Re: TRUE and FALSE values in the relational lattice

On Jun 19, 2:51 pm, Marshall <marshall.spi..._at_gmail.com> wrote:
> On Jun 19, 2:06 pm, Vadim Tropashko <vadimtro_inva..._at_yahoo.com>
> wrote:
> > Therefore, associating any other relation than 01 and 00 with
> > TRUE and FALSE, correspondingly, doesn't seem right.
>
> I agree with the feeling; it is why I was saying 00 and 01
> previously....
> Well, I just wrote up an explanation and it wasn't at all convincing.
> I'll have to review my notes. Perhaps I have been overly influenced
> by reading lattice logic papers that don't have 00.

The confusing part is that there are many homomorphisms of relational lattice into various boolean algebras. And in boolean algebra we always have TRUE as the top lattice element and the FALSE is the bottom. Figure 2 of
http://arxiv.org/ftp/cs/papers/0603/0603044.pdf shows 4. One of them is the 16 element boolean algebra highlighted in blue, which has the top element 10, and the bottom element 11.

The other important homomorhism is concerned whether relation is empty or not, so that the lattice is mapped into the boolean algebra of the two elements 00 (top) and 01 (bottom).

Recently, in the "relational lattice complement" thread you discovered something, which turned out to be yet another homomorphism:

R -> (R \/ 00) /\ (R \/ 11)

The target boolean algebra has the top element coinsident with the top lattice one 10, and the bottom element coinsident with the bottom lattice element 01.

> > Assuming the domain x = {a,b,c,d,...} the relation "exists(x) R(x,y)"
> > evaluates to
>
> > R(a,y) \/ R(b,y) \/ R(c,y) \/ R(d,y) \/ ...
>
> > I don't see any emptiness here.
>
> It seems you are defining existential quantification as something
> that is relation valued, but I am used to thinking of it as truth-
> valued.
>
> Assuming
>
> R(a1,b1)

Let me doublecheck that a1 and b1 are supposed to be some values here.

> Is your definition any different than
>
> R /\ `x=a1`
>
> Here a1 is an attribute name and x is free, aka a parameter
> of the expression.

Now I'm confused. If "a1" is an attribute name (I thought it is value), what is then "x"?

In the epression from my earlier post:

R(a1,y) \/ R(a2,y) \/ R(a3,y) \/ R(a4,y) \/ ...

each unary relation R(ak,y) is defined as

R(ak,y) =def= (R /\ `x=ak`) \/ `y`

so I kind of see your "R /\ `x=a1`" as a fragment of this definition... Received on Wed Jun 20 2007 - 01:16:13 CEST