Re: Why is "group by" obligatory in SQL?

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> >> What are the fundamental limitations of binary systems one should take into considerations when representing and operating relations.
> I'd say the very first one is that while we can 'see' or interpret a
> relation in the mind's eye, the machine representation, whatever form it
> takes, is by nature (and by definition where an implementation is
> concerned), idiosyncratic.
Yes but I was not refering only to informal limitations but also to formal limitations such as how to represent a relation in a Ncoordinate  referential, be it a vectorial referential, a fractal based referential or even a trigonometric referential.

Some of these limitations can be mathematically evaluated to quantify numerically the logical weight of operations. Such knowledge can for instance allow the quantification a direct image system relation representation vs a column store relation representation in function of degree and cardinality. This is the core of my work.

> >> How to effectively represent disjointness between two sets
> > ...
>
> Yes, given that representation is very much the basic decision for
> implementation.  And what 'principles' (maybe axioms is a better word)
> could be adopted to interpret disjointness?
I would not call an effective algorhythmics for effectively representing disjoint sets an axiom but I hear you. It is also very difficult to exchange online on such complex subject, but let me try express some premices I used so far.

--> A formal system that requires a logical ordering of elements to represent disjointedness or some other property between 2 sets is a direct image system.
--> Properties of disjointedness between 2 sets can only be expressed if the 2 sets have common constraints that allow to define these sets. --> Unordered representations must allow a numeric quantification of logical IO's (a logical IO being the elementary constitution of a single tuple). This quantification is a mathematical function that involves relation cardinality and degree as well as the number of permutations required to establish the property of disjointedness. --> Disjointedness and other set level properties should have equal logical IO's whether the members of the composing set ordered or not.

> How practical are the
> variious choices?  
Unfortunately, practicality is a highly subjective concept. My guess is that, o a fundamental level, the practical aspect depends on the familiarity one has with mathematical models used to optimize the establishment of the property of disjointedness. I have used 3 models so far: vectorial representations, fractal representations and trigonometrical representations. While I find fractals very practical to make effective permutations for unordered sets, I prefer trigonometric representations to express symetry between disjointedness and its opposite intersections.

> Should an implementation assume disjoint sets that
> are eligible subsets of  some relation are necessarily contrary assertions?
I am not totally sure what you mean by contrary assertion but if you mean it as constraints defining the subset then the answer would be no. One can only assume that the product of their intersection is the empty set. Received on Fri Jul 24 2009 - 20:49:48 CEST