There is a set called atoms:
atoms Ì {alphabet defined by ISO/IEC 10646}*.
The alphabet defines a language - the RDF string values.
There is a set called Nodes:
Nodes = `C È`P.
The following steps result in a definition of `C and `P.
Note that "resources"
and "property types" have to be semantically defined.
- C_{0} = { "resources" - certain semantically defined elements from atoms }.
- C_{i} = È_{all a Î
Ci-1} [T_{a,2}
(C_{i-1})]. (Note T is defined below, in terms of
C_{i-1}).
- There is a mapping prop(i) : C_{i} ® Nodes, s.t.
(prop(i))(n) = { "property types" of n - certain semantically defined elements from atoms }
for n Î C_{i}.
- Define P_{i} := (prop(i)(C_{i}).
- `P = È_{all i} P_{i}.
- Define the operator T:
T (C_{k}, i) : C_{k} ®
P_{i} ´ C_{k} ´
C_{k}, such that,
T (C_{k}, i) = {(t_{1}, t_{2}, t_{3}) |
t_{1} ÎP_{i}.
t_{2} ÎC_{k},
t_{3} ÎC_{k}}.
Note that for any (k,i) pair, T maps to a partial function. This function maps onto sets of triples.
- Then T_{a,j} (C_{k}), a subset of
T (C_{k}, k), is given by:
T_{a,j} (C_{k}) =
{(t_{1}, t_{2}, t_{3}) | t_{j} = a,
(t_{1}, t_{2}, t_{3}) Î T (C_{k}, k)},
(if j = 1 then a Î P_{k},
if j = 2 or j = 3 then a Î C_{k}).
- `C = È_{all i} C _{i}.
A description of a node n, a special case of T_{n,a} (`C), is defined to be:
T_{n,2} (`C).
Note that substituting k := i - 1 in point 2 implies:
T_{a,k}(C_{k}) Ì C_{k + 1},
so that descriptions of a node from C_{k} are contained within C_{k + 1}, i.e.
T_{n,2} : C_{k} ® C_{k + 1}.
Example:
- A description of the resource n:
- T_{n,2} (C_{0})
- A meta-description of the resource n:
- T_{ Tn,2 (C0) ,2} (C_{1}),
- which is equivalent to a description of the node T_{n,2} (C_{0}).
In the above
C_{0} and
C_{1} may be replaced by
`C. As is,
the notation reflects that only resources will be contained within
C_{0} and descriptions (not meta-descriptions) within
C_{1}.
Let S be the set of schemas.
For each i > 0 there is a surjective function schema(i), that is an instance of the operator schema that maps onto
S, i.e. schema(i) : P_{i} ® S_{i}.
This defines the pair (p_{i}, s_{i}), i.e. each property of a (possibly multi-level) description corresponds to exactly one schema.