url - Universität zu Lübeck

7.2. INTERSECTION OF TWO PATH EXPRESSIONS 129
7.2.2 Automaton for Mod(p)
The general idea of the intersection algorithm is to build two finite automata A
and A ′ with A accepting Mod(p) and A ′ accepting Mod(p ′ ) with p, p ′ ∈ P labs two absolute
linear path expressions.
Having A and A ′ we build the product automaton B . The emptiness of the intersection
of p and p’ is a property of B.
Unfortunately, finite automata are defined on the basis of a finite alphabet. In
contrast, path expressions operate on XML data with an infinite alphabet because
the node labels are not limited.
In section 3.2.1 we showed that a linear path expression p ∈ P labs can be transformed
into a regular expression r ∈ REG Σ,α with Σ = Σ(p) and α ∉ Σ an arbitrary
new symbol.
Lemma 3 For any regular expression r ∈ REG Σ,α , respectively its language L r , one
can construct a finite automaton A that decides whether an input string is a word
of L r or not.
The proof of the lemma is omitted here because it is basic knowledge in theory of
finite automata. The proof can be found, for instance, in [52].
When reading an XML data as input for a finite automaton we have the following
problem: The automaton expects a string of several symbols in a defined order.
In tree-like XML data a node may have several children so that the next symbol
(element label) is not defined unambigiously.
Therefore, we define a function path leaf : T → P(string) that extracts all paths
(sequences of nodes) from the root node to each leaf element node in the XML
data. Text nodes are ignored as they are not affected by linear path expressions.
The paths are returned as strings built from the labels of the contained nodes.
The function is defined as follows:
Definition 28 (Function path leaf )
path leaf (t) = path(t.root)
{
path(n) =
n.label : n.children = ∅
n.label + ”; ” + {path(c)|c ∈ n.children} : otherwise
with t ∈ T and n ∈ N; + denotes the concatenation of strings with a + ”; ”{b, c, d} =
{a; b, a; c, a; d}. The semicolon is a delimiter used to distinguish different element
label (e.g. a; b ≠ ab).