# Re: no names allowed, we serve types only

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.
*

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 - 20:36:49 CET