Re: RL notation

On Feb 7, 2:04 pm, Tegiri Nenashi <TegiriNena..._at_gmail.com> wrote:
> On Feb 7, 1:10 pm, Marshall <marshall.spi..._at_gmail.com> wrote:
>
> > I'm not even absolutely sure there are two different
> > equalities, though. Maybe there is just the one equality,
> > and we define the relational comparison with it. Vice versa
> > also works. But we need *something* besides just the
> > two lattice operators or we can't compare relations.
>
> I'm not so concerned about syntactic sugar as conceptual flaws

I am concerned with both. :-)

> Can we
> have just one more constant: Universal Equality Relation (denoted as
> E)? Then x=y is defined E projected to attributes x and y.

What are the advantages and disadvantages? It doesn't seem much different from having = as a logical operator. But maybe it is, and equality (and hence substitutibility) exists only in the metalanguage. I guess one advantage would be that E would then be subject to the regular axioms as any other relation is. I hadn't thought of that before ...

> What abouy
> incompatible domains, though? If x and z are incompatible, we can't
> define E projected to x,z as empty. We have to (somewhat counter
> intuitively) to define x=z as a cartesian product of the domains!

Can you be specific with what you mean by incompatible domains?

I would think the domain of the attributes of E would be the universal set of whatever is allowed to be in a tuple. (If the theory allows nested relations, then it would contain all relations; if the theory admits things that aren't relations, then it would contain all such things.)

I don't see this as an issue.

> So we have 5 constants: 00, 01, 10, 11 and E -- sounds too many.
> Although, 10 and 01 are the least intersting elements of RL, so it is
> really only 00, 11 and E that matter.

You at least need 10 and 01 as the lattice top and bottom elements, which makes each one the fixpoint and identity for meet and join. If we are going to axiomatize the RL then we need axioms to say at least this much.

> > Yes, there is some concern about relation variables
> > and attribute names appearing at the same lexical
> > level in expressions, but my tendency is to believe the
> > brevity is worth the risk...
>
> Again, syntactic sugar doesn't bother me much.

It is of no import mathematically, but the above issue is a large problem for programming languages in the real world, hence it concerns me.

My syntax is not designed for math.

> I'd suggest operating
> RL expressions in completely attribute free fascion. Whenever there is
> an expression and there is a relation with some specific constraints
> (e.g. having attribute x, or being empty), then it could be rewritten
> in more general way without these constraints. In principle generality
> should lead to simplicity....

I agree this is desirable.

I think that approach may place some limits on how expressive the resulting algebra can be. Exactly to what extent this is true will be a result of the axiomatization.

Marshall Received on Thu Feb 07 2008 - 23:29:42 CET