# Re: Idempotence and "Replication Insensitivity" are equivalent ?

Date: 21 Sep 2006 09:09:15 -0700

Message-ID: <1158854955.673605.116210_at_m73g2000cwd.googlegroups.com>

Chris Smith ha scritto:

*> <pamelafluente_at_libero.it> wrote:
*

> > for an idempotent / assoc / comm function we have

*> >
**> > f( 4 3 3 3 3) = f( 4 f(3 3 3 3) ) = f(4 3)
**> >
**> > but adding another 4 and using the above result
**> >
**> > f( 4 4 3 3 3 3) = f( 4 f( 4 3 3 3 3) ) = f ( 4 f(4 3) )
**> >
**> > In general, f(4 3) and f ( 4 f(4 3) ) can be different.
**> >
**> > For a duplication insensitive function this cannot happen.
**> >
**> > Therefore I meant
**> >
**> > f( 4 3 3 3 3) = f( 4 4 3 3 3 3) holds for duplication insensitive
**> > f( 4 3 3 3 3) = f( 4 4 3 3 3 3) may not hold for idempotence
**> >
**> > Therefore
**> >
**> > duplication insensitivity does not imply idempotent (n>3)
**> >
**> > Make sense?
**>
**> No. There is an additional requirement that the aggregate function be
**> well-defined. We are not considering binary functions that do not yield
**> well-defined aggregate functions. If f(4 f(4 3)) is not equal to f(4
**> 3), then the aggregate function is not well-defined.
**>
**> (It turns out that, for your definition, being associative and
**> commutative *is* a necessary and sufficient condition for the aggregate
**> function being well-defined. So the fact that f(4 f(4 3)) must be equal
**> to f(4 3) for the aggregate function to be well defined manifests itself
**> in that if the function is associative and commutative *and* idempotent,
**> then f(4 f(4 3)) = f(f(4 4) 3) [associative property] = f(4 3) [due to
**> idempotence].)
*

Ok. Very good!

We still have a good argument for the case associ/comm are missing.

At last we should be writing an article on this stuff :)

*>
*

> See elsethread for the proofs. If I have time, I'll write them for a

*> consistent notation and set of definitions.
**>
**> --
**> Chris Smith
*

Received on Thu Sep 21 2006 - 18:09:15 CEST