Re: Principle of Orthogonal Design(B

On Feb 7, 6:38 pm, Marshall <marshall.spi..._at_gmail.com> wrote:
> On Feb 6, 4:24 pm, JOG <j..._at_cs.nott.ac.uk> wrote:
>
>
>
> > On Feb 6, 7:12 pm, Marshall <marshall.spi..._at_gmail.com> wrote:
>
> > > > In fact, if one considers the full enumeration of the binary
> > > > predicate (denoted as S), one can state reflexivity as:
>
> > > > $B"O(Bx { a:x, b:x } $B":(B S
>
> > > > Which I find kind of neat - that sort of notation allows one to get
> > > > completely set-theoretic on a data model ass.
>
> > > Yeah, that's sort of the direction I want to go in.
>
> > > But you've still got tuple-level constructs in there, which
> > > I think we can do without.
>
> > Is a tuple level construct not a ... proposition?
>
> Fer sure.
>
> > And is that not
> > exactly what a data model should be concerned with? Perhaps I am
> > missing your point.
>
> It's not a point so much as a style of a preference for one
> formalism over another equivalent one. I am interested if
> it is possible to avoid any tuple-level constructs. I think it
> is, in much the same way that it is possible to formalize
> set theory with the subset operator instead of the
> is-a-member-of operator.

Interesting.

>
> > > Let's see:
>
> > > X(x) -- the domain
> > > R(a, b) -- the relation
>
> > > R is reflexive if the domain extended to a and b is a subset of R
>
> > > X & x=a & x=b => R
>
> > I can't follow the above I'm afraid - I'm unclear how x=a and x=b for
> > starters, but that is no doubt because I haven't read the lattice
> > theory papers you refer to. If you have a reference I will certainly
> > have a look when I get chance, or if you want to break it down to
> > brass tacks for chimps like myself, that is also welcome...
>
> Brass tacks:
>
> X(x) -- the domain
> R(a, b) -- the relation
>
> X & (x=a) & (x=b) => R
>
> X is a relation with one attribute, x.
> R is a relation with attributes a and b, taken from
> the members of X
>
> The phrase "(x=a)" denotes the infinite relation of attributes
> a and x where a = x. In set builder notation:
>
> {(x, a) | x in X and x = a}
>
> "&" is natural join. So
>
> X & (x=a)
>
> joins X with (x=a), creating a new relation with attributes x and a.

How does that differ from {(x, a) | x in X and x = a} which you already defined (x=a) as?

>
> X & (x=a) & (x=b)
>
> does the same thing, so now we have
>
> {(x, a, b) | x in X and x = a and x = b}

Would this not be simply (x=a) & (x=b)?

>
> "=>" is the generalized subset operator. It takes two relation
> operands, and evaluates to true if the left operand has a
> superset of the attributes, and a subset of the elements
> of the right operand.
>
> The thing on the left has attributes {x, a, b} so it clearly
> has a superset of the attributes of R. Its projection over
> {a, b} is {(a, b) | a in X and a = b} which is exactly the
> reflexive criteria for X, cast in the attribute names of
> R. So has all such elements in it, R is reflexive.
>
> Did that make any sense?

pretty much, although I do find the notation touch unpalatable. Where does it originate from?

>
> Marshall
Received on Thu Feb 07 2008 - 23:08:33 CET