### Genericity

In the grand scheme of things, functional programming is about abstracting things out. Skimming over Jeremy Gibbons’s 2006 book Datatype-Generic Programming, I found a nice overview.

Generic programming is about making programming languages more flexible without compromising safety.

#### Genericity by value

One of the first and most fundamental techniques that any programmer learns is how to parametrize computations by values

``````def triangle4: Unit = {
println("*")
println("**")
println("***")
println("****")
}
``````

We can abstract out 4 into a parameter:

``````def triangle(side: Int): Unit = {
(1 to side) foreach { row =>
(1 to row) foreach { col =>
println("*")
}
}
}
``````

#### Genericity by type

`List[A]` is a polymorphic datatype parametrized by another type, the type of list elements. This enables parametric polymorphism.

``````def head[A](xs: List[A]): A = xs(0)
``````

The above function would work for all proper types.

#### Genericity by function

Higher-order programs are programs parametrized by other programs.

For example `foldLeft` can be used to `append` two lists:

``````def append[A](list: List[A], ys: List[A]): List[A] =
list.foldLeft(ys) { (acc, x) => x :: acc }

append(List(1, 2, 3), List(4, 5, 6))
// res0: List[Int] = List(3, 2, 1, 4, 5, 6)
``````

Or it can also be used to add numbers:

``````def sum(list: List[Int]): Int =
list.foldLeft(0) { _ + _ }
``````

#### Genericity by structure

“Generic programming” embodied in the sense of a collection library, like Scala Collections. In the case of C++’s Standard Template Library, the parametric datatypes are called containers, and various abstractions are provided via iterators, such as input iterators and forward iterators.

The notion of the typeclass fits in here too.

``````trait Read[A] {
}

def reads(s: String): Option[A] = f(s)
}
}

try {
Some(s.toInt)
} catch {
case e: NumberFormatException => None
}
}
}

// res1: Option[Int] = Some(value = 1)
``````

The typeclass captures the requirements required of types, called typeclass contract. It also lets us list the types providing these requirements by defining typeclass instances. This enables ad-hoc polymorphism because `A` in `Read[A]` is not universal.

#### Genericity by property

In Scala Collection library, some of the concepts promised are more elaborate than the list of operations covered by the type.

as well as signatures of operations, the concept might specify the laws these operations satisfy, and non-functional characteristics such as the asymptotic complexities of the operations in terms of time and space.

Typeclasses with laws fit in here too. For example `Monoid[A]` comes with the monoid laws. The laws need to be validated for each instance using property-based testing tools.

#### Genericity by stage

Various flavors of metaprogramming can be though of as the development or program that construct or manipulate other programs. This could include code generation and macros.

#### Genericity by shape

Let’s say there’s a polymorphic datatype of binary trees:

``````sealed trait Btree[A]
object Btree {
case class Tip[A](a: A) extends Btree[A]
case class Bin[A](left: Btree[A], right: Btree[A]) extends Btree[A]
}
``````

Let’s write `foldB` as a way of abstracting similar programs.

``````def foldB[A, B](tree: Btree[A], b: (B, B) => B)(t: A => B): B =
tree match {
case Btree.Tip(a)      => t(a)
case Btree.Bin(xs, ys) => b(foldB(xs, b)(t), foldB(ys, b)(t))
}
``````

The next goal is to abstract `foldB` and `foldLeft`.

In fact, what differs between the two fold operators is the shape of the data on which they operate, and hence the shape of the programs themselves. The kind of parametrization required is by this shape; that is, by the datatype or type constructor (such as `List` or `Tree`) concerned. We call this datatype genericity.

For example, `fold` apparently could be expressed as

``````import cats._, cats.data._, cats.syntax.all._

trait Fix[F[_,_], A]
def cata[S[_,_]: Bifunctor, A, B](t: Fix[S, A])(f: S[A, B] => B): B = ???
``````

In the above, `S` represents the shape of the datatype. By abstracting out the shapes, we can construct parametrically datatype-generic programs. We’ll come back to this later.

Alternatively, such programs might be ad-hoc datatype-generic, when the behaviour exploits that shape in some essential manner. Typical examples of the latter are pretty printers and marshallers.

The example that fits in this category might be Scala Pickling. Pickling defines picklers for common types, and it derives pickler instances for different shapes using macro.

This approach to datatype genericity has been variously called polytypism, structural polymorphism or typecase , and is the meaning given to ‘generic programming’ by the Generic Haskell team. Whatever the name, functions are defined inductively by case analysis on the structure of datatypes ….

We consider parametric datatype genericity to be the ‘gold standard’, and in the remainder of these lecture notes, we concentrate on parametric datatype-generic definitions where possible.

In Scala, shapeless is focused on abstracting out the shape.