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Re: completeness of the relational lattice

From: Jan Hidders <hidders_at_gmail.com>
Date: Sat, 23 Jun 2007 13:53:28 -0000
Message-ID: <1182606808.002178.145700@n2g2000hse.googlegroups.com>


On 23 jun, 02:53, Vadim Tropashko <vadimtro_inva..._at_yahoo.com> wrote:
>
>
> I cleared my confusion. What would be the result of join betwwen empty
> relation R and empty relation S with disjoint attributes? 00, of
> course.

Sure? I would describe that as [H1] /\ [H2] wich is equivalent to [H3] where H3 is the union of H1 and H2, wich would not be empty.

Btw. let me give a rough sketch of what my plan of attack is going to be. It takes very big steps and ignores important technical issues, but it should give you an idea of what my ideas are. The proof is going to proceed in several stages with several normal forms. The syntax I'll start from is:

E ::= R | E /\ E | E \/ E | 11 | [H] | 'x=y'

I allow H to be empty, so I omitted 00 = [], 01 = 11 \/ [] and 10 = 11 /\ []

The first stage will be a normalization to a "union normal form" that I already talked about. This normal form means that the expression is in a form r_1 \/ ... \/ r_n with the r_i such that the contain no \/ except in the form s \/ [H]. I will refer to these r_i's as the disjuncts.

Conjecture: we have enough rewrite rules to achieve this normal form.

Some of the disjuncts will be conjunctive queries (CQ) and some will not. There are three reasons why some are not in CQ: (1) the result may be always empty, (2) the result is (infinitely) wide and (3) the result is (infinitely) long.

Conjecture: if a disjunct always returns an empty result it can be rewritten to []

That would take care disjuntionts with problem (1). For the other two problems I conjecture the following:

Conjecture: if a disjunct returns a wide or long result it can be rewritten to r /\ W or r /\ W /\ 11 where r is a query in CQ and W a conjunction of 'x=y's such that x and y are not in A(r).

If this holds then we can probably split the problem in showing completeness for queries in CQ and for special queries such as W.

Conjecture: the rewrite rules are complete for boolean combinations of 'x=y's

So we can the concentrate on expressions that represent normal CQ's. We can think of those queries as expressed by Horn clauses or:

  { (c=x d=y) | R(a=x, b=x) /\ S(a=x, b=y) /\ T(a=y, b=z) }

In fact you might read that as a shorthand for our algebra where for example
- S(a=x, b=y) denotes (S /\ 'a=x' /\ 'b=y') \/ [x,y] - { (c=x d=y) | r } denotes (r /\ 'c=x' /\ 'd=y') \/ [c,d]

Conjecture: all disjuncts in CQ can be written in this form

For completeness we need to show that if two such queries subsume each other we can perform absorption. This is enough because for CQ's it holds that disjunctions of CQ's between which there is no subsumption that they are equivalent iff it describes the same set of CQ's. So what is needed for this? We need to be able to rename the variables, e.g., in the above example we should be able to replace x, y, z with u, v an w, for example. Of course this is only possible because these are projected out afterwards.

Conjecture: the variables can be renamed with the rewriting rules.

Probably this requires that we have or can derive a rule like:

  r /\ [H] = (r /\ 'x=y') \/ [H] where x and y not in H, and x in A(r) and y not in A(r)

We should also be able to make two variables the same:

  r /\ [H] = ((r /\ 'x=y') \/ r) \/ [H]

Also needed is a rule to merge two queries:

  { (c=x d=y) | r1 } \/ { (c=x d=y) | r2 } = { (c=x d=y) | r1 \/ r2 }

These would allow us to generate from a disjunct versions with renamed variables and versions where more variables are required to be equal, which are exactly all queries that are subsumed. So if another disjunct is subsumed we can do the reverse, i.e., have absorption.

Until sofar. I hope this made some sense. If have to move to other work now, so I might be less present in the coming days.

Received on Sat Jun 23 2007 - 08:53:28 CDT

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