Input Products for Weighted Extended Top-down Tree Transducers Andreas Maletti Universitat Rovira i Virgili Tarragona, Spain andreas.maletti@urv.cat London, ON — August 17, 2010 Input Products for WXTT A. Maletti 1 ·
Machine translation Schema Machine Input − → translation − → Output system Question How does the system handle input sentences containing “system”? Some answer take regular language L = ∗ system ∗ turn into a regular tree language use forward application Input Products for WXTT A. Maletti 2 ·
Machine translation Schema Input − → WXTT − → Output Question How does the system handle input sentences containing “system”? Some answer take regular language L = ∗ system ∗ turn into a regular tree language use forward application Input Products for WXTT A. Maletti 2 ·
Machine translation Schema Input − → WXTT − → Output Question How does the system handle input sentences containing “system”? Some answer take regular language L = ∗ system ∗ turn into a regular tree language use forward application Input Products for WXTT A. Maletti 2 ·
Machine translation Schema Input − → WXTT − → Output Question How does the system handle input sentences containing “system”? Some answer take regular language L = ∗ system ∗ turn into a regular tree language use forward application Input Products for WXTT A. Maletti 2 ·
Machine translation (cont’d) Question How does the system handle input sentences containing “system”? Forward application Problem: we obtain only output trees ⇒ not informative enough Another answer regular language, regular tree language as before input product restricts input to regular tree language ⇒ retains full translation information Input Products for WXTT A. Maletti 3 ·
Machine translation (cont’d) Question How does the system handle input sentences containing “system”? Forward application Problem: we obtain only output trees ⇒ not informative enough Another answer regular language, regular tree language as before input product restricts input to regular tree language ⇒ retains full translation information Input Products for WXTT A. Maletti 3 ·
Input product Applications parsing (one-sided, both-sided) translation forward application (input product + domain projection) regular look-ahead computation of interesting parameters (inside/outside weights) Input Products for WXTT A. Maletti 4 ·
Input product Applications parsing (one-sided, both-sided) translation forward application (input product + domain projection) regular look-ahead computation of interesting parameters (inside/outside weights) Input Products for WXTT A. Maletti 4 ·
Input product Applications parsing (one-sided, both-sided) translation forward application (input product + domain projection) regular look-ahead computation of interesting parameters (inside/outside weights) Input Products for WXTT A. Maletti 4 ·
Contents Motivation 1 Weighted Tree Automaton 2 Weighted Extended Top-down Tree Transducer 3 Input Product 4 Input Products for WXTT A. Maletti 5 ·
Weight structure Definition Commutative semiring ( C , + , · , 0 , 1 ) if ( C , + , 0 ) and ( C , · , 1 ) commutative monoids · distributes over finite (incl. empty) sums Idempotent if c + c = c Example B OOLEAN semiring ( { 0 , 1 } , max , min , 0 , 1 ) (idempotent) Semiring ( N , + , · , 0 , 1 ) of natural numbers Tropical semiring ( N ∪ {∞} , min , + , ∞ , 0 ) (idempotent) Any field, ring, etc. Input Products for WXTT A. Maletti 6 ·
Weight structure Definition Commutative semiring ( C , + , · , 0 , 1 ) if ( C , + , 0 ) and ( C , · , 1 ) commutative monoids · distributes over finite (incl. empty) sums Idempotent if c + c = c Example B OOLEAN semiring ( { 0 , 1 } , max , min , 0 , 1 ) (idempotent) Semiring ( N , + , · , 0 , 1 ) of natural numbers Tropical semiring ( N ∪ {∞} , min , + , ∞ , 0 ) (idempotent) Any field, ring, etc. Input Products for WXTT A. Maletti 6 ·
Weighted tree automaton Definition (B ERSTEL , R EUTENAUER 1982) Weighted tree automaton (WTA) A = ( Q , Σ , I , δ ) with rules σ c q . . . · q 1 · q k q , q 1 , . . . , q k ∈ Q are states c ∈ C is a weight σ ∈ Σ k is a k -ary input symbol Input Products for WXTT A. Maletti 7 ·
Run S NP VP JJ NNS VBP ADVP Colorless ideas sleep RB furiously Input Products for WXTT A. Maletti 8 ·
Run S q NP q ′ VP q 1 JJ q ′ NNS q ′′ VBP q ′ ADVP q 2 1 Colorless w ideas w sleep w RB q 2 furiously w Input Products for WXTT A. Maletti 8 ·
Run S . 4 q NP . 2 VP . 4 q 1 q ′ VBP . 2 JJ . 3 NNS . 3 ADVP . 3 q ′ q ′′ q ′ q 2 1 RB . 2 Colorless . 1 ideas . 1 sleep . 1 q 2 w w w furiously . 1 w Definition Weight wt ( r ) of a run r = product of its weights Input Products for WXTT A. Maletti 8 ·
Run S . 4 q NP . 2 VP . 4 q 1 q ′ VBP . 2 JJ . 3 NNS . 3 ADVP . 3 q ′ q ′′ q ′ q 2 1 RB . 2 Colorless . 1 ideas . 1 sleep . 1 q 2 w w w furiously . 1 w Example (Weight of the run) wt ( r ) = 0 . 4 · 0 . 2 · 0 . 3 · 0 . 1 · 0 . 3 · 0 . 1 · 0 . 4 · 0 . 2 · 0 . 1 · 0 . 3 · 0 . 2 · 0 . 1 Input Products for WXTT A. Maletti 8 ·
Semantics Definition The weight A ( t ) of input tree t = sum of weights of all runs ending in initial state � A ( t ) = wt ( r ) r run on t root ( r ) ∈ I Note Weighted tree language regular if computable by WTA Input Products for WXTT A. Maletti 9 ·
Semantics Definition The weight A ( t ) of input tree t = sum of weights of all runs ending in initial state � A ( t ) = wt ( r ) r run on t root ( r ) ∈ I Note Weighted tree language regular if computable by WTA Input Products for WXTT A. Maletti 9 ·
Contents Motivation 1 Weighted Tree Automaton 2 Weighted Extended Top-down Tree Transducer 3 Input Product 4 Input Products for WXTT A. Maletti 10 ·
Syntax Definition (A RNOLD , D AUCHET 1976, G RAEHL , K NIGHT 2004) Weighted extended top-down tree transducer (WXTT) M = ( Q , Σ , ∆ , I , R ) with finitely many rules q ∆ c Σ → . . . q ′ ( x i ) p ( x j ) . . . x 1 x k q , q ′ , p ∈ Q are states i , j ∈ { 1 , . . . , k } Input Products for WXTT A. Maletti 11 ·
Syntax (cont’d) Definition (R OUNDS 1970, T HATCHER 1970) Weighted top-down tree transducer (WTT) if all rules q ∆ c σ → . . . x 1 x k . . . q ′ ( x i ) p ( x j ) Input Products for WXTT A. Maletti 12 ·
Semantics Example States { q S , q V , q NP } of which only q S is initial q S q V q NP S ′ q V q NP 0 . 4 q V q NP q NP 1 1 S VP VP → → → x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 2 x 1 x 2 Derivation q S S ′ S ′ S ′ q V q NP q NP q V q NP q NP S q V q NP q NP 0 . 4 1 1 ⇒ ⇒ ⇒ t 1 t 1 t 2 t 1 VP VP VP VP t 2 t 1 t 3 t 2 t 3 t 2 t 3 t 2 t 3 t 2 t 3 Input Products for WXTT A. Maletti 13 ·
Semantics (cont’d) Definition Computed transformation ( t ∈ T Σ and u ∈ T ∆ ): � M ( t , u ) = c 1 · . . . · c n q ∈ I c 1 cn q ( t ) ⇒ u ⇒··· left-most derivation Input Products for WXTT A. Maletti 14 ·
Contents Motivation 1 Weighted Tree Automaton 2 Weighted Extended Top-down Tree Transducer 3 Input Product 4 Input Products for WXTT A. Maletti 15 ·
Input product Definition Given WTA A and WTT M , their input product is WTT N with N ( t , u ) = M ( t , u ) · A ( t ) Notes Input product . . . is special composition is like regular look-ahead can be used for parsing can be used for preservation of regularity Input Products for WXTT A. Maletti 16 ·
Input product Definition Given WTA A and WTT M , their input product is WTT N with N ( t , u ) = M ( t , u ) · A ( t ) Notes Input product . . . is special composition is like regular look-ahead can be used for parsing can be used for preservation of regularity Input Products for WXTT A. Maletti 16 ·
Nondeletion Example q S q V q NP S ′ q V q NP 0 . 4 q V q NP q NP 1 1 S VP VP → → → x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 2 x 1 x 2 nondeleting linear linear Definition WTT M is nondeleting if var ( l ) = var ( r ) for all rules l → r linear if no variable appears twice in r for all rules l → r Input Products for WXTT A. Maletti 17 ·
Nondeletion Example q S q V q NP S ′ q V q NP 0 . 4 q V q NP q NP 1 1 S VP VP → → → x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 2 x 1 x 2 nondeleting deletes x 2 deletes x 1 Definition all-copies nondeleting = nondeleting = every copy of an input subtree is fully explored some-copy nondeleting = one copy of each input subtree is fully explored Input Products for WXTT A. Maletti 17 ·
Nondeletion (cont’d) Example q S q V q NP S ′ q V q NP 0 . 4 1 1 q V q NP q NP S VP VP → → → x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 2 x 1 x 2 is not some-copy nondeleting Input Products for WXTT A. Maletti 18 ·
Nondeletion (cont’d) Example q S q V q NP S ′ VP q V 0 . 4 1 1 q NP q V q V q NP q NP S VP VP → → → x 1 x 1 x 2 x 1 x 2 x 1 x 2 x 2 x 1 x 2 x 1 x 2 can be some-copy nondeleting Input Products for WXTT A. Maletti 18 ·
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