Tree Transformations and Dependencies
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1 Tree Transformations and Deendencies Andreas Maletti Institute for Natural Language Processing University of Stuttgart Nara Setember 6, 2011 A Maletti MOL
2 Contents 1 Motivation 2 Extended to-down tree transducer 3 Extended multi bottom-u tree transducer 4 Synchronous tree-seuence substitution grammar A Maletti MOL
3 Warning! This won t be a regular survey talk A Maletti MOL
4 Warning! This won t be a regular survey talk Rather I want to show a useful techniue A Maletti MOL
5 Context-free Language Question How do you show that a language is not context-free? A Maletti MOL
6 Context-free Language Question How do you show that a language is not context-free? Answers uming lemma (BAR-HILLEL lemma) semi-linearity (PARIKH s theorem) A Maletti MOL
7 Context-free Language Hardly ever Let us assume that there is a CFG (long case distinction on the shae of its rules) Contradiction! A Maletti MOL
8 Tree Transformation Alication areas NLP XML rocessing syntax-directed semantics Examle t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 1 t 2 t 3 t 4 t n 3 t n 2 t n 1 t n A Maletti MOL
9 Tree Transducer Definition A formal model comuting a tree transformation A Maletti MOL
10 Tree Transducer Definition A formal model comuting a tree transformation Examle to-down or bottom-u tree transducer synchronous grammar (SCFG, STSG, STAG, etc) multi bottom-u tree transducer bimorhism A Maletti MOL
11 Tree Transducer Question How do you show that a tree transformation cannot be comuted by a class of tree transducers? A Maletti MOL
12 Tree Transducer Question How do you show that a tree transformation cannot be comuted by a class of tree transducers? Answers uming lemma (????) size and height deendencies domain and image roerties roerties under comosition (??) A Maletti MOL
13 Tree Transducer Question How do you show that a tree transformation cannot be comuted by a class of tree transducers? Tyical aroach Assume that it can be comuted Contradiction! A Maletti MOL
14 Examle t 1 t 2 t 3 t n 4 t n 3 t 2 t 1 t 4 t 3 t n 2 t n 3 t n 2 t n 1 t n t n 1 t n Question Can this tree transformation be comuted by a linear, nondeleting extended to-down tree transducer (XTOP)? A Maletti MOL
15 Examle t 1 t 2 t 3 t n 4 t n 3 t 2 t 1 t 4 t 3 t n 2 t n 3 t n 2 t n 1 t n t n 1 t n Question Can this tree transformation be comuted by a linear, nondeleting extended to-down tree transducer (XTOP)? Answer [Arnold, Dauchet 1982] No! (via indirect roof) A Maletti MOL
16 Another Examle γ γ t 1 t 2 t 3 t 1 t 2 t 3 Question Can this tree transformation be comuted by an XTOP? A Maletti MOL
17 Another Examle γ γ t 1 t 2 t 3 t 1 t 2 t 3 Question Can this tree transformation be comuted by an XTOP? Answer [, Graehl, Hokins, Knight 2009] No! (via indirect roof) A Maletti MOL
18 Yet Another Examle β β γ Question Can this tree transformation be comuted by an MBOT? β γ β γ γ A Maletti MOL
19 Yet Another Examle β β γ β β γ γ Question Can this tree transformation be comuted by an MBOT? MBOT linear, nondeleting multi bottom-u tree transducer γ A Maletti MOL
20 Yet Another Examle Question β β γ Can this tree transformation be comuted by an MBOT? β γ β γ MBOT linear, nondeleting multi bottom-u tree transducer γ Answer We ll know at the end A Maletti MOL
21 Contents 1 Motivation 2 Extended to-down tree transducer 3 Extended multi bottom-u tree transducer 4 Synchronous tree-seuence substitution grammar A Maletti MOL
22 Extended To-down Tree Transducer Definition (XTOP) tule (Q, Σ,, I, R) Q: states Σ, : inut and outut symbols I Q: initial states [ARNOLD, DAUCHET: Morhismes et bimorhismes d arbres Theor Comut Sci 1982] A Maletti MOL
23 Extended To-down Tree Transducer Definition (XTOP) tule (Q, Σ,, I, R) Q: states Σ, : inut and outut symbols I Q: initial states R T Σ (Q) Q T (Q): finite set of rules l and r are linear in Q var(l) = var(r) for every l r R [ARNOLD, DAUCHET: Morhismes et bimorhismes d arbres Theor Comut Sci 1982] A Maletti MOL
24 Examle XTOP Examle XTOP (Q, Σ, Σ, {}, R) with Q = {,,, r} and Σ = {,,, } r r x x x x A Maletti MOL
25 Sentential Form L = {S S N N } A Maletti MOL
26 Sentential Form L = {S S N N } Definition (Sentential form) ξ, D, ζ T Σ (Q) L T (Q) such that v os(ξ) and w os(ζ) for every (v, w) D A Maletti MOL
27 Link Structure Definition rule l r R, ositions v, w N links v,w (l r) = {(vv, ww ) v os (l), w os (r)} Q A Maletti MOL
28 Link Structure Definition rule l r R, ositions v, w N links v,w (l r) = {(vv, ww ) v os (l), w os (r)} Q A Maletti MOL
29 Link Structure Definition rule l r R, ositions v, w N links v,w (l r) = {(vv, ww ) v os (l), w os (r)} Q A Maletti MOL
30 Derivation Definition ξ, D, ζ ξ[l] v, D links v,w (l r), ζ[r] w if rule l r R osition v os (ξ) osition w os (ζ) such that (v, w) D A Maletti MOL
31 Derivation Rules r r x x x x A Maletti MOL
32 Derivation Rules r r x x x x A Maletti MOL
33 Derivation Rules r r x x x x A Maletti MOL
34 Derivation Rules r r x x x x A Maletti MOL
35 Derivation Rules r r x x x x A Maletti MOL
36 Derivation Rules r r x x x x A Maletti MOL
37 Derivation Rules r r x x x x r r A Maletti MOL
38 Derivation Rules r r x x x x r r r r A Maletti MOL
39 Derivation Rules r r x x x x r r A Maletti MOL
40 Semantics of XTOP Definition the deendencies de(m) { t, D, u T Σ L T I :, {(ε, ε)}, M t, D, u } the tree transformation M = {(t, u) t, D, u de(m)} A Maletti MOL
41 Semantics of XTOP Definition the deendencies de(m) { t, D, u T Σ L T I :, {(ε, ε)}, M t, D, u } the tree transformation M = {(t, u) t, D, u de(m)} A Maletti MOL
42 Semantics of XTOP A Maletti MOL
43 Semantics of XTOP Examle D = {(ε, ε), (1, 1), (2, 2), (21, 21), (211, 211), (22, 221), (23, 222), (231, 2221), (232, 22221), (233, 22222)} A Maletti MOL
44 Semantics of XTOP Examle D = {(ε, ε), (1, 1), (2, 2), (21, 21), (211, 211), (22, 221), (23, 222), (231, 2221), (232, 22221), (233, 22222)} A Maletti MOL
45 Non-closure under Comosition t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 1 t 2 t 3 t 4 t n 3 t n 2 t 2 t 1 t 4 t 3 t n 2 t n 3 t n 1 t n t n 1 t n [ARNOLD, DAUCHET: Morhismes et bimorhismes d arbres Theor Comut Sci 1982] A Maletti MOL
46 Secial Linking Structure Definition D L is inut hierarchical if for all (v 1, w 1 ), (v 2, w 2 ) D v 1 < v 2 imlies w 2 w 1 and w 1 w 2 : (v 1, w 1 ) D A Maletti MOL
47 Secial Linking Structure Definition D L is inut hierarchical if for all (v 1, w 1 ), (v 2, w 2 ) D v 1 < v 2 imlies w 2 w 1 and w 1 w 2 : (v 1, w 1 ) D v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
48 Secial Linking Structure Definition D L is inut hierarchical if for all (v 1, w 1 ), (v 2, w 2 ) D v 1 < v 2 imlies w 2 w 1 and w 1 w 2 : (v 1, w 1 ) D v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
49 Secial Linking Structure Definition D L is strictly inut hierarchical if for all (v 1, w 1 ), (v 2, w 2 ) D v 1 < v 2 imlies w 2 w 1 and w 1 w 2 : (v 1, w 1 ) D w 1 w 2 imlies v 1 v 2 v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
50 Secial Linking Structure Definition D L is strictly inut hierarchical if for all (v 1, w 1 ), (v 2, w 2 ) D v 1 < v 2 imlies w 2 w 1 and w 1 w 2 : (v 1, w 1 ) D w 1 w 2 imlies v 1 v 2 v v 1 w 1 w 2 v 2 w 1 w 2 A Maletti MOL
51 Deendencies Examle This linking structure is strictly (inut and outut) hierarchical A Maletti MOL
52 Deendencies v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
53 Deendencies v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
54 Deendencies v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
55 Deendencies v v 1 w 1 w 2 v 2 w 1 w 2 A Maletti MOL
56 Another Proerty Definition D 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 Illustration: v > k vw A Maletti MOL
57 Another Proerty Definition D 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 Illustration: k v vw vw A Maletti MOL
58 Another Proerty Definition D 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 roerty for range) A Maletti MOL
59 Bounded Distance Examle inut side: bounded by 1 outut side: bounded by 2 A Maletti MOL
60 Main Result Theorem The deendencies comuted by an XTOP are: strictly (inut and outut) hierarchical bounded distance A Maletti MOL
61 Comatibility Definition ξ, D, ζ is comatible with ξ, D, ζ if D D A Maletti MOL
62 Comatibility Definition ξ, D, ζ is comatible with ξ, D, ζ if D D set L is comatible with L if for all ξ, _, ζ L (some comutation) exists ξ, D, ζ L (imlemented comutation) exists comatible ξ, D, ζ L (comatible comutation) A Maletti MOL
63 Comatibility Illustration Some comutation A Maletti MOL
64 Comatibility Illustration Imlemented comutation A Maletti MOL
65 Comatibility Illustration Comatible comutation A Maletti MOL
66 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n A Maletti MOL
67 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n use boundedness A Maletti MOL
68 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n use boundedness A Maletti MOL
69 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n A Maletti MOL
70 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
71 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
72 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n v 1 w 2 v 1 w 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
73 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
74 Back to the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n Theorem The deendencies above are incomatible with any XTOP A Maletti MOL
75 A First Instance t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t 4 t 3 t n 2 t n 3 t n 2 t n 1 t n t n 1 t n Theorem (ARNOLD, DAUCHET 1982) The transformation above cannot be comuted by any XTOP A Maletti MOL
76 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 A Maletti MOL
77 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 A Maletti MOL
78 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 use boundedness A Maletti MOL
79 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 use boundedness A Maletti MOL
80 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 A Maletti MOL
81 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 Theorem The deendencies above are incomatible with any XTOP A Maletti MOL
82 A Second Instance γ t 3 γ t 1 t 2 t 3 t 1 t 2 Theorem (, GRAEHL, HOPKINS, KNIGHT 2009) The transformation above cannot be comuted by any XTOP A Maletti MOL
83 Contents 1 Motivation 2 Extended to-down tree transducer 3 Extended multi bottom-u tree transducer 4 Synchronous tree-seuence substitution grammar A Maletti MOL
84 Extended Multi Bottom-u Tree Transducer Definition (MBOT) tule (Q, Σ,, I, R) Q: ranked alhabet of states Σ, : inut and outut symbols I Q 1 : unary initial states A Maletti MOL
85 Extended Multi Bottom-u Tree Transducer Definition (MBOT) tule (Q, Σ,, I, R) Q: ranked alhabet of states Σ, : inut and outut symbols I Q 1 : unary initial states R T Σ (Q) Q T (Q) : finite set of rules l is linear in Q rk() = r var( r) var(l) for every (l,, r) R A Maletti MOL
86 Examle MBOT Examle (Q, Σ, Σ, {f }, R) with Q = { (2), (1), (1), r (1), f (1) } and Σ = {,,, } f x x x x r r A Maletti MOL
87 New Linking Structure Definition rule l r R, ositions v, w 1,, w n N w = w 1 w n with n = rk() links v, w (l r) = Q i=1 n {(vv, w i w i ) v os (l), w i os (r i )} A Maletti MOL
88 New Linking Structure Definition rule l r R, ositions v, w 1,, w n N w = w 1 w n with n = rk() links v, w (l r) = Q i=1 n {(vv, w i w i ) v os (l), w i os (r i )} Illustration: A Maletti MOL
89 New Linking Structure Definition rule l r R, ositions v, w 1,, w n N w = w 1 w n with n = rk() links v, w (l r) = Q i=1 n {(vv, w i w i ) v os (l), w i os (r i )} Illustration: A Maletti MOL
90 Derivation Definition ξ, D, ζ ξ[l] v, D links v, w (l r), ζ[ r] w if rule l r R with rk() = n osition v os (ξ) w = w 1 w n os (ζ) with w 1 w n {w 1,, w n } = {w (v, w) D} A Maletti MOL
91 Derivation Rules f x x x x r r A Maletti MOL
92 Derivation Rules f x x x x r r A Maletti MOL
93 Derivation Rules f x x x x r r A Maletti MOL
94 Derivation Rules f x x x x r r A Maletti MOL
95 Derivation Rules f x x x x r r A Maletti MOL
96 Derivation Rules f x x x x r r A Maletti MOL
97 Derivation Rules f x x x x r r A Maletti MOL
98 Derivation Rules f x x x x r r r r A Maletti MOL
99 Derivation Rules f x x x x r r r r A Maletti MOL
100 Semantics of MBOT Definition MBOT M comutes deendencies de(m) { t, D, u T Σ L T I :, {(ε, ε)}, M t, D, u } the tree transformation M = {(t, u) t, D, u de(m)} A Maletti MOL
101 Semantics of MBOT Definition MBOT M comutes deendencies de(m) { t, D, u T Σ L T I :, {(ε, ε)}, M t, D, u } the tree transformation M = {(t, u) t, D, u de(m)} A Maletti MOL
102 Semantics of MBOT Examle 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)} A Maletti MOL
103 Semantics of MBOT A Maletti MOL
104 Semantics of MBOT Examle Above deendencies are inut hierarchical strictly outut hierarchical A Maletti MOL
105 Semantics of MBOT Examle Above deendencies are inut hierarchical but not strictly strictly outut hierarchical A Maletti MOL
106 Semantics of MBOT v v 1 w 1 w 2 v 2 w 1 w 2 A Maletti MOL
107 Main Result Theorem The deendencies comuted by an MBOT are: inut hierarchical strictly outut hierarchical bounded distance A Maletti MOL
108 Link Structure for the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n A Maletti MOL
109 Link Structure for the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n A Maletti MOL
110 Link Structure for the Examle t 1 t 2 t 1 t 2 t 3 t n 4 t n 3 t n 2 t n 1 t n t 4 t 3 t n 2 t n 3 t n 1 t n Theorem Above deendencies are comatible with an MBOT A Maletti MOL
111 Another MBOT Examle (Q, Σ,, {f }, R) with Q = { (3), (3), r (3), f (1) } Σ = {,, β, γ} and = Σ {} f β β γ r r r r r γ r r r r r A Maletti MOL
112 Another MBOT β β γ γ β β γ γ taken from Examle 45 of [RADMACHER: An automata theoretic aroach to the theory of rational tree relations TR AIB , RWTH Aachen, 2008] {( l (β m (γ n ())), ( l (), β m (), γ n ())) l, m, n N} A Maletti MOL
113 Another MBOT Theorem β β γ β β γ γ These inverse deendencies are not comatible with any MBOT (because the dislayed deendencies are not strictly inut hierarchical) v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 γ A Maletti MOL
114 Another MBOT Theorem β β γ β β γ γ These inverse deendencies are not comatible with any MBOT (because the dislayed deendencies are not strictly inut hierarchical) v 1 w 2 v 1 v 2 w 1 v 2 w 1 w 2 γ Theorem This tree transformation cannot be comuted by any MBOT A Maletti MOL
115 Contents 1 Motivation 2 Extended to-down tree transducer 3 Extended multi bottom-u tree transducer 4 Synchronous tree-seuence substitution grammar A Maletti MOL
116 Synchronous Tree-Seuence Substitution Grammar Definition (STSSG) tule (Q, Σ,, I, R) with Q: doubly ranked alhabet Σ, : inut and outut symbols I Q 1,1 : doubly unary initial states [ZHANG et al: A tree seuence alignment-based tree-to-tree translation model Proc ACL, 2008] A Maletti MOL
117 Synchronous Tree-Seuence Substitution Grammar Definition (STSSG) tule (Q, Σ,, I, R) with Q: doubly ranked alhabet Σ, : inut and outut symbols I Q 1,1 : doubly unary initial states R T Σ (Q) Q T (Q) : finite set of rules rk() = ( l, r ) for every ( l,, r) R [ZHANG et al: A tree seuence alignment-based tree-to-tree translation model Proc ACL, 2008] A Maletti MOL
118 Yet Another Linking Structure Definition rule l r R, ositions v 1,, v m, w 1,, w n N with rk() = (m, n) links v, w ( l r) = m n {(v j v j, w iw i ) v j os (l j ), w i os (r i )} Q j=1 i=1 where v = v 1 v m and w = w 1 w n A Maletti MOL
119 Derivation Definition if rule l ξ, D, ζ ξ[ l] v, D links v, w ( l r R r), ζ[ r] w v = v 1 v m os (ξ) and w = w 1 w n os (ζ) v 1 v m, w 1 w n, and rk() = (m, n) linked ositions {w 1,, w n } = m j=1 {w (v j, w) D} {v 1,, v m } = n i=1 {v (v, w i) D} A Maletti MOL
120 A Final Theorem β β γ β β γ γ Examle This transformation can be comuted by an STSSG (because it is an inverse MBOT) γ A Maletti MOL
121 A Final Theorem β β γ β β γ γ Examle This transformation can be comuted by an STSSG (because it is an inverse MBOT) γ A Maletti MOL
122 A Final Theorem β β γ β β γ γ Examle This transformation can be comuted by an STSSG (because it is an inverse MBOT) γ Theorem Deendencies comuted by STSSG are (inut and outut) hierarchical bounded distance A Maletti MOL
123 A Final Instance β γ γ β Theorem These deendencies are not comatible with any STSSG v 1 w 2 v 1 w 1 β γ β v 2 w 1 v 2 w 1 w 2 γ A Maletti MOL
124 A Final Instance γ Theorem γ These deendencies are not comatible with any STSSG β β v 1 w 2 v 1 w 1 β γ β v 2 w 1 Theorem v 2 w 1 w 2 γ This transformation cannot be comuted by any STSSG A Maletti MOL
125 Summary Comuted deendencies by device inut outut XTOP strictly hierarchical strictly hierarchical A Maletti MOL
126 Summary Comuted deendencies by device inut outut XTOP strictly hierarchical strictly hierarchical MBOT hierarchical strictly hierarchical A Maletti MOL
127 Summary Comuted deendencies by device inut outut XTOP strictly hierarchical strictly hierarchical MBOT hierarchical strictly hierarchical STSSG hierarchical hierarchical A Maletti MOL
128 Summary Comuted deendencies by device inut outut XTOP strictly hierarchical strictly hierarchical MBOT hierarchical strictly hierarchical STSSG hierarchical hierarchical STAG A Maletti MOL
129 That s all, folks! Thank you for your attention! A Maletti MOL
130 References ARNOLD, DAUCHET: Morhismes et bimorhismes d arbres Theor Comut Sci 20, 1982 ENGELFRIET, LILIN, MALETTI: Extended multi bottom-u tree transducers Acta Inf 46, 2009 MALETTI, GRAEHL, HOPKINS, KNIGHT: The ower of extended to-down tree transducers SIAM J Comut 39, 2009 RADMACHER: An automata theoretic aroach to rational tree relations Proc SOFSEM 2008 RAOULT: Rational tree relations Bull Belg Math Soc Simon Stevin 4, 1997 ZHANG, JIANG, AW, LI, TAN, LI: A tree seuence alignment-based tree-to-tree translation model Proc ACL 2008 A Maletti MOL
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