On Identity and Equality
For some time my email signature read:
Identity is:
- the difference between 'this' and 'that'
- not subject to change
- absolutely necessary
I was reading the second edition of The Third Manifesto around that time and I emailed Hugh Darwen on the subject of Identity and Equality in The Third Manifesto. A lengthy discussion on this and related topics followed.
This discussion caused me to write A critical reading of The Third Manifesto amended here and The Third Manifesto Revisited detailing my findings.
Sometime later while enjoying the pleasantries of a warm summer day in the south of Holland, Adriaan van Os of Microbizz fame told me about an interesting type of binary trees which he baptized as differential trees. Adriaan has kindly allowed me use his inventiveness to illustrate a few key characteristics of identity, which contribute to my opinion that the relational model according to Codd is to be preferred to modeled pitched as such by Date and Darwen.
Consider the following example of a classical binary tree.
It is used to store a few integer numbers in an arrangement which proves quite handy for different purposes in computer science.
Differential trees
The following binary tree, called a differential tree, stores the same numbers as the one above. In other words they both trees store the set integer numbers: 2,3 4,5,7, 8, 9, 10, 12, 14.
As Adriaan explains, differential trees store values corresponding with differences between identities stored in the tree. So, for example number 4 is stored as 4 - 8 = -4 and the number 5 is stored as 5 - 4 = 5 - (8 - 4) = +1.
Here I want to point out that even though both trees store the same set of numbers, differential binary trees are unique in that these trees can store numbers whose values make no appearance in the tree. In fact the particular value stored to represent a specific number depends on the values already stored in the differential binary tree.
With a quick wink to the Bourne Identity I will refer to data structures based on characteristics of differential binary trees as data structures employing Van Os Identity.
Van Os Identity and Databases
The relevance of the above in the context of relational databases becomes apparent when we consider that the relational model according to Date en Darwen does not recognize the requirement for a system generated, immutable and unique tuple identifier while to Codd does.
Date and Darwen, claim that the value of a tuple represents the identity of a tuple, thus rendering it unneeded to recognize the
identity of a tuple separate from value of a tuple.
Van Os identity establishes that the identity of a value is a tree (or a relation for that matter) is not necesarily related to the value of said tuple.
Yes, this is a big deal! Stay tuned to see how these tuple identifiers can be used to, in the words of Codd, allow the relational model to capture more meaning.