Re: [Haskell-cafe] 'Associative' order of calling

Kim-Ee, I think you are overcomplicating things (and on the specific question of 2. vs. 3.: no, what I described was specifically meant to capture Charles's 3., not 2. point; note that my "application trees" could be balanced in any way, rather than being completely left-nested). In fact, the discussion of "arbitrary data structures" (in this thread in general) distracts a lot from the properties under consideration, and from even first finding out whether they are properties of foldLike or of f in foldLike f. So, let's specialize. Let's consider only the case of non-empty lists. Then, candidate functions for "foldlike functions" will be functions of this type: foldLike : (a -> a -> a) -> [a] -> a Here are some candidates (I give only an equation for four element lists in each case, but I assume everyone has enough imagination to see how these could be extended to other lists in an intended way): foldLike1 f [a,b,c,d] = [] -- throws away stuff foldLike2 f [a,b,c,d] = f a (f b (f c d)) -- we have all ancountered this one foldLike3 f [a,b,c,d] = f (f (f a b) c) d -- also looks familiar foldLike4 f [a,b,c,d] = f (f a b) (f c d) -- that's a "new" one, but looks good foldLike5 f [a,b,c,d] = f a a -- probably not a very popular one foldLike6 f [a,b,c,d] = f (f c a) (f b d) -- a reasonable one, for example there's a Traversable instance that leads to this one; but still, it's not one that Charles would like, I think So now we can ask which of these satisfy Charles's 1., 2., 3. points. Can't we? There was:

1. Promises to call f on all data (does not have any guarantees on order)

This is satisfied by foldLike2, foldLike3, foldLike4, and foldLike6, but not by the others.

2. Promises to call f on all data in order (like a left fold)

This is satisfied by foldLike3, but not by the others.

3. Promises to call f "associatively" (perhaps can be formalized as an in order break down of the data into tree structures)

This is satisfied by foldLike2, foldLike3, and foldLike4, but not by the others. Since I am able to tell, for a given foldLike candidate, whether or not it satisfies 3. (for example, I could specifically see that foldLike6 does not satisfy 3., while it does satisfy 1.), it cannot be maintained that 3. has no meaning. Note: Nothing in the above makes any assumptions about f. Whether or not f is an associative function is irrelevant for what is asked here. 2015-10-25 4:19 GMT+01:00 Kim-Ee Yeoh :

On Sun, Oct 25, 2015 at 1:12 AM, Janis Voigtländer < janis.voigtlaender@gmail.com> wrote:

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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.

Isn't this what Charles meant by his 2nd property:

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2. Promises to call f on all data in order (like a left fold)

What about his 3rd?

Do you agree that what I describe above is a property of a given fold-like

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function, not of the f handed to that fold-like function?

Before discussing a property of X, isnt it worth asking what X even means?

The folds whose meanings are crystal clear are the arrows out of initial objects in the category of F-algebras.

They are crystal clear because they couple as one with the data definition spelled out in full.

In the quest for useful generalizations of catamorphisms, that coupling with the data definition continues to be assumed.

Observe, for instance:

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a) takes all the content elements from a data structure, say x1,...,xn,

Does a foliar data structure have a canonical flattening out of its leaves? Are there symmetric canonicalizations? How is one selected over the others?

Is the meaning of "all" referentially transparent? That turns out to be a subtle point, as this convo shows:

http://haskell.1045720.n5.nabble.com/A-Proposed-Law-for-Foldable-tp5765339.h...

With the theory of F-algebras, we started with precise notions of data and folds came for free.

But data can be overspecified. And also, the folds among swathes of data suggest useful generalizations.

So now, a raft of proto-precise and necessarily psychological notions of Foldable waded in, and since then it's been fun playing sorting games with shape blocks and holes to squeeze them into.

Fun is good. It's a stage in the journey to knowledge.

And do you agree that what I describe above is a property that is weaker

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than (and so, in particular different from) for example the property "this fold-like function is foldl or foldr".

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2015-10-24 19:55 GMT+02:00 Kim-Ee Yeoh :

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On Sun, Oct 25, 2015 at 12:42 AM, Matteo Acerbi

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wrote:

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For what concerns question 3, I didn't understand the idea of calling a function "associatively".

This. Associativity is a property of binary operators. It's not a property of the catamorphism 'calling' on a given binary operator.

Also, when Charles writes: "Then it goes on to use f in "thisFold f [0,1,2]" like "f (1 (f 0 2))""

commutativity appears to raise its head.

-- Kim-Ee