Linking Theorems for Tree Transducers

Linking Theorems for Tree Transducers Andreas Maletti maletti@ims.uni-stuttgart.de peyer October 1, 2015 Andreas Maletti Linking Theorems for MBOT Theorietag 2015 1 / 32

tatistical Machine Translation w kana ynzran -BJ PP-CLR PP-MNR Aly b h $kl mdhk him a funny way at looking PP-CLR in PP they were And -BJ

tatistical Machine Translation ynzran w kana -BJ PP-CLR Aly h PP-MNR b $kl mdhk h him b b $kl in $kl a. way mdhk mdhk funny w w And Aly Aly at kana kana -BJ they. were ynzran ynzran looking him at looking PP-CLR a funny way in PP they were And -BJ

tatistical Machine Translation ynzran w kana -BJ PP-CLR PP-MNR Aly b $kl mdhk h him b b $kl in $kl a. way mdhk mdhk funny w w And Aly Aly at kana kana -BJ they. were ynzran ynzran looking $kl mdhk $kl Aly b ynzran PP-CLR PP kana w kana

tatistical Machine Translation ynzran w kana -BJ PP-CLR PP-MNR Aly b $kl mdhk h him b b $kl in $kl a. way mdhk mdhk funny w w And Aly Aly at kana kana -BJ they. were ynzran ynzran looking PP-CLR PP-CLR PP-CLR Aly Aly Aly $kl mdhk $kl b $kl mdhk $kl mdhk $kl ynzran PP-CLR PP kana w kana

tatistical Machine Translation ynzran w kana -BJ PP-CLR PP-MNR b h him b b $kl in $kl a. way mdhk mdhk funny w w And Aly Aly at kana kana -BJ they. were ynzran ynzran looking ynzran PP-CLR PP kana b PP-CLR PP-CLR PP-CLR Aly Aly $kl mdhk $kl mdhk $kl w kana

tatistical Machine Translation Extracted rules w w kana kana. kana $kl mdhk $kl mdhk $kl PP-MNR PP-MNR b ynzran b -BJ PP h w w And PP-CLR PP-MNR him b b $kl mdhk in $kl a. way mdhk funny kana kana they. were ynzran ynzran looking Aly Aly at PP-CLR PP-CLR ynzran PP-CLR PP-MNR Aly PP-CLR Aly

Linear Multi Tree Transducer MBOT linear multi tree transducer (Q, Σ, I, R) finite set Q alphabet Σ states input and output symbols I Q initial states finite set R T Σ (Q) Q T Σ (Q) rules each Q occurs at most once in l (l,, r) R each Q that occurs in r also occurs in l (l,, r) R

Linear Multi Tree Transducer yntactic properties MBOT (Q, Σ, I, R) is linear tree transducer with regular look-ahead (XTOP R ) if r 1 linear tree transducer (XTOP) if r = 1 (l,, r) R (l,, r) R

Linear Multi Tree Transducer yntactic properties MBOT (Q, Σ, I, R) is linear tree transducer with regular look-ahead (XTOP R ) if r 1 linear tree transducer (XTOP) if r = 1 ε-free if l / Q (l,, r) R (l,, r) R (l,, r) R

Linear Multi Tree Transducer Extracted rules w w kana kana. kana $kl mdhk $kl mdhk $kl PP-MNR PP-MNR b ynzran b -BJ PP h w w And PP-CLR PP-MNR him b b $kl mdhk in $kl a. way mdhk funny kana kana they. were ynzran ynzran looking Aly Aly at PP-CLR PP-CLR ynzran PP-CLR PP-MNR Aly PP-CLR Aly Properties XTOP R : XTOP: ε-free:

Another Example Textual example MBOT M = (Q, Σ, {}, R) Q = {,, id, id } Σ = {,, γ, α} the following rules in R: (, ) (, ) (, ) (id, id ), (id, id ) γ(id) id,id γ(id) α id,id α

Another Example Graphical representation, id id id id γ id id,id γ id α id,id α Properties XTOP R : XTOP: ε-free:

emantics Rules, id id id id γ id id,id γ id α id,id α

emantics Rules, id id id id γ id id,id γ id α id,id α id id id id

emantics Rules, id id id id γ id id,id γ id α id,id α α α α α α α α α

emantics Rules, id id id id γ id id,id γ id α id,id α α α γ α α α α α α α

emantics id id id id entential forms t, A, D, u t T Σ (Q) A N N D N N u T Σ (Q) input tree active links (red) disabled links (gray) output tree

emantics M w w M w kana w kana kana M w And kana kana kana M w And they kana were

emantics Dependencies and relation state-computed dependencies: M = { t, D, u t, u T Σ,, {(ε, ε)},, M t,, D, u } computed dependencies: dep(m) = I M

emantics Dependencies and relation state-computed dependencies: M = { t, D, u t, u T Σ,, {(ε, ε)},, M t,, D, u } computed dependencies: dep(m) = I M 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 uery rewriting

Contents 1 Basics 2 Linking techniue

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 1 w 2 w 1 2 (v 1, w 1 ) D with w 1 w 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 1 w 2 w 1 2 (v 1, w 1 ) D with w 1 w 2 for all (v 1, w 1 ), (v 2, w 2 ) D with v 1 < v 2 strictly input hierarchical if 1 v 1 < v 2 implies w 1 w 2 2 v 1 = v 2 implies w 1 w 2 or w 2 w 1 for all (v 1, w 1 ), (v 2, w 2 ) D

Dependencies Distance properties A dependency t, D, u is input link-distance bounded by b N if for all (v 1, w 1 ), (v 1 v, w 2 ) D with v > b (v 1 v, w 3 ) 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 (v 1, w 1 ), (v 1 v, w 2 ) D with v > b (v 1 v, w 3 ) D such that v < v and 1 v b strict input link-distance bounded by b if for all v 1, v 1 v pos(t) with v > b (v 1 v, w 3 ) D such that v < v and 1 v b

Dependencies α α γ α α α α α 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 output input output XTOP R strictly strictly strictly MBOT strictly strictly

Linking Theorem Theorem Let M 1,..., M k be ε-free XTOP R over Σ such that {(c[t 1,..., t n ], c [t 1,..., t n ]) t 1,..., t n T } τ M1 ; ; τ Mk for some contexts c, c C Σ (X n ) and special T T Σ. 1 i k, 1 j n t j T, u i 1, D i, u i dep(m i ), (v ji, w ji ) D i such that u 0 = c[t 1,..., t n ] and u k = c [t 1,..., t n ] pos xj (c ) w jk v ji w j(i 1) if i 2 pos xj (c) v j1

Linking Theorem Corollary [Arnold, Dauchet, TC 1982] Illustrated relation τ cannot be computed by any ε-free XTOP R t 2 t 1 t 1 t 4 t 3 t 3 t 2 t n 3 t n 2 t n 3 t n 4 t n t n 1 t n t n 1 t n 2

Linking Theorem Corollary [M. et al., ICOMP 2009] Illustrated relation τ cannot be computed by any ε-free XTOP R γ γ s t u s 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 PRP VBD AD it rained NN RB night yesterday, NN yesterday NN night PRP it VBD rained

Topicalization On the tree level PRP we VBD toiled DT all NN day NN yesterday PP IN at DT the NN restaurant BAR WH WDT that VBZ charges JJ extra PP IN for JJ clean NN plates PP IN at DT the NN restaurant BAR WH WDT that VBZ charges JJ extra PP IN for JJ clean NN plates, PRP we VBD toiled DT all NN day NN yesterday

Topicalization t 2 t 1 t 3 t 2 t n 1 t 3 t n t 1 t n 1 t n Theorem Topicalization is in rp-mbot

Topicalization Theorem Topicalization cannot be computed by any composition of ε-free XTOP R u 1 u 2 u 0 u 3 t 2 t 3? (1) (2). (3). t 1 t 2 t n 1 t n t 1 v 12 v 13 t 3 t n 1 t n v (n 1)2 v n2 v n3 v (n 1)3 3 ε-free XTOP R sufficient to simulate any composition of ε-free XTOP R

Topicalization Corollary (XTOP R ) rp-mbot

Linking Theorem Theorem Let M = (Q, Σ, I, R) be an ε-free MBOT such that {(c[t 1,..., t n ], c [t 1,..., t n ]) t 1,..., t n T } τ M for some contexts c, c C Σ (X n ) and special T T Σ. 1 j n, t j T, u, D, u dep(m), (v j, w j ) D with u = c[t 1,..., t n ] and u = c [t 1,..., t n ] pos xj (c) v j pos xj (c ) w j

Linking Theorem Corollary Inverse of topicalization cannot be computed by any ε-free MBOT t 1 t 2 t 2 t 3 t 3 t n 1 t n 1 t n t n t 1

ummary & References ummary 1 (XTOP R ) rp-mbot 2 rp-mbot not closed under inverses 3 What happens to invertable MBOT?

ummary & References ummary 1 (XTOP R ) 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, LNC 7907, 2013 P. Koehn: tatistical machine translation. Cambridge Univ. Press, 2009, J. Graehl, M. Hopkins, K. Knight: The power of extended top-down tree transducers. IAM J. Comput., 2009