Re: Idempotence and "Replication Insensitivity" are equivalent ?

Chris Smith wrote:
> 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.

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.

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

Did we ever see a counterexample to the

  idempotence <=> duplicate-insensitive

equation? I've been spotty about reading the last day or two.

Marshall Received on Thu Sep 21 2006 - 03:13:05 CEST