Linking Theorems for Tree Transducers Andreas Maletti - - PowerPoint PPT Presentation

Linking Theorems for Tree Transducers

Andreas Maletti maletti@ims.uni-stuttgart.de Speyer — October 1, 2015

Andreas Maletti Linking Theorems for MBOT Theorietag 2015 1 / 32

Statistical Machine Translation

S w VP kAnA VP ynZrAn NP-SBJ ⋆ PP-CLR Aly NP h PP-MNR b NP $kl mDHk S And NP-SBJ they VP were VP looking PP-CLR at NP him PP in NP a funny way

Statistical Machine Translation

S w VP kAnA VP ynZrAn NP-SBJ ⋆ PP-CLR Aly NP h PP-MNR b NP $kl mDHk S And NP-SBJ they VP were VP looking PP-CLR at NP him PP in NP a funny way

Statistical Machine Translation

S w VP kAnA VP ynZrAn NP-SBJ ⋆ PP-CLR Aly NP h PP-MNR b NP $kl mDHk S And NP-SBJ they VP were VP looking PP-CLR at NP him PP in NP a funny way

NP h

qNP

— NP him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And Aly

qAly

— at kAnA

qkAnA

— NP-SBJ they . were ynZrAn

qynZrAn

— looking

Statistical Machine Translation

S qw VP qkAnA VP qynZrAn NP-SBJ ⋆ PP-CLR qAly qNP PP-MNR qb NP q$kl qmDHk S qw qkAnA VP qkAnA VP qynZrAn PP-CLR qAly qNP PP qb NP q$kl qmDHk q$kl

NP h

qNP

— NP him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And Aly

qAly

— at kAnA

qkAnA

— NP-SBJ they . were ynZrAn

qynZrAn

— looking

Statistical Machine Translation

S qw VP qkAnA VP qynZrAn NP-SBJ ⋆ PP-CLR qAly qNP PP-MNR qb NP q$kl qmDHk S qw qkAnA VP qkAnA VP qynZrAn PP-CLR qAly qNP PP qb NP q$kl qmDHk q$kl

NP h

qNP

— NP him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And Aly

qAly

— at kAnA

qkAnA

— NP-SBJ they . were ynZrAn

qynZrAn

— looking PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl

Statistical Machine Translation

S qw VP qkAnA VP qynZrAn NP-SBJ ⋆ qPP-CLR PP-MNR qb qNP S qw qkAnA VP qkAnA VP qynZrAn qPP-CLR PP qb qNP

NP h

qNP

— NP him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And Aly

qAly

— at kAnA

qkAnA

— NP-SBJ they . were ynZrAn

qynZrAn

— looking PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl

Statistical Machine Translation

S qw VP qkAnA VP qynZrAn NP-SBJ ⋆ qPP-CLR PP-MNR qb qNP S qw qkAnA VP qkAnA VP qynZrAn qPP-CLR PP qb qNP

NP h

qNP

— NP him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And Aly

qAly

— at kAnA

qkAnA

— NP-SBJ they . were ynZrAn

qynZrAn

— looking PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP

Statistical Machine Translation

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP

Linear Multi Tree Transducer

MBOT

linear multi tree transducer (Q, Σ, I, R)

• finite set Q

states

• alphabet Σ

input and output symbols

• I ⊆ Q

initial states

• finite set R ⊆ TΣ(Q) × Q × TΣ(Q)∗

rules

– each q ∈ Q occurs at most once in ℓ (ℓ, q, r) ∈ R – each q ∈ Q that occurs in r also occurs in ℓ (ℓ, q, r) ∈ R

Linear Multi Tree Transducer

Syntactic properties

MBOT (Q, Σ, I, R) is

• linear tree transducer with regular look-ahead (XTOPR)

if | r| ≤ 1 ∀(ℓ, q, r) ∈ R

• linear tree transducer (XTOP)

if | r| = 1 ∀(ℓ, q, r) ∈ R

Linear Multi Tree Transducer

Syntactic properties

MBOT (Q, Σ, I, R) is

• linear tree transducer with regular look-ahead (XTOPR)

if | r| ≤ 1 ∀(ℓ, q, r) ∈ R

• linear tree transducer (XTOP)

if | r| = 1 ∀(ℓ, q, r) ∈ R

• ε-free if ℓ /

∈ Q ∀(ℓ, q, r) ∈ R

Linear Multi Tree Transducer

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP

Properties

XTOPR: ✗ XTOP: ✗ ε-free: ✓

Another Example

Textual example

MBOT M = (Q, Σ, {⋆}, R)

• Q = {⋆, q, id, id′}
• Σ = {σ, δ, γ, α}
• the following rules in R:

σ(⋆, q)

⋆

− → σ(⋆, q) σ(⋆, q)

q

− → q δ(id, id′)

⋆,q

− → δ(id, id′) γ(id)

id,id′

− → γ(id) α

id,id′

− → α

Another Example

Graphical representation

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

Properties

XTOPR: ✓ XTOP: ✓ ε-free: ✓

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

⋆ ⋆

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ ⋆ q σ ⋆ q

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ ⋆ q σ ⋆ q

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ id id′ q σ δ id id′ q

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ id id′ q σ δ id id′ q

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ α α σ ⋆ δ α α σ δ α α δ α α

Semantics

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ α α σ γ α δ α α σ δ α α δ α α

Semantics

σ δ id id′ q σ δ id id′ q

Sentential forms

t, A, D, u

• t ∈ TΣ(Q)

input tree

• A ⊆ N∗ × N∗

active links (red)

• D ⊆ N∗ × N∗

disabled links (gray)

• u ∈ TΣ(Q)
• utput tree

Semantics

q — q ⇒M S qw qVP — S qw qVP qVP ⇒M S qw VP qkAnA qVP — S qw qkAnA VP qkAnA qVP ⇒M S w VP qkAnA qVP — S And qkAnA VP qkAnA qVP ⇒M S w VP kAnA qVP — S And they VP were qVP

Semantics

Dependencies and relation

• state-computed dependencies:

Mq = {t, D, u | t, u ∈ TΣ, q, {(ε, ε)}, ∅, q ⇒∗

M t, ∅, D, u}

• computed dependencies:

dep(M) =

• q∈I

Mq

Semantics

Dependencies and relation

• state-computed dependencies:

Mq = {t, D, u | t, u ∈ TΣ, q, {(ε, ε)}, ∅, q ⇒∗

M t, ∅, D, u}

• computed dependencies:

dep(M) =

• q∈I

Mq

• computed transformation:

τM = {(t, u) | t, D, u ∈ dep(M)}

Further Properties

Regularity-preserving

transformation τ ⊆ TΣ × TΣ preserves regularity if τ(L) = {u | (t, u) ∈ τ, t ∈ L} is regular for all regular L ⊆ TΣ rp-MBOT = regularity preserving transformations computable by MBOT

Compositions

• τ1 ; τ2 = {(s, u) | ∃t : (s, t) ∈ τ1, (t, u) ∈ τ2}
• support modular development
• allow integration of external knowledge sources
• occur naturally in query rewriting

Contents

1

Basics

2

Linking technique

Dependencies

Recent research

• Bojańczyk, ICALP 2014
• Maneth et al., ICALP 2015
• n models with dependencies

Dependencies

Hierarchical properties

A dependency t, D, u is

• input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

Dependencies

Hierarchical properties

A dependency t, D, u is

• input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

Dependencies

Hierarchical properties

A dependency t, D, u is

• input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

• strictly input hierarchical if

1

v1 < v2 implies w1 ≤ w2

2

v1 = v2 implies w1 ≤ w2 or w2 ≤ w1

for all (v1, w1), (v2, w2) ∈ D

Dependencies

Distance properties

A dependency t, D, u is

• input link-distance bounded by b ∈ N

if for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Dependencies

Distance properties

A dependency t, D, u is

• input link-distance bounded by b ∈ N

if for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Dependencies

Distance properties

A dependency t, D, u is

• input link-distance bounded by b ∈ N

if for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

• strict input link-distance bounded by b

if for all v1, v1v′ ∈ pos(t) with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Dependencies

σ δ α α σ γ α δ α α σ δ α α δ α α

Dependencies

Dependencies

strictly input hierarchical

Dependencies

strictly input hierarchical and strictly output hierarchical

Dependencies

strictly input hierarchical and strictly output hierarchical with strict input link-distance 2

Dependencies

strictly input hierarchical and strictly output hierarchical with strict input link-distance 2 and strict output link-distance 1

Dependencies

hierarchical link-distance bounded Model \ Property input

• utput

input

• utput

XTOPR strictly strictly ✓ strictly MBOT ✓ strictly ✓ strictly

Linking Theorem

Theorem

Let M1, . . . , Mk be ε-free XTOPR over Σ such that {(c[t1, . . . , tn] , c′[t1, . . . , tn]) | t1, . . . , tn ∈ T} ⊆ τM1 ; · · · ; τMk for some contexts c, c′ ∈ CΣ(Xn) and special T ⊆ TΣ. ∀1 ≤ i ≤ k, ∀1 ≤ j ≤ n ∃tj ∈ T, ∃ui−1, Di, ui ∈ dep(Mi), ∃(vji, wji) ∈ Di such that

• u0 = c[t1, . . . , tn] and uk = c′[t1, . . . , tn]
• posxj(c′) ≤ wjk
• vji ≤ wj(i−1) if i ≥ 2
• posxj(c) ≤ vj1

Linking Theorem

Corollary [Arnold, Dauchet, TCS 1982]

Illustrated relation τ cannot be computed by any ε-free XTOPR

σ σ σ δ tn tn−1 tn−3 tn−2 t4 t3 t2 t1 δ σ σ σ tn tn−1 tn−2 tn−3 tn−4 t3 t2 t1

Linking Theorem

Corollary [M. et al., SICOMP 2009]

Illustrated relation τ cannot be computed by any ε-free XTOPR

δ γ γ δ s t u δ s δ t u

Topicalization

Example

• It rained yesterday night.

Topicalized: Yesterday night, it rained.

Topicalization

Example

• It rained yesterday night.

Topicalized: Yesterday night, it rained.

• We toiled all day yesterday at the restaurant that charges extra for

clean plates. Topicalized: At the restaurant that charges extra for clean plates, we toiled all day yesterday.

Topicalization

On the tree level

S NP PRP it VP VBD rained ADVP NP NN yesterday RB night S NP NN yesterday NN night , S NP PRP it VP VBD rained

Topicalization

On the tree level

S NP PRP we VP VBD toiled NP DT all NN day NP NN yesterday PP IN at NP NP DT the NN restaurant SBAR WHNP WDT that S VP VBZ charges NP JJ extra PP IN for NP JJ clean NNS plates S PP IN at NP NP DT the NN restaurant SBAR WHNP WDT that S VP VBZ charges NP JJ extra PP IN for NP JJ clean NNS plates , S NP PRP we VP VBD toiled NP DT all NN day NP NN yesterday

Topicalization

δ t2 δ t3 δ tn−1 δ tn t1 — δ t1 δ t2 δ t3 δ tn−1 tn

Theorem

Topicalization is in rp-MBOT

Topicalization

Theorem

Topicalization cannot be computed by any composition of ε-free XTOPR

u0 δ t2 δ t3 δ tn−1 δ tn t1 . . . (3) . . . (2) ? (1) — u1 v12 v(n−1)2 vn2 — u2 vn3 v(n−1)3 v13 — u3 δ t1 δ t2 δ t3 δ tn−1 tn 3 ε-free XTOPR sufficient to simulate any composition of ε-free XTOPR

Topicalization

Corollary

(XTOPR)∗ rp-MBOT

Linking Theorem

Theorem

Let M = (Q, Σ, I, R) be an ε-free MBOT such that {(c[t1, . . . , tn] , c′[t1, . . . , tn]) | t1, . . . , tn ∈ T} ⊆ τM for some contexts c, c′ ∈ CΣ(Xn) and special T ⊆ TΣ. ∀1 ≤ j ≤ n, ∃tj ∈ T, ∃u, D, u′ ∈ dep(M), ∃(vj, wj) ∈ D with

• u = c[t1, . . . , tn] and u′ = c′[t1, . . . , tn]
• posxj(c) ≤ vj
• posxj(c′) ≤ wj

Linking Theorem

Corollary

Inverse of topicalization cannot be computed by any ε-free MBOT δ t1 δ t2 δ t3 δ tn−1 tn — δ t2 δ t3 δ tn−1 δ tn t1

Summary & References

Summary

1 (XTOPR)∗ rp-MBOT 2 rp-MBOT not closed under inverses 3 What happens to invertable MBOT?

Summary & References

Summary

1 (XTOPR)∗ rp-MBOT 2 rp-MBOT not closed under inverses 3 What happens to invertable MBOT?

References

• J. Engelfriet, E. Lilin, ∼: Extended multi bottom-up tree transducers —

Composition and decomposition. Acta Inf., 2009

• Z. Fülöp, ∼: Composition closure of ε-free linear extended top-down tree
• transducers. Proc. 17th DLT, LNCS 7907, 2013
• P. Koehn: Statistical machine translation. Cambridge Univ. Press, 2009
• ∼, J. Graehl, M. Hopkins, K. Knight: The power of extended top-down tree
• transducers. SIAM J. Comput., 2009