Re: Idempotence and "Replication Insensitivity" are equivalent ?

<pamelafluente_at_libero.it> wrote:
> What confuses me a little is that I thought that math is a (kind of)
> exact science. But here everything (proofs and definitions) seems that
> can be subject to opinion and I do not see resolutive statements !

Then I've failed in being clear. Certainly, math is more or less exact. The ambiguous bit here is how we define what we're talking about.

Basically, the story so far is that I tried to provide a more precise definition for what Marshall meant when he said that your "replication insensitive" functions were equivalent to the set of aggregate functions that are defined by idempotent binary functions.

First, I came up with a way fo defining aggregate functions from binary functions that I thought was sufficient. For the functions I defined, it *is* true that replication sensitivity is equivalent to idempotence of a binary generating function. However, it turns out that the functions I initially defined don't include everything that's commonly called an aggregate function.

So I then made a second attempt to define aggregate functions. Marshall also made an attempt, which I believe is equivalent to my second attempt. This second attempt does seem to include all interesting aggregate functions, but I believe that one of the possible definitions of your NOR function is a counter-example to the equivalence of binary idempotence and replication insensitivity. Specifically, that equivalence is only true when x_0 (which Marshall calls e) is an identity for the binary function.

We have been talking about (and even proving) statements about both of these two mathematical structures. The only ambiguity relates to which best captures the very imprecise discussion that was going on before. For example, early in this thread, Marshall wrote "If the binary form of the aggregate function is idempotent, the aggregate will return the same value even if values are repeated arbitrarily." -- and that initially raised the question of what is meant by the binary form of an aggregate function. Unfortunately, mathematics has no answer to the question of what Marshall meant when he wrote "the binary form of the aggregate function", so that's why the conversation isn't so cut and dry.

-- 
Chris Smith