Re: no names allowed, we serve types only

On 17 feb, 21:00, Tegiri Nenashi <tegirinena..._at_gmail.com> wrote:
> On Feb 17, 11:36 am, Nilone <rea..._at_gmail.com> wrote:
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> > On Feb 17, 8:51 pm, Tegiri Nenashi <tegirinena..._at_gmail.com> wrote:
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> > > On Feb 17, 10:14 am, David BL <davi..._at_iinet.net.au> wrote:
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> > > > On Feb 17, 9:15 pm, Nilone <rea..._at_gmail.com> wrote:
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> > > > > On Feb 17, 1:29 pm, David BL <davi..._at_iinet.net.au> wrote:
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> > > > > > 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").
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> > > > > 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?
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> > > > 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.
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> > > > 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.
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> > > > 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.
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> > > > Alternatively a relation can be identified with its graph.  In that
> > > > case the domains are determined as the projection of each attribute.
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> > > > 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.
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> > > > 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 :-).
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> > > > 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.
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> > > > 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.
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> > > > 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.
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> > > 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.
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> > 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.
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> > 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?
>
> I'm not sure what hidden types or actor-like model of delegating the
> work to the operand, and why it is undesirable. Predicates such as
> "Plus" do have a set of functional dependencies, so why not to allow
> these dependencies be implemented in procedural language? These could
> be considered implementation details, pretty much as indexes belonging
> to physical layer of relational model. This idea is explored in
>
> http://vadimtropashko.wordpress.com/relational-programming-with-qbql/...

You seem to be linking being untyped and representing functions/ operations with predicates. Can I ask why? Would you not agree that one can have one without the other and vice versa?

• Jan Hidders