[PPT] - Unidirectional Derivation Semantics for Synchronous Tree-Adjoining PowerPoint Presentation

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Unidirectional Derivation Semantics for Synchronous Tree-Adjoining Grammars

Matthias Büchse1 and Andreas Maletti2 and Heiko Vogler1

1 Faculty of Computer Science

Technische Universität Dresden 01062 Dresden, Germany

2 Institute for Natural Language Processing

Universität Stuttgart 70569 Stuttgart, Germany maletti@ims.uni-stuttgart.de

Taipei — August 17, 2012

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Tree-Adjoining Grammars

Motivation [JOSHI]

mildly context-sensitive formalism
local dependencies in rules

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Tree-Adjoining Grammars

Motivation [JOSHI]

mildly context-sensitive formalism
local dependencies in rules
but global dependencies in derivation

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Tree-Adjoining Grammars

Motivation [JOSHI]

mildly context-sensitive formalism
local dependencies in rules
but global dependencies in derivation

Applications

TAG for English [XTAG GROUP 2001]
TAG for German [KALLMEYER et al. 2010]

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Tree-Adjoining Grammars

Definition (JOSHI et al. 1969)

Tree-adjoining grammar (TAG) has a finite set of

substitution rules
adjunction rules

Substitution rule (rules of a regular tree grammar):

NP NP

f

NP

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Tree-Adjoining Grammars

S

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Tree-Adjoining Grammars

S NP VP

Used substitution rule

S NP VP

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Tree-Adjoining Grammars

S NP N children VP

Used substitution rule

NP N children

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Tree-Adjoining Grammars

S NP N children VP V NP

Used substitution rule

VP V NP

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Tree-Adjoining Grammars

S NP N children VP V like NP

Used substitution rule

V like

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Tree-Adjoining Grammars

S NP N children VP V like NP N

Used substitution rule

NP N

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Tree-Adjoining Grammars

S NP N children VP V like NP N candies

Used substitution rule

N candies

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Tree-Adjoining Grammars

Definition (JOSHI et al. 1969)

Tree-adjoining grammar (TAG) has a finite set of

substitution rules
adjunction rules

Adjunction rule:

N ADJ

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Tree-Adjoining Grammars

S NP N children VP V like NP N candies

Used adjunction rule

N ADJ

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Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ

N

candies

Used adjunction rule

N ADJ

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Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ

N

candies

Used adjunction rule

N ADJ

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Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ N candies

Used adjunction rule

N ADJ

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Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ red N candies

Used adjunction rule

N ADJ

Used substitution rule

ADJ red

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Synchronous Tree-Adjoining Grammars

Definition (SHIEBER and SCHABES 1990)

Synchronous tree-adjoining grammar (STAG) consists of two synchronized TAG Substitution rule:

NP NP

f

NP — NP NP NP

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Synchronous Tree-Adjoining Grammars

Definition (SHIEBER and SCHABES 1990)

Synchronous tree-adjoining grammar (STAG) consists of two synchronized TAG Adjunction rule:

N ADJ

—

N

ADJ

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Synchronous Tree-Adjoining Grammars

S S

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Synchronous Tree-Adjoining Grammars

S NP VP S NP VP

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Synchronous Tree-Adjoining Grammars

S NP N children VP S NP DET les N enfants VP

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Synchronous Tree-Adjoining Grammars

S NP N children VP V NP S NP DET les N enfants VP V NP

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Synchronous Tree-Adjoining Grammars

S NP N children VP V like NP S NP DET les N enfants VP V aiment NP

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Synchronous Tree-Adjoining Grammars

S NP N children VP V like NP N S NP DET les N enfants VP V aiment NP DET les N

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Synchronous Tree-Adjoining Grammars

S NP N children VP V like NP N candies S NP DET les N enfants VP V aiment NP DET les N bonbons

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Synchronous Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ N candies S NP DET les N enfants VP V aiment NP DET les N N bonbons ADJ

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Synchronous Tree-Adjoining Grammars

S NP N children VP V like NP N ADJ red N candies S NP DET les N enfants VP V aiment NP DET les N N bonbons ADJ rouges

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ α β t

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ x1 x2 α σ x1 γ α β t tα0

1

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ α β t tα0

1

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ α β t tα0

1 = tx1/α0

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 x2 α t tα0

1 = tx1/α0

tγ(, β)1

2

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ

α

β t tα0

1 = tx1/α0

tγ(, β)1

2

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ

α

β t tα0

1 = tx1/α0

tγ(, β)1

2

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ α β t tα0

1 = tx1/α0

tγ(, β)1

2

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Tree Substitution

Types

first-order: tu0

v replaces leaf at v in t by u

second-order: tu1

v replaces unary node at v in t by u

(with the subtree at v1 substituted into u) σ x1 x2 α σ α x2 α σ x1 γ α β t tα0

1 = tx1/α0

tγ(, β)1

2 = tx2/γ(, β)1

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Monadic Doubly Ranked Alphabet

Definition

Alphabet Q with a mapping rk: Q → {0, 1}2

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Monadic Doubly Ranked Alphabet

Definition

Alphabet Q with a mapping rk: Q → {0, 1}2

Notes

input and output rank rk1 and rk2
rank 0 → first-order substitution t· · ·0
rank 1 → second-order substitution t· · ·1

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Monadic Doubly Ranked Alphabet

Definition

Alphabet Q with a mapping rk: Q → {0, 1}2

Notes

input and output rank rk1 and rk2
rank 0 → first-order substitution t· · ·0
rank 1 → second-order substitution t· · ·1
Q(i,j) = {q ∈ Q | rk(q) = (i, j)}

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Synchronous Tree-Adjoining Grammar

Definition (BÜCHSE, NEDERHOF, VOGLER 2011)

(Q, Σ, q0, R) synchronous tree-adjoining grammar (STAG) if

Q monadic doubly-ranked alphabet

states

Σ alphabet

terminals

q0 ∈ Q(0,0)

initial state

R finite set of elements

rules

f the form q → ζζ′, q1 · · · qm

– ζ, ζ′ trees over Σ ∪ {x1, . . . , xm} ∪ {} – occurs according to rank of q in (ζ, ζ′) – xj occurs exactly once in ζ and ζ′ – rank of xj in (ζ, ζ′) equals rank of qj

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Synchronous Tree-Adjoining Grammar

Example

(Q, Σ, q0, R) with q0 ∈ Q(0,0) and q ∈ Q(1,1) q0 →

x1

S # x2 # x1 S # x2 #

, qq0

q0 →
#

# , ε

q →
S

a x1 S b

c

d S a d x1 S b c

, q
q →
, ε
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Unidirectional Derivation Semantics

Definition

ξ1

ρ

⇒ ξ2 with ρ = q → ζζ′, q1 · · · qm if there are (i) minimal redex position v in ξ1 and (ii) trees t1, . . . , tm

1

ccurs in tj according to rk1(qj)

2 ξ1(v) = (q, ζθ1 · · · θm) with θj = xj/tjrk1(qj) 3 ξ2 = ξ1ζ′θ′ 1 · · · θ′ mrk2(q) v

with θ′

j =

xj/(qj, tj)0

if rk2(qj) = 0 xj/(qj, tj)()1

therwise

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

Example (Used rule)

q0 →

x1

S # x2 # x1 S # x2 #

, qq0

(ρ1)

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

Example (Used rule)

q0 →

x1

S # x2 # x1 S # x2 #

, qq0

(ρ1)

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

Example (Used rule)

q0 →

x1

S # x2 # x1 S # x2 #

, qq0

(ρ1)

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Unidirectional Derivation Semantics

d

⇒ =

ρ1

⇒ ; · · · ;

ρn

⇒ with d = ρ1 · · · ρn

Definition

STAG G derivation-induces κG = {(s, t) | ∃d ∈ R∗ : (q0, s) d ⇒ t}

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

ρ3

⇒

S a d (q, ) S b c S # (q0, #) #

ρ4ρ2

= ⇒

S a d S b c S # # #

Example (Rules)

q0 →

x1

S # x2 # x1 S # x2 #

, qq0

(ρ1)

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

ρ3

⇒

S a d (q, ) S b c S # (q0, #) #

ρ4ρ2

= ⇒

S a d S b c S # # #

Example (Rules)

q →

S

a x1 S b

c

d S a d x1 S b c

, q
(ρ3)

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Unidirectional Derivation Semantics

q0,

S a S b S # # # c d

ρ1

⇒

q,

S a S b

c

d

S

# (q0, #) #

ρ3

⇒

S a d (q, ) S b c S # (q0, #) #

ρ4ρ2

= ⇒

S a d S b c S # # #

Example (Rules)

q0 →

#

# , ε

(ρ4)

q →

, ε
(ρ2)

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Extended Top-down Tree Transducer

Definition

STAG (Q, Σ, q0, R) is a (linear and nondeleting) extended top-down tree transducer (XTOP) if Q = Q(0,0)

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Explicit Substitution

Σ = Σ ∪ {·[·], } where

·[·] binary substitution symbol
nullary substitution site symbol

Definition

Evaluation ·E : TΣ → TΣ∪{}

E =
σ(t1, . . . , tk)E = σ(tE

1 , . . . , tE k )

·[·](t, u)E = tE[ ← uE]

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XTOP using Explicit Substitution

Example

q0 →

·[·]

x1 S # x2 # ·[·] x1 S # x2 #

, qq0

q0 →
#

# , ε

q →
S

a ·[·] x1 S b

c

d S a d ·[·] x1 S b c

, q
q →
, ε
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Definition

Tree t ∈ TΣ is well-behaved (under ·E) if

tE ∈ TΣ
tE

1 ∈ CΣ for every subtree of the form ·[·](t1, t2) in t

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Definition

Tree t ∈ TΣ is well-behaved (under ·E) if

tE ∈ TΣ
tE

1 ∈ CΣ for every subtree of the form ·[·](t1, t2) in t

Lemma

Well-behaved trees form a regular tree language

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Well-behaved XTOP

Example

q0 →

·[·]

x1 S # x2 # ·[·] x1 S # x2 #

, qq0

q0 →
#

# , ε

q →
S

a ·[·] x1 S b

c

d S a d ·[·] x1 S b c

, q
q →
, ε
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Main Theorem

κG and κM: unidirectional derivation semantics

Theorem

For every STAG G there is a well-behaved XTOP M such that κG = (κM)E and vice versa

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Bimorphism Semantics

Note

The bimorphism semantics is

taken from [BÜCHSE, NEDERHOF, VOGLER 2011]
similar (and equivalent) to the synchronous derivation

semantics

written as τG

Theorem (Theorem 4 of [MALETTI 2007])

τM = κM for every XTOP M

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Bimorphism Semantics

q0 q0

q0 →

x1

S # x2 # x1 S # x2 #

, qq0

(ρ1)

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Bimorphism Semantics

q S # q0 # q S # q0 #

q →

S

a x1 S b

c

d S a d x1 S b c

, q
(ρ3)

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Bimorphism Semantics

S a q S b S # q0 # c d S a d q S b c S # q0 #

q0 →

#

# , ε

(ρ4)

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Bimorphism Semantics

S a q S b S # # # c d S a d q S b c S # # #

q →

, ε
(ρ2)

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Bimorphism Semantics

S a S b S # # # c d S a d S b c S # # #

q →

, ε
(ρ2)

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Main Theorem

τG and τM: bimorphism semantics

Theorem

For every STAG G there is a well-behaved XTOP M such that τG = (τM)E and vice versa

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Uniform STAG

Definition

STAG (Q, Σ, q0, R) uniform if

Q = Q(1,1) ∪ {q0}
q0 does not occur in the right-hand sides

Theorem

For every STAG G there is a uniform STAG G′ with τG = τG′

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Summary

κG and κM: unidirectional derivation semantics τG and τM: bimorphism semantics

Corollary

For a tree transformation τ, the following are equivalent:

1 ∃ STAG G with τ = κG 2 ∃ well-behaved XTOP M with τ = (κM)E 3 ∃ well-behaved XTOP M with τ = (τM)E 4 ∃ STAG G with τ = τG 5 ∃ uniform STAG G with τ = τG

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References

BÜCHSE, NEDERHOF, VOGLER: Tree parsing with synchronous

tree-adjoining grammars. In Parsing Technologies (IWPT 2011)

JOSHI, KOSARAJU, YAMADA: String adjunct grammars. In Switching and

Automata Theory (SWAT 1969)

KALLMEYER: A lexicalized tree-adjoining grammar for a fragment of

German focussing on syntax and semantics. Emmy-Noether group 2010

MALETTI: Compositions of extended top-down tree transducers.
Inform. Comput. 206(9–10), 2008
MALETTI: A tree transducer model for synchronous tree-adjoining
grammars. In Association for Computational Linguistics (ACL 2010)
SHIEBER, SCHABES: Synchronous tree-adjoining grammars.

In Computational Linguistics (CoLing 1990)

XTAG GROUP: A lexicalized tree adjoining grammar for English.
Techn. Report IRCS-01-03. University of Pennsylvania, 2001

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