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

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

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

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

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

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

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

Statistical Machine Translation Extracted rules S S VP VP NP NP q q VP — q kAnA . q NP — — q w q VP q w q VP q VP q kAnA q VP q kAnA q VP q $kl q mDHk q $kl q mDHk q $kl NP q $kl q mDHk q NP q b $kl — a . way mDHk — funny — him b — in PP-MNR PP q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w w — And kAnA — they . were ynZrAn — looking Aly — at VP VP PP-CLR PP-CLR q ynZrAn q VP q PP-CLR q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆

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 ⊆ T Σ ( Q ) × Q × T Σ ( Q ) ∗ rules – each q ∈ Q occurs at most once in ℓ ( ℓ, q ,� r ) ∈ R – each q ∈ Q that occurs in � r ) ∈ R r also occurs in ℓ ( ℓ, q ,�

Linear Multi Tree Transducer Syntactic properties MBOT ( Q , Σ , I , R ) is • linear tree transducer with regular look-ahead (XTOP R ) 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 (XTOP R ) 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 S VP VP NP NP q q VP — q kAnA . q NP — — q w q VP q w q VP q VP q kAnA q VP q kAnA q VP q $kl q mDHk q $kl q mDHk q $kl NP q $kl q mDHk q NP q b $kl — a . way mDHk — funny PP — him b — in PP-MNR q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w w — And kAnA — they . were ynZrAn — looking Aly — at VP VP PP-CLR PP-CLR q ynZrAn q VP q PP-CLR q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆ Properties XTOP R : 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 : q ⋆ − → σ ( ⋆, q ) − → q σ ( ⋆, q ) σ ( ⋆, q ) id , id ′ id , id ′ ⋆, q δ ( id , id ′ ) → δ ( id , id ′ ) − γ ( id ) − → γ ( id ) α − → α

Another Example Graphical representation σ σ δ δ ⋆, q ⋆ − → − → q q ⋆ ⋆ id id ′ id id ′ γ γ σ q → q id , id ′ id , id ′ − − → α − → α q ⋆ id id Properties XTOP R : XTOP: ε -free: ✓ ✓ ✓

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id ⋆ ⋆

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ q q ⋆ ⋆

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ q q ⋆ ⋆

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ q q δ δ id ′ id ′ id id

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ q q δ δ id ′ id ′ id id

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ σ δ δ δ α α ⋆ δ α α α α α α

Semantics Rules γ γ σ σ δ δ σ ⋆, q → q q id , id ′ id , id ′ ⋆ − → − → − − → − → α α q q q ⋆ ⋆ ⋆ id ′ id ′ id id id id σ σ σ δ δ δ γ α α δ α α α α α α α

Semantics σ σ q q δ δ id ′ id ′ id id Sentential forms � t , A , D , u � • t ∈ T Σ ( Q ) input tree • A ⊆ N ∗ × N ∗ active links (red) • D ⊆ N ∗ × N ∗ disabled links (gray) • u ∈ T Σ ( Q ) output tree

Semantics S S S S q — q ⇒ M ⇒ M — q w — q w q kAnA VP VP q w q VP q w q VP q VP q kAnA q VP q kAnA q VP S S S S ⇒ M ⇒ M — q kAnA w — w VP they VP And VP And VP q kAnA q VP q kAnA q VP q VP q VP kAnA were

Semantics Dependencies and relation • state-computed dependencies: M q = {� t , D , u � | t , u ∈ T Σ , � q , { ( ε, ε ) } , ∅ , q � ⇒ ∗ M � t , ∅ , D , u �} • computed dependencies: � dep ( M ) = M q q ∈ I

Semantics Dependencies and relation • state-computed dependencies: M q = {� t , D , u � | t , u ∈ T Σ , � q , { ( ε, ε ) } , ∅ , q � ⇒ ∗ M � t , ∅ , D , u �} • computed dependencies: � dep ( M ) = M q q ∈ I • computed transformation: τ M = { ( t , u ) | � t , D , u � ∈ dep ( M ) }

Further Properties Regularity-preserving transformation τ ⊆ T Σ × T Σ preserves regularity if τ ( L ) = { u | ( t , u ) ∈ τ, t ∈ L } is regular for all regular L ⊆ T Σ rp-MBOT = regularity preserving transformations computable by MBOT Compositions • τ 1 ; τ 2 = { ( s , u ) | ∃ t : ( s , t ) ∈ τ 1 , ( t , u ) ∈ τ 2 } • support modular development • allow integration of external knowledge sources • occur naturally in query rewriting

Contents Basics 1 Linking technique 2

Dependencies Recent research • Bojańczyk, ICALP 2014 • Maneth et al., ICALP 2015 on models with dependencies

Dependencies Hierarchical properties A dependency � t , D , u � is • input hierarchical if w 2 � < w 1 1 ∃ ( v 1 , w ′ 1 ) ∈ D with w ′ 1 ≤ w 2 2 for all ( v 1 , w 1 ) , ( v 2 , w 2 ) ∈ D with v 1 < v 2

Dependencies Hierarchical properties A dependency � t , D , u � is • input hierarchical if w 2 � < w 1 1 ∃ ( v 1 , w ′ 1 ) ∈ D with w ′ 1 ≤ w 2 2 for all ( v 1 , w 1 ) , ( v 2 , w 2 ) ∈ D with v 1 < v 2