Re: What is an automorphism of a database instance?

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena..._at_gmail.com> wrote:
> On Jan 9, 11:13 am, Jan Hidders <hidd..._at_gmail.com> wrote:
>
>
>
> > On 9 jan, 19:10, Tegiri Nenashi <TegiriNena..._at_gmail.com> wrote:
>
> > > On Jan 9, 12:29 am, Kira Yamato <kira..._at_earthlink.net> wrote:
>
> > > > On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd..._at_gmail.com> said:
>
> > > > > On 28 dec 2007, 06:15, Kira Yamato <kira..._at_earthlink.net> wrote:
> > > > >> I need help in understanding what is an automorphism of a database instanc
> > > > > e.
>
> > > > >> The following definition is from the book Relational Database Theory by
> > > > >> Atzeni and De Antonellis:
>
> > > > >> Definition: An automorphism of a database instance r is a partial function
>
> > > > >>         h : D --> D
> > > > >> where D is the domain of the database r such that
> > > > >> 1) the partial function h is a permutation of the active domain D_r, and
> > > > >> 2) when we extend its definition to tuples, relations, and database
> > > > >> instances, we obtain a function on instances that is the identity on r,
> > > > >> namely
> > > > >>         h(r) = r.
>
> > > > >> I can understand 1), but I cannot understand 2).
>
> > > > >> In mathematics, an automorphism is a 1-1 mapping that preserves the
> > > > >> structure of an underlying set.  For example, if in some set whose
> > > > >> members x, y and z obeys
> > > > >>         z = x + y,
> > > > >> then we expect an automorphism f on that set to also obey
> > > > >>         f(z) = f(x) + f(y).
> > > > >> So, the structure of "addition" is preserved.
>
> > > > >> Now, back to relational database theory, what "structure" is being
> > > > >> preserved by 2)?  Can someone explain the formalization in 2) more
> > > > >> carefully?
>
> > > > > I only just saw your posting so I wondered if you still needed help
> > > > > with this.
>
> > > > Thanks for the follow-up.  The notion is still somewhat ambiguous in my
> > > > mind.  I sort of feel where I want to end up, but it is somewhat
> > > > difficult to formulate it in rigorous formalism.
>
> > > > What I want to formalize is the notion that two databases are
> > > > "essentially" containing the "same information" modulo a difference in
> > > > labelings of the names of the relations/attributes/values.
>
> > > > The difficulty is in formalizing the term "essentially" and "same information."
>
> > > I suggest defining automorphism of database instance (where "database
> > > instance" is understood to be a set of relations) algebraically as a
> > > mapping f such that for any relations Q and R
>
> > > f(Q /\ R ) = f(Q) /\ f(R)
> > > f(Q \/ R ) = f(Q) \/ f(R)
>
> > > this is general enough to cover both domain value permutations and
> > > column/relation renamings.
>
> > How do you know that it is not too general?
>
> I don't:-(
>
> > Btw. didn't you mean "homomorphism" rather than "automorphism"?
>
> Automorphism is a homomorphism of a database instanse into itself,
> isn't it?

It's usually defined as a kind of isomorphism.

• Jan Hidders