Comonad example in Scala

The scalaz library provides a ComonadStore which extends the property of Comonad. It is defined like this:

trait ComonadStore[F[_], S] extends Comonad[F] { self =>

  def pos[A](w: F[A]): S
  def peek[A](s: S, w: F[A]): A

  def peeks[A](s: S => S, w: F[A]): A =
    peek(s(pos(w)), w)

  def seek[A](s: S, w: F[A]): F[A] =
    peek(s, cojoin(w))

  def seeks[A](s: S => S, w: F[A]): F[A] =
    peeks(s, cojoin(w))

  def experiment[G[_], A](s: S => G[S], w: F[A])(implicit FG: Functor[G]): G[A] =
    FG.map(s(pos(w)))(peek(_, w))

}

And a Store( which is analogous to (S => A, S)) has an instance of Comonad. You can look at this question that explains what it is more specifically.

You also have the Coreader and Cowriter Comonads that are the dual of the Reader and Writer Monads, here is an excellent blog post that talks about it in Scala.

Well, monads allow you to add values to them, change them based on a computation from a non-monad to a monad. Comonads allow you to extract values from them, and change them based on a computation from a comonad to a non-comonad.

The natural intuition is that they'll usually appear where you have a CM[A] and want to extract A.

See this very interesting post that touches on comonads a bit casually, but, to me at least, making them very clear.

What follows is a literal translation of code from this blog post.

case class U[X](left: Stream[X], center: X, right: Stream[X]) {
  def shiftRight = this match {
    case U(a, b, c #:: cs) => U(b #:: a, c, cs)
  }

  def shiftLeft = this match {
    case U(a #:: as, b, c) => U(as, a, b #:: c)
  }
}

// Not necessary, as Comonad also has fmap.
/*
implicit object uFunctor extends Functor[U] {
  def fmap[A, B](x: U[A], f: A => B): U[B] = U(x.left.map(f), f(x.center), x.right.map(f))
}
*/

implicit object uComonad extends Comonad[U] {
  def copure[A](u: U[A]): A = u.center
  def cojoin[A](a: U[A]): U[U[A]] = U(Stream.iterate(a)(_.shiftLeft).tail, a, Stream.iterate(a)(_.shiftRight).tail)
  def fmap[A, B](x: U[A], f: A => B): U[B] = U(x.left.map(f), x.center |> f, x.right.map(f))
}

def rule(u: U[Boolean]) = u match {
  case U(a #:: _, b, c #:: _) => !(a && b && !c || (a == b))
}

def shift[A](i: Int, u: U[A]) = {
  Stream.iterate(u)(x => if (i < 0) x.shiftLeft else x.shiftRight).apply(i.abs)
}

def half[A](u: U[A]) = u match {
  case U(_, b, c) => Stream(b) ++ c
}

def toList[A](i: Int, j: Int, u: U[A]) = half(shift(i, u)).take(j - i)

val u = U(Stream continually false, true, Stream continually false)

val s = Stream.iterate(u)(_ =>> rule)

val s0 = s.map(r => toList(-20, 20, r).map(x => if(x) '#' else ' '))

val s1 = s.map(r => toList(-20, 20, r).map(x => if(x) '#' else ' ').mkString("|")).take(20).force.mkString("\n")

println(s1)

Output:

 | | | | | | | | | | | | | | | | | | | |#| | | | | | | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#| | | | | | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| |#| | | | | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#|#|#| | | | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| | | |#| | | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#| | |#|#| | | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| |#| |#| |#| | | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#|#|#|#|#|#|#| | | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| | | | | | | |#| | | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#| | | | | | |#|#| | | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| |#| | | | | |#| |#| | | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#|#|#| | | | |#|#|#|#| | | | | | | |
 | | | | | | | | | | | | | | | | | | | |#| | | |#| | | |#| | | |#| | | | | | |
 | | | | | | | | | | | | | | | | | | | |#|#| | |#|#| | |#|#| | |#|#| | | | | |
 | | | | | | | | | | | | | | | | | | | |#| |#| |#| |#| |#| |#| |#| |#| | | | |
 | | | | | | | | | | | | | | | | | | | |#|#|#|#|#|#|#|#|#|#|#|#|#|#|#|#| | | |
 | | | | | | | | | | | | | | | | | | | |#| | | | | | | | | | | | | | | |#| | |
 | | | | | | | | | | | | | | | | | | | |#|#| | | | | | | | | | | | | | |#|#| |
 | | | | | | | | | | | | | | | | | | | |#| |#| | | | | | | | | | | | | |#| |#|
 | | | | | | | | | | | | | | | | | | | |#|#|#|#| | | | | | | | | | | | |#|#|#|#