1. Tree Automata First: A reminder on Automata on words Typing semistructured data

2. Finite state automata on words Alphabet State Initial state Accepting states Transitions Typing semistructured data

3. q0q0 Nondeterministic automaton: Example a b a a b - a b a - q0q0 q1q1 q0q0 q0q0 q1q1 q0q0 q1q1 q0q0 q0q0 q1q1 q0q0 q0q0 q2q2 q1q1 q0q0 KO OK

4. Deterministic – No  transition – No alternative transitions such as Determinization – It is possible to obtain an equivalent deterministic automaton – State of new automaton = set of states of the original one – Possible exponential blow-up Minimization Limitations – cannot do – Context-free languages Essential tool – e.g., lexical analysis Reminder

5. Reminder (2) L(A) = set of words accepted by automata A Regular languages Can be described by regular expressions, e.g. a(b+c)*d Closed under complement Closed under union, intersection – Product automata with states (s,s’) where s is from A and s’ is from A’

6. Automata on words versus trees a bba a b b a b b ab a Left to right Right to left No difference Bottom upBottom up Top downTop down Differences

7. Automata Automata on ranked trees Typing semistructured data

8. Binary tree automata Parallel evaluation For leaves: For other nodes: a b b a b ab a Bottom upBottom up q q’ b q” q1q” q2 qqq’ Typing semistructured data

9. Bottom-up tree automata Bottom-up: if a node labeled a has its children in states q, q’ then the node moves nondeterministically to state r or r’ Accepts is the root is in some state in F Not deterministic if alternatives or  -transitions:

10. Example: deterministic bottom-up

11. Boolean circuit evaluation v v v 1 v v 1 1 0 v 0 1 1 OK

12. Regular tree language = set of trees accepted by a bottom-up tree automaton Typing semistructured data

13. Regular tree languages Theorem: the following are equivalent – L is a regular tree language – L is accepted by a nondeterministic bottom-up automaton – L is accepted by a deterministic bottom-up automaton – L is accepted by a nondeterministic top-down automaton Deterministic top-down is weaker

14. Top-down tree automata Top-down: if a node labeled a is in state q”, then its left child moves to state q, right to q’ Accepts is all leaves are in states in F Not deterministic if

15. Why deterministic top-down is weaker? Consider the language – L = {,,, ) } It can be accepted by a bottom-up TA – Exercise: write a BUTA A such that L = L(A) Suppose that B is a deterministic top-down TA that accepts both trees in L – Exercise: Show that B also accepts – A contradiction Fact: No deterministic top-down tree automata accepts exactly L

16. Ranked trees automata: Properties Like for words Determinization Minimization Closed under – Complement – Intersection – Union

17. But… XML documents are unranked: book (intro,section*,conclusion)

18. Automata Automata on unranked tree Typing semistructured data

19. Unranked tree automata Issue: represent an infinite set of transitions Solution: a regular language

20. Rule: Meaning: if the states of the children of some node labeled a form a word in L(Q), this node moves to some state in {r 1,…,r m } Unranked tree automata (2)

21. Building on ranked trees a b b b b ab ab a b b b b ab ab Ranked tree: FirstChild-NextSibling F: encoding into a ranked tree F is a bijection F -1 : decoding

22. Building on bottom-up ranked trees (2) For each Unranked TA A, there is a Ranked TA accepting F(L(A)) For each Ranked TA A, there is an unranked TA accepting F -1 (L(A)) Both are easy to construct Consequence: Unranked TA are closed under union, intersection, complement Determinaztaion also possible, a bit more tricky

23. Top-down? This is more delicate Transition  (a,q)=A(a,q) – The state of the automata A(a,q) when reading the labels of the children of a node labeled a determines the states of the children of that node – Accepts if all the leaves are in accepting state

24. Boolean circuit evaluation v v v 1 v 1 1 01 0 v 1 1 1 1 10 0 v v v A tree is accepted if, for some possible run, the states of all leaves are final