Equivalence of Deterministic Nested Word to Word Transducers Gregoire Laurence (Joint work with Slawek Staworko, Aurelien Lemay, Joachim Niehren) INRIA Lille, Mostrare University of Lille

Mostrare, 06 May 2009

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Motivation



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Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Overview

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Outline

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata book

pers

name

pers

adr

name

adr

Visibly Pushdown Automata [Alur et al’04] Streaming Tree Automata [Gauwin et al’08]

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata

book

pers

name

pers

adr

name

adr

< book > < pers > < name > < /name > < adr > < /adr > < /pers > < pers > < name > < /name > < adr > < /adr > < /pers > < /book > Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

5/1

Nested Word Automata op book:γb

book

pers

name

pers

adr

name

adr

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

Mostrare, 06 May 2009

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Nested Word Automata 0 op book:γb

book

pers

name

pers

adr

name

adr

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

Mostrare, 06 May 2009

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Nested Word Automata 0 op book:γb

γb 1 book

pers

name

pers

adr

name

adr

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

< book >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

5/1

Nested Word Automata 0 op book:γb

γb 1 book

2

name

γp pers

pers

adr

name

adr

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

< book > < pers >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

5/1

Nested Word Automata 0 op book:γb

γb 1 book

2

γn 3 name

γp pers

pers

adr

name

adr

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

< book > < pers > < name >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

5/1

Nested Word Automata 0 op book:γb

γb 1 book

2

γp pers

γn 3 name 4

pers

adr

name

adr

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

< book > < pers > < name > < /name >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

5/1

Nested Word Automata 0 op book:γb

γb 1 book 6

2

γp pers

γn 3 name 4

3

1

2

γa adr

5

γp pers

γn 3 name 4

3

1

γa adr

5

0 −−−−−−→ 1 op pers:γp 1 −−−−−−→ 2 op name:γn 2 −−−−−−→ 3 cl name:γn 3 −−−−−−→ 4 op adr :γa 4 −−−−−→ 3 cl adr :γa 3 −−−−−→ 5 cl pers:γp 5 −−−−−−→ 1 cl book:γb 1 −−−−−−→ 6

< book > < pers > < name > < /name > < adr > < /adr > < /pers > < pers > < name > < /name > < adr > < /adr > < /pers > < /book > Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata ↓ 0 op book:6

6 1 book 6

2

1 pers

4 3 name 4

3

1

2

5 adr

5

1 pers

4 3 name 4

3

1

5 adr

5

0 −−−−−−→ 1 op pers:1 1 −−−−−→ 2 op name:4 2 −−−−−−→ 3 cl name:4 3 −−−−−−→ 4 op adr :5 4 −−−−−→ 3 cl adr :5 3 −−−−−→ 5 cl pers:1 5 −−−−−→ 1 cl book:6 1 −−−−−−→ 6

< book > < pers > < name > < /name > < adr > < /adr > < /pers > < pers > < name > < /name > < adr > < /adr > < /pers > < /book > Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata na Definition T = (Σ, states, stack, rules, initial, final) op a:γ

Rules of the form q −−−→ q 0

cl a:γ

q −−−→ q 0 .

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata na Definition T = (Σ, states, stack, rules, initial, final) op a:γ

Rules of the form q −−−→ q 0

cl a:γ

q −−−→ q 0 .

Determinism (dna) op a:γ

In a opening transition q −−−→ q 0 , q and a determines γ and q 0 . cl a:γ

In a closing transition q −−−→ q 0 , q, a and γ determines q 0 .

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Automata na Definition T = (Σ, states, stack, rules, initial, final) op a:γ

Rules of the form q −−−→ q 0

cl a:γ

q −−−→ q 0 .

Determinism (dna) op a:γ

In a opening transition q −−−→ q 0 , q and a determines γ and q 0 . cl a:γ

In a closing transition q −−−→ q 0 , q, a and γ determines q 0 .

Top Down(na↓ ) stack symbols = states cl a:q 0

all closing rules have the form q −−−−→ q 0

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducers

Nested Word to Word (n2w) Definition T = (Σ,states, stack, rules, initial, final) op a

:γ

Rules of the form q −−−−−→ q 0

cl a

:γ

q −−−−−→ q 0

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducers

Nested Word to Word (n2w) Definition T = (Σ,∆,states, stack, rules, initial, final) op a/u:γ

Rules of the form q −−−−−→ q 0

cl a/u:γ

q −−−−−→ q 0

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducers

Nested Word to Word (n2w) Definition T = (Σ,∆,states, stack, rules, initial, final) op a/u:γ

Rules of the form q −−−−−→ q 0 JT K : TΣ → ∆∗

cl a/u:γ

q −−−−−→ q 0

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducers

Nested Word to Word (n2w) Definition T = (Σ,∆,states, stack, rules, initial, final) op a/u:γ

Rules of the form q −−−−−→ q 0 JT K : TΣ → ∆∗

cl a/u:γ

q −−−−−→ q 0

Determinism and Top Down Same definition as na.

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

a

b

b

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

0 3

1

a

1

b

1

b

3

2

2

2

2

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

2

2 < b >< /b >

2

2

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

2

2 < b >< /b >

2

2

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

c

2

2 < b >< /b >

2

2

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

8/1

Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

c

2

2 < b >< /b >

2

2

b

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

c

2

2 < b >< /b >

2

2

b

b

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

8/1

Nested Word Transducer Example

0 3

< c >1

a

ε

1

b

ε

1

b

3 < a >< /a >< /c >

c

2

2 < b >< /b >

2

2

b

b

a

< b >< /b >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Equivalence Problem

Equivalence of dn2ws Two dn2ws T1 , T2 are equivalent iff they are defined on the same domain and for each tree t in this domain we have JT1 K(t) = JT2 K(t). nondeterministic equivalence is undecidable [Griffiths’68]

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Outline

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Morphism Equivalence on cfg A (word) morphism M : Σ → ∆∗ M(v1 · v2 · · · vn ) = M(v1 ) · M(v2 ) · · · M(vn )

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

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Morphism Equivalence on cfg A (word) morphism M : Σ → ∆∗ M(v1 · v2 · · · vn ) = M(v1 ) · M(v2 ) · · · M(vn )

Equivalence on cfg Two morphisms M1 , M2 are equivalent on a cfg G iff M1 (w ) = M2 (w ) for all w ∈ L(G ).

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Morphism Equivalence on cfg A (word) morphism M : Σ → ∆∗ M(v1 · v2 · · · vn ) = M(v1 ) · M(v2 ) · · · M(vn )

Equivalence on cfg Two morphisms M1 , M2 are equivalent on a cfg G iff M1 (w ) = M2 (w ) for all w ∈ L(G ).

Theorem[Plandowski’94] The morphism equivalence problem for context-free languages can be solved in polynomial time.

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Outline

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓

Proposition Morphism equivalence on cfgs can be reduced in quadratic time to dn2w↓ -equivalence.

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓

Proposition Morphism equivalence on cfgs can be reduced in quadratic time to dn2w↓ -equivalence. Given a cfg G and a morphism M, we construct a dn2w↓ T : input : (extended) parse tree t of w ∈ L(G ) output : JT K(t) = M(w )

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M M(a) = ab M(b) = b

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M

S

R

A

B

a

b

M(a) = ab M(b) = b

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M

r2

r3

r4

r5

a

b

M(a) = ab M(b) = b

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

(r2 , 0) r2

r3

Morphism M

r4

r5

a

b

M(a) = ab M(b) = b

r2 :f

o −−−→ (r2 , 0) Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M

(r2 , 0) r2

(r3 , 0) r3

(r4 , 0) r4

r5

a

b

M(a) = ab M(b) = b

op r3 :(r2 ,1)

(r2 , 0) −−−−−−→ (r3 , 0)

op r4 :(r3 ,1)

(r3 , 0) −−−−−−→ (r4 , 0)

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

(r2 , 0) r2

(r3 , 0) r3

Morphism M

(r4 , 0) r4

r5

a

b

M(a) = ab M(b) = b d op a:(r4 ,1)

(r4 , 0) −−−−−−→ d Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

14 / 1

From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

(r2 , 0) r2

(r3 , 0) r3

Morphism M

(r4 , 0) r4

r5

M(a) = ab M(b) = b d

a (r4 , 1)

b

cl a:(r4 ,1)

d −−−−−−→ (r4 , 1) Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

14 / 1

From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M

(r2 , 0) r2

(r3 , 0) r3 (r2 , 1)

(r4 , 0) r4 (r3 , 1)

(r5 , 0) r5 (r3 , 2)

M(a) = ab M(b) = b a (r4 , 1)

d op r5 :(r3 ,2)

(r3 , 1) −−−−−−→ (r5 , 0)

d

b (r5 , 1)

cl r5 :(r3 ,2)

(r5 , 1) −−−−−−→ (r3 , 2)

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

14 / 1

From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

(r2 , 0) r2

f

(r3 , 0) r3 (r2 , 1)

Morphism M

(r4 , 0) r4 (r3 , 1)

(r5 , 0) r5 (r3 , 2)

M(a) = ab M(b) = b d

a (r4 , 1)

d

b (r5 , 1)

cl r2 :f

(r2 , 1) −−−−−→ f Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

14 / 1

From Morphisms to dn2w↓ o

cfg G r 1: r 2: r 3: r 4: r 5:

S → RS S →R R → AB A→a B→b

Morphism M

ε (r2 , 0) r2

f

ε

ε (r3 , 0) r3 (r2 , 1) ε

ε (r4 , 0) r4 (r3 , 1) ε ε (r5 , 0) r5 (r3 , 2) ε

M(a) = ab M(b) = b a (r4 , 1) ε b d

ab d op a/M(a):(r4 ,1)

(r4 , 0) −−−−−−−−−−→ d

b (r5 , 1) ε

op b/M(b):(r5 ,1)

(r5 , 0) −−−−−−−−−−→ d

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms

Proposition dn2w-equivalence can be reduced in polynomial time to morphism equivalence on cfgs.

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms

Proposition dn2w-equivalence can be reduced in polynomial time to morphism equivalence on cfgs. For dn2w T1 and T2 , we define : a cfg G which recognizes successfull parallel runs, L(G ) ⊆ (rulesT1 × rulesT2 )∗ . two morphisms M1 and M2 s.t.: For all s ∈ L(G ), M1 (s) = JT1 K(t) and M2 (s) = JT2 K(t).

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1 (init = {0}, fin = {3})

(init = {00 }, fin = {40 }) op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1 r2 : 1 −−−−−→ 1 cl b/:γ2

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

r4 : 2 −−−−−−−−−−−−−−−→ 3

op b/:γ 0

cl a/:γ 0

1 40 r30 : 30 −−−−−−−−−−−−→

cl b/:γ 0

2 30 r40 : 20 −−−−−−−−−−−→

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1 (init = {0}, fin = {3})

(init = {00 }, fin = {40 }) op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1 r2 : 1 −−−−−→ 1 cl b/:γ2

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

r4 : 2 −−−−−−−−−−−−−−−→ 3 (r1 , r10 )

a (r4 , r30 )

(r2 , r20 )

b (r3 , r40 )

op b/:γ 0

cl a/:γ 0

1 40 r30 : 30 −−−−−−−−−−−−→

cl b/:γ 0

2 30 r40 : 20 −−−−−−−−−−−→

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1 (init = {0}, fin = {3})

(init = {00 }, fin = {40 }) op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1 r2 : 1 −−−−−→ 1 cl b/:γ2

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

r4 : 2 −−−−−−−−−−−−−−−→ 3 (r1 , r10 )

a (r4 , r30 )

(r2 , r20 )

b (r3 , r40 )

op b/:γ 0

cl a/:γ 0

1 40 r30 : 30 −−−−−−−−−−−−→

cl b/:γ 0

2 30 r40 : 20 −−−−−−−−−−−→

[S] → (r1 , r10 ) · [(1, 2), (10 , 30 )] · (r4 , r30 )

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1 (init = {0}, fin = {3})

(init = {00 }, fin = {40 }) op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1 r2 : 1 −−−−−→ 1 cl b/:γ2

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

r4 : 2 −−−−−−−−−−−−−−−→ 3 (r1 , r10 )

a (r4 , r30 )

(r2 , r20 )

b (r3 , r40 )

op b/:γ 0

cl a/:γ 0

1 40 r30 : 30 −−−−−−−−−−−−→

cl b/:γ 0

2 30 r40 : 20 −−−−−−−−−−−→

[S] → (r1 , r10 ) · [(1, 2), (10 , 30 )] · (r4 , r30 ) [(1, 2), (10 , 30 )] → (r2 , r20 ) · [(1, 1), (20 , 20 )] · (r3 , r40 )

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1

(init = {00 }, fin = {40 })

(init = {0}, fin = {3})

op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1

op b/:γ 0

r2 : 1 −−−−−→ 1

cl a/:γ 0

cl b/:γ2

1 40 r30 : 30 −−−−−−−−−−−−→

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

cl b/:γ 0

r4 : 2 −−−−−−−−−−−−−−−→ 3 (r1 , r10 )

a (r4 , r30 )

(r2 , r20 )

b (r3 , r40 )

2 30 r40 : 20 −−−−−−−−−−−→

[S] → (r1 , r10 ) · [(1, 2), (10 , 30 )] · (r4 , r30 ) [(1, 2), (10 , 30 )] → (r2 , r20 ) · [(1, 1), (20 , 20 )] · (r3 , r40 ) [(1, 1), (20 , 20 )] → ε

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms T2

T1

(init = {00 }, fin = {40 })

(init = {0}, fin = {3})

op a/:γ 0

op a/:γ1

1 10 r10 : 00 −−−−−−−→

op b/ε:γ2

2 r20 : 10 −−−−−−−−→ 20

r1 : 0 −−−−−−−→ 1

op b/:γ 0

r2 : 1 −−−−−→ 1

cl a/:γ 0

cl b/:γ2

1 40 r30 : 30 −−−−−−−−−−−−→

r3 : 1 −−−−−−−−−−−→ 2 cl a/:γ1

cl b/:γ 0

r4 : 2 −−−−−−−−−−−−−−−→ 3 (r1 , r10 )

a (r4 , r30 )

(r2 , r20 )

b (r3 , r40 )

2 30 r40 : 20 −−−−−−−−−−−→

[S] → (r1 , r10 ) · [(1, 2), (10 , 30 )] · (r4 , r30 ) [(1, 2), (10 , 30 )] → (r2 , r20 ) · [(1, 1), (20 , 20 )] · (r3 , r40 ) [(1, 1), (20 , 20 )] → ε

M1 ((r3 , r40 )) =< b >< /b >

M2 ((r3 , r40 )) =< /b >< a >

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms parallel run O(|T1 |2 × |T2 |2 )

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Other Models

Top Down Ranked Tree to Word (dr2w↓ ) q(a(x1 , . . . , xk )) → u0 · q1 (x1 ) · u1 · . . . · uk−1 · qk (xk ) · uk .

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

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Other Models

Top Down Ranked Tree to Word (dr2w↓ ) q(a(x1 , . . . , xk )) → u0 · q1 (x1 ) · u1 · . . . · uk−1 · qk (xk ) · uk .

Deterministic Bottom Up Ranked Tree to Word (dr2w↑ ) a(q1 (v1 ), . . . , qk (vk )) → q(u0 · v1 · u1 · · · vk · uk ).

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Other Models parallel run O(|T1 |2 × |T2 |2 )

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Other Models parallel run O(|T1 |2 × |T2 |2 )

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓ first child next sibling O(|Σ|2 ∗ |statesT |2 )

dr2w↓

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Other Models parallel run O(|T1 |2 × |T2 |2 )

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓ first child next sibling O(|Σ|2 ∗ |statesT |2 )

ranked ⊂ unranked O(|S| ∗ n)

dr2w↓

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Other Models parallel run O(|T1 |2 × |T2 |2 )

Morphism

dn2w extended parse tree trivial

O(|M| + |G |2 )

dn2w↓ first child next sibling O(|Σ|2 ∗ |statesT |2 )

ranked ⊂ unranked O(|S| ∗ n)

dr2w↓

dr2w↑

parallel run |S1 | ∗ |S2 | Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Outline

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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Conclusion

Summary new model of deterministic nested word transducers. equivalence problem on various classes. relation to morphisms equivalence on cfg.

Future work other problems on dn2w. grammatical inference.

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From Morphisms to dn2w↓ r ∈ rulesG

lhs(r ) ∈ initialG

op r /ε:f

o −−−−→ (r , 0)

r ∈ rulesG

(r , 0) −−−−−−−−−→ d

cl r /ε:f

cl a/ε:(r ,1)

(r , |r |) −−−−→ f r , r 0 ∈ rulesG

rhs(r ) = a

op a/M(a):(r ,1)

d −−−−−−−→ (r , 1)

rhs(r ) = q1 · · · qk

1 ≤ j ≤ |r |

lhs(r 0 ) = qj

op r 0 /ε:(r ,j)

(r , j−1) −−−−−−−→ (r 0 , 0) cl r 0 /ε:(r ,j)

(r 0 , |r 0 |) −−−−−−−→ (r , j)

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

Mostrare, 06 May 2009

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From dn2w to Morphisms r1 , r10 ∈ rulesT1 r2 , r20 ∈ rulesT2

op a/u1 :γ1

cl a/u 0 :γ1

p1 ∈ initialT1

q10 ∈ finalT1

r2 = p2 −−−−−−→ q2 r20 = p20 −−−−−−→ q20 p2 ∈ initialT2 o → (r1 , r2 ) · ((q1 , p10 ), (q2 , p20 )) · (r10 , r20 )

q20 ∈ finalT2

r1 = p1 −−−−−−→ q1 op a/u2 :γ2

r1 , r10 ∈ rulesT1

1 r10 = p10 −−−−− −→ q10

cl a/u20 :γ2

op a/u1 :γ1

r1 = p1 −−−−−−→ q1

cl a/u 0 :γ1

1 r10 = p10 −−−−− −→ q10 ,

op a/u2 :γ2

cl a/u 0 :γ2

2 r2 , r20 ∈ rulesT2 r2 = p2 −−−−−−→ q2 r20 = p20 −−−−− −→ q20 0 0 0 0 0 0 ((p1 , q1 ), (p2 , q2 )) → (r1 , r2 ) · ((q1 , p1 ), (q2 , p2 )) · (r1 , r2 )

p1 , p10 , q1 ∈ statesT1 p2 , p20 , q2 ∈ statesT2 ((p1 , q1 ), (p2 , q2 )) → ((p1 , p10 ), (p2 , p20 )) · ((p10 , q1 ), (p20 , q2 ))

Gr´ egoire Laurence ( INRIA Lille, Mostrare, University of Equivalence Lille ) of dn2ws

q1 ∈ statesT1 , q2 ∈ statesT2 ((q1 , q1 ), (q2 , q2 )) → ε

Mostrare, 06 May 2009

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