Re: no names allowed, we serve types only

From: Nilone <reaanb_at_gmail.com>
Date: Wed, 17 Feb 2010 11:36:49 -0800 (PST)
Message-ID: <71826aa9-776c-4cdf-8ffe-8d8eb4311309_at_q21g2000yqm.googlegroups.com>



On Feb 17, 8:51 pm, Tegiri Nenashi <tegirinena..._at_gmail.com> wrote:
> On Feb 17, 10:14 am, David BL <davi..._at_iinet.net.au> wrote:
>
>
>
> > On Feb 17, 9:15 pm, Nilone <rea..._at_gmail.com> wrote:
>
> > > On Feb 17, 1:29 pm, David BL <davi..._at_iinet.net.au> wrote:
>
> > > > Operators can be formalised without a type system too.  Simply
> > > > formalise an operator as a function defined on some domain, where a
> > > > domain is merely a set (not a "type").
>
> > > Thanks for the introduction, I haven't seen the typeless model
> > > before.  I don't see how such a system would handle arithmetic
> > > operators (e.g. + and <) and string operators like concatenation and
> > > search - could you perhaps give an example?
>
> > In a typeless system a unary function could for example be formalised
> > as a triple (D,C,G) where D is the domain, C is the co-domain and G is
> > the graph of the function (a subset of DxC).  This is typeless in the
> > sense that a function value doesn't formally have any concept of a
> > defined type.  Rather the domain and co-domain are formally part of
> > the function's value as a triple (D,C,G).  For example two functions
> > can have the same domain and graph but different co-domains.  That
> > makes them distinct.  This is actually conventional, as when one
> > determines whether a given function is surjective (i.e. its range
> > equals its co-domain).  It wouldn't make sense to ask whether a
> > function is surjective if its co-domain isn't part of its value.
>
> > Alternatively a typeless system could instead formalise a unary
> > function as a set of pairs (i.e. what we above called its graph).  In
> > that case the domain and range is determined from the graph using
> > projection, but there is no concept of a co-domain.
>
> > Similarly a typeless system could formalise a relation in two
> > different ways.  One allows for attributes to have domains specified
> > independently of the graph (or "body") of the relation, and these
> > domains represent part of the relation's value.  That means that two
> > distinct relations can have exactly the same graph.
>
> > Alternatively a relation can be identified with its graph.  In that
> > case the domains are determined as the projection of each attribute.
>
> > In a typeless formalism one is very clear about what it means for two
> > functions or two relations to be equal.   It seems to require more
> > effort to understand what equality means in a typed formalism.
>
> > In a D&D type system, a value has a MST, but this actually depends on
> > the prevailing type system.  E.g. two different databases could use
> > different type systems.  Putting it another way, the MST of a value
> > depends on who you ask :-).
>
> > D&D want to ensure that equality of values is independent of declared
> > types.  That's why they say that a selector for an ellipse value that
> > happens to specify equal width and height actually returns a value
> > which has an MST of circle.  It's like a "magic downcast".  They point
> > out that OO systems don't normally work that way.  E.g. an OO
> > constructor for ellipse never returns a circle.
>
> > I think D&D end up treating relations with the same body and attribute
> > names as equal.  i.e. in essence the declared attribute domains are
> > not part of the relation's value.  I think they define subtyping of
> > relation types accordingly.
>
> > It seems to me that D&D spend a lot of effort discussing ideas that
> > are either eliminated or trivialised in a typeless formalism of the
> > RM.
>
> Formalization is less of an issue here. I interpret the question as
> how to make a working system operating predicates such as Plus(x,y,z)
> and Concat(x,y,z). Logical programming provides sort of an answer.

You're right. I'm a programmer rather than a mathematician. As such, infinite sets can only be approximated and every value has a cost in space and time. So I'm interested in how operators would be effectively (and hopefully, efficiently) defined in a software version of such a model.

The operators in a typed system are based on inspecting and manipulation the representation of values. I don't see how anything similar is possible in an untyped relational model. There's exhaustively generating all operands and results, which is impractical. With a successor operator defined (again, exhaustively?), we can define plus inductively, which would be highly inefficient. Is there a way to define these operators without resorting to hidden types or an actor-like model of delegating the work to the operand? Received on Wed Feb 17 2010 - 13:36:49 CST

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