures what it means that your behavior "does something". In most libraries is something like this:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
An F-coalgebra A is essentially just a function from A to F[A]:
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
Now, a terminal F-coalgebra T is an F-coalgebra which additionally has a property that from every other F-coalgebra A there is a mediating morphism A => T (such that everything commutes, blah blah):
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
Can we implement it for arbitrary F? It turns out we can:
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)))
For the sake of a concrete example, let's see what that contraption does for the simplest imaginable functor Option:
given Functor[Option] with
def map[A, B](fa: Option[A])(f: A => B): Option[B] = fa.map(f)
type ConaturalNumber = TerminalFCoalgCarrier[Option]
It turns out that the terminal F-coalgebra for Option are the so-called conatural numbers. These are basically the natural numbers, plus countable infinity. These things are nicely suitable for representing lengths of potentially infinite "clicking" processes.
Let's try it on a finite behavior:
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)
Now, the above machinery automatically gives us a universal way to translate those counting words into conatural numbers. Let's add a helper method for describing conatural numbers (approximately):
def describe(c: ConaturalNumber): String =
var counter = 0
var curr = c
while true do
curr.step() match
case None => return s"${counter}"
case Some(next) =>
if counter > 42 then
return "probably infinite"
else {
counter = 1
curr = next
}
throw new Error("We have counted to infinity, yay! :D")
What does it show for the Welsh counting words?
@main def demo(): Unit =
for w <- WelshCounting.values do
val conat = welshMediatingMorphism(w)
println(s"${w} -> ${describe(conat)}")
// Eeny -> 0
// Meeny -> 1
// Miny -> 2
// Moe -> 3
Ok, that's neat. Let's try an infinitely clicking process with just one state that is successor of itself:
object LoopForever extends FCoalg[Option, Unit]:
def apply(u: Unit) = Some(())
val loopForeverMediatingMorphism =
summon[TerminalFCoalg[Option, TerminalFCoalgCarrier[Option]]]
.mediate(LoopForever)
How would it now describe the single state ()?
println(s"${()} -> ${describe(loopForeverMediatingMorphism(()))}")
// () -> probably infinite
Seems to work.
Full code:
trait Functor[F[_]]:
def map[A, B](fa: F[A])(f: A => B): F[B]
trait FCoalg[F[_]: Functor, A]:
def apply(a: A): F[A]
trait TerminalFCoalg[F[_]: Functor, T] extends FCoalg[F, T]:
def mediate[A](coalg: FCoalg[F, A]): A => T
case class TerminalFCoalgCarrier[F[_]: Functor](
step: () => F[TerminalFCoalgCarrier[F]]
)
given tfcImpl[F[_]: Functor]: TerminalFCoalg[F, TerminalFCoalgCarrier[F]] with
def apply(a: TerminalFCoalgCarrier[F]): F[TerminalFCoalgCarrier[F]] = a.step()
def mediate[A](coalg: FCoalg[F, A]): A => TerminalFCoalgCarrier[F] = a =>
TerminalFCoalgCarrier(() => summon[Functor[F]].map(coalg(a))(mediate(coalg)))
given Functor[Option] with
def map[A, B](fa: Option[A])(f: A => B): Option[B] = fa.map(f)
type ConaturalNumber = TerminalFCoalgCarrier[Option]
def describe(c: ConaturalNumber): String =
var counter = 0
var curr = c
while true do
curr.step() match
case None => return s"${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(()))}")
CodePudding user response：
Expanding that alias will lead to a function of undefined order i.e. S => S => S => ....
A high order function will return another function. The type of a function needs to be well defined hence you need to know the precise order of the returned function.
If you plan to make a function that returns another high order function (possibly itself) you could use currying:
def makeFOrder1(arg: Int): Int => Int =
arg2 => arg2 arg
def makeFOrder2(arg: Int): Int => Int => Int =
arg2 => arg3 => arg arg2 arg3
def makeFOrder3(arg: Int): Int => Int => Int => Int =
arg2 => arg3 => arg4 => arg arg2 arg3 arg4
//They behave like curried functions
makeFOrder1(1)(2)
makeFOrder2(1)(2)(3)
makeFOrder3(1)(2)(3)(4)
//Use currying directly
def makeF(arg: Int)(arg2: Int)(arg3: Int)(arg4: Int): Int =
arg arg2 arg3 arg4
makeF(1)(2)(3)(4)
val partiallyApplied1 = makeF(1)
val partiallyApplied2 = partiallyApplied1(2)(3)
val partiallyApplied3 = partiallyApplied1(4)