Hierarchies of Tree Series TransducersRevisited 1

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1 Hierarchies of Tree Series TransducersRevisited 1 Andreas Maletti 2 Technische Universität Dresden Fakultät Informatik June 27, Financially supported by the Friends and Benefactors of TU Dresden 2 Research funded by German Research Foundation (DFG GK 334) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

2 1 Machine Translation 2 Tree Series Transducers 3 Results Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

3 Machine Translation Problem Translate text of language X into grammatical text of language Y. 1 Preserve meaning 2 Preserve connotation 3 Preserve style Approaches traditional phrase-based systems (e.g. CARMEL) upcoming syntax-based systems (e.g. Tiburon) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

4 Syntax-based Machine Translation System Input sentence parser Parse trees translator Parse trees Output sentences language model translated sentences yield Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

5 Syntax-based Model [Yamada, Knight 01] PRP 1 2 PRP 2 1 He adores He adores listening NN NN listening to music music to PRP 2 1 PRP 2 1 kare ha ga daisuki desu He ha ga adores desu NN kiku no NN listening no ongaku wo music to kare ha ongaku wo kiku no ga daisuki desu 3 phases: (i) Reorder, (ii) Insert, (iii) Translate Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

6 1 Machine Translation 2 Tree Series Transducers 3 Results Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

7 Tree Series Transducers Overview tree series transducer weighted tree automaton weighted transducer tree transducer weighted automaton tree automaton generalized sequential machine string automaton History Introduced in [Kuich 99] Extended to full generality in [Engelfriet, Fülöp, Vogler 02] Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

8 Hasse Diagram for Compositions [Engelfriet 82] BOT n+1 P n+1 BOT n P n BOT 2 P 2 BOT 1 P 1 Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

9 Semiring Denition (A, +,, 0, 1) semiring, if (A, +, 0) commutative monoid (A,, 1) monoid distributes (both sided) over + 0 is absorbing for (a 0 = 0 = 0 a) Example Reals (R, +,, 0, 1) Probabilities ([0, 1], max,, 0, 1) Subsets (P(A),,,, A) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

10 Tree Series Denition Tree series is mapping ψ : T Σ (X ) A Set of tree series A T Σ (X ) Conventions A usually endowed with semiring structure 0 is tree series that maps every tree to 0 ψ(t) written as (ψ, t) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

11 Tree Series Denition Let ψ A T Σ (X ) supp(ψ) = {t T Σ (X ) (ψ, t) 0} i.e. set of nonzero-weighted trees Convention tree series ψ written as t TΣ(X )(ψ, t) t (ψ + ϕ, t) = (ψ, t) + (ϕ, t) (a ψ, t) = a (ψ, t) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

12 Tree Series Transducer [Engelfriet et al 02] Denition Top-down tree series transducer (Q, Σ,, A, I, R) where Q nite set of states Σ and input and output ranked alphabet A = (A, +,, 0, 1) semiring I Q set of initial states R nite set of rules of the form where t T (Q(X k )) q(σ(x 1,..., x k )) a t Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

13 Tree Series Transducer [Engelfriet et al 02] Denition Bottom-up tree series transducer (Q, Σ,, A, F, R) where Q nite set of states Σ and input and output ranked alphabet A = (A, +,, 0, 1) semiring F Q set of nal states R nite set of rules of the form where t T (X k ) σ(q 1 (x 1 ),..., q k (x k )) a q(t) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

14 Tree Series Transducer Top-down Rules q σ x 1 x 2 a γ q 1 δ δ q 1 q 2 x 1 x 2 x 2 Bottom-up Rules σ q 1 q 2 x 1 x 2 a x 1 q σ σ σ α x 2 x 1 Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

15 Implementation of Phase (i): Reorder Implementation by top-down tree series transducer TT (vb) (0.723) TT (prp) TT (vb2) TT (vb1) PRP 1 2 PRP 2 1 He adores He adores listening NN listening NN to music to music vb((x 1, x 2, x 3 )) (prp(x 1 ), vb2(x 3 ), vb1(x 2 )) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

16 Implementation of Phase (i): Reorder (0.723) (0.723) TT (prp) TT (vb2) TT (vb1) PRP (1) TT (vb2) TT (vb1) PRP 2 1 He 2 1 He adores adores listening NN listening NN to music to music prp(prp(x 1 )) 1 PRP(x 1 ) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

17 Implementation of Phase (i): Reorder (0.723) (0.723) PRP (1) TT (vb2) TT (vb1) PRP (1) 2 (0.749) TT (vb1) He 2 1 He TT (to) TT (vb) 1 adores adores listening NN NN listening to music to music vb2(2(x 1, x 2 )) (to(x 2 ), vb(x 1 )) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

18 Implementation of Phase (i): Reorder (0.723) PRP (1) 2 (0.749) TT (vb1) He TT (to) TT (vb) 1 adores to NN music listening (0.723) PRP (1) 2 (0.749) 1 (1) He (0.893) (1) adores NN (1) music (1) to listening Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

19 Implementation of Phase (i): Reorder (0.723) PRP 1 2 PRP (1) 2 (0.749) 1 (1) He adores He (0.893) (1) adores listening NN NN (1) (1) listening to music music to The above reordering has probability: = Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

20 Classes of Transformations Denition denotation class of transformations computed by P(A) top-down tree series transducers BOT(A) bottom-up tree series transducers Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

21 1 Machine Translation 2 Tree Series Transducers 3 Results Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

22 Composition of Transformations Denition Let ϕ: T Σ T A ψ : T T Γ A Composition of ϕ and ψ (ϕ ; ψ): T Σ T Γ A (t, v) ϕ(t, u) ψ(u, v) u T Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

23 Required Semiring Properties Denition A = (A, +,, 0, 1) commutative, if a b = b a idempotent, if = 1 zero-divisor free, if a b = 0 implies a = 0 or b = 0 zero-sum free, if a + b = 0 implies a = 0 positive, if zero-divisor free and zero-sum free Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

24 Hasse Diagram for Compositions [Fülöp et al 05] A commutative, positive idempotent semiring BOT n+1 (A) P n+1 (A) BOT n (A) P n (A) BOT 2 (A) P 2 (A) BOT 1 (A) P 1 (A) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

25 Homomorphic Images Denition χ: A {0, 1} χ(a) = { 0 if a = 0 1 otherwise Theorem A positive semiring χ(p(a)) = P(B) and χ(bot(a)) = BOT(B) χ(τ 1 ; τ 2 ) = χ(τ 1 ) ; χ(τ 2 ) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

26 Hasse Diagram for Compositions A commutative, positive semiring (e.g. R + ) BOT n+1 (A) P n+1 (A) BOT n (A) P n (A) BOT 2 (A) P 2 (A) BOT 1 (A) P 1 (A) Andreas Maletti (TU Dresden) Hierarchies June 27, / 26

27 References Joost Engelfriet, Zoltán Fülöp, and Heiko Vogler. Bottom-up and top-down tree series transformations. J. Autom. Lang. Combin., 7(1):1170, Zoltán Fülöp, Zsolt Gazdag, and Heiko Vogler. Hierarchies of tree series transformations. Theoret. Comput. Sci., 314(3):387429, Werner Kuich. Tree transducers and formal tree series. Acta Cybernet., 14(1):135149, Jonathan May and Kevin Knight. Tiburon: A weighted tree automata toolkit. In Proc. 11th Int. Conf. Implementation and Application of Automata, volume 4094 of LNCS. Springer, Andreas Maletti (TU Dresden) Hierarchies June 27, / 26