# Re: Does Codd's view of a relational database differ from that ofDate&Darwin?[M.Gittens]

Date: 17 Jun 2005 07:16:19 -0700

Message-ID: <1119017779.378065.148070_at_g49g2000cwa.googlegroups.com>

Jon Heggland wrote:

> I understand him as saying that some "join paths" should be considered

*> special, and others forbidden, thus simplifying the formulation of some
**> queries (by not requiring the joins to be stated), but disallowing
**> others altogether. In other words, the functionality that views provide,
**> but with additional limitations.
**>
**> A more generous reading might be that the other queries are not
**> disallowed, but that their joins must be stated explicitly.
**>
*

Thank you.

> > Now, a more interesting question is what exacly AS's model is. It appears

*> > that it's different from concept lattices and I utterly failed to get a
**> > concise explanation from the author. Would you be able to explain the
**> > difference, by any chance ?
**>
**> Sorry, but I am not familiar with concept lattices.
*

As far as I understand, the FCA (concept lattices) approach is to view the world as consisting of 'objects' and 'attributes'.

Let's define a 'formal context' as a triple (O, A, R) where O and M are
sets of 'objects' and 'attributes' respectively and R is a binary
relation defined on OxA. Further, for X <= O let's define two sets:
X' := {a in A | (o, a) in R forall o in X}
and, similarly, for Y <= A

Y' := {o in O | (o, a) in R forall a in Y}

Then a pair (X, Y) is a 'formal concept' iff X<=O, Y<=A, X'=Y and X=Y'. X is called the 'extent' of the concept and Y the 'intent'. (Some authors, rather confusingly, say that X is the 'intent' of Y and Y is the 'extent' of X).

One can define an order amongst 'formal concepts' for given context as:

(X1,Y1) 'le' (X2,Y2) iff X1 <= X2 (or iff Y2 <= Y1)

[ Some call (X1, Y1) a 'formal subconcept' and (X2,Y2) a 'formal superconcept'.]

It can be shown then that the ordered set of all formal concepts of the formal context (O, A, R) form a complete lattice which is called the 'concept lattice' of (O,A,R).

Note that the FCA folks nowadays carefully use the adjective 'formal' probably after having been bitten by controversial philosophical associations in the past. In earlier works, they just freely talked about concepts, contexts and such.

Thanks.

vc

*> --
**> Jon
*

Received on Fri Jun 17 2005 - 16:16:19 CEST