Re: Idempotence and "Replication Insensitivity" are equivalent ?

Marshall <marshall.spight_at_gmail.com> wrote:
> Chris Smith wrote:
> > <pamelafluente_at_libero.it> wrote:

> > > "A predicate in logic equivalent to the composition
> > > NOT OR that yields false (F) if ANY condition is true,
> > > and true (T) if ALL conditions are false. "
> >
> > Hmm. In that case, the aggregate you want can be defined in two ways:
> >
> > 1.
> > x_0 = T
> > g(F,x) = F
> > g(T,F) = T
> > g(T,T) = F
> >
> > So in this case, it appears that you are right, under my (most correct)
> > formalism. In this case, g is not idempotent and yet f is duplication-
> > insensitive.
>
> Not quite; see my earlier post on the T,T,F case. Also you have
> a mistake above: g(F,F) = T. If g is NOR, the aggregate is duplication
> sensitive.

No, that wasn't a mistake. I'm not defining an aggregate over binary NOR, which I agree would behave as you suggest. I am instead defining the aggregate function that Pamela wanted, which is that the function is true if *none* of the members of the multiset are true, and false otherwise. (Also, I wrote this in my earlier notation, not your notation, so note that the order of arguments to my g is the opposite of the order of arguments to your f.)

> > This one works. The difference is that in the latter case, x_0 is an
> > identity of g. That condition is sufficient for idempotence to be
> > equivalent to duplication-insensitivity. Unfortunately, not all
> > aggregate functions can be converted into such a definition. For
> > example, there's no clean way to define COUNT in that manner.
>
>
> COUNT:
>
> x_0 = 0
> g(x, y) = y+1

(Inverting the arguments...) Perhaps I should have been more wordy. By "in that manner", I meant in a manner such that x_0 is an identity of g. In your definition, x_0 is 0, but g(0,5) is 1, not 5. So 0 is not an identity for g(y,x) = y+1.

-- 
Chris Smith