Avoiding bidimensional computing relation representation

The following is a part of an empirical effort for designing a computing model for optimizing relation physical represention and manipulation.

Building a relational system involves the necessity of somehow representing relations in an effective manner so that they could be truly algebrically operated. We notice however that most current physical representations still make the *implicit* assumption that the relations are operated as 2D tables.

For instance, I have found little indication onto how the computing evaluation of such representation could equate or compare to some other relation *as a relation*. But, I have hard time conceiving that a system built on relational algebric principles could do without an degree independent equality operator allowing to retun a specific value if 2 relations are to be compared. Such comparaison should of course take in consideration the unordered nature of sets and would be probably based on an effective evaluation of cardinality.

In such spirit, I thought about defining a what we could call a property named *Tuple-arity* of a relation relatively to another.

*Tuple-arity* would be algebrically expressed as

Considering relations R1 and R2 (in advance, apologies for any misunderstanding caused by anotation)

Tuple-arity of R1 to R2 written symbolically as T(R1, R2) would be defined as

=> Cardinality of (R1 /\ R2) = Cardinality (R1) --> T(R1, R2) would
evaluate as = 1

=> Cardinality of (R1 /\ R2) < Cardinality (R1) --> T(R1, R2) would
evaluate as = -1

We can then write T(R1, R2) = -T(R2, R1) = 1 (or -(-1))

(-1<1 are of course arbitrary values). I see an interest into using the property of Tuple-arity as it is degree independent and is anticommutative. It may help to express complex operations in another form than one implicitely assuming 2D representation based on keys. It may for instance allow to make use of cardinalities more efficiently to optimize relation and relation complex operations. I am curious if there have been some similar work done before out of the traditional TABLE_DEE TABLE_DUM. Moreover I would like to know the opinions about taking such path. I guess, I am afraid such path has already be taken and led nowhere.

Note: This is a purely empirical effort to determine to think about algebric tools that may help improve relation computing representation and therefore make its manipulation closer to its logical counterpart. It is neither a self promoting effort nor a logical rethinking of RM.

In advance, thank you to men of good faith for helping me on that. Received on Tue Feb 13 2007 - 13:49:32 CET