Property testing of Tree Regular Languages

1. Property testing of Tree Regular Languages Frédéric Magniez, LRI, CNRS Michel de Rougemont, LRI , University Paris II

2. Property testing of Tree Regular Languages • Tester for regular words with the Edit Distance with Moves 2. Tester for ranked regular trees with the Tree-Edit Distance with Moves,

3. Testers on a class K Let F be a property on a class K of structures U An ε -tester for F is a probabilistic algorithm A such that: • If U |= F, A accepts • If U is ε far from F, A rejects with high probability • Time(A) independent of n. (Goldreich, Golwasser, Ron 1996 , Rubinfeld, Sudan 1994) Tester usually implies a linear time corrector.

4. History of Testers Self-testers and correctors for Linear Algebra ,Blum & Kanan 1989 Robust characterizations of polynomials, R. Rubinfeld, M. Sudan, 1994 Testers for graph properties : k-colorability, Goldreich and al. 1996 graph properties have testers, Alon and al. 1999 Regular languages have testers, Alon and al. 2000s Testers for Regular tree languages , Mdr and Magniez, ICALP 2004

5. Edit distance on Words • Classical Edit Distance: Insertions, Deletions, Modifications • Edit Distance with moves 0111000011110011001 0111011110000011001 3. Edit Distance with Moves generalizes to Trees

6. Testers on words Simpler proof which generalizes to regular trees. L is a regular language and A an automaton for L. Admissible Z= A word W is Z-feasible if there are two states accept init

7. The Tester Tester. Input : W,A, ε For every admissible path Z: else REJECT. Theorem: Tester(W,A, ε ) is an ε -tester for L(A).

8. Proof schema of the Tester Theorem: Regular words are testable. Robustness lemma: If W is ε-far from L, then for every admissible path Z, there exists such that the number of Z-infeasible subwords Splitting lemma: if W is far from L there are many disjoint infeasible subwords. Amplifying lemma: If there are many infeasible words, there are many short ones.

9. Merging Merging lemma: Let Z be an admissible path, and let F be a Z-feasible cut of size h’ . Then C C C C C C Take each word and split it along its connected components, removing single letters. Rearrange all the words of the same component in its Z-order. Add gluing words to obtain W’ in L:

10. Splitting Splitting lemma: If Z is an admissible path, W a word s.t. dist(W,L) > h, then W has Proof by contraposition:

11. Tree-Edit-Distance a b Deletion Edge a e c b a b e c d Insertion Node and Label f e e d c Tree Edit distance with moves: a a 1 move b b e e c d c d Distance Problem is NP-complete, non-approximable.

12. Tree-Edit-Distance on binary trees Binary trees : Distance with moves allows permutations Distance(T1,T2) =4 m-Distance (T1,T2) =2

13. Tree automata (q1,q1)q2 (q1,q0)q2 (q2,-) q2 (-,q2) q2 • (q0, q0)  q1 • (q0,q1)  q1 q1 q1 q0 q1 q0 q2 q1 q0 q1 q1 q0 q0 q0 q0 q0 q0

14. Infeasible subtrees Fact . If then the number of infeasible subtrees of constant size is O(n).

15. Tester for regular Trees Tester. Input : T,A, Theorem: Tester(T,A, ε ) is an ε -tester for L(A).

16. Proof schema of the Tester Theorem: Regular trees are testable. Robustness lemma: If T is ε-far from L, then for every admissible path Z, there exists such that the number of Z-infeasible i-subtrees Splitting lemma: if T is far from L there are many disjoint infeasible subtrees. Amplifying lemma: If there are many infeasible subtrees, there are many small ones.

17. Splitting and Merging Splitting and Merging on words: C C C C C C Splitting and Merging on trees:

18. Splitting and Merging trees E C C Connected Components Corrected tree C D D

19. Conclusion • Verification is hard. • Approximate verification can be feasible. • Testers and Correcters for regular words • Tester for regular trees • Corrector for regular trees • Unranked trees: XML files • Applications: Constant algorithm for Edit Distance with moves (Fischer, Magniez, Mdr)