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A Tree Transducer Model for Synchronous Tree-Adjoining Grammars

Andreas Maletti

Universitat Rovira i Virgili Tarragona, Spain andreas.maletti@urv.cat

Uppsala, Sweden — July 13, 2010

A Tree Transducer Model for STAG

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Synchronous Tree Substitution Grammar

S S

A Tree Transducer Model for STAG

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Synchronous Tree Substitution Grammar

S S CONJ wa S

Used rule

S — S CONJ wa S

A Tree Transducer Model for STAG

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Synchronous Tree Substitution Grammar

S NP 1 VP V NP 2 S CONJ wa S V NP 1 NP 2

Used rule

S NP VP V NP — S V NP NP

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Synchronous Tree Substitution Grammar

S NP 1 VP V saw NP 2 S CONJ wa S V ra’aa NP 1 NP 2

Used rule

V saw — V ra’aa

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Synchronous Tree Substitution Grammar

S NP DT the N VP V saw NP S CONJ wa S V ra’aa NP N NP

Used rule

NP DT the N — NP N

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Synchronous Tree Substitution Grammar

S NP DT the N boy VP V saw NP S CONJ wa S V ra’aa NP N atefl NP

Used rule

N boy — N atefl

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Synchronous Tree Substitution Grammar

S NP DT the N boy VP V saw NP DT the N S CONJ wa S V ra’aa NP N atefl NP N

Used rule

NP DT the N — NP N

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Synchronous Tree Substitution Grammar

S NP DT the N boy VP V saw NP DT the N door S CONJ wa S V ra’aa NP N atefl NP N albab

Used rule

N door — N albab

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Synchronous Tree Substitution Grammar (cont’d)

Advantages

simple and natural model easy to train (from linguistic resources) symmetric

Implementation

extended top-down tree transducer in TIBURON [MAY, KNIGHT ’06]

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Synchronous Tree Substitution Grammar (cont’d)

Synchronous tree substitution grammar rule: S NP 1 VP V NP 2

w

— S V NP 1 NP 2 Corresponding extended top-down tree transducer rule: qS S x1 VP x2 x3

w

− → S qV x2 qNP x1 qNP x3

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Synchronous Tree-Adjoining Grammar

S S

A Tree Transducer Model for STAG

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Synchronous Tree-Adjoining Grammar

S NP VP S NP VP

Used substitution rule

S NP VP — S NP VP

A Tree Transducer Model for STAG

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Synchronous Tree-Adjoining Grammar

S NP 1 VP V NP 2 S NP 1 VP V NP 2

Used substitution rule

VP V NP — VP V NP

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Synchronous Tree-Adjoining Grammar

S NP 1 VP V likes NP 2 S NP 1 VP V aime NP 2

Used substitution rule

V likes — V aime

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Synchronous Tree-Adjoining Grammar

S NP VP V likes NP N S NP VP V aime NP DT les N

Used substitution rule

NP N — NP DT les N

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Synchronous Tree-Adjoining Grammar

S NP VP V likes NP N candies S NP VP V aime NP DT les N bonbons

Used substitution rule

N candies — N bonbons

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Synchronous Tree-Adjoining Grammar

S NP VP V likes NP N ADJ N candies S NP VP V aime NP DT les N N bonbons ADJ

Used adjunction rule

N ADJ N⋆ — N N⋆ ADJ

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Synchronous Tree-Adjoining Grammar

S NP VP V likes NP N ADJ red N candies S NP VP V aime NP DT les N N bonbons ADJ rouges

Used substitution rule

ADJ red — ADJ rouges

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Main Question

Theorem

Every STSG is an STAG.

Question

Are they further related?

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Roadmap

1

Motivation

2

Explicit Substitution

3

Synchronous Tree-Adjoining Grammar

4

Main Result

5

Application

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First-Order Substitution

Definition

t[v1 ← t1, . . . , vk ← tk] denotes the result obtained by replacing (in parallel) all occurrences of leaves labelled vi in t by ti.

Example

S NP VP V saw NP NP DT the N S NP DT the N VP V saw NP DT the N t u t[NP ← u]

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Second-Order Substitution

Example

·[NP ← ·] S NP VP V saw NP NP DT the N

Explicit substitution

keep an explicit representation of substitutions in tree any number of substitutions allowed at any level

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Second-Order Substitution

Example

·[NP ← ·] S NP VP V saw NP NP DT the N

Evaluation

eval(·[x ← ·](t, u)) = eval(t)[x ← eval(u)] eval(σ(t1, . . . , tk)) = σ(eval(t1), . . . , eval(tk))

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Second-Order Substitution

Example

·[NP ← ·] S NP VP V saw NP NP DT the N

Evaluation

S NP DT the N VP V saw NP DT the N

Evaluation

eval(·[x ← ·](t, u)) = eval(t)[x ← eval(u)] eval(σ(t1, . . . , tk)) = σ(eval(t1), . . . , eval(tk))

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Roadmap

1

Motivation

2

Explicit Substitution

3

Synchronous Tree-Adjoining Grammar

4

Main Result

5

Application

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Tree-Adjoining Grammar

Intuition

A TAG has two types of rules: substitution rules (as in TSG) adjunction rules

Example (Adjunction)

NP DT les N bonbons N N⋆ ADJ rouges NP DT les N N bonbons ADJ rouges derived tree auxiliary tree adjunction

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Tree-Adjoining Grammar (cont’d)

Simplifications (see [SHIEBER ’06])

no substitution rules adjunction mandatory (if possible) each adjunction spot used at most once root nodes of auxiliary trees are never adjunction spots

Definition

A TAG is a finite set of derived trees (initial trees) and auxiliary trees (those containing a starred node)

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Tree-Adjoining Grammar (cont’d)

Simplifications (see [SHIEBER ’06])

no substitution rules adjunction mandatory (if possible) each adjunction spot used at most once root nodes of auxiliary trees are never adjunction spots

Definition

A TAG is a finite set of derived trees (initial trees) and auxiliary trees (those containing a starred node)

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Tree-Adjoining Grammar (cont’d)

Example

S T c S a S S⋆ a initial tree auxiliary tree S b S S⋆ b S S⋆ auxiliary tree auxiliary tree

Derivation

S T c ⇒ S a S S T c a ⇒ S a S b S S S T c a b ⇒ S a S b S S S S T c a b

String language

{wcw | w ∈ {a, b}∗}

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Synchronous Tree-Adjoining Grammar

Example

S T c — S T c S S a S⋆ a — S a S S⋆ a initial tree pair auxiliary tree pair S S⋆ — S S⋆ S S b S⋆ b — S b S S⋆ b auxiliary tree pair auxiliary tree pair

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Synchronous Tree-Adjoining Grammar (cont’d)

Example

S T c — S T c S S a S T c a — S a S S T c a S S S b S a S T c a b — S a S b S S S T c a b

String translation

{(wcwR, wcw) | w ∈ {a, b}∗}

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Roadmap

1

Motivation

2

Explicit Substitution

3

Synchronous Tree-Adjoining Grammar

4

Main Result

5

Application

A Tree Transducer Model for STAG

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Simulation

Question

Can we simulate an STAG by some STSG?

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Simulation of Adjunction

Example (TAG)

S T c S a S S⋆ a initial tree auxiliary tree S b S S⋆ b S S⋆ auxiliary tree auxiliary tree

Correspondence (TSG)

·[S⋆ ← ·] S S T c S a ·[S⋆ ← ·] S S S⋆ a S b ·[S⋆ ← ·] S S S⋆ b S S⋆

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Simulation of Adjunction (cont’d)

Example

·[S⋆ ← ·] S S T c ⇒ ·[S⋆ ← ·] S a ·[S⋆ ← ·] S S S⋆ a S T c ⇒ ·[S⋆ ← ·] S a ·[S⋆ ← ·] S b ·[S⋆ ← ·] S S S⋆ b S S⋆ a S T c ⇒ ·[S⋆ ← ·] S a ·[S⋆ ← ·] S b ·[S⋆ ← ·] S S⋆ S S⋆ b S S⋆ a S T c

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Simulation of Adjunction (cont’d)

TSG result

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b ·[S⋆ ← ·] S S⋆ S S⋆ b S S⋆ a S T c

Evaluation

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b S S S⋆ b S S⋆ a S T c

Note

coincides with the result obtained by TAG

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Simulation of Adjunction (cont’d)

TSG result

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b ·[S⋆ ← ·] S S⋆ S S⋆ b S S⋆ a S T c

Evaluation

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b S S S⋆ b S S⋆ a S T c ·[S⋆ ← ·] S a S b S S S S⋆ a b S T c

Note

coincides with the result obtained by TAG

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Simulation of Adjunction (cont’d)

TSG result

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b ·[S⋆ ← ·] S S⋆ S S⋆ b S S⋆ a S T c

Evaluation

·[S⋆ ← ·] S a ·[S⋆ ← ·] S b S S S⋆ b S S⋆ a S T c ·[S⋆ ← ·] S a S b S S S S⋆ a b S T c S a S b S S S S T c a b

Note

coincides with the result obtained by TAG

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Main Result

Theorem

For every TAG G there exists a TSG G′ such that L(G) = {eval(t) | t ∈ L(G′)}

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Main Result

Theorem

For every TAG G there exists a TSG G′ such that L(G) = {eval(t) | t ∈ L(G′)}

Theorem

For every STAG G there exists a STSG G′ such that T(G) = {(eval(t), eval(u)) | (t, u) ∈ T(G′)}

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Main Result

Theorem

For every STAG G there exists a STSG G′ such that T(G) = {(eval(t), eval(u)) | (t, u) ∈ T(G′)}

Proof.

HOM HOM eval eval STSG STAG EMB EMB

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Main Result

Theorem

For every STAG G there exists a STSG G′ such that T(G) = {(eval(t), eval(u)) | (t, u) ∈ T(G′)}

Proof.

HOM HOM eval eval STSG STAG EMB EMB

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Main Result

Theorem

For every STAG G there exists a STSG G′ such that T(G) = {(eval(t), eval(u)) | (t, u) ∈ T(G′)}

Proof.

HOM HOM eval eval STSG STAG EMB EMB

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Main Result

Theorem

For every STAG G there exists a STSG G′ such that T(G) = {(eval(t), eval(u)) | (t, u) ∈ T(G′)}

Proof.

HOM HOM eval eval STSG STAG EMB EMB

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Roadmap

1

Motivation

2

Explicit Substitution

3

Synchronous Tree-Adjoining Grammar

4

Main Result

5

Application

A Tree Transducer Model for STAG

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Application

Overview

run an STAG in TIBURON (which can run STSGs) translate STSG algorithms to STAGs (factorization, etc.) integrate explicit substitution into semantics separate “context-free” and “context-sensitive” behavior

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References

ARNOLD, DAUCHET: Morphismes et bimorphismes d’arbres.

• Theoret. Comput. Sci. 20. 1982

CHIANG, KNIGHT: An introduction to synchronous grammars. Tutorial at ACL. 2006 ENGELFRIET, VOGLER: Macro tree transducers. J. Comput. Syst.

• Sci. 31. 1985

MAY, KNIGHT: TIBURON — a weighted tree automata toolkit. In CIAA, LNCS 4094. 2006 NEDERHOF: Weighted parsing of trees. In IWPT. 2009 SHIEBER, SCHABES: Synchronous tree-adjoining grammars. Computational Linguistics 3. 1990 SHIEBER: Unifying synchronous tree adjoining grammars and tree transducers via bimorphisms. In EACL. 2006

Thank you for your attention!

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