Re: Interpretation of Relations
Date: Tue, 23 Jan 2007 21:31:52 GMT
Message-ID: <2007012407313043042-usenet_at_thurboncom>
On 2007-01-24 00:00:23 +1000, "vc" <boston103_at_hotmail.com> said:
>> >> Depending on how you interpret relations into predicates, I would say >> that JOIN and PROJECT are kinds of inferencing rules. But they seem >> quite different to modus ponens.
>
> They have got nothing to do with modus ponens and certainly are not
> inference rules.
>
> The simplest way to see that RA is a subset of FOL is to rewrite RA
> expressions as Horn clauses:
>
> Natural join: nj(X,Y,Z) <- r1(X,Y), r2(Y,Z).
> Projection: proj(X) <- r1(X, Y).
> Selection: sel(X,Y) <- r1(X, 123).
These look like inference rules to me, as long as you consider them as axiom schema rather than sentences in the logic.
r1(a, b)
r2(b, c)
r1_nj_r2(a, b, c)
It's an inferencing rule in the sense that given a bunch of relations, you can use natural join to infer some other ones.
But I don't really want to get bogged down in terminological debates, and the horn clause interpretation is one that I'm familiar with. So thanks for the different way to think about it.
>
>
>> I have other questions, too, of course. What does it mean to close a >> set of relations under consequence? (Is is the repeated application of >> JOIN and PROJECT?)
>
> "to close a set of relations under consequence" means nothing. The
> consequence relation is defined on a set of logical formulas (e.g.
> A,B,C |= D).
I'm being loose with the terminology, granted. But what it means is 'given a set of relations, what is the set of facts that they represent'. For example, you might just define it as 'the relations, and all the ground terms represented by your nj and proj definitions above'.
>
>> What is the analog of, say, material implication?
>
> The usual.
>
>> What is a valid implication?
>
> What's your definition of a "valid implication" ?
Sorry, I should have said 'valid inferences'. Or perhaps 'what is the proof theory of the relational calculus'.
>
>> >> What parts of logical consequence do I >> lose when I represent my knowledge in a relational form.
>
> What's your definition of logical consequence ?
The usual. This thread started with me (perhaps clumsily) asking how I should interpret relations as logical statements. If I understand that translation, and what the JOIN, PROJECT, etc operations map to (thanks for your input above), then I know what subset of logic (and additions, like type information) I get when I look at a relational theory.
Cheers,
Joe
Received on Tue Jan 23 2007 - 22:31:52 CET