[PPT] - Composition Closure of Linear Extended Top-down Tree Transducers PowerPoint Presentation

SLIDE 1

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

SLIDE 2

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 Aly NP h PP-MNR b NP $kl mDHk S And NP-SBJ they VP were VP looking PP-CLR at NP him PP in NP a funny way

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 3

The problem Upper bounds Linking technique Lower bounds

Syntax-based Statistical Machine Translation

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 4

The problem Upper bounds Linking technique Lower bounds

Syntax-based Statistical Machine Translation

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP

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

SLIDE 5

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

SLIDE 6

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

τ1 ; τ2 = {(s, u) | ∃t : (s, t) ∈ τ1, (t, u) ∈ τ2}
support modular development
allow integration of external knowledge sources
occur naturally in query rewriting

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 7

The problem Upper bounds Linking technique Lower bounds

Problem

Question: given a class C of transformations, is there n ∈ N such that Cn =

k≥1

Ck Ck = C ; · · · ; C

k times

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 8

The problem Upper bounds Linking technique Lower bounds

Problem

Question: given a class C of transformations, is there n ∈ N such that Cn =

k≥1

Ck Ck = C ; · · · ; C

k times

Note

Ck ⊆ Ck+1 for our classes C

→ we search least n such that Cn = Cn+1 (if it exists)

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 9

The problem Upper bounds Linking technique Lower bounds

Linear Extended Multi Bottom-up Tree Transducer

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 10

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 ⊆ TΣ(Q) × Q × TΣ(Q)∗

rules

– each q ∈ Q occurs at most once in ℓ (ℓ, q, r) ∈ R – each q ∈ Q that occurs in r also occurs in ℓ (ℓ, q, r) ∈ R

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 11

The problem Upper bounds Linking technique Lower bounds

Linear Extended Multi Bottom-up Tree Transducer

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 12

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 (XTOPR) 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

SLIDE 13

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 (XTOPR) if | r| ≤ 1 ∀(ℓ, q, r) ∈ R

linear extended top-down tree transducer (XTOP) if |

r| = 1

linear top-down tree transducer (TOP/TOPR)

if XTOP/XTOPR and ℓ contains exactly one element of Σ

ε-free (resp. strict) if ℓ /

∈ Q (resp. r / ∈ Q+)

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 14

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 (XTOPR) if | r| ≤ 1 ∀(ℓ, q, r) ∈ R

linear extended top-down tree transducer (XTOP) if |

r| = 1

linear top-down tree transducer (TOP/TOPR)

if XTOP/XTOPR and ℓ contains exactly one element of Σ

ε-free (resp. strict) if ℓ /

∈ Q (resp. r / ∈ Q+)

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

SLIDE 15

The problem Upper bounds Linking technique Lower bounds

Linear Extended Multi Bottom-up Tree Transducer

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP

Properties XTOPR: ✗ XTOP: ✗ TOPR: ✗ TOP: ✗

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 16

The problem Upper bounds Linking technique Lower bounds

Linear Extended Multi Bottom-up Tree Transducer

Extracted rules

S qw qVP

q

— S qw qVP qVP VP qkAnA qVP

qVP

— qkAnA . VP qkAnA qVP NP q$kl qmDHk

qNP

— NP q$kl qmDHk q$kl PP-MNR qb qNP

qPP-MNR

— PP qb qNP NP h

qNP

— him b

qb

— in $kl

q$kl

— a . way mDHk

qmDHk

— funny w

qw

— And kAnA

qkAnA

— they . were ynZrAn

qynZrAn

— looking Aly

qAly

— at VP qynZrAn NP-SBJ ⋆ qPP-CLR qPP-MNR

qVP

— VP qynZrAn qPP-CLR qPP-MNR PP-CLR qAly qNP

qPP-CLR

— PP-CLR qAly qNP

Properties XTOPR: ✗ XTOP: ✗ TOPR: ✗ TOP: ✗ ε-free: ✓ strict: ✓ delabeling: ✗ nondeleting: ✓

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 17

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:

σ(⋆, q)

⋆

− → σ(⋆, q) σ(⋆, q)

q

− → q δ(id, id′)

⋆,q

− → δ(id, id′) γ(id)

id,id′

− → γ(id) α

id,id′

− → α

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 18

The problem Upper bounds Linking technique Lower bounds

Another Example

Graphical representation σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α Properties XTOPR: ✓ XTOP: ✓ TOPR: ✓ TOP: ✓

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 19

The problem Upper bounds Linking technique Lower bounds

Another Example

Graphical representation σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α Properties XTOPR: ✓ XTOP: ✓ TOPR: ✓ TOP: ✓ ε-free: ✓ strict: ✗ delabeling: ✓ nondeleting: ✗

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 20

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Discussion

typical semantics: derivation semantics; input-driven

⋆ σ δ α α σ α δ α α ⇒ σ ⋆ δ α α q σ α δ α α ⇒2 σ δ id α id′ α q δ α α ⇒∗ σ δ α α δ α α

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 21

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Discussion

typical semantics: derivation semantics; input-driven

⋆ σ δ α α σ α δ α α ⇒ σ ⋆ δ α α q σ α δ α α ⇒2 σ δ id α id′ α q δ α α ⇒∗ σ δ α α δ α α

unsuitable for our purposes
input and output fragments should always be visible

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 22

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

⋆ ⋆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 23

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ ⋆ q σ ⋆ q

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 24

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ ⋆ q σ ⋆ q

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 25

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ id id′ q σ δ id id′ q

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 26

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ id id′ q σ δ id id′ q

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 27

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ α α σ ⋆ δ α α σ δ α α δ α α

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 28

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Rules

σ ⋆ q

⋆

− → σ ⋆ q δ id id′

⋆,q

− → δ id id′ σ ⋆ q

q

− → q γ id

id,id′

− → γ id α

id,id′

− → α

σ δ α α σ γ α δ α α σ δ α α δ α α

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 29

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Definition (Generation step) t, A, D, u ⇒M t′, A′, D′, u′ if and only if ∃q ∈ Q, ∃v ∈ pos(t) labeled by q, and ∃ℓ

q

→ r ∈ P

|

r| = |A(v)| and w =

A(v)
t′ = t[ℓ]v and u′ = u[

r]

w

A′ = (A \ L) ∪ linksv,

w(ℓ q

→ r) and D′ = D ∪ L with L = {(v, w) | w ∈ A(v)}

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 30

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

q — q ⇒M S qw qVP — S qw qVP qVP ⇒M S qw VP qkAnA qVP — S qw qkAnA VP qkAnA qVP ⇒M S w VP qkAnA qVP — S And qkAnA VP qkAnA qVP ⇒M S w VP kAnA qVP — S And they VP were qVP

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 31

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Definition

state-computed dependencies:

Mq = {t, D, u | t, u ∈ TΣ, q, {(ε, ε)}, ∅, q ⇒∗

M t, ∅, D, u}

computed dependencies: dep(M) =

q∈I Mq

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 32

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Definition

state-computed dependencies:

Mq = {t, D, u | t, u ∈ TΣ, q, {(ε, ε)}, ∅, q ⇒∗

M t, ∅, D, u}

computed dependencies: dep(M) =

q∈I Mq

computed transformation:

τM = {(t, u) | t, D, u ∈ dep(M)}

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 33

The problem Upper bounds Linking technique Lower bounds

Semantics — Synchronous Generation

Definition

state-computed dependencies:

Mq = {t, D, u | t, u ∈ TΣ, q, {(ε, ε)}, ∅, q ⇒∗

M t, ∅, D, u}

computed dependencies: dep(M) =

q∈I Mq

computed transformation:

τM = {(t, u) | t, D, u ∈ dep(M)}

(can be made to) coincide with traditional semantics

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 34

The problem Upper bounds Linking technique Lower bounds

Known Relations

(n)-MBOT XTOP∗ e-XTOP∗ XTOP2 e-XTOP2 n-XTOP2 XTOPR ne-XTOP2 XTOP e-XTOP n-XTOP ne-XTOP

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 35

The problem Upper bounds Linking technique Lower bounds

Known Results on Composition Closure

TOP XTOP MBOT ε-free, strict, nondeleting 1 1 ε-free, strict 2 1 ε-free 2 1

therwise (without delabeling)

2 1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 37

The problem Upper bounds Linking technique Lower bounds

Known Results on Composition Closure

TOP XTOP MBOT ε-free, strict, nondeleting 1 2 1 ε-free, strict 2 ? 1 ε-free 2 ? 1

therwise (without delabeling)

2 ? 1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 38

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem switch delabeling from back to front: e[s]-XTOPR ; [s]d-TOPR ⊆ e[s]-XTOPR ⊆ [s]d-TOPR ; esn-XTOP

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 39

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem switch delabeling from back to front: e[s]-XTOPR ; [s]d-TOPR ⊆ e[s]-XTOPR ⊆ [s]d-TOPR ; esn-XTOP

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 40

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem switch delabeling from back to front: e[s]-XTOPR ; [s]d-TOPR ⊆ e[s]-XTOPR ⊆ [s]d-TOPR ; esn-XTOP Notes

other transducer becomes strict and nondeleting

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 41

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem switch delabeling from back to front: e[s]-XTOPR ; [s]d-TOPR ⊆ e[s]-XTOPR ⊆ [s]d-TOPR ; esn-XTOP Notes

other transducer becomes strict and nondeleting
other transducer looses look-ahead

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 42

The problem Upper bounds Linking technique Lower bounds

ε-free and Look-ahead

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem (e[s]-XTOPR)n ⊆ [s]d-TOPR ; esn-XTOP2 ⊆ (e[s]-XTOPR)3

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 43

The problem Upper bounds Linking technique Lower bounds

ε-free and Look-ahead

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem (e[s]-XTOPR)n ⊆ [s]d-TOPR ; esn-XTOP2 ⊆ (e[s]-XTOPR)3 Proof. (e[s]-XTOPR)n+1 ⊆ ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 44

The problem Upper bounds Linking technique Lower bounds

ε-free and Look-ahead

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem (e[s]-XTOPR)n ⊆ [s]d-TOPR ; esn-XTOP2 ⊆ (e[s]-XTOPR)3 Proof. (e[s]-XTOPR)n+1 ⊆ e[s]-XTOPR ; [s]d-TOPR ; esn-XTOP2 ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 45

The problem Upper bounds Linking technique Lower bounds

ε-free and Look-ahead

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem (e[s]-XTOPR)n ⊆ [s]d-TOPR ; esn-XTOP2 ⊆ (e[s]-XTOPR)3 Proof. (e[s]-XTOPR)n+1 ⊆ e[s]-XTOPR ; [s]d-TOPR ; esn-XTOP2 ⊆ [s]d-TOPR ; esn-XTOP3 ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 46

The problem Upper bounds Linking technique Lower bounds

ε-free and Look-ahead

e = ε-free; d = delabeling s = strict; n = nondeleting

Theorem (e[s]-XTOPR)n ⊆ [s]d-TOPR ; esn-XTOP2 ⊆ (e[s]-XTOPR)3 Proof. (e[s]-XTOPR)n+1 ⊆ e[s]-XTOPR ; [s]d-TOPR ; esn-XTOP2 ⊆ [s]d-TOPR ; esn-XTOP3 ⊆ [s]d-TOPR ; esn-XTOP2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 47

The problem Upper bounds Linking technique Lower bounds

ε-free, but no Look-ahead

Corollary e[s]-XTOPn ⊆ QR ; [s]d-TOP ; esn-XTOP2 ⊆ e[s]-XTOP4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 48

The problem Upper bounds Linking technique Lower bounds

ε-free, but no Look-ahead

Corollary e[s]-XTOPn ⊆ QR ; [s]d-TOP ; esn-XTOP2 ⊆ e[s]-XTOP4 Proof. uses only standard encoding of look-ahead

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 49

The problem Upper bounds Linking technique Lower bounds

Results so far

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 strict 2 look-ahead 1 — 2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 50

The problem Upper bounds Linking technique Lower bounds

Results so far

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 strict 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 51

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around Even More

Theorem delabeling homomorphism moving from front to back: sd-HOM ; es-XTOP ⊆ es-XTOP ⊆ esn-XTOP ; sd-HOM

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 52

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around Even More

Theorem delabeling homomorphism moving from front to back: sd-HOM ; es-XTOP ⊆ es-XTOP ⊆ esn-XTOP ; sd-HOM Notes

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 53

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around Even More

Theorem delabeling homomorphism moving from front to back: sd-HOM ; es-XTOP ⊆ es-XTOP ⊆ esn-XTOP ; sd-HOM Notes

other transducer becomes nondeleting

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 54

The problem Upper bounds Linking technique Lower bounds

Delabelings Move Around Even More

Theorem delabeling homomorphism moving from front to back: sd-HOM ; es-XTOP ⊆ es-XTOP ⊆ esn-XTOP ; sd-HOM Notes

other transducer becomes nondeleting
other transducer needs to be strict and have no look-ahead

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 55

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 56

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ ⊆ ⊆ ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 57

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ esn-XTOP ; sd-HOM ; es-XTOP2 ⊆ ⊆ ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 58

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ esn-XTOP ; sd-HOM ; es-XTOP2 ⊆ esn-XTOP3 ; sd-HOM ⊆ ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 59

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ esn-XTOP ; sd-HOM ; es-XTOP2 ⊆ esn-XTOP3 ; sd-HOM ⊆ esn-XTOP2 ; sd-HOM ⊆ ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 60

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ esn-XTOP ; sd-HOM ; es-XTOP2 ⊆ esn-XTOP3 ; sd-HOM ⊆ esn-XTOP2 ; sd-HOM ⊆ esn-XTOP ; es-XTOPR ⊆

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 61

The problem Upper bounds Linking technique Lower bounds

ε-free and Strict

Theorem (es-XTOPR)n ⊆ esn-XTOP ; es-XTOP ⊆ es-XTOP2 Proof. (es-XTOPR)n+1 ⊆ (es-XTOPR)n ; es-XTOP ⊆ esn-XTOP ; sd-HOM ; es-XTOP2 ⊆ esn-XTOP3 ; sd-HOM ⊆ esn-XTOP2 ; sd-HOM ⊆ esn-XTOP ; es-XTOPR ⊆ esn-XTOP ; es-XTOP

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 62

The problem Upper bounds Linking technique Lower bounds

Upper Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 strict 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 63

The problem Upper bounds Linking technique Lower bounds

Upper Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 ≤ 2 strict 2 ≤ 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 64

The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Hierarchy properties) A dependency t, D, u is

input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 66

The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Hierarchy properties) A dependency t, D, u is

input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 67

The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Hierarchy properties) A dependency t, D, u is

input hierarchical if

1

w2 < w1

2

∃(v1, w ′

1) ∈ D with w ′ 1 ≤ w2

for all (v1, w1), (v2, w2) ∈ D with v1 < v2

strictly input hierarchical if

1

v1 < v2 implies w1 ≤ w2

2

v1 = v2 implies w1 ≤ w2 or w2 ≤ w1

for all (v1, w1), (v2, w2) ∈ D

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Distance properties) A dependency t, D, u is

input link-distance bounded by b ∈ N if

for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Distance properties) A dependency t, D, u is

input link-distance bounded by b ∈ N if

for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Properties of Dependencies

Definition (Distance properties) A dependency t, D, u is

input link-distance bounded by b ∈ N if

for all (v1, w1), (v1v′, w2) ∈ D with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

strict input link-distance bounded by b if for all

v1, v1v′ ∈ pos(t) with |v′| > b ∃(v1v, w3) ∈ D such that v < v′ and 1 ≤ |v| ≤ b

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Dependencies

σ δ α α σ γ α δ α α σ δ α α δ α α

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 72

The problem Upper bounds Linking technique Lower bounds

Dependencies

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 73

The problem Upper bounds Linking technique Lower bounds

Dependencies

strictly input hierarchical

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 74

The problem Upper bounds Linking technique Lower bounds

Dependencies

strictly input hierarchical and strictly output hierarchical

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 75

The problem Upper bounds Linking technique Lower bounds

Dependencies

strictly input hierarchical and strictly output hierarchical with strict input link-distance 2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 76

The problem Upper bounds Linking technique Lower bounds

Dependencies

strictly input hierarchical and strictly output hierarchical with strict input link-distance 2 and strict output link-distance 1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Theorem on Dependency Properties

hierarchical link-distance bounded Model \ Property input

utput

input

utput

n-XTOP strictly strictly strictly strictly XTOPR strictly strictly ✓ strictly MBOT ✓ strictly ✓ strictly

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free XTOPR

Theorem Let M1, . . . , Mk be ε-free XTOPR over Σ such that {(c[t1, . . . , tn] , c′[t1, . . . , tn]) | t1, . . . , tn ∈ T} ⊆ τM1 ; · · · ; τMk for some contexts c, c′ ∈ CΣ(Xn) and special T ⊆ TΣ. ∀1 ≤ i ≤ k, ∀1 ≤ j ≤ n ∃tj ∈ T, ∃ui−1, Di, ui ∈ dep(Mi), ∃(vji, wji) ∈ Di such that

u0 = c[t1, . . . , tn] and uk = c′[t1, . . . , tn]
posxj(c′) ≤ wjk
vji ≤ wj(i−1) if i ≥ 2
posxj(c) ≤ vj1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free XTOPR

Corollary [see Sect. 3.4 in [Arnold, Dauchet 1982]] The illustrated tree transformation τ cannot be computed by any ε-free XTOPR

σ σ σ δ tn tn−1 tn−3 tn−2 t4 t3 t2 t1 δ σ σ σ tn tn−1 tn−2 tn−3 tn−4 t3 t2 t1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free XTOPR

Corollary [see Thm. 5.2 in [Maletti et al. 2009]] The illustrated tree transformation τ cannot be computed by any ε-free XTOPR

δ γ γ δ s t u δ s δ t u

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free XTOPR

Inverse of topicalization

δ t1 δ t2 δ t3 δ tn−1 tn — δ t2 δ t3 δ tn−1 δ tn t1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free XTOPR

Corollary Topicalization cannot be computed by any composition chain of ε-free XTOPR

u0 δ t2 δ t3 δ tn−1 δ tn t1 . . . (3) . . . (2) ? (1) — u1 v12 v(n−1)2 vn2 — u2 vn3 v(n−1)3 v13 — u3 δ t1 δ t2 δ t3 δ tn−1 tn

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free MBOT

Theorem Let M = (Q, Σ, I, R) be an ε-free MBOT such that {(c[t1, . . . , tn] , c′[t1, . . . , tn]) | t1, . . . , tn ∈ T} ⊆ τM for some contexts c, c′ ∈ CΣ(Xn) and special T ⊆ TΣ. ∀1 ≤ j ≤ n, ∃tj ∈ T, ∃u, D, u′ ∈ dep(M), ∃(vj, wj) ∈ D with

u = c[t1, . . . , tn] and u′ = c′[t1, . . . , tn]
posxj(c) ≤ vj
posxj(c′) ≤ wj

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Linking Theorem for ε-free MBOT

Corollary Inverse of topicalization cannot be computed by any ε-free MBOT

δ t1 δ t2 δ t3 δ tn−1 tn — δ t2 δ t3 δ tn−1 δ tn t1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Known Result

Theorem [see [Maletti et al. 2009]] es-XTOP es-XTOPR es-XTOP2 = (es-XTOPR)2

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Known Result

Theorem [see [Maletti et al. 2009]] es-XTOP es-XTOPR es-XTOP2 = (es-XTOPR)2 Proof.

look-ahead adds power at first level
none of the basic classes is closed under composition

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Lower Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 ≤ 2 strict 2 ≤ 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 89

The problem Upper bounds Linking technique Lower bounds

Lower Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 2 strict 2 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Main theorem

Theorem e-XTOP2 ⊆ (e-XTOPR)2 e-XTOP3 ⊆ (e-XTOPR)3

t = σ1 σ1 σ1 σ2 tn tn−1 σ2 ti+2 ti+1 c σ2 ti ti−1 σ2 t2 t1 σ1 σ1 σ1 σ1 σ1 tn σ2 tn−1 tn−2 σ2 ti+1 ti σ2 ti−1 ti−2 σ2 t3 t2 t1 = u s = si+1 si si−1 v′ v vi−1 vi vi+1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Main theorem

Theorem e-XTOP2 ⊆ (e-XTOPR)2 e-XTOP3 ⊆ (e-XTOPR)3 v vi−1 and v vi and v vi+1

t = σ1 σ1 σ1 σ2 tn tn−1 σ2 ti+2 ti+1 c σ2 ti ti−1 σ2 t2 t1 σ1 σ1 σ1 σ1 σ1 tn σ2 tn−1 tn−2 σ2 ti+1 ti σ2 ti−1 ti−2 σ2 t3 t2 t1 = u s = si+1 si si−1 v′ v vi−1 vi vi+1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Main theorem

Theorem e-XTOP2 ⊆ (e-XTOPR)2 e-XTOP3 ⊆ (e-XTOPR)3 v vi−1 and v vi and v vi+1

t = σ1 σ1 σ1 σ2 tn tn−1 σ2 ti+2 ti+1 c σ2 ti ti−1 σ2 t2 t1 σ1 σ1 σ1 σ1 σ1 tn σ2 tn−1 tn−2 σ2 ti+1 ti σ2 ti−1 ti−2 σ2 t3 t2 t1 = u s = si+1 si si−1 v′ v vi−1 vi vi+1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Main theorem

Theorem e-XTOP2 ⊆ (e-XTOPR)2 e-XTOP3 ⊆ (e-XTOPR)3 v vi−1 and v vi and v vi+1 v′ vi−1 and v′ vi and v′ vi+1

t = σ1 σ1 σ1 σ2 tn tn−1 σ2 ti+2 ti+1 c σ2 ti ti−1 σ2 t2 t1 σ1 σ1 σ1 σ1 σ1 tn σ2 tn−1 tn−2 σ2 ti+1 ti σ2 ti−1 ti−2 σ2 t3 t2 t1 = u s = si+1 si si−1 v′ v vi−1 vi vi+1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 94

The problem Upper bounds Linking technique Lower bounds

Lower Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 2 strict 2 2 look-ahead 1 ≤ 3 — 2 ≤ 4

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

SLIDE 95

The problem Upper bounds Linking technique Lower bounds

Lower Bounds

TOP e-XTOP strict, nondeleting 1 2 strict, look-ahead 1 2 strict 2 2 look-ahead 1 3 — 2 3–4 (4)

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Missing Cases

TOP XTOP XTOPR ε-free, nondeleting 1 ∞ ∞ strict 2 ∞ ∞ nondeleting 1 ∞ ∞ strict, nondeleting 1 ∞ ∞ — 2 ∞ ∞ Proof.

completely different technique [Fülöp, Maletti, 2013]

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Summary

TOP XTOP XTOPR MBOT ε-free, strict, nondeleting 1 2 2 1 ε-free, strict 2 2 2 1 ε-free 2 4 3 1

therwise (without delabeling)

2 ∞ ∞ 1

Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP

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The problem Upper bounds Linking technique Lower bounds

Summary

(n)-MBOT XTOP∞ e-XTOP4 e-XTOP3 XTOP2 e-XTOP2 n-XTOP2 XTOPR ne-XTOP2 XTOP e-XTOP n-XTOP ne-XTOP ? ? Zoltán Fülöp and Andreas Maletti Composition Closure of Linear XTOP