Is there one (or a few) canonical/ specific use cases for "functions returning functions" (beyond "decorators")?

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Is there one (or a few) canonical/ specific use cases for "functions returning functions" (beyond "decorators")?

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What can you do with "functions returning functions" that you cannot (easily or elegantly or expressively) do with functions as variables ("functions taking functions as their arguments") and/or composing functions, as in :(f(g(x)).

A common reason to return a function is when programming with iterators. Specifically, when you have a data structure (such as a list or set) and you want to return an iterator over it.

In a pure functional style, an iterator is an initial state init: S together with a function

next: S -> Option

where S is the state of the iterator and A is the output type of the iterator.
It takes in a current state and returns either None or a value and a new state.

So, if you want to define an iterator over a list, you need to write a function

iterate_over_list: List A -> (S, S -> Option)

where S is defined appropriately -- depending on your implementation it could either be (List A, int) (the list together with an index into it) or just List A if you consume the list while iterating over it.

There's no way to avoid returning a function here because an iterator is something that you call repeatedly (every time you want a new element, you ask the iterator to generate a new element).

To give an even simpler example along the same lines, sometimes you want an iterator over all integers starting at a given integer. Then you would write a function

iterate_starting_from: int -> (S, S -> Option)

where S in this case would just be int (the state of the iterator is the current integer to be returned). The implementation of this is something like (in pseudocode):

iterate_starting_from(i):
    let init = i
    let next = lambda x: Some((x, x+1))
    return (init, next)

StackExchange Computer Science Q#143427, answer score: 5