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

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

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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. Testers on a class K

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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 graph properties have testers, Alon and al Regular languages have testers, Alon and al. 2000s Testers for Regular tree languages, Mdr and Magniez, ICALP 2004

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1.Classical Edit Distance: Insertions, Deletions, Modifications 2.Edit Distance with moves Edit Distance with Moves generalizes to Trees Edit distance on Words

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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 init accept

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

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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.

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Merging Merging lemma: Let Z be an admissible path, and let F be a Z- feasible cut of size h’. Then C CC 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:

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

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

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

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(q0, q0) q1 (q0,q1) q1 Tree automata q0 q1 q0 q1 q2 (q1,q1) q2 (q1,q0) q2 (q2,-) q2 (-,q2) q2

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

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

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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.

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Splitting and Merging C CC C C C Splitting and Merging on words: Splitting and Merging on trees:

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

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Conclusion Verification is hard. Approximate verification can be feasible. 1.Testers and Correcters for regular words 2.Tester for regular trees 3.Corrector for regular trees 4.Unranked trees: XML files 5.Applications: Constant algorithm for Edit Distance with moves (Fischer, Magniez, Mdr)

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