Re: A different definition of MINUS, Part 3

vadimtro@xxxxxxxxx wrote:
....

Purely algebraic approach,,,, all the way to go. Consider a linear (or
more general polynomial system), e.g.

x + y = t
x - y = s

It is a mapping of input set of variables (vector) into an output set.
To extend the analogy to view updates, we also have an input delta
vector mapped into the output delta. In this particular example, it is
easy to establish that the view update problem (that is calculating
input delta vector from output one) is well defined.

Contrast this to relational model where a view is a mapping from a set
of base relations into some output relation variable. We have two
differences
i. The algebra
ii. The number of equations, because a view is defined as a single
equation

I never understood why view update problem is scoped to a single view.
In a typical usage scenario -- database schema evolution -- a set of
base tables is substituted with a set of views, and the user can see
the whole update transaction as affecting multiple views. Therefore,
it is not unreasonable to generalize view updates to a *system* of
view equations. Why do we want to do that? Two reasons:
1. Database constraints are equations, and this generalization is a
natural way to encompass them.
2. Information preservation. This one is easier to explain by the
familiar linear system example. If there is not enough (linearly
independent) equations, then there is a fundamental ambiguity of the
inverse map that calculates input delta vector from the output.



I made some more equations and tried some different cases. As long as I adjust for different headers, it all seems determinate.. Where the fun begins is when R = A <AND> B and A and B have equal headings and different extensions. This case seems to be indeterminate, which I find somehow reassuring, because when they are combined into one I find that the tuples that are not joined involve exclusive OR. I could venture that McGoveran might say that this case is trying to join contradictory propositions! This particular result somehow increases my confidence in the purely algebraic analysis. (Also the remaining case where A and B have equal headings and equal extensions, where it looks like the algebra has only one solution, deleting from both A and B, gives evidence that the pure algebra is pretty consistent and rather faithful.)

Would you like try to your hand on the case where A and B have equal headings and extensions to see if you get what I get?






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