I don't think I see what you are getting at, Tom. Let's consider the special case of non-empty lists. One fold-like function that has the property I think Charles meant by 3., would be one that works as follows:

foldBalanced :: (a -> a -> a) -> [a] -> a
foldBalanced f [x] = x
foldBalanced f [x,y] = f x y
foldBalanced f [x,y,z] = f x (f y z)
foldBalanced f [x,y,z,u] = f (f x y) (f z u)
... -- I hope you can see the pattern (building f-application trees that are as balanced as possible)

I don't see how this function can be written as g . foldl f z for some g and z.


2015-10-25 0:55 GMT+02:00 Tom Ellis <tom-lists-haskell-cafe-2013@jaguarpaw.co.uk>:
On Sat, Oct 24, 2015 at 08:12:11PM +0200, Janis Voigtländer wrote:
> It has already been established in this thread what Charles meant by 3.
>
> He meant that a fold-function that has the property he is after would
> guarantee that it:
>
> a) takes all the content elements from a data structure, say x1,...,xn,
>
> b) builds an application tree with the to-be-folded, binary operation f in
> the internal nodes of a binary tree, whose leafs, read from left to right,
> form exactly the sequence x1,...,xn,
>
> c) evaluates that application tree.
>
> Do you agree that what I describe above is a property of a given fold-like
> function, not of the f handed to that fold-like function?
>
> And do you agree that what I describe above is a property that is weaker
> than (and so, in particular different from) for example the property "this
> fold-like function is foldl or foldr".

I do agree.  I would be interested whether you think such a property could
differ from my earlier proposed property:

    "the function factors through `foldl f`", i.e.  is `g . foldl f` for
    some `g`.

(Actually when I wrote that I suppose I meant `g . foldl f z` for some `g`
and `z`)

Tom
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