url - Universität zu Lübeck

3.2. A MODEL FOR PATH EXPRESSIONS 51
Definition 12 (Model Mod(p))
It is obvious that Mod(p) ⊆ T .
Mod(p) ::=
{
}
t ∈ T | p(t) ≠ ∅
Lemma 2 Mod(p) = ∅ or Mod(p) is infinite.
If p is not satisfiable (e.g. 1 ) Mod(p) = ∅. On the other hand, if we have t 1 ∈ Mod(p)
we are able to create t 2 by adding an arbitrary node (element) to t 1 . Then t 2 will
also be ∈ Mod(p) because p will select the same nodes in t 2 as in t 1 . Therefore, the
set Mod(p) is infinite (without proof).
An algorithm that decides whether an XML data t is ∈ Mod(p) is introduced in
section 7.2.2.
In order to process path expressions with automaton theory, in section 7.2.2 we
need access to the nodes of a path expression.
Definition 13 (Function nodes(p) : P l → N Pl )
The function nodes(p) returns a set consisting of all node tests of the linear path
expression p. The set of all path expression nodes is denoted with N Pl .
Analogously to nodes in the XML model the functions
• p.root returns the root node of a linear path expression p;
• n.label with n ∈ N Pl returns the label of n;
• n.children returns the one child node of n if it exists or ∅ otherwise;
• n.descendant returns the one descendant node of n if it exists or ∅ otherwise;
Example 6 p = /a//b/ ∗ /b//a with p ∈ P l has the following properties:
nodes(p) = {a 1 , a 2 , b 1 , b 2 , ∗ 1 }
root(p) = a 1
a 1 .label = a
a 2 .label = a
b 1 .label = b
b 2 .label = b
1 p=//a[b and not(b)]