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Source file: no-neutral-element.fut

# 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 semigroup into a monoid, 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:

``````def 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.

``````def 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.