Re: Constraints and Functional Dependencies: Notation

On Mar 2, 3:57 pm, mAsterdam <mAster..._at_vrijdag.org> wrote: [Snipped]
>
> What, in words, are you saying with
> "∀a ∈ R,∀R(a)" ?
>
> > ... I was trying to point out some
> > additional elements making the formalism closer to what math requires
> > to gives one the right to exploit a definition in further
> > demonstrating perspective:
>
> > 1) It is preferable to express all domain of values for any element
> > present in the definition.
>
> > For instance speaking of R(a) or S(b) one *must* define what a, b, R
> > and S. In set theory, such definition can be done in two possible
> > ways : *naively* using symbols ∀, ∃, ∃!, ∈ ...(+ others such as
> > include etc...) such as or *explicitely* using formal operation
> > definition such as *| a = R(a) /\ S(b) = somevalue*.
>
> ∩, intersection (Unicode 2229), right?
Correct.

> > I personally favor the latter.
> > Bourbaki's naive operators have been strongly criticized for
> > some vagueness they induce in further demonstration.
>
> Remarks like this don't help me.
> Is it a sidetrack? - it doesn't look like one.
No just a friendly word of caution. A matter of stepping back from computing and knowing the mathematical context around Bourbakism which generated the symbols used. Understanding the mathematical limitations of Bourbakism helps understand the limitations of attribute based formalism as it was the primary source of Codd's inspiration for RM. (who could blame him? he was starting from pretty much nothing or very little)

> What is the gist? Do not use these: ∀, ∃, ∃!, ∈ ...?
> Do not use them naively? Just: use them with care?
Using them knowing their primary role in set theory: formulation or description. Their limitation reside in establishing formal demonstration based on equations/inequations. For formal demonstration, one should favor use of operations ∩, associated with diverse axioms and an appropriate structuralism.

> > It was the purpose of completing the definition stated.
>
> Back to:
>
> > 1) It is preferable to express all domain of values for any element
> > present in the definition.
>
> The OP stated that R and S are relations, and
> (not to my liking) that a and b are used both for denoting an
> attribute and for denoting an attribute value.
>
> > We can use the existing attribute
> > names as the names of the logic variables.
>
> In
> R(a) レ S(b) ≡
> ∀R(a): ∃S(b)| a=b
>
> For a=b (values) to be possible, the domains of
> a and b (attributes) must have a non-empty intersection.
Yes. You just need to express mathematically such fact by an equation. That price must be paid for precision's sake. You will find out that expressing a non empty intersection is pretty much (except for symbols) is pretty much the alternate way presented. The reason why nobody understands is because it does not assume an attribute based structuralism but it pretty much says the same thing more explicitely.

> > 2) Second, math rigor dictates that a definition must be first
> > expressed and systematically demonstrated (or negated).
> > In other words, a definition should be considered a hypothesis and
> > demonstrated thanks to set theory axiomatic (extensionnality etc...)
> > and set theory operations. Only when demonstrated, it becomes
> > acceptable.
>
> I think you'll likehttp://us.metamath.org/mpegif/mmset.html

> > ... demonstration should
> > be done not using attribute based structuralism but domain based
> > structuralism...
>
> > In such direction, R is a relation and a is relation variable
> > representing an NTuple set as opposed to current's structuralism.
>
> A third, completely different meaning for 'a'?
> That, at this point, introduces /more/ possible
> misunderstandings instead of less.
A difficult question. I am sorry I do not have an answer yet for this question. What does introduce most misunderstandings: a structuralism that is more easy to formulate in computing perspective (and concentrate more brain power and consensus) with less demonstrative opportunities or a structuralism that is less computing bound and that has allows more possibilities for formal mathematical demonstration.

> > In other words,when using R(a) - a being an attribute, one often ends
> > up with tons of undemonstrated definitions based on naive operators.
> > Oppositely, defining R(x) - x being a relation variable that may
> > *filled* by NTuples set values helps getting easier to demonstrate
> > (mostly by equations, inequations resolutions) new definitions.
>
> Sorry, I do not understand this. I get the feeling you are using the
> term 'relation variable' in a (to me) strange way - but there may be
> something else blocking my understanding.
I conceive relation variable in a prime mathematical sense as an N tuple set that would be to a relation what a variable is to a function. I have good reasons to believe that a relation is a just a special kind of function while the traditional consensus is that all functions would necessarily be relations. I have demonstrated that some functions can be something else than relations by contradicting some of the traditional set theory axioms.

In such sense, when I write R(x), I mean relation *R* with relation variable *x* R(x) = y means that R(x) equates to relation value *y* I have not found so far proofs that would make believe that such expression and formalism would be in conflict with traditional attribute based formalim. Quite contrary it is complementary and more adapted to RA equation resolution.

> > This is in no sense a conclusion but
> > an observation after several hundred
> > attempts going the way you described.
>
> You mean the way Marshall described, right?
Yes. One must note that Marshall's formulation is (almost) perfectly correct according to current structuralism but its usefulness is limited to communication and formulation rather than formal RA demonstration.

[Snipped] Received on Fri Mar 02 2007 - 19:15:55 CET