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Multiple tree automata a new model of tree automata Gwendal Collet (TU Wien), Julien David (LIPN) S eminaire CALIN, 24 mars 2015 Outline Introduction to automata: definitions and motivation 1 Description of the model: Multiple Tree Automata

  1. Multiple tree automata a new model of tree automata Gwendal Collet (TU Wien), Julien David (LIPN) S´ eminaire CALIN, 24 mars 2015

  2. Outline Introduction to automata: definitions and motivation 1 Description of the model: Multiple Tree Automata 2 Minimization 3 Closure properties 4 Yield of a MTA: Link with language theory 5

  3. Introduction: Regular Word Automata Finite alphabet: a, b, c... Set of transitions: ∆ ⊂ Q × Σ × Q A = (Σ , Q, I, F, δ ) Finite set of states: initial, final... i i ∈ I, r, s ∈ F a c b ( i, b, r ) , ( q, a, q ) , . . . ∈ ∆ q r L A = ( bc ) ⋆ (1 + a + b ) a e.g.: bcaaab ∈ L A b s

  4. Introduction: Regular Tree Automata Finite ranked alphabet: a (0) , b (1) , c (1) , d (2) ... A = (Σ= ∪ k ≥ 0 Σ k , Q, I, ∆) Set of transitions: ∆ ⊂ ∪ k ≥ 0 Q × Σ k × Q k Finite set of states: initial, final... i a ∈ Σ 0 (leaf) , b, c ∈ Σ 1 , d ∈ Σ 2 i ∈ I c b d ( i, b, r ) , ( q, a, ǫ ) , ( s, d, ( q, q )) , . . . ∈ ∆ q s r L A = ( b ( c ( . . . b ( c ( d ( a, d ( a, a ))))))) b ⋆ a d c e.g.: ∈ L A d a d a a

  5. Introduction: Regular Tree Automata Finite ranked alphabet: a (0) , b (1) , c (1) , d (2) ... A = (Σ= ∪ k ≥ 0 Σ k , Q, I, ∆) Set of transitions: ∆ ⊂ ∪ k ≥ 0 Q × Σ k × Q k Finite set of states: initial, final... i a ∈ Σ 0 (leaf) , b, c ∈ Σ 1 , d ∈ Σ 2 i ∈ I c b d ( i, b, r ) , ( q, a, ǫ ) , ( s, d, ( q, q )) , . . . ∈ ∆ q s r L A = ( b ( c ( . . . b ( c ( d ( a, d ( a, a ))))))) b ⋆ a d c Independence e.g.: ∈ L A d a d a a

  6. Introduction: Regular Tree Automata Finite ranked alphabet: a (0) , b (1) , c (1) , d (2) ... A = (Σ= ∪ k ≥ 0 Σ k , Q, I, ∆) Set of transitions: ∆ ⊂ ∪ k ≥ 0 Q × Σ k × Q k Finite set of states: initial, final... i What if we could handle c b d dependencies between children? q s r a d Independance

  7. Introduction: Motivation Random sampling of trees controlling the number of occurrences of a given pattern d d d Pattern a c c b d 2 occurrences a d d b a a b b a a

  8. Introduction: Motivation Random sampling of trees controlling the number of occurrences of a given pattern Pattern d When reading the tree top-down: d b Dependencies between nodes at a same height a b a a Idea (C., David, Jacquot 2014): • Use refined tree automata which count occurrences of a given pattern → need to handle dependencies • Translate the associated tree grammar into a system of equations on generating series • Design a bivariate Boltzmann sampler with the GS

  9. Multiple Tree Automata (MTA) Finite ranked alphabet: a (0) , b (1) , c (1) , d (2) ... A = (Σ = ∪ k ≥ 0 Σ k , Q = ∪ ℓ ≥ 1 Q ℓ , I, ∆) Initial states ∈ Q 1 Finite ranked set of states Set of transitions: ∆ ⊂ ∪ ℓ ≥ 1 Q ℓ × Σ ℓ × Part × Q ⋆ ( q, ( a 1 , . . . , a ℓ ) , P = ( p 1 , . . . , p r ) , ( q 1 , . . . , q r )) q such that: | P | = � ℓ i =1 rank ( a i ) ∀ 1 ≤ j ≤ r, rank ( q j ) = | p j | ( q, ( ) , {{ 2 } , { 1 , 3 , 4 , 6 } , { 5 , 7 }} , ( q 1 , q 2 , q 3 )) 1 2 3 4 5 6 7 q 1 q 2 q 3

  10. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  11. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  12. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  13. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  14. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  15. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  16. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  17. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

  18. Multiple Tree Automata (MTA)             i               A               t 1           L A =       n ≥ 0 n ≥ 0         ’  m ≥ 0  q             t 3 t 2                                 r s t t 6 t 7 t 8 t 4 t 5

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