Re: Idempotence and "Replication Insensitivity" are equivalent ?

Marshall <marshall.spight_at_gmail.com> wrote:
> Oh, okay. I think I see what you're trying to get at. We have an n-ary
> function which we wish to express as an aggregate, that is n-ary
> NOR. That is, if all operands are F, return T, otherwise return F.
> So we could define this n-ary NOR as this aggregate:
>
> not(fold(or,F))
>
> OR() is idempotent, n-ary NOR is duplication insensitive. So
> my proposed equivalence holds.

Yes, it does for this definition. However, the other possible definition of the same function was to plug the following into your framework from an earlier post. I'm converting this to your notation, since there were gratuitous differences and it makes more sense to adopt your answer than to patch the holes in mine.

   e = true

   f(x,y) = false, if y is false
          = false, if y is true and x is true
          = true , if y is true and x is false

To be clear, this is intended for your original model; the one where there is an e.

> But the above function has some problems. For one thing,
> it's not NOR.

I don't think it's at all obvious how to define NOR as anything other than a binary relation. As such, I am willing to accept anyone else's specification for how such a function should behave. Pamela had a function which is well-specified, and I provided two possible definitions.

> For another, it's not commutative, so it's
> going to evaluate to different things depending on the
> order in which it is supplied operands.

True, the binary function is not commutative. However, the aggregate function doesn't evaluate to different things depending on the order of evaluation. Do you disagree? If so, can you provide a sample of input that evaluates to different things depending on the order?

> But even if we're generous and call it well-defined, while not
> idempotent, it's duplicate-sensitive, so it's not a counter to the
> equivalence.

How is it duplicate-sensitive? I don't think it is.

-- 
Chris Smith