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Reducing or scanning without a neutral element

The reduce and scan functions expect you to provide a neutral element, such as 0 for addition or 1 for multiplication. But sometimes there may not be an obvious neutral element. In mathematics, such a structure is called a semigroup, while those with neutral elements are called monoids.

We can always turn any monoid into a semigroup, simply by adding a distinct new value to serve as the neutral element.

type with_neutral 't = #neutral | #val t

The operator must also be augmented to handle the neutral element:

let f_with_neutral 't (f: t -> t -> t)
                      (x: with_neutral t)
                      (y: with_neutral t)
                      : with_neutral t =
  match (x, y)
  case (#val x, #val y) -> #val (f x y)
  case (#neutral, _) -> y
  case (_, #neutral) -> x

We can then define a variant of reduce that does not require a neutral element to be provided. If the input array is empty, it will return the #neutral value.

let reduce1 't (f: t -> t -> t) (ts: []t) : with_neutral t =
  reduce (f_with_neutral f) #neutral (map (\t -> #val t) ts)

Try it out in the REPL:

> reduce1 (+) (iota 100)
#val 4950i32
> reduce1 (+) (iota 0)
#neutral

reduce1 is less efficient than reduce due to the baggage of carrying around #neutral, as well as the extra control flow. It is always better if a neutral element is more naturally available, but this technique will do in a pinch.

Sadly, no similar trick exists for turning a non-associative function associative.