Towards a definition of atomic
- From: David BL <davidbl@xxxxxxxxxxxx>
- Date: Fri, 1 Feb 2008 05:30:25 -0800 (PST)
AFAIK the conventional wisdom is that no absolute definition of atomic
exists for domain types. Throwing caution to the wind, in this post I
wish to conjecture a definition of atomic that, perhaps with some more
effort at its formalisation, can provide some absolute meaning for a
given attribute within a given RDB schema.
The examples are a little contrived, but are only meant to be
illustrative.
Example 1:
"Einstein discovered the formula E = mc^2"
"Newton discovered the formula F = ma"
Example 2:
"Bill is a parent of { Mary, John }"
"Mary is a parent of { Don, Alex, Sue }"
In example 1, in Prolog we can define a predicate 'discovered' to
represent the two facts as follows
discovered(einstein, eq(var("E"), prod(var("m"),
pow(var("c"),num(2))))).
discovered(newton, eq(var("F),prod(var("m"),var("a")))).
In previous threads I have discussed how it is not possible to
decompose the information in nested expressions into a set of
propositions about the nodes without the introduction of node
identifiers.
By contrast, in example 2 it is straightforward to map the two facts
into five (by decomposing the sets of children) as follows
parent(bill,mary).
parent(bill,john).
parent(mary,don).
parent(mary,alex).
parent(mary,sue).
I think we could make the meaning of "atomic" more tangible if we can
define what decompositions of an attribute are valid.
A decomposition of an attribute involves translation from an original
DB schema to a new DB schema with alternative relations. Of course
the original attribute (and therefore its containing relation) isn't
present in the new DB schema.
Firstly ISTM that a valid attribute decomposition must be *non-
trivial*, and perhaps this could be formalised somehow using entropy
(by saying that the new attribute(s) have less states available than
the original attribute). Although I'm not sure exactly how to state
this mathematically, I expect one would find general agreement on what
a non-trivial decomposition means in practice.
Secondly (and this is where the examples are relevant), a valid
decomposition must coincide with a defined bijection that maps a DB
state in the original schema to a DB state in the new schema. This
is where those node identifiers in the first example come to play,
because they seem to be at odds with defining such a bijection.
Putting it more simply, it seems that the node identifiers aren't
functionally dependent on the original DB state. It is for this
reason that one may claim that such a decomposition is unreasonable -
in the sense of not achieving information equivalence as a set of
propositions.
Unfortunately, it seems that one could be tricky and come up with a
bijection that makes the node identifiers functionally dependent on
the underlying DB state; by defining some unique ordering on the
identifiers (one could then use integers according to ordinal
position). I say this is unfortunate because it upsets the proposal
for a simple meaning of atomic. However, I wonder whether things can
be salvaged at the expense of a complicated definition of atomic, by
introducing a constraint on the bijection that it not be pathological,
in the sense that addition of information shouldn't be able to cause
widespread reassignment of identifiers. The very fact that this can
happen points to their arbitrary or meaningless nature, and more
specifically to the fact that they identify a sub-value rather than an
entity in the UoD.
Continuing with example 2, note that no further decomposition allowing
information equivalence is possible. For example, a person's name is
represented as a string domain type, and this is atomic because any
attempt at decomposing the string into its individual characters
forces the introduction of additional identifiers.
.
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