 # Exponentiation operator for iterated functions?

Would it make sense to define the `^` operator for iterated functions? Something like:

``````^(f::Function, n::Integer) = ∘(fill(f,n)...)
``````

Then we can do things like

``````typeof(1.0) |> supertype^3
``````

``````typeof(1.0) |> supertype |> supertype |> supertype
``````

A possible downside: people might expect `(sin^2)(x)` to mean `sin(x)^2` instead of `sin(sin(x))` (although they should not IMO ).

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I think not. Whatever works for functions should work for all callables, and some types can be callables and at the same time define methods for `^` (eg as an operator on some algebra).

The basic issue is that `^` in Julia is for iteration of `*`. So it would be inconsistent (and poorly composable as @Tamas_Papp notes) to overload it for iteration of `∘`.

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Ah right it makes sense. Maybe at some point another operator will prove adequate…

In case you’re not aware, you can do it for your own functions if you want!

``````julia> supertype(T) = Base.supertype(T)
supertype (generic function with 1 method)

julia> Base.:^(::typeof(supertype), n::Integer) = n == 1 ? supertype : supertype ∘ supertype^(n-1)

julia> (supertype^3)(Float64)
Number
``````

Thanks, my original post actually included a working implementation, but it’s nice to see another one using recursion. It makes me realize that my version only works for n>=2.

The difference is that your original version commits type piracy if it’s not in `Base` because it applies to all `Function`. That’s (one of) the reason(s) why you were suggesting its inclusion, I thought.

Mine applies only to the function that I own. That’s also the reason why I created a new function (with the same name) in the first place.

Does that make sense? I should have made the distinction clearer.

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Ah sorry I read your message too fast and missed the point, it makes perfect sense.

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``````function Base.:^(x::Any, exponent::Tuple{Function,Integer})
op, p = exponent
p == 1 ? x : reduce(op, fill(x, p))
end
``````
``````julia> typeof(1.0) |> supertype^(∘,3)
Number

julia> 3^(+,3) # Iterated addition -> Scalar multiplication
9

julia> 3^(^, 3) # Tetration 😄
19683
``````

Implementation and syntax is obviously up for debate: For example, closure could work too e.g. `3 ^(+) 3`, but I couldn’t figure out how to do it; I don’t know how to return an anonymous function that behaves like an operator.

I feel like a more general exponentiation operator has several benefits:

• Convenient and intuitive iteration of any closed binary operator (e.g. the main post) without having to implement a new `^` method for custom types
• When you have more than one product-like operation defined, you can eliminate ambiguity as to which operation your exponentiation is referring to.
• If you want to be more strict, instead of overloading `^` for custom types to mean iteration of some non-`*` operator, you can instead specify exactly what you mean without much loss in convenience.
• Code golf As for how useful it would be in practice…I’m not the person to ask.

My dream is for regular exponentiation `x^p` to just be a special case via

``````(^)(x::Any, p::Integer) = x^(*, p)
`````` I don’t think this is a very common use case for the typical Julia user.

It may be relevant algebra and similar, but it’s perfectly fine to have a package define some syntax for this (without type piracy, of course), perhaps using one of the available operators.

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Interesting generalization… It’s basically providing a shorter `reduce` syntax for the case where the collection is a repetition of the same element. Then I would rather provide nice syntax for the collection itself:

``````julia> ↑(el,n::Integer) = fill(el,n)

julia> reduce(*, 2↑10)
1024

julia> typeof(1.0) |> ∘(supertype↑3...)
Number
``````

We could also define a reciprocal operator for the reduction:

``````↓(itr, op::Function) = reduce(op, itr)

julia> (2↑10)↓*
1024

julia> typeof(1.0) |> (supertype↑3)↓∘
Number
``````

But the definition of `↓` should probably define the associativity of reduction (using `foldl` or `foldr`), and it’s not clear which it should be.

In any case I think @Tamas_Papp is right, it’s better to define this in a package using another operator.

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