Re: Understanding Logics with Aggregate Operators

On Jan 26, 8:01 am, "Jan Hidders" <hidd..._at_gmail.com> wrote:
> On Jan 25, 11:06 pm, "arthernan" <arther..._at_hotmail.com> wrote:
>
>
>
>
>
> > I agree with the saying "go to the source". So I did. I do database
> > programming for a living. And I want to know what else is out there,
> > that is based in formal methods. I just hear that there is all this
> > great research already done and we are not using it.
>
> > Aggregates seem to be one missing piece in most logic description
> > languages. So I decided to look up articles in the subject and I found
> > this one.
>
> > Logics with Aggregate Operators. (2000) (Lauri Hella).
>
> > I read the first four pages just fine, but I am stuck (for now) on page
> > 5. Anyway here is my problem:
>
> > Notes: 1)The link to the original PDF article is at the bottom. 2) The
> > symbol φ in case it's hard to read here is the Greek letter phi)
>
> > I'm not sure what this formula means
>
> > t(x) = y.φ(x,y)      (x and y have a vector sign on top of them)
>
> > And then from there I think I am also missing.
>
> > Aggr f y.(φ(x,y),t(x,y))
>
> > I don't think I know which one is the free variable.The direct answer to that question is "the variables in x (which is a
> vector of variables)".
>
> I'll try to explain a bit more. Note that they are inductively defining
> what the formulas in the logic are, and as part of that they are
> definiting here terms and in particular terms of the second sort, i.e.,
> terms that evaluate to a number. So things like "1" or "1 + 2" or "1 -
> x" (if x is of the second sort).
>
> Since aggregation operations also return a number they can also be used
> to construct a term. For example, from the formula
>
>   R(x1,x2, y1, y2) AND x1>x2
>
> we can construct
>
>   # (y1,y2) . R(x1,x2, y1, y2) AND x1>x2
>
> which counts the number of pairs (y1,y2) such that the formula R(x1,x2,
> y1, y2) AND x1>x2 holds. Clearly its result is defined if you know R
> and the values of x1 and x2, so this is a term with free variables x1
> and x2. Observe that this term can be used in a formula in any place
> where a number is expected. So, the following is a valid formula:
>
>   5  =  # (y1,y2) . R(x1,x2, y1, y2) AND x1>x2
>
> The case for other aggregation operators is analogous.
>
> Hope that helped,
>
> -- Jan Hidders-

It does, thank you very much.

The last part of the puzzle for now, is the # ___ . ___ notation.

What does it mean?

Arturo Hernandez Received on Sat Jan 27 2007 - 01:14:06 CET