Re: Is a function a relation?


"David BL" <davidbl@xxxxxxxxxxxx> wrote in message
news:52c7c97b-56f7-4486-a2a6-139dac691470@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jun 29, 5:11 pm, "Brian Selzer" <br...@xxxxxxxxxxxxxxxxxxx> wrote:
"David BL" <davi...@xxxxxxxxxxxx> wrote in message


The definition of 'variable' (relevant to this discussion) doesn't
constrain physical implementation choices in any way. The sense in
which a variable is deemed to hold a value can be arbitrarily
indirect, and for example may involve physical storage of deltas.

The use of relvars does constrain physical implementation--at least
indirectly. The decision to use relvars relegates delete, insert, and
update to being shortcuts for assignment.

I disagree. Variables can have many distinct ways of being
reassigned. This can be modelled by representing changes to variables
as values of various types!

Can you please elaborate?

My area of research (Operational Transform) concerns the recording of
locally generated deltas on a replicated, distributed database as
values (called "operations") that can be propagated efficiently in
order to achieve synchronisation without any need for distributed
transactions.

It sounds like a very interesting area of research. I can only imagine what
it entails.

I certainly don't think of operations as just shortcuts
for assignments. This is not just a physical distinction, it is a
logical one. In fact a logical representation of an operation allows
for the concept of intention preservation when operations are
transformed against each other in such a way that they can be applied
in different orders at different sites, yet still allow all sites in
the distributed system to converge to the same final state.

Let the execution of operation O on state S result in state S' denoted
by S' = S+O. In the OT literature two concurrent operations O1,O2
that are applied on the same initial state S are said to be equivalent
if S+O1 = S+O2. This is not a sufficient condition for O1=O2.

I'm cringing a bit here. This reminds me of the problems associated with
Sql Server replicating updates as delete/insert pairs.

This formalises the sense in which it is possible to say that delete,
insert and update can be equivalent to assignment, but not the same as
assignment.

I'm cringing again, because this implies that descriptions must be rigid
under the intended interpretation. I think that in order to be equivalent,
the result has to be not just the same syntactically, but also semantically.
When I issue an update, I am asserting that something appears different, but
when I issue a delete and an insert, I am asserting that something ceased to
exist while at the same time something else came into existence; therefore,
even though the results of two operations may be identical syntactically,
they are not necessarily semantically equivalent.

This is regardless of whether one treats the database state as the
abstract value of a database variable.


Abstract, well defined mathematical objects cannot change, so a
formalism
that admits only abstract, well defined mathematical objects cannot be
used
to model things that can change.

I already pointed out the flaw in that statement. Physical database
systems are variables not values.

I don't think it is flawed since those variables can only contain
abstract,
well defined mathematical objects that cannot change.

I don't see what the problem is.

When you say a value means different things at different times it
seems like you're thinking a value is what C.Date calls an
appearance
of a value, which actually has more to do with a variable that
exists
in time and space.

Please don't misrepresent me: I didn't say that a value means
different
things at different times; I said that a term in a formal language can
mean
different things at different times.

Also, a variable is a container for a value. How can it exist in time
and
space?

You're going to have to expand on that.

What do you want to know?

When someone uses a debugger to view values of integer variables on
the stack frame, are you saying they aren't variables, or that they
don't exist in time and space?

Now you're talking about a physical manifestation of the abstract.

I don't consider relation values to have names.

That doesn't surprise me. I suppose that you would attribute the same
meaning to a relation that is constructed by explicitly stating the
heading
and tuples, as a relation consisting of the exact same set of tuples
that
is
the result of a union of two relations in the database?

Relation values aren't constructed (rather they are selected). You
make it sound like they can suddenly spring into existence.

They can in D. See <relation selector inv> on page 110 of TTM 3rd
Edition.

That is a *selection* of a relation value. It doesn't cause a
relation value to come into existence. D&D are careful about this -
that's why they use the term selection not construction.

I don't know about you, but the right hand side of the assignment,

PQ := RELATION { TUPLE { P# P#('P1'), QTY QTY( 600) } ,
TUPLE { P# P#('P5'), QTY QTY( 500) } ,
TUPLE { QTY QTY(1000), P# P#('P2') } ,
TUPLE { P# P#('P4'), QTY QTY( 500) } ,
TUPLE { QTY QTY(400), P# P#('P3') } ,
TUPLE { P# P#('P6'), QTY QTY( 100) } } ;

From page 153, TTM 3rd Edition, looks to me like a relation that just sprang
into existence!

Since relation values don't exist in space or time it doesn't make
sense to think that they in themselves record facts about the real
world. They only do that when they appear as encoded values in a
physical database (as the current value of a relation variable).

Not sure what you mean by this.

Given a relation type (i.e. schema), there are a number (possibly
infinite) of relation values that conform to that type. The type and
the values of that type are all abstract and carry almost no
information.

Consider the analogy of a textual string. If you could look at a vast
number of random strings, you might occasionally find some interesting
ones like Shakespeare's Macbeth. Information comes from the
*selection* of useful values from amongst the multitude. An analogy
is how a sculptor uncovers a work of art in a block of stone even
though it was there all along.

The amount of information in a value can be quantified by the size of
the smallest program that can generate that value. A program to
generate the set of all strings is trivial (at least on an abstract
machine that's equivalent to a Turing machine with its infinite
tape!), whereas a program to output only one selected string is
arbitrarily large.



.

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