Normalization of Sequential Top-Down Tree-to-Word Transducers Gregoire Laurence joint work with Aurelien Lemay, Joachim Niehren, Slawek Staworko, and Marc Tommasi Mostrare Project INRIA Lille - Nord Europe University of Lille

May 27, 2011

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

1 / 25

Why STWs? XSLT

Gregoire L. (MOSTRARE - INRIA Lille)



Normalization of STWs

May 27, 2011

2 / 25

Why STWs? XSLT



Tree-to-word transformations

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

2 / 25

Why STWs? XSLT



Tree-to-word transformations

XSLT framework root left right

label

⋯ left ⋯

right ⋯

concatenation on the output XSLT is TURING COMPLET

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

2 / 25

Why STWs?

XSLT inspired framework left middle right

root label

⋯ left

right

⋯ middle ⋯

ranked input finite states machine (template = state) deterministic (equivalence undecidable for non det string transducers [Griffiths 68])

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

3 / 25

Related work

Top-down tree transducers [Maneth et Al. PODS 2010] Tree-to-tree transducer Learning results Normalization algorithm Allow copy

Visibly pushdown transducers [Raskin et Al. ALP 2008] Tree-to-words transducer Works on stream of events

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

4 / 25

Why the normalization problem?

Learning Grammatical inference

Classic problem String transducers [Choffrut, 1978] Top-down transducers [Engelfriet et Al. JCSS 2009] Bottom-up transducers [Friese et Al. DLT 2010] Macro tree transducers (open problem)

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

5 / 25

Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

6 / 25

Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

7 / 25

Example of Transformation f f

bbb a

a f babb

g

a

a ⋯

g ⎧ ⎪ ⎪ ⎪ n⎨ ⋮ ⎪ ⎪ ⎪ ⎩ g

g

a

a

⎫ ⎪ ⎪ ⎪ ⋮ ⎬ m ↦ b ⋅ an ⋅ b ⋅ am ⋅ b ⎪ ⎪ ⎪ g ⎭

f g

g babaab

f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

g

a

a ⋯

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

8 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

g

g

a

g

a

f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal Deterministic Each node visited exactly once

g

g

a

g

a

f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal

b

Deterministic

T’ g

Each node visited exactly once

b b

g T’

Output concatenation T (f (t1 , t2 )) = b ⋅ T ′ (t1 ) ⋅ b ⋅ T ′ (t2 ) ⋅ b

a

g

a

f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal

b

b b

Deterministic g

Each node visited exactly once Output concatenation T (f (t1 , t2 )) = b ⋅ T ′ (t1 ) ⋅ b ⋅ T ′ (t2 ) ⋅ b T ′ (g (t1 )) = a ⋅ T ′ (t1 )

g ε

a

ε

a g

a

ε

a a

f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal

b

b b

Deterministic g

Each node visited exactly once Output concatenation T (f (t1 , t2 )) = b ⋅ T ′ (t1 ) ⋅ b ⋅ T ′ (t2 ) ⋅ b T ′ (g (t1 )) = a ⋅ T ′ (t1 ) T (a) = ε

g ε

a a ε

ε

a g

ε

a

′

a ε f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal

b

b b

Deterministic g

Each node visited exactly once Output concatenation T (f (t1 , t2 )) = b ⋅ T ′ (t1 ) ⋅ b ⋅ T ′ (t2 ) ⋅ b T ′ (g (t1 )) = a ⋅ T ′ (t1 ) T (a) = ε

g ε

a a ε

ε

a g

ε

a

′

a ε f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Sequential Top-Down Tree-to-Word Transducers (STWs)

Machine definition

f

Top Down tree traversal

b

b b

Deterministic g

Each node visited exactly once Output concatenation q(f (x 1 , x 2 )) = b ⋅ q ′ (x 1 ) ⋅ b ⋅ q ′ (x 2 ) ⋅ b q ′ (g (x 1 )) = a ⋅ q ′ (x 1 ) q (a) = ε

g ε

a a ε

ε

a g

ε

a

′

a ε f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

9 / 25

Expressivity

Input domain Regular tree language Ranked Path-closed (recognizable by deterministic top-down automata)

Output CFL

Transformation no reordering no copy

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

10 / 25

Problem

What’s the most adapted normal form for such machine? Does every STWs can be normalized in such form? What’s the cost of this operation?

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

11 / 25

Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

12 / 25

On the road to an earliest form f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b earliest

another definition

f

f

b

ε

b

ε

b

b

g

g ε

a a ε

g ε

a

ε

g

a b

ε

a a ε

ε

a g

ε

a a b

q0 (f (x1 , x2 )) → b ⋅ q1 (x1 ) ⋅ b ⋅ q1 (x2 ) ⋅ b q1 (g (x1 )) → a ⋅ q1 (x1 ) q1 (a) → ε

Gregoire L. (MOSTRARE - INRIA Lille)

g a

Normalization of STWs

q0 (f (x1 , x2 )) → q2 (x1 ) ⋅ b ⋅ q1 (x2 ) q1 (g (x1 )) → a ⋅ q1 (x1 ) q2 (g (x1 )) → q2 (x1 ) ⋅ a q1 (a) → b q2 (a) → b May 27, 2011

13 / 25

On the road to an earliest form f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b earliest

another definition

f

f

b

ε

b

ε

b

b

g

g ε

a a ε

g ε

a

ε

g

a b

ε

a

g a

ε

a g

ε

a

a ε

a b

as soon as possible ▸ ▸

no clear semantic

top-down left-to-right

bigger definition

smallest definition Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

13 / 25

On the road to an earliest form f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b earliest STW (wrt input)

earliest wrt output (VPT)

f

f

b

b

ε

b ε

b g

g ε

a a ε

g ε

a g

a b

ε

a a ε

ε

a g

ε

a a b

q0 (f (x1 , x2 )) → b ⋅ q1 (x1 ) ⋅ b ⋅ q1 (x2 ) ⋅ b q1 (g (x1 )) → a ⋅ q1 (x1 ) q1 (a) → ε

Gregoire L. (MOSTRARE - INRIA Lille)

g ε

a

Normalization of STWs

q0 (f (x1 , x2 )) → b ⋅ q1 (x1 ) ⋅ q1 (x2 ) q1 (g (x1 )) → a ⋅ q1 (x1 ) q1 (a) → b

May 27, 2011

13 / 25

On the road to an earliest form f (g n (a), g m (a)) ↦ b ⋅ an ⋅ b ⋅ am ⋅ b earliest STW (wrt input)

earliest wrt output (VPT)

f

f

b

b

ε

b ε

b g

g ε

a a ε

g ε

a g

a b

ε

a

g ε

a

g

a b

as soon as possible ▸

as soon as possible

top-down left-to-right

▸

depth first

same size

same size Gregoire L. (MOSTRARE - INRIA Lille)

ε

a

a ε

▸

ε

a

Normalization of STWs

May 27, 2011

13 / 25

Bad news for earliest wrt output Reverse Transformation

⎧ ⎪ ⎪ ⎪ ⋯ , ⎨ ⎪ ⎪ ⎪ ⎩

b

a b #

ba , ⋯ ,

a c

⎫ ⎪ ⎪ cab , ⋯ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

#

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

14 / 25

Bad news for earliest wrt output Reverse Transformation

⎧ ⎪ ⎪ ⎪ ⋯ , ⎨ ⎪ ⎪ ⎪ ⎩

b

a b #

ba , ⋯ ,

a c

⎫ ⎪ ⎪ cab , ⋯ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

# ε ε ε ε

a b a c

ε ε ε

caba

ε

# caba

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

14 / 25

Bad news for earliest wrt output Reverse Transformation

⎧ ⎪ ⎪ ⎪ ⋯ , ⎨ ⎪ ⎪ ⎪ ⎩

b

a ba , ⋯ ,

b

a c

#

⎫ ⎪ ⎪ cab , ⋯ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

# ε ε ε ε

a b a c

ε ε ε

caba

ε

#

qε (a(x1 )) → qa (x1 ) qa (b(x1 )) → qba (x1 ) ⋯ qw (a(x1 )) → qa⋅w (x1 ) qw (b(x1 )) → qb⋅w (x1 ) qw (c(x1 )) → qc⋅w (x1 ) ⋯ qw (#) → w ⋯

caba Infinite number of states Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

14 / 25

Bad news for earliest wrt output Reverse Transformation

ε ε ε ε

a b a c

a

b

a ba , ⋯ ,

b

a c

#

⎫ ⎪ ⎪ cab , ⋯ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

#

b a

⎧ ⎪ ⎪ ⎪ ⋯ , ⎨ ⎪ ⎪ ⎪ ⎩

caba

q(a(x1 )) → q(x1 ) ⋅ a q(b(x1 )) → q(x1 ) ⋅ b q(c(x1 )) → q(x1 ) ⋅ c q(#) → ε

c

# ε

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

14 / 25

Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

15 / 25

Push up output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

f ε

q0 (f (x1 , x2 )) → q1 (x1 ) ⋅ b ⋅ q2 (x2 )

ε b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

g

q1 (a) → b

ε

q2 (g (x1 )) → a ⋅ q2 (x1 )

g a

g

q2 (a) → b

ε

a

ε

a a b

a b Output as soon as possible following the top-down order

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

16 / 25

Push up output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

f ε

q0 (f (x1 , x2 )) → q1 (x1 ) ⋅ b ⋅ q2 (x2 )

ε b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

g

q1 (a) → b

ε

q2 (g (x1 )) → a ⋅ q2 (x1 )

g a

g

q2 (a) → b

ε

a

ε

a a b

a b Output as soon as possible following the top-down order

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

16 / 25

Push up output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

f b

q0 (f (x1 , x2 )) → b ⋅ q1 (x1 ) ⋅ b ⋅ q2 (x2 ) ⋅ b

b b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

g

q1 (a) → ε

ε

q2 (g (x1 )) → a ⋅ q2 (x1 )

g a

g

q2 (a) → ε

ε

a

ε

a a ε

a ε Output as soon as possible following the top-down order

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

16 / 25

Push left output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

f b

q0 (f (x1 , x2 )) → b ⋅ q1 (x1 ) ⋅ b ⋅ q2 (x2 ) ⋅ b

b b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

g

q1 (a) → ε

ε

q2 (g (x1 )) → a ⋅ q2 (x1 )

g a

g

q2 (a) → ε

ε

a

ε

a a ε

a ε Output as soon as possible following the left-to-right preorder

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

17 / 25

Push left output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

q0 (f (x1 , x2 )) → b ⋅ q3 (x1 ) ⋅ b ⋅ q2 (x2 ) ⋅ b

f b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

b b

q1 (a) → ε

g

q2 (g (x1 )) → a ⋅ q2 (x1 ) q2 (a) → ε

g ε

a g

q3 (g (x1 )) → a ⋅ q2 (x1 )

ε

a

ε

a a ε

a ε Output as soon as possible following the left-to-right preorder

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

17 / 25

Push left output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

q0 (f (x1 , x2 )) → b ⋅ q3 (x1 ) ⋅ b ⋅ q2 (x2 ) ⋅ b

f b

q1 (g (x1 )) → q1 (x1 ) ⋅ a

b b

q1 (a) → ε

g

q2 (g (x1 )) → a ⋅ q2 (x1 ) q2 (a) → ε

g ε

a g

q3 (g (x1 )) → a ⋅ q3 (x1 )

ε

a

q3 (a) → ε

ε

a a ε

a ε Output as soon as possible following the left-to-right preorder

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

17 / 25

Push left output annotations f (g m (a), g n (a)) ↦ b ⋅ (a)m ⋅ b ⋅ (a)n ⋅ b

f b q0 (f (x1 , x2 )) → b ⋅ q2 (x1 ) ⋅ b ⋅ q2 (x2 ) ⋅ b q2 (g (x1 )) → a ⋅ q2 (x1 )

b b

g

q2 (a) → ε

g ε

a g

ε

a

ε

a a ε

a ε Output as soon as possible following the left-to-right preorder

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

17 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ (ab)m ⋅ ac ⋅ (ab)n q0 (f (x1 , x2 )) → q1 (x1 ) ⋅ a ⋅ q2 (x2 ) q1 (g (x1 )) → ab ⋅ q1 (x1 )

f ε

q1 (a) → ε q2 (g (x1 )) → q2 (x1 ) ⋅ ab q2 (a) → c

ε a

g

g ε

ab g

Applying an operation can lead to:

ε

ab

ε

ab a c

a ε

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

18 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ a ⋅ (ba)m ⋅ c ⋅ (ab)n q0 (f (x1 , x2 )) → q1 (x1 ) ⋅ a ⋅ q2 (x2 ) q1 (g (x1 )) → ab ⋅ q1 (x1 )

f ε

q1 (a) → ε q2 (g (x1 )) → q2 (x1 ) ⋅ ab q2 (a) → c

ε a

g

g ε

ab g

Applying an operation can lead to:

ε

ab

ε

ab a c

a ε

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

18 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ a ⋅ (ba)m ⋅ c ⋅ (ab)n q0 (f (x1 , x2 )) → a ⋅ q1 (x1 ) ⋅ q2 (x2 ) q1 (g (x1 )) → ba ⋅ q1 (x1 )

f ε

q2 (g (x1 )) → q2 (x1 ) ⋅ ab q2 (a) → c

ε

a

q1 (a) → ε

g

g ε

ba g

Applying an operation can lead to: modify other output annotations

ε

ba

ε

ab a c

a ε

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

18 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ a ⋅ (ba)m ⋅ c ⋅ (ab)n q0 (f (x1 , x2 )) → a ⋅ q1 (x1 ) ⋅ c ⋅ q2 (x2 ) q1 (g (x1 )) → ba ⋅ q1 (x1 )

f c

q2 (g (x1 )) → q2 (x1 ) ⋅ ab q2 (a) → ε

ε

a

q1 (a) → ε

g

g ε

ba g

Applying an operation can lead to: modify other output annotations

ε

ba

ε

ab a ε

a ε

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

18 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ a ⋅ (ba)m ⋅ c ⋅ (ab)n q0 (f (x1 , x2 )) → a ⋅ q1 (x1 ) ⋅ c ⋅ q2 (x2 ) q1 (g (x1 )) → ba ⋅ q1 (x1 )

f c

q2 (g (x1 )) → ab ⋅ q2 (x1 ) q2 (a) → ε

ε

a

q1 (a) → ε

g

g ε

ba g

Applying an operation can lead to: modify other output annotations reapply other operations

Gregoire L. (MOSTRARE - INRIA Lille)

ε

ba

ε

ab a ε

a ε

Normalization of STWs

May 27, 2011

18 / 25

Combinatorial problems

f (g m (a), g n (a)) ↦ a ⋅ (ba)m ⋅ c ⋅ (ab)n q0 (f (x1 , x2 )) → a ⋅ q1 (x1 ) ⋅ c ⋅ q2 (x2 ) q1 (g (x1 )) → ba ⋅ q1 (x1 )

f c

q2 (g (x1 )) → ab ⋅ q2 (x1 ) q2 (a) → ε

ε

a

q1 (a) → ε

g

g ε

ba g

Applying an operation can lead to: modify other output annotations reapply other operations

Gregoire L. (MOSTRARE - INRIA Lille)

ε

ba

ε

ab a ε

a ε

Normalization of STWs

May 27, 2011

18 / 25

Normalization results

Theorem For an STW M, we can construct an equivalent earliest STW M ′ in time polynomial in the size of M ′ . ′ M size is at most doubly-exponential in the size of M.

Proof techniques Constructive Combinatory on words

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

19 / 25

Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

20 / 25

Explosion of the size of rules input: a perfect (well-balanced) binary tree of height n output: concatenation of all leaves, erasing inner nodes f ε

ε ε f

ε

f ε

ε

ε

ε

ε ⋮

f ε

f ε

ε

a a

a a

ε a a

ε

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

For 0 ≤ i < n: qi (f (x1 , x2 )) → qi+1 (x1 ) ⋅ qi+1 (x2 ) qn (a) → a

ε

output: a2

Gregoire L. (MOSTRARE - INRIA Lille)

a a

n

Normalization of STWs

May 27, 2011

21 / 25

Explosion of the size of rules input: a perfect (well-balanced) binary tree of height n output: concatenation of all leaves, erasing inner nodes f a2

n

ε ε f

ε

f ε

ε

ε

ε

For 0 < i < n:

ε

n

q0 (f (x1 , x2 )) → a2 ⋅ q1 (x1 ) ⋅ q1 (x2 )

⋮ f ε

ε

ε

a ε

a ε

ε a ε

qi (f (x1 , x2 )) → qi+1 (x1 ) ⋅ qi+1 (x2 )

f ε

qn (a) → ε

ε a ε

Size of the first rule is exponential in the size of the initial transducer Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

21 / 25

Explosion of the number of states 0 ε

0 ε

ε

1 ε

ε input: (0∣1)n = x binary encoding followed by a or b

1 a2

ε

a2

output: start always by ax 0 ε

0 ε

ε

1 ε

ε

For 0 ≤ i < n: qi (0(x1 )) → qi+1 (x1 )

1 a

8

ε

a

a ε

b a ⋅#

a10

a16 ⋅ # ⋅ a10

Gregoire L. (MOSTRARE - INRIA Lille)

i

qi (1(x1 )) → qi+1 (x1 ) ⋅ a2

8

qn (a) → ε n

qn (b) → a2 ⋅ #

16

n states

Normalization of STWs

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Explosion of the number of states 0 ε

0 ε

ε

1

output: start always by ax 0

ε

ε

1

ε

For 0 ≤ i < n: k qik (0(x1 )) → qi+1 (x1 )

1 ε

a8

ε

a2

0 ε

input: (0∣1)n = x binary encoding followed by a or b

1 ε

a2

ε

a8

i

a ε

b a6 ⋅ # ⋅ a10

a10

a16 ⋅ # ⋅ a10

i

k+2 qik (1(x1 )) → a2 ⋅ qi+1 (x1 )

ε

qnk (a) → ε qnk (b) → a2

n

−k

⋅ # ⋅ ak

n i ∑i=0 2 states

Obtained number of states is exponential on the number of the initial transducer states

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

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Outline

1

Sequential Top-Down Tree-to-Word Transducers

2

Earliest STWs

3

Normalization

4

Complexity bounds

5

Minimization

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

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Minimization

Theorem Minimization of earliest STWs is in PTIME.

Lemma Between two earliest STWs, states can be duplicated but keep sames rules. The equivalence test on earliest STWs is in PTIME [Staworko et Al. FCT 2009]

Lemma Minimization of arbitrary STWs is NP-complete.

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

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Conclusion

Result We have defined the class of STW : nice expressivity with good properties ▸ ▸

decidable equivalence in PTIME normalization algorithm

Perspectives learning algorithm allow swapping

Gregoire L. (MOSTRARE - INRIA Lille)

Normalization of STWs

May 27, 2011

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