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The Power of Tree Series Transducers Andreas Maletti 1 Technische Universitt Dresden Fakultt Informatik June 15, 2006 1 Research funded by German Research Foundation (DFG GK 334) Andreas Maletti (TU Dresden) The Power of Tree Series

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SLIDE 1

The Power of Tree Series Transducers

Andreas Maletti1

Technische Universität Dresden Fakultät Informatik

June 15, 2006

1Research funded by German Research Foundation (DFG GK 334) Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 1 / 26

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SLIDE 2

1 Motivation 2 Definition of Tree Series Transducers 3 Results

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 2 / 26

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SLIDE 3

Machine Translation

Overview

Subfield of computational linguistics Automatic translation shall assist (human) translator Offer several (likely) alternatives

History

1954: High prospects and expectations after Georgetown Experiment 1966: “Perfect translation” failed (ALPAC report) 1993: Statistical machine translation system [Brown et al 93]

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 3 / 26

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SLIDE 4

Machine Translation

Problem

Translate text of language X into grammatical text of language Y .

1 Preserve meaning 2 Preserve connotation 3 Preserve style

Relaxed Problem

Transform text of language X into text of language Y such that

1 the result is grammatical 2 expert for X and Y can discern original sentence Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 4 / 26

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SLIDE 5

Tree-based Model [Yamada, Knight 01]

VB PRP He VB1 adores VB2 VB listening TO TO to NN music ⇒ VB PRP He VB2 TO NN music TO to VB listening VB1 adores ⇓ VB PRP kare ha VB2 TO NN

  • ngaku

TO wo VB kiku no ga VB1 daisuki desu ⇐ VB PRP He ha VB2 TO NN music TO to VB listening no ga VB1 adores desu kare ha ongaku wo kiku no ga daisuki desu

3 phases: (i) Reorder, (ii) Insert, (iii) Translate

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 5 / 26

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SLIDE 6

Implementation of Phase (i): Reorder

Implementation by top-down tree series transducer

TT (vb) VB PRP He VB1 adores VB2 VB listening TO TO to NN music ⇒ VB (0.723) TT (prp) PRP He TT (vb2) VB2 VB listening TO TO to NN music TT (vb1) VB1 adores vb(VB(x1, x2, x3))

0.723

→ VB(prp(x1), vb2(x3), vb1(x2))

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 6 / 26

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SLIDE 7

Implementation of Phase (i): Reorder

VB (0.723) TT (prp) PRP He TT (vb2) VB2 VB listening TO TO to NN music TT (vb1) VB1 adores ⇒ VB (0.723) PRP (1) He TT (vb2) VB2 VB listening TO TO to NN music TT (vb1) VB1 adores prp(PRP(x1))

1

→ PRP(x1)

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 7 / 26

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SLIDE 8

Implementation of Phase (i): Reorder

VB (0.723) PRP (1) He TT (vb2) VB2 VB listening TO TO to NN music TT (vb1) VB1 adores ⇒ VB (0.723) PRP (1) He VB2 (0.749) TT (to) TO TO to NN music TT (vb) VB listening TT (vb1) VB1 adores vb2(VB2(x1, x2))

0.749

→ VB2(to(x2), vb(x1))

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 8 / 26

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SLIDE 9

Implementation of Phase (i): Reorder

VB (0.723) PRP (1) He VB2 (0.749) TT (to) TO TO to NN music TT (vb) VB listening TT (vb1) VB1 adores ⇒∗ VB (0.723) PRP (1) He VB2 (0.749) TO (0.893) NN (1) music TO (1) to VB (1) listening VB1 (1) adores

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 9 / 26

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SLIDE 10

Implementation of Phase (i): Reorder

VB PRP He VB1 adores VB2 VB listening TO TO to NN music ⇒∗ VB (0.723) PRP (1) He VB2 (0.749) TO (0.893) NN (1) music TO (1) to VB (1) listening VB1 (1) adores

The above reordering has probability: 0.723 · 0.749 · 0.893 = 0.484

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 10 / 26

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SLIDE 11

Implementation Details

Rules

Original Reordered Probability PRP VB1 VB2 PRP VB1 VB2 0.074 PRP VB2 VB1 0.723 VB1 PRP VB2 0.061 VB1 VB2 PRP 0.037 VB2 PRP VB1 0.083 VB2 VB1 PRP 0.021 VB TO VB TO 0.251 TO VB 0.749 TO NN TO NN 0.107 NN TO 0.893

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 11 / 26

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SLIDE 12

Tiburon [May, Knight 06]

Overview

Implements top-down weighted tree automata and top-down tree series transducers over the probability semiring Operations WTA: intersection, weighted determinization, pruning Operations TST: application, composition, training

Applications

Used to implement Yamada-Knight model (custom implementation took > 1 year, implementation in Tiburon 2 days) Used to implement Japanese transliteration [Knight, Graehl 98]

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 12 / 26

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SLIDE 13

1 Motivation 2 Definition of Tree Series Transducers 3 Results

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 13 / 26

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SLIDE 14

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) The Power of Tree Series Transducers June 15, 2006 14 / 26

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SLIDE 15

Semiring

Definition

(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

Natural numbers (N, +, ·, 0, 1) Probabilities ([0, 1], max, ·, 0, 1) Subsets (P(A), ∪, ∩, ∅, A)

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 15 / 26

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Top-down Tree Series Transducer [Engelfriet et al 02]

Definition

Polynomial top-down tree series transducer (Q, Σ, ∆, A, I, R) where Q finite set of states Σ and ∆ input and output ranked alphabet A = (A, +, ·, 0, 1) semiring I ⊆ Q set of initial states R finite set of rules of the form q(σ(x1, . . . , xk))

a

→ t where t ∈ T∆(Q(Xk))

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 16 / 26

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SLIDE 17

Properties of TST

Definition

(Q, Σ, ∆, A, I, R) top-down TST deterministic, if there is at most one rule with a given left hand side and at most one initial state linear, if (for every rule) every variable appears at most once in the right hand side nondeleting, if (for every rule) variables that occur in the left hand side also occur in the right hand side

Note

Bottom-up TST process input tree from leaves toward root.

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 17 / 26

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Classes of Transformations

Definition

denotation class of transformations computed by substitution x-TOPε(A) top-down TST with properties x ε-subst. x-TOPo(A) top-down TST with properties x

  • -subst.

x-BOTε(A) bottom-up TST with properties x ε-subst. x-BOTo(A) bottom-up TST with properties x

  • -subst.

In diagram: x-TOPω(A) abbreviated to x⊤

ω

x-BOTω(A) abbreviated to x⊥

ω

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 18 / 26

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SLIDE 19

1 Motivation 2 Definition of Tree Series Transducers 3 Results

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 19 / 26

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SLIDE 20

Hasse Diagram for Deterministic TST

Probability Semiring and Semiring of Natural Numbers

d⊥

ε

d⊥

  • dt⊥

ε

dl⊥

ε

dn⊥

ε

dn⊥

  • dl⊥
  • dt⊥
  • h⊥

ε

dlt⊥

ε

dnt⊥

ε

dnl⊥

=

dnt⊥

  • dlt⊥
  • h=
  • hl⊥

ε

hn⊥

ε

dnlt⊥

=

hn=

  • hl=
  • hnl=

=

dnlt⊤

=

dnl⊤

=

dnt⊤

=

dlt⊤

=

dn⊤

=

dl⊤

=

dt⊤

=

d⊤

=

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 20 / 26

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SLIDE 21

Hasse Diagram for Deterministic TST

Semiring of Subsets

d⊥

=

d⊤

=

dn⊥

=

dl⊥

=

dt⊥

=

dt⊤

=

dn⊤

=

dl⊤

=

dnl⊥

=

dnt⊥

=

dlt⊥

=

h⊤

=

h⊥

  • h⊥

ε

dnt⊤

=

dlt⊤

=

dnl⊤

=

dnlt⊥

=

hn=

=

hl⊤

=

hl⊥

  • hl⊥

ε

dnlt⊤

=

hnl=

=

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 21 / 26

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SLIDE 22

Composition of Transformations

Definition

Let ϕ: TΣ × T∆ → A ψ: T∆ × TΓ → A Composition of ϕ and ψ (ϕ ; ψ): TΣ × TΓ → A (t, v) →

  • u∈T∆

ϕ(t, u) · ψ(u, v)

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 22 / 26

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SLIDE 23

Composition Results

Theorem (see [Kuich 99] and [Engelfriet et al 02])

A commutative semiring nlp-BOT(A) ; p-BOT(A) = p-BOT(A) p-BOT(A) ; bdth-BOT(A) = p-BOT(A)

Theorem

A commutative semiring lp-BOT(A) ; p-BOT(A) = p-BOT(A) p-BOT(A) ; bd-BOT(A) = p-BOT(A) bdt-TOP(A) ; lp-TOP(A) ⊆ p-TOP(A)

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 23 / 26

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SLIDE 24

References

Joost Engelfriet, Zoltán Fülöp, and Heiko Vogler. Bottom-up and top-down tree series transformations.

  • J. Autom. Lang. Combin., 7(1):11–70, 2002.

Werner Kuich. Full abstract families of tree series I. In Jewels Are Forever, pages 145–156. Springer, 1999. Werner Kuich. Tree transducers and formal tree series. Acta Cybernet., 14(1):135–149, 1999. Jonathan May and Kevin Knight. Tiburon: A weighted tree automata toolkit. In Proc. 11th Int. Conf. Implementation and Application of Automata,

  • LNCS. Springer, 2006.

to appear.

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 24 / 26

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Publications

Conferences

Myhill-Nerode theorem for sequential transducers over unique GCD-monoids.

  • Proc. 9th Int. Conf. Implementation and Application of Automata

Relating Tree Series Transducers and Weighted Tree Automata.

  • Proc. 8th Int. Conf. Developments in Language Theory

Compositions of bottom-up tree series transformations.

  • Proc. 11th Int. Conf. Automata and Formal Languages

The power of tree series transducers of type I and II.

  • Proc. 9th Int. Conf. Developments in Language Theory

Hierarchies of tree series transformations—Revisited.

  • Proc. 10th Int. Conf. Developments in Language Theory

The substitution vanishes. (with A. Kühnemann)

  • Proc. 11th Int. Conf. Algebraic Methodology and Software Technology

Does o-substitution preserve recognizability?

  • Proc. 11th Int. Conf. Implementation and Application of Automata

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 25 / 26

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SLIDE 26

Publications

Journals

Relating tree series transducers and weighted tree automata.

  • Int. J. of Foundations of Computer Science 16, 2005

Hasse diagrams for classes of deterministic bottom-up tree-to-tree-series transformations. Theoretical Computer Science 339, 2005 Cut sets as recognizable tree languages. (with B. Borchardt, B. Šešelja, A. Tepavčević, H. Vogler) Fuzzy Sets and Systems 157, 2006 Incomparability results for classes of polynomial tree series transformations. (with H. Vogler)

  • J. Automata, Languages and Combinatorics, to appear, 2006

Bounds for tree automata with polynomial costs. (with B. Borchardt, Zs. Gazdag, Z. Fülöp)

  • J. Automata, Languages and Combinatorics, to appear, 2006

Compositions of tree series transformations. Theoretical Computer Science, to appear, 2006

Andreas Maletti (TU Dresden) The Power of Tree Series Transducers June 15, 2006 26 / 26