Composition Closure of Linear Extended Top-down Tree Transducers Zoltán Fülöp and Andreas Maletti maletti@ims.uni-stuttgart.de Leipzig — April 8, 2014 Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Syntax-based Statistical Machine Translation Input data 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 Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Syntax-based 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 NP q b q $kl q mDHk — him b — in $kl — a . way mDHk — funny PP-MNR PP q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w kAnA — they . were ynZrAn — looking w — And Aly — at VP VP PP-CLR PP-CLR q VP q PP-CLR q ynZrAn q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Syntax-based 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 NP q b q $kl q mDHk — him b — in $kl — a . way mDHk — funny PP-MNR PP q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w kAnA — they . were ynZrAn — looking w — And Aly — at VP VP PP-CLR PP-CLR q VP q PP-CLR q ynZrAn q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆ • for a tree-to-tree transformation device = tree transducer • here: for a linear extended multi bottom-up tree transducer Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Motivation Tree transducer • used in statistical machine translation [Knight, Graehl 2005] • used in XML query processing [Benedikt et al. 2013] Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Motivation Tree transducer • used in statistical machine translation [Knight, Graehl 2005] • used in XML query processing [Benedikt et al. 2013] 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 Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Problem Question: given a class C of transformations, is there n ∈ N such that � C n = C k = C ; · · · ; C C k � �� � k ≥ 1 k times Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Problem Question: given a class C of transformations, is there n ∈ N such that � C n = C k = C ; · · · ; C C k � �� � k ≥ 1 k times Note • C k ⊆ C k + 1 for our classes C → we search least n such that C n = C n + 1 (if it exists) Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up 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 NP q b q $kl q mDHk — him b — in $kl — a . way mDHk — funny PP-MNR PP q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w kAnA — they . were ynZrAn — looking w — And Aly — at VP VP PP-CLR PP-CLR q VP q PP-CLR q ynZrAn q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up Tree Transducer Definition (MBOT) linear extended multi bottom-up 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 r ) ∈ R ( ℓ, q ,� – each q ∈ Q that occurs in � ( ℓ, q ,� r ) ∈ R r also occurs in ℓ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up 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 NP q b q $kl q mDHk — him b — in $kl — a . way mDHk — funny PP-MNR PP q PP-MNR h — q b q NP q b q NP q kAnA q ynZrAn q Aly q w kAnA — they . were ynZrAn — looking w — And Aly — at VP VP PP-CLR PP-CLR q VP q PP-CLR q ynZrAn q PP-CLR q PP-MNR NP-SBJ — — q ynZrAn q Aly q Aly q PP-CLR q PP-MNR q NP q NP ⋆ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up Tree Transducer Definition (Syntactic properties) MBOT ( Q , Σ , I , R ) is • linear extended top-down tree transducer with regular look-ahead (XTOP R ) if | � r | ≤ 1 ∀ ( ℓ, q ,� r ) ∈ R • linear extended top-down tree transducer (XTOP) if | � r | = 1 Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up Tree Transducer Definition (Syntactic properties) MBOT ( Q , Σ , I , R ) is • linear extended top-down tree transducer with regular look-ahead (XTOP R ) if | � r | ≤ 1 ∀ ( ℓ, q ,� r ) ∈ R • linear extended top-down tree transducer (XTOP) if | � r | = 1 • linear top-down tree transducer (TOP/TOP R ) if XTOP/XTOP R and ℓ contains exactly one element of Σ ∈ Q + ) • ε -free (resp. strict) if ℓ / ∈ Q (resp. � r / Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up Tree Transducer Definition (Syntactic properties) MBOT ( Q , Σ , I , R ) is • linear extended top-down tree transducer with regular look-ahead (XTOP R ) if | � r | ≤ 1 ∀ ( ℓ, q ,� r ) ∈ R • linear extended top-down tree transducer (XTOP) if | � r | = 1 • linear top-down tree transducer (TOP/TOP R ) if XTOP/XTOP R and ℓ contains exactly one element of Σ ∈ Q + ) • ε -free (resp. strict) if ℓ / ∈ Q (resp. � r / • delabeling if it is a TOP and � r contains at most one element of Σ • nondeleting if the same elements of Q occur in ℓ and � r Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up 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 NP q b q $kl q mDHk $kl — a . way mDHk — funny PP-MNR PP — him b — in 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 PP-CLR q PP-MNR q Aly q NP q Aly q NP ⋆ Properties XTOP R : TOP R : ✗ XTOP: ✗ ✗ TOP: ✗ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Linear Extended Multi Bottom-up 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 NP q b q $kl q mDHk $kl — a . way mDHk — funny PP-MNR PP — him b — in 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 PP-CLR q PP-MNR q Aly q NP q Aly q NP ⋆ Properties XTOP R : TOP R : ✗ XTOP: ✗ ✗ TOP: ✗ ε -free: strict: delabeling: nondeleting: ✓ ✓ ✗ ✓ Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
The problem Upper bounds Linking technique Lower bounds Another Example Example (textual) 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 ) ⋆, q id , id ′ id , id ′ δ ( id , id ′ ) → δ ( id , id ′ ) − γ ( id ) − → γ ( id ) α − → α Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP
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