Language, Automata and Logic for Finite Trees Olivier Gauwin UMons Feb/March 2010 Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 1 / 66 Languages, Automata, Logic Example for regular word languages: Σ∗.a.b.Σ∗ Complexity a, b a, b a b ∃x. ∃y. lab (x) S A B Automata Logic a ∧ labb(y) ∧ succ(x, y) Grammars / Expressions S → aS S → aB X → aX X → ǫ S → bS B → bX X → bX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 2 / 66 Languages, Automata, Logic Example for regular word languages: Σ∗.a.b.Σ∗ Example for regular tree languages: all trees with an a-node having a b-child Complexity a, b a, b a b ∃x. ∃y. lab (x) S A B Automata Logic a ∧ labb(y) ∧ succ(x, y) (S, X ) → S ∃x. ∃y. laba(x) (X , S) → S ∧ labb(y) ∧ child(x, y) a(B, X ) → S a(X , B) → S Grammars / Expressions ... S → aS S → aB X → aX X → ǫ b → B S → bS B → bX X → bX ... S → (S, X ) S → a(B, X ) B → b(X , X ) X → (X , X ) S → (X , S) S → a(X , B) B → b X → Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 2 / 66 References The main reference for this talk is the TATA book [CDG+07]: Tree Automata, Techniques and Applications by Hubert Comon, Max Dauchet, R´emi Gilleron, Christof L¨oding, Florent Jacquemard, Denis Lugier, Sophie Tison, Marc Tommasi. Other references will be mentionned progressively. Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 3 / 66 1 Ranked Trees Trees on Ranked Alphabet Tree Automata Tree Grammars Logic 2 Unranked Trees Unranked Trees Automata Logic Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 4 / 66 1 Ranked Trees Trees on Ranked Alphabet Tree Automata Tree Grammars Logic 2 Unranked Trees Unranked Trees Automata Logic Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 5 / 66 Trees on Ranked Alphabet Ranked alphabet Ranked alphabet = finite alphabet + arity function Σr Σ= {a, b, c} ar:Σ → N ar(a) = 2 ar(b) = 2 ar(c) = 0 Ranked trees over Σr TΣr , the set of ranked trees, is the smallest set of terms f (t1,..., tk ) such that: f ∈ Σr , k = ar(f ), and ti ∈TΣr for all 1 ≤ i ≤ k. b a b b c c c c c A tree language T is a set of trees: T ⊆TΣr . Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 6 / 66 Trees as relational structures We will sometimes consider a ranked tree t as a relational structure t t t t t t (nodes , {laba, labb, labc , ch1, ch2}). b ǫ a b 1 2 t laba = {1} b c c c t 1.1 1.2 2.1 2.2 labb = {ǫ, 1.1, 2} labt = {1.1.1, 1.1.2, 1.2, 2.1, 2.2} c c c 1.1.1 1.1.2 t ch1 = {(ǫ, 1), (1, 1.1), (2, 2.1) . } t ch2 = {(ǫ, 2), (1, 1.2), (2, 2.2) . } For convenience we write lab(π) for the label of node π. Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 7 / 66 1 Ranked Trees Trees on Ranked Alphabet Tree Automata Tree Grammars Logic 2 Unranked Trees Unranked Trees Automata Logic Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 8 / 66 Tree Automata over Ranked Trees Definition A tree automaton (TA) over Σr = (Σ, ar) is a tuple A = (Q, F , ∆, Σr ) where: Q is a finite set of states, F ⊆ Q a set of final states, ∆ are rules of type: a(q1,..., qk ) → q with a ∈ Σ, k = ar(a) and q, q1,..., qk ∈ Q Runs A run ρ of A on t is a function ρ : nodest → Q such that: t if π ∈ nodes with children π1,...,πk and label a then a(ρ(π1), . , ρ(πk )) → ρ(π) ∈ ∆ Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 9 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S b(qX , qS ) → qS a a(qB , qX ) → qS b Rules: a(qX , qB ) → qS b(qX , qX ) → qB b c c c a(qX , qX ) → qX b(qX , qX ) → qX c c c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S b(qX , qS ) → qS a a(qB , qX ) → qS b Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S b(qX , qS ) → qS a a(qB , qX ) → qS b Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qB qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S b(qX , qS ) → qS a q q q a b ( B , X ) → S qS Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qB qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S b(qX , qS ) → qS a q q q a b ( B , X ) → S qS qX Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qB qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S qS b(qX , qS ) → qS a q q q a b ( B , X ) → S qS qX Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qB qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Bottom-up view a(q1,..., qk ) → q is a bottom-up point of view: a(qS , qX ) → qS a(qX , qS ) → qS b q q q b ( S , X ) → S qS b(qX , qS ) → qS a q q q a b ( B , X ) → S qS qX Rules: a(qX , qB ) → qS b(q , q ) → q b c c c X X B qB qX qX qX a(q , q ) → q X X X b(q , q ) → q X X X cq c q X X c → qX A run of A on t is accepting if ρ(ǫ) ∈ F . L(A)= {t | there exists an accepting run ρ of A on t} Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 10 / 66 Bottom-up vs top-down Top-down view We could have written rules this way: a(q) → (q1,..., qk ), and name F the initial states. This corresponds to a top-down definition: Initial: {qS } a(qS ) → (qS , qX ) a(qS ) → (qX , qS ) b b q q q ( S ) → ( S , X ) b(qS ) → (qX , qS ) a b a(qS ) → (qB , qX ) Rules: a(qS ) → (qX , qB ) b c c c b(qB ) → (qX , qX ) a(qX ) → (qX , qX ) c c b(qX ) → (qX , qX ) c(qX ) Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 11 / 66 Bottom-up vs top-down Top-down view We could have written rules this way: a(q) → (q1,..., qk ), and name F the initial states. This corresponds to a top-down definition: Initial: {qS } a(qS ) → (qS , qX ) a(qS ) → (qX , qS ) bq b q q q S ( S ) → ( S , X ) b(qS ) → (qX , qS ) a b a(qS ) → (qB , qX ) Rules: a(qS ) → (qX , qB ) b c c c b(qB ) → (qX , qX ) a(qX ) → (qX , qX ) c c b(qX ) → (qX , qX ) c(qX ) Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 11 / 66 Bottom-up vs top-down Top-down view We could have written rules this way: a(q) → (q1,..., qk ), and name F the initial states. This corresponds to a top-down definition: Initial: {qS } a(qS ) → (qS , qX ) a(qS ) → (qX , qS ) b qS b(qS ) → (qS , qX ) b(qS ) → (qX , qS ) a b a(q ) → (q , q ) qS qX S B X Rules: a(qS ) → (qX , qB ) b c c c b(qB ) → (qX , qX ) a(qX ) → (qX , qX ) c c b(qX ) → (qX , qX ) c(qX ) Olivier Gauwin (UMons) Finite Tree Automata Feb/March 2010 11 / 66 Bottom-up vs top-down Top-down view We could have written rules this way: a(q) → (q1,..., qk ), and name F the initial states.