Re: Codd's Information Principle

> > ... In traditional algebra, valid quantifiers are values not operations.  ...
>
> I don't think there is universally agreed concept of quantifier for
> algebra.
> Carrying over quantifiers from logic, one may suggest that
> summation (http://en.wikipedia.org/wiki/Summation), product, infimum,
> and supremum are quantifiers (they are essentially generalizations of
> binary operations: addition, multiplication, meet, and join,
> correspondingly). It is common in algebra to represent qunatified
> operation in terms of binary ones; example:
>
> 1 + 2 + 3 + 4 + ... + n = n/(1-n)
You are correct. Perhaps a more appropriate term would have been *function* instead of values. But there is a *significant* difference between functions and operations since the first can be used as a variable while the other not: an operation is an operation and a variable is a variable. I do not see the clarity is an algebra that uses a similar tool both for operating and quantifying variables. And that applies to Summation as well. If we'd view Summation as an algebric expression it is easy to dispute the fact that the variable sigma is *not* the operation.

> Likewise, relational calculus quantified expression
>
> exists y : R(x,y)
I do not recall talking about calculus. Was exclusively refering to algebra.

> is essentially a disjunction
>
> R(x,1) <OR> R(x,2) <OR> R(x,3) <OR> ...
>
> (assuming positive integers domain {1,2,3,...} for y). This repeated
> application of binary operation evaluates to binary operation: set
> intersection join:
>
> D(y) set_intersect R(x,y)
>
> where D(y) is domain of y (which we assumed earlier to be
> {1,2,3,...}). The last expression evaluates to projection which is
> well known fact, but misses the big idea that universal and
> existential quantifiers are dual quantifiers. Logical quantifiers in
> algebraic form are set joins (which in some cases evaluate to
> projection and relational division).
Yes. But that does not take away the possibility of having more effective quantifiers than set joins (while keeping set joins as operations).

Since combinatory analysis allows to establish that it is impossible to build a Turing complete machine using such toolset, I particularily doubt the soundness of goiing such path in the current state of relational theory need for clarification. These are the kind of conclusions one reaches when trying to build a computing model for RM.

Regards... Received on Sat Nov 07 2009 - 00:00:19 CET