Re: What is an automorphism of a database instance?
Date: Wed, 9 Jan 2008 11:13:50 -0800 (PST)
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?
- Jan Hidders