monoids.scala

// Sketches of various meta-encodings of monoids

// Naive typeclass trait
{
  trait Monoid[M] {
    def zero: M
    def append(m1: M, m2: M): M
  }

  implicit object DoubleSumMonoid extends Monoid[Double] {
    def zero = 1
    def append(m1: Double, m2: Double) = m1 + m2
  }

  implict object DoubleProductMonoid extends Monoid[Double] {
    def zero = 0
    def append(m1: Double, m2: Double) = m1 * m2
  }
}

// Typeclass trait with monoid selector
{
  trait Monoid[M, S] {
    def zero: M
    def append(m1: M, m2: M): M
  }

  trait Sum
  trait Product

  implicit object DoubleSumMonoid extends Monoid[Double, Sum] {
    def zero = 1
    def append(m1: Double, m2: Double) = m1 + m2
  }

  implicit object DoubleProductMonoid extends Monoid[Double, Product] {
    def zero = 0
    def append(m1: Double, m2: Double) = m1 * m2
  }
}

// Typeclass trait with typed lookup
{
  trait InstanceAt[M, S] { def get: M }

  trait Monoid[M] {
    def zero: M
    def append(m1: M, m2: M): M
  }

  object DoubleSumMonoid extends Monoid[Double] {
    def zero = 1
    def append(m1: Double, m2: Double) = m1 + m2
  }

  object DoubleProductMonoid extends Monoid[Double] {
    def zero = 0
    def append(m1: Double, m2: Double) = m1 * m2
  }

  trait Sum
  trait Product

  implicit val doubleSum = InstanceAt[Sum](DoubleSumMonoid)
  implicit val doubleProduct = InstanceAt[Product](DoubleProductMonoid)
}

// Free Monoid, naive
{
  sealed trait Rep[M]
  case class Zero extends Rep[M]
  case class Append(m1: Rep[M], m2: Rep[M])
}

// Free Monoid, selector
{
  sealed trait Rep[M, S]
  case class Zero extends Rep[M, S]
  case class Append(m1: Rep[M, S], m2: Rep[M, S]) extends Rep[M, S]
}

// Tagless, naive
{
  trait Monoidal[Rep[_], M] {
    def zero: Rep[M]
    def append(m1: Rep[M], m2: Rep[M]): Rep[M]
  }
}

// Tagless, selector
{
  trait Monoidal[Rep[_, _], M, S] {
    def zero: Rep[M, S]
    def append(m1: Rep[M, S], m2: Rep[M, S]): Rep[M, S]
  }
}

// Tagless, naive, polymorphic
{
  trait Monoidal[Rep[_]] {
    def zero[M]: Rep[M]
    def apend[M](m1: Rep[M], m2: Rep[M]): Rep[M]
  }
}


// Tagless, selector, polymorphic
{
  trait Monoidal[Rep[_, _]] {
    def zero[M, S]: Rep[M, S]
    def append[M, S](m1: Rep[M, S], m2: Rep[M, S]): Rep[M, S]
  }
}