url - Universität zu Lübeck

132 CHAPTER 7. THE XML INDEX UPDATE PROBLEM
3. ∀x ∈ nodes(p) : x.descendant = y , A has |Σ(p) ∪ α| transitions (q y ; s) → q y for
all s ∈ Σ(p) ∪ α. These transitions build loops on q y
4. A has one transition (q 0 , e.label) → q e with q 0 the bottommost state that has
no incoming transition.
These transitions lead to a nondeterministic finite automaton. The terminal state
is q p.root . The initial state is q 0 .
Example 23 The linear path expression p = /a/ ∗ //c//d leads to an automaton
A with Q = {q a ,q ∗ ,q c ,q d ,q 0 } and σ = {t 0 ... t 14 }. The alphabet is Σ(p) = {a, c, d}
The following transitions are built:
t 0 = (q 0 ; ”d”) → q d ,
t 1 = (q d ; ”a”) → q d ,
t 2 = (q d ; ”c”) → q d ,
t 3 = (q d ; ”d”) → q d ,
t 4 = (q d ; ”α”) → q d ,
t 5 = (q d ; ”c”) → q c ,
t 6 = (q c ; ”a”) → q c ,
t 7 = (q c ; ”c”) → q c ,
t 8 = (q c ; ”d”) → q c ,
t 9 = (q c ; ”α”) → q c ,
t 10 = (q c ; ”a”) → q ∗ ,
t 11 = (q c ; ”c”) → q ∗ ,
t 12 = (q c ; ”d”) → q ∗ ,
t 13 = (q c ; ”α”) → q ∗ ,
t 14 = (q ∗ ; ”a”) → q a ,
t 15 = (q 0 ; ”a”) → q 0 ,
t 16 = (q 0 ; ”c”) → q 0 ,
t 17 = (q 0 ; ”d”) → q 0 ,
t 18 = (q 0 ; ”α”) → q 0 .
Figure 7.2: Automaton accepting Mod(p) with p = /a/ ∗ //c//d
For an XML data t A processes all strings of the set rename(path leaf (t)) bottom-up,
i.e. from a leaf node to the root. A visual representation of A is on the right side
of figure 7.2. The initial state is q 0 ; q a is the one final state. Please note that the
transitions are not deterministic (e.g. t 2 , t 5 ). Analogously to tree-automata the
finite automaton processes the XML data bottom-up. Basically, it is possible to
build an equivalent automaton that processes the input top-down.