Re: What is an automorphism of a database instance?
Date: Wed, 9 Jan 2008 01:29:02 -0800 (PST)
On 9 jan, 09:29, 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 notion of auto/isomorfism as presented by Atzeni et al. assumes that you only relabel values. If you also want to relabel relation names you extend the function h such that it also permutes relation names and attribute names, and generalize its application to a database accordingly.
Whether a formalization is appropriate depends on what you want to use the notion for. What did you have in mind?
- Jan Hidders