two variances in Scala
Jan. 3rd, 2016 08:12 pmscala> val s = Set("a", "b", "c") s: scala.collection.immutable.Set[String] = Set(a, b, c) scala> val t = s.map(_ + ":)") t: scala.collection.immutable.Set[String] = Set(a:), b:), c:)) scala> val s = Set("a1", "a2", "a3") s: scala.collection.immutable.Set[String] = Set(a1, a2, a3) scala> val t = s.map(_ take 1) t: scala.collection.immutable.Set[String] = Set(a) scala> val u:Set[Any] = s map identity u: Set[Any] = Set(a1, a2, a3) scala> val v:Set[Any] = s <console>:8: error: type mismatch; found : scala.collection.immutable.Set[String] required: Set[Any] Note: String <: Any, but trait Set is invariant in type A. You may wish to investigate a wildcard type such as `_ <: Any`. (SLS 3.2.10) val v:Set[Any] = s ^
From a categorist's p.o.v, wtf, if we have
map
, we have a covariant functor. But wow, it's "type theory", covariance here means only covariance w.r.t. subtyping. So, big deal, map with identity, no? I mean, not being a typist, I don't even understand the problem. Do you?You know the stupid problem with Java/Scala collections: their
Well, a natural transformation from a covariant functor to a contravariant is not only possible, it's a relatively popular thing. So, does it mean we can make
Questions? :)
I think it's cool; category theory applied directly to solve a code design problem.
contains()
method takes any type; the explanation being that for this method to be type-dependent, we need contravariance, while collections are (mostly) covariant. So there.Well, a natural transformation from a covariant functor to a contravariant is not only possible, it's a relatively popular thing. So, does it mean we can make
contains()
method type-dependent? The answer is YES!trait MyList[+T] { // declare a simple collection, e.g. list; it is covariant def head:T def tail:MyList[T] def ::[U>:T](x:U) = HeadAndTail(x, this) def size: Int } case object EmptyList extends MyList[Nothing] { // it's okay that it's a list of Nothing def head = error("This list is empty") def tail = error("this list is empty") def size = 0 } case class HeadAndTail[T](head:T, tail: MyList[T]) extends MyList[T] { def size = 1 + tail.size } // All the stuff above allows us to treat lists in a covariant way, like this: case class A(name: String) {override def toString = "A(" + name + ")"} case class B(override val name: String) extends A(name) {override def toString = "B(" + name + ")"} val listB = B("1") :: B("2") :: EmptyList listB.size must_== 2 val listA = A("3") :: listB listA.size must_== 3 val listC: MyList[Any] = listA // See, MyList is covariant, so if B is a subtype of A, List[B] is a subtype of List[A] And now... introduce a contravariant trait: trait Container[-T] { def contains(t:T): Boolean } // and the natural transformation from MyList[+T] to Container[-T]: implicit def asContainer[T](list:MyList[T]): Container[T] = new Container[T] { def contains(t:T) = list.size > 0 && (list.head == t || asContainer(list.tail).contains(t)) } And it works. I can draw the appropriate commutative diagrams later, if it is not obvious. listA contains A("3") must beTrue listA contains B("1") must beTrue // listA contains "abracadabra" is a compilation error listC contains "abracadabra" must beFalse listC contains B("2") must beTrue
Questions? :)
I think it's cool; category theory applied directly to solve a code design problem.
монада из двух сопряженных функторов
Jul. 20th, 2012 07:18 pmТут мы с
sassa_nf обсуждали вопрос, как это получается, что оно монада.
В принципе, это есть в МакЛейне, но у МакЛейна всё так мудрено. А надо-то на пальцах.
В другом месте - т.наз. функторы Hom участвуют - а у меня к ним прямо гомофобия; такую неприязнь чувствую к функтору Hom, кушать не могу.
Так вот, напишу-ка я тут вкрадце, шо за сопряженные функторы и как они связаны с монадой.
Для программистов напишу, так что никаких когомологий, кобордизмов, 2-категорий, расширений Кана, а всё на пальцах, лишь бы не лень читать было.
( Read more... )
![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
В принципе, это есть в МакЛейне, но у МакЛейна всё так мудрено. А надо-то на пальцах.
В другом месте - т.наз. функторы Hom участвуют - а у меня к ним прямо гомофобия; такую неприязнь чувствую к функтору Hom, кушать не могу.
Так вот, напишу-ка я тут вкрадце, шо за сопряженные функторы и как они связаны с монадой.
Для программистов напишу, так что никаких когомологий, кобордизмов, 2-категорий, расширений Кана, а всё на пальцах, лишь бы не лень читать было.
( Read more... )
Subclassing errors, OPP, and practically checkable rules to prevent them
P.S. I believe the problem is that with stateful objects, we do not know exactly which category are we dealing with; roughly speaking, for class A subclassing class B, there are actually two monads, with a natural transformation from one to another; and we think we have a functor from, not sure yet, one Kleisli category to another, or from one Eilenberg-Moore category to another, or even an interesting relationship between two categories of adjoints generated by the monads.
Have to look closer; maybe this explains the problem with "Liskov substitution".
P.S. I believe the problem is that with stateful objects, we do not know exactly which category are we dealing with; roughly speaking, for class A subclassing class B, there are actually two monads, with a natural transformation from one to another; and we think we have a functor from, not sure yet, one Kleisli category to another, or from one Eilenberg-Moore category to another, or even an interesting relationship between two categories of adjoints generated by the monads.
Have to look closer; maybe this explains the problem with "Liskov substitution".