Tree Transformations and Dependencies · 4 Synchronous tree-sequence substitution grammar A....
Transcript of Tree Transformations and Dependencies · 4 Synchronous tree-sequence substitution grammar A....
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Tree Transformations and Dependencies
Andreas Maletti
Institute for Natural Language ProcessingUniversity of Stuttgart
Nara — September 6, 2011
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Contents
1 Motivation
2 Extended top-down tree transducer
3 Extended multi bottom-up tree transducer
4 Synchronous tree-sequence substitution grammar
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Warning!
This won’t be a regular survey talk.
Rather I want to show a useful technique.
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Warning!
This won’t be a regular survey talk.
Rather I want to show a useful technique.
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Context-free Language
QuestionHow do you show that a language is not context-free?
Answers
• pumping lemma (BAR-HILLEL lemma)• semi-linearity (PARIKH’s theorem)• ...
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Context-free Language
QuestionHow do you show that a language is not context-free?
Answers
• pumping lemma (BAR-HILLEL lemma)• semi-linearity (PARIKH’s theorem)• ...
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Context-free Language
Hardly ever ...
Let us assume that there is a CFG ...(long case distinction on the shape of its rules)Contradiction!
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Tree Transformation
Application areas
• NLP• XML processing• syntax-directed semantics
Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
7→
δ
t1 δ
t2 δ
t3 δ
t4 δ
tn−3 δ
tn−2 δ
tn−1 tn
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Tree Transducer
DefinitionA formal model computing a tree transformation
Example
• top-down or bottom-up tree transducer• synchronous grammar (SCFG, STSG, STAG, etc.)• multi bottom-up tree transducer• bimorphism
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Tree Transducer
DefinitionA formal model computing a tree transformation
Example
• top-down or bottom-up tree transducer• synchronous grammar (SCFG, STSG, STAG, etc.)• multi bottom-up tree transducer• bimorphism
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Tree Transducer
QuestionHow do you show that a tree transformation cannot be computedby a class of tree transducers?
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Tree Transducer
QuestionHow do you show that a tree transformation cannot be computedby a class of tree transducers?
Answers
• pumping lemma (????)• size and height dependencies• domain and image properties• properties under composition (??)
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Tree Transducer
QuestionHow do you show that a tree transformation cannot be computedby a class of tree transducers?
Typical approach
• Assume that it can be computed .... Contradiction!
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Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
7→
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
QuestionCan this tree transformation be computed by a linear,nondeleting extended top-down tree transducer (XTOP)?
Answer [Arnold, Dauchet 1982]
No! (via indirect proof)
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Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
7→
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
QuestionCan this tree transformation be computed by a linear,nondeleting extended top-down tree transducer (XTOP)?
Answer [Arnold, Dauchet 1982]
No! (via indirect proof)
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Another Example
σ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
QuestionCan this tree transformation becomputed by an XTOP?
Answer [∼, Graehl, Hopkins, Knight 2009]
No! (via indirect proof)
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Another Example
σ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
QuestionCan this tree transformation becomputed by an XTOP?
Answer [∼, Graehl, Hopkins, Knight 2009]
No! (via indirect proof)
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Yet Another Exampleα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
7→
QuestionCan this tree transformation becomputed by an MBOT?
MBOTlinear, nondeleting multibottom-up tree transducer
AnswerWe’ll know at the end
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Yet Another Exampleα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
7→
QuestionCan this tree transformation becomputed by an MBOT?
MBOTlinear, nondeleting multibottom-up tree transducer
AnswerWe’ll know at the end
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Yet Another Exampleα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
7→
QuestionCan this tree transformation becomputed by an MBOT?
MBOTlinear, nondeleting multibottom-up tree transducer
AnswerWe’ll know at the end
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Contents
1 Motivation
2 Extended top-down tree transducer
3 Extended multi bottom-up tree transducer
4 Synchronous tree-sequence substitution grammar
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Extended Top-down Tree Transducer
Definition (XTOP)
tuple (Q,Σ,∆, I,R)
• Q: states• Σ, ∆: input and output symbols• I ⊆ Q: initial states
• R ⊆ TΣ(Q)×Q × T∆(Q): finite set of rules– l and r are linear in Q– var(l) = var(r)
for every l q— r ∈ R
[ARNOLD, DAUCHET: Morphismes et bimorphismes d’arbres. Theor. Comput.Sci. 1982]
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Extended Top-down Tree Transducer
Definition (XTOP)
tuple (Q,Σ,∆, I,R)
• Q: states• Σ, ∆: input and output symbols• I ⊆ Q: initial states• R ⊆ TΣ(Q)×Q × T∆(Q): finite set of rules
– l and r are linear in Q– var(l) = var(r)
for every l q— r ∈ R
[ARNOLD, DAUCHET: Morphismes et bimorphismes d’arbres. Theor. Comput.Sci. 1982]
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Example XTOP
Example
XTOP (Q,Σ,Σ, {?},R) with Q = {?,p,q, r} and Σ = {σ, δ, α, ε}
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
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Sentential Form
L = {S | S ⊆ N∗ × N∗}
Definition (Sentential form)
〈ξ,D, ζ〉 ∈ TΣ(Q)× L× T∆(Q) such that
• v ∈ pos(ξ) and w ∈ pos(ζ) for every (v ,w) ∈ D.
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Sentential Form
L = {S | S ⊆ N∗ × N∗}
Definition (Sentential form)
〈ξ,D, ζ〉 ∈ TΣ(Q)× L× T∆(Q) such that
• v ∈ pos(ξ) and w ∈ pos(ζ) for every (v ,w) ∈ D.
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Link Structure
Definitionrule l q— r ∈ R, positions v ,w ∈ N∗
linksv ,w (l q— r) =⋃
p∈Q
{(vv ′,ww ′) | v ′ ∈ posp(l),w ′ ∈ posp(r)}
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Link Structure
Definitionrule l q— r ∈ R, positions v ,w ∈ N∗
linksv ,w (l q— r) =⋃
p∈Q
{(vv ′,ww ′) | v ′ ∈ posp(l),w ′ ∈ posp(r)}
σ
p q ??—
δ
p δ
q ?
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Link Structure
Definitionrule l q— r ∈ R, positions v ,w ∈ N∗
linksv ,w (l q— r) =⋃
p∈Q
{(vv ′,ww ′) | v ′ ∈ posp(l),w ′ ∈ posp(r)}
σ
p q ??—
δ
p δ
q ?
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Derivation
Definition
〈ξ,D, ζ〉 ⇒ 〈ξ[l]v , D ∪ linksv ,w (l q— r), ζ[r ]w 〉
if• rule l q— r ∈ R• position v ∈ posq(ξ)
• position w ∈ posq(ζ) such that (v ,w) ∈ D
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
? ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
? ?
δ
p ?
δ
p ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
p ?
δ
p ?
δ
p σ
p q ?
δ
p δ
p δ
q ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
p σ
p q ?
δ
p δ
p δ
q ?
δ
ε σ
p q ?
δ
ε δ
p δ
q ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
ε σ
p q ?
δ
ε δ
p δ
q ?
δ
ε σ
α
p
q ?
δ
ε δ
α
p
δ
q ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
ε σ
α
p
q ?
δ
ε δ
α
p
δ
q ?
δ
ε σ
α
ε
q ?
δ
ε δ
α
ε
δ
q ?
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
ε σ
α
ε
q ?
δ
ε δ
α
ε
δ
q ?
δ
ε σ
α
ε
q σ
p q r
δ
ε δ
α
ε
δ
q δ
p δ
q r
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
ε σ
α
ε
q σ
p q r
δ
ε δ
α
ε
δ
q δ
p δ
q r
δ
ε σ
α
ε
ε σ
p q r
δ
ε δ
α
ε
δ
ε δ
p δ
q r
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Derivation
Rules
σ
p q ??—
δ
p δ
q ?
σ
p q r?—
δ
p δ
q r
δ
p ?
?—
δ
p ?
α
xx—
α
xε x— ε
δ
ε σ
α
ε
ε σ
p q r
δ
ε δ
α
ε
δ
ε δ
p δ
q r
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Semantics of XTOP
Definition
• the dependencies dep(M)
{〈t ,D,u〉 ∈ TΣ×L×T∆ | ∃q ∈ I : 〈q, {(ε, ε)},q〉 ⇒∗M 〈t ,D,u〉}
• the tree transformation M = {(t ,u) | 〈t ,D,u〉 ∈ dep(M)}
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Semantics of XTOP
Definition
• the dependencies dep(M)
{〈t ,D,u〉 ∈ TΣ×L×T∆ | ∃q ∈ I : 〈q, {(ε, ε)},q〉 ⇒∗M 〈t ,D,u〉}
• the tree transformation M = {(t ,u) | 〈t ,D,u〉 ∈ dep(M)}
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
7→
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Semantics of XTOP
Example
D = {(ε, ε), (1,1), (2,2), (21,21), (211,211), (22,221), (23,222),
(231,2221), (232,22221), (233,22222)}
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Semantics of XTOP
Example
D = {(ε, ε), (1,1), (2,2), (21,21), (211,211), (22,221), (23,222),
(231,2221), (232,22221), (233,22222)}
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
7→
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Semantics of XTOP
Example
D = {(ε, ε), (1,1), (2,2), (21,21), (211,211), (22,221), (23,222),
(231,2221), (232,22221), (233,22222)}
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Non-closure under Composition
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
7→
δ
t1 δ
t2 δ
t3 δ
t4 δ
tn−3 δ
tn−2 δ
tn−1 tn
7→
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
[ARNOLD, DAUCHET: Morphismes et bimorphismes d’arbres. Theor. Comput.Sci. 1982]
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Special Linking Structure
DefinitionD ∈ L is input hierarchical if for all (v1,w1), (v2,w2) ∈ D• v1 < v2 implies w2 6< w1 and ∃w ′1 ≤ w2 : (v1,w ′1) ∈ D
• w1 6≤ w2 implies v1 6≤ v2
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Special Linking Structure
DefinitionD ∈ L is input hierarchical if for all (v1,w1), (v2,w2) ∈ D• v1 < v2 implies w2 6< w1 and ∃w ′1 ≤ w2 : (v1,w ′1) ∈ D
• w1 6≤ w2 implies v1 6≤ v2
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Special Linking Structure
DefinitionD ∈ L is input hierarchical if for all (v1,w1), (v2,w2) ∈ D• v1 < v2 implies w2 6< w1 and ∃w ′1 ≤ w2 : (v1,w ′1) ∈ D
• w1 6≤ w2 implies v1 6≤ v2
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Special Linking Structure
DefinitionD ∈ L is strictly input hierarchical if for all (v1,w1), (v2,w2) ∈ D• v1 < v2 implies w2 6< w1 and ∃w ′1 ≤ w2 : (v1,w ′1) ∈ D• w1 6≤ w2 implies v1 6≤ v2
v1 w2
v2 w1
v1
v2 w1 w2
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Special Linking Structure
DefinitionD ∈ L is strictly input hierarchical if for all (v1,w1), (v2,w2) ∈ D• v1 < v2 implies w2 6< w1 and ∃w ′1 ≤ w2 : (v1,w ′1) ∈ D• w1 6≤ w2 implies v1 6≤ v2
v
w1 w2
v1
v2 w1 w2
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Dependencies
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
Example
This linking structure is strictly (input and output) hierarchical
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Dependencies
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Dependencies
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Dependencies
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
v1 w2
v2 w1
v1
v2 w1 w2
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Dependencies
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
v
w1 w2
v1
v2 w1 w2
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Another Property
DefinitionD ⊆ L has bounded distance if ∃k ∈ N such that ∀D ∈ D• if v , vw ∈ dom(D) with |w | > k , then ∃w ′ ≤ w with
– vw ′ ∈ dom(D)– |w ′| ≤ k
• (symmetric property for range)
Illustration:
v vw
> k
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Another Property
DefinitionD ⊆ L has bounded distance if ∃k ∈ N such that ∀D ∈ D• if v , vw ∈ dom(D) with |w | > k , then ∃w ′ ≤ w with
– vw ′ ∈ dom(D)– |w ′| ≤ k
• (symmetric property for range)
Illustration:
v vw ′ vw
≤ k
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Another Property
DefinitionD ⊆ L has bounded distance if ∃k ∈ N such that ∀D ∈ D• if v , vw ∈ dom(D) with |w | > k , then ∃w ′ ≤ w with
– vw ′ ∈ dom(D)– |w ′| ≤ k
• (symmetric property for range)
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Bounded Distance
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
Example
• input side: bounded by 1• output side: bounded by 2
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Main Result
TheoremThe dependencies computed by an XTOP are:• strictly (input and output) hierarchical• bounded distance
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Compatibility
Definition
• 〈ξ,D, ζ〉 is compatible with 〈ξ,D′, ζ〉 if D ⊆ D′
• set L is compatible with L′ iffor all 〈ξ,_, ζ〉 ∈ L (some computation)
– exists 〈ξ,D, ζ〉 ∈ L (implemented computation)– exists compatible 〈ξ,D′, ζ〉 ∈ L′ (compatible computation)
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Compatibility
Definition
• 〈ξ,D, ζ〉 is compatible with 〈ξ,D′, ζ〉 if D ⊆ D′
• set L is compatible with L′ iffor all 〈ξ,_, ζ〉 ∈ L (some computation)
– exists 〈ξ,D, ζ〉 ∈ L (implemented computation)– exists compatible 〈ξ,D′, ζ〉 ∈ L′ (compatible computation)
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Compatibility
Illustration
Some computation
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Compatibility
Illustration
Implemented computation
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Compatibility
Illustration
Compatible computation
δ
ε σ
α
ε
ε σ
ε ε ε
δ
ε δ
α
ε
δ
ε δ
ε δ
ε ε
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
use boundedness
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
use boundedness
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
v1 w2
v2 w1
v1
v2 w1 w2
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
v1 w2
v2 w1
v1
v2 w1 w2
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Back to the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
TheoremThe dependencies above are incompatible with any XTOP
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A First Instance
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
7→
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
Theorem (ARNOLD, DAUCHET 1982)
The transformation above cannot be computed by any XTOP
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
use boundedness
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
use boundedness
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
v1 w2
v2 w1
v1
v2 w1 w2
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
TheoremThe dependencies above are incompatible with any XTOP
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A Second Instanceσ
γ
...
γ
σ
t1 t2
t3
7→δ
t1 t2 t3
Theorem (∼, GRAEHL, HOPKINS, KNIGHT 2009)
The transformation above cannot be computed by any XTOP
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Contents
1 Motivation
2 Extended top-down tree transducer
3 Extended multi bottom-up tree transducer
4 Synchronous tree-sequence substitution grammar
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Extended Multi Bottom-up Tree Transducer
Definition (MBOT)
tuple (Q,Σ,∆, I,R)
• Q: ranked alphabet of states• Σ, ∆: input and output symbols• I ⊆ Q1: unary initial states
• R ⊆ TΣ(Q)×Q × T∆(Q)∗: finite set of rules– l is linear in Q– rk(q) = |~r |– var(~r) ⊆ var(l)
for every (l ,q,~r) ∈ R
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Extended Multi Bottom-up Tree Transducer
Definition (MBOT)
tuple (Q,Σ,∆, I,R)
• Q: ranked alphabet of states• Σ, ∆: input and output symbols• I ⊆ Q1: unary initial states• R ⊆ TΣ(Q)×Q × T∆(Q)∗: finite set of rules
– l is linear in Q– rk(q) = |~r |– var(~r) ⊆ var(l)
for every (l ,q,~r) ∈ R
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Example MBOT
Example
(Q,Σ,Σ, {f},R) with Q = {?(2),p(1),q(1), r (1), f (1)} andΣ = {σ, δ, α, ε}
δ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
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New Linking Structure
Definitionrule l q— ~r ∈ R, positions v ,w1, . . . ,wn ∈ N∗~w = w1 · · ·wn with n = rk(q)
linksv ,~w (l q— ~r) =⋃
p∈Q
n⋃i=1
{(vv ′,wiw ′i ) | v ′ ∈ posp(l),w ′i ∈ posp(ri)}
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New Linking Structure
Definitionrule l q— ~r ∈ R, positions v ,w1, . . . ,wn ∈ N∗~w = w1 · · ·wn with n = rk(q)
linksv ,~w (l q— ~r) =⋃
p∈Q
n⋃i=1
{(vv ′,wiw ′i ) | v ′ ∈ posp(l),w ′i ∈ posp(ri)}
Illustration:σ
p q ??— p
σ
? q ?
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New Linking Structure
Definitionrule l q— ~r ∈ R, positions v ,w1, . . . ,wn ∈ N∗~w = w1 · · ·wn with n = rk(q)
linksv ,~w (l q— ~r) =⋃
p∈Q
n⋃i=1
{(vv ′,wiw ′i ) | v ′ ∈ posp(l),w ′i ∈ posp(ri)}
Illustration:σ
p q ??— p
σ
? q ?
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Derivation
Definition
〈ξ,D, ζ〉 ⇒ 〈ξ[l]v , D ∪ linksv ,~w (l q— ~r), ζ[~r ]~w 〉
if• rule l q— ~r ∈ R with rk(q) = n• position v ∈ posq(ξ)
• ~w = w1 · · ·wn ∈ posq(ζ)∗ with– w1 @ · · · @ wn– {w1, . . . ,wn} = {w | (v ,w) ∈ D}
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
? ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
? ?
δ
p ?
σ
? p ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
p ?
σ
? p ?
δ
ε ?
σ
? ε ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε ?
σ
? ε ?
δ
ε σ
p q ?
σ
p ε σ
? q ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε σ
p q ?
σ
p ε σ
? q ?
δ
ε σ
α
p
q ?
σ
α
p
ε σ
? q ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε σ
α
p
q ?
σ
α
p
ε σ
? q ?
δ
ε σ
α
ε
q ?
σ
α
ε
ε σ
? q ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε σ
α
ε
q ?
σ
α
ε
ε σ
? q ?
δ
ε σ
α
ε
ε ?
σ
α
ε
ε σ
? ε ?
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε σ
α
ε
ε ?
σ
α
ε
ε σ
? ε ?
δ
ε σ
α
ε
ε σ
p q r
σ
α
ε
ε σ
p ε δ
q r
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Derivation
Rulesδ
p ?f—
σ
? p ?
α
xx—
α
xε x— ε
σ
p q ??— p
σ
? q ?
σ
p q r?— p
δ
q r
δ
ε σ
α
ε
ε σ
p q r
σ
α
ε
ε σ
p ε δ
q r
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
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Semantics of MBOT
DefinitionMBOT M computes• dependencies dep(M)
{〈t ,D,u〉 ∈ TΣ×L×T∆ | ∃q ∈ I : 〈q, {(ε, ε)},q〉 ⇒∗M 〈t ,D,u〉}
• the tree transformation M = {(t ,u) | 〈t ,D,u〉 ∈ dep(M)}
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Semantics of MBOT
DefinitionMBOT M computes• dependencies dep(M)
{〈t ,D,u〉 ∈ TΣ×L×T∆ | ∃q ∈ I : 〈q, {(ε, ε)},q〉 ⇒∗M 〈t ,D,u〉}
• the tree transformation M = {(t ,u) | 〈t ,D,u〉 ∈ dep(M)}
δ
ε σ
α
ε
ε σ
ε ε ε
7→
σ
α
ε
ε σ
ε ε δ
ε ε
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Semantics of MBOT
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
Example
D = {(ε, ε), (1,2), (2,1), (2,3), (21,1), (211,11), (22,32),
(23,31), (23,33), (231,31), (232,331), (233,332)}
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Semantics of MBOT
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
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Semantics of MBOT
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
Example
Above dependencies are• input hierarchical• strictly output hierarchical
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Semantics of MBOT
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
Example
Above dependencies are• input hierarchical but not strictly• strictly output hierarchical
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Semantics of MBOT
δ
ε σ
α
ε
ε σ
ε ε ε
σ
α
ε
ε σ
ε ε δ
ε ε
v
w1 w2
v1
v2 w1 w2
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Main Result
TheoremThe dependencies computed by an MBOT are:• input hierarchical• strictly output hierarchical• bounded distance
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Link Structure for the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
TheoremAbove dependencies are compatible with an MBOT
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Link Structure for the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
TheoremAbove dependencies are compatible with an MBOT
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Link Structure for the Example
δ
t1 σ
t2 t3 σ
tn−4 tn−3 σ
tn−2 tn−1 tn
σ
t2 t1 σ
t4 t3 σ
tn−2 tn−3 δ
tn−1 tn
TheoremAbove dependencies are compatible with an MBOT
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Another MBOT
Example
(Q,Σ,∆, {f},R) with Q = {p(3),q(3), r (3), f (1)}Σ = {ε, α, β, γ} and ∆ = Σ ∪ {σ}
p f—
σ
p p p
α
pp—
α
pp p
β
qq— q
β
γ
rr— r r
γ
r
q p— q q q r q— r r r ε r— ε ε ε
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Another MBOTα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
taken fromExample 4.5 of [RADMACHER: An automatatheoretic approach to the theory of rationaltree relations. TR AIB-2008-05, RWTHAachen, 2008]
{(α`(βm(γn(ε))), σ(α`(ε), βm(ε), γn(ε))) | `,m,n ∈ N}
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Another MBOTα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
TheoremThese inverse dependencies are notcompatible with any MBOT(because the displayed dependenciesare not strictly input hierarchical)
v1 w2
v2 w1
v1
v2 w1 w2
TheoremThis tree transformation cannot becomputed by any MBOT
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Another MBOTα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
7→
TheoremThese inverse dependencies are notcompatible with any MBOT(because the displayed dependenciesare not strictly input hierarchical)
v1 w2
v2 w1
v1
v2 w1 w2
TheoremThis tree transformation cannot becomputed by any MBOT
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Contents
1 Motivation
2 Extended top-down tree transducer
3 Extended multi bottom-up tree transducer
4 Synchronous tree-sequence substitution grammar
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Synchronous Tree-Sequence Substitution Grammar
Definition (STSSG)
tuple (Q,Σ,∆, I,R) with• Q: doubly ranked alphabet• Σ, ∆: input and output symbols• I ⊆ Q1,1: doubly unary initial states
• R ⊆ TΣ(Q)∗ ×Q × T∆(Q)∗: finite set of rules– rk(q) = (|~l |, |~r |) for every (~l ,q,~r) ∈ R
[ZHANG et al.: A tree sequence alignment-based tree-to-tree translation model.Proc. ACL, 2008]
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Synchronous Tree-Sequence Substitution Grammar
Definition (STSSG)
tuple (Q,Σ,∆, I,R) with• Q: doubly ranked alphabet• Σ, ∆: input and output symbols• I ⊆ Q1,1: doubly unary initial states• R ⊆ TΣ(Q)∗ ×Q × T∆(Q)∗: finite set of rules
– rk(q) = (|~l |, |~r |) for every (~l ,q,~r) ∈ R
[ZHANG et al.: A tree sequence alignment-based tree-to-tree translation model.Proc. ACL, 2008]
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Yet Another Linking Structure
Definitionrule~l q— ~r ∈ R, positions v1, . . . , vm,w1, . . . ,wn ∈ N∗with rk(q) = (m,n)
links~v ,~w (~l q— ~r)
=⋃
p∈Q
m⋃j=1
n⋃i=1
{(vjv ′j ,wiw ′i ) | v ′j ∈ posp(lj),w ′i ∈ posp(ri)}
where ~v = v1 · · · vm and ~w = w1 · · ·wn
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Derivation
Definition
〈ξ,D, ζ〉 ⇒ 〈ξ[~l]~v , D ∪ links~v ,~w (~l q— ~r), ζ[~r ]~w 〉
if• rule~l q— ~r ∈ R• ~v = v1 · · · vm ∈ posq(ξ)∗ and ~w = w1 · · ·wn ∈ posq(ζ)∗
• v1 @ · · · @ vm, w1 @ · · · @ wn, and rk(q) = (m,n)
• linked positions{w1, . . . ,wn} =
⋃mj=1{w | (vj ,w) ∈ D}
{v1, . . . , vm} =⋃n
i=1{v | (v ,wi) ∈ D}
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A Final Theoremα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
7→
Example
This transformation can be computedby an STSSG(because it is an inverse MBOT)
TheoremDependencies computed by STSSG are• (input and output) hierarchical• bounded distance
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A Final Theoremα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
Example
This transformation can be computedby an STSSG(because it is an inverse MBOT)
TheoremDependencies computed by STSSG are• (input and output) hierarchical• bounded distance
A. Maletti MOL 2011 121
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A Final Theoremα
...
α
β
...
β
γ
...
γ
ε
σ
α
...
α
ε
β
...
β
ε
γ
...
γ
ε
Example
This transformation can be computedby an STSSG(because it is an inverse MBOT)
TheoremDependencies computed by STSSG are• (input and output) hierarchical• bounded distance
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A Final Instance
α
...
α
β
...
β
γ
...
γ
ε
γ
...
γ
β
...
β
α
...
α
ε
TheoremThese dependencies are notcompatible with any STSSG
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
TheoremThis transformation cannot becomputed by any STSSG
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A Final Instance
α
...
α
β
...
β
γ
...
γ
ε
γ
...
γ
β
...
β
α
...
α
ε
7→
TheoremThese dependencies are notcompatible with any STSSG
v1 w2
v2 w1
v1 w ′1
v2 w1 w2
TheoremThis transformation cannot becomputed by any STSSG
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Summary
Computed dependencies by device
input output
XTOP strictly hierarchical strictly hierarchical
MBOT hierarchical strictly hierarchicalSTSSG hierarchical hierarchical
STAG — —
A. Maletti MOL 2011 125
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Summary
Computed dependencies by device
input output
XTOP strictly hierarchical strictly hierarchicalMBOT hierarchical strictly hierarchical
STSSG hierarchical hierarchicalSTAG — —
A. Maletti MOL 2011 126
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Summary
Computed dependencies by device
input output
XTOP strictly hierarchical strictly hierarchicalMBOT hierarchical strictly hierarchical
STSSG hierarchical hierarchical
STAG — —
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Summary
Computed dependencies by device
input output
XTOP strictly hierarchical strictly hierarchicalMBOT hierarchical strictly hierarchical
STSSG hierarchical hierarchicalSTAG — —
A. Maletti MOL 2011 128
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That’s all, folks!
Thank you for your attention!
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References• ARNOLD, DAUCHET: Morphismes et bimorphismes d’arbres.
Theor. Comput. Sci. 20, 1982• ENGELFRIET, LILIN, MALETTI:
Extended multi bottom-up tree transducers.Acta Inf. 46, 2009
• MALETTI, GRAEHL, HOPKINS, KNIGHT:The power of extended top-down tree transducers.SIAM J. Comput. 39, 2009
• RADMACHER: An automata theoretic approach to rational treerelations.Proc. SOFSEM 2008
• RAOULT: Rational tree relations.Bull. Belg. Math. Soc. Simon Stevin 4, 1997
• ZHANG, JIANG, AW, LI, TAN, LI: A tree sequencealignment-based tree-to-tree translation model.Proc. ACL 2008
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