An algebraic characterization of unary 2-way transducers Christian - - PowerPoint PPT Presentation

An algebraic characterization of unary 2-way transducers

Christian Choffrut1 and Bruno Guillon1,2

1LIAFA - Université Paris-Diderot, Paris 7 2Dipartimento di Informatica - Università degli studi di Milano

Septembre 17, 2014 ICTCS - Perugia - 2014 work published in MFCS 2014

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2-way automaton over Σ

⊲ t h e i n p u t w o r d ⊳

Automaton

READ

← →

left endmarker right endmarker WRITE

→ A

• Q, q-, F, δ
• transition set: ⊂ Q × Σ⊲,⊳ × {−1, 0, 1} × Q

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2-way automaton over Σ

⊲ t h e i n p u t w o r d ⊳

Automaton

READ

← →

left endmarker right endmarker WRITE

→ A

• Q, q-, F, δ
• transition set: ⊂ Q × Σ⊲,⊳ × {−1, 0, 1} × Q

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2-way transducer over Σ, Γ

⊲ t h e i n p u t w o r d ⊳

Automaton

READ

← →

left endmarker right endmarker t h e

• u

t p u t WRITE

→ (A, φ)

• Q, q-, F, δ
• production function: δ → Rat(Γ∗)

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A simple example: SQUARE = {(w, ww) | w ∈ Σ∗}

⊲ a b a c c a b ⊳

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A simple example: SQUARE = {(w, ww) | w ∈ Σ∗}

⊲ a b a c c a b ⊳ a b a c c a b ◮ copy the input word

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A simple example: SQUARE = {(w, ww) | w ∈ Σ∗}

⊲ a b a c c a b ⊳ a b a c c a b ◮ copy the input word ◮ rewind the input tape

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A simple example: SQUARE = {(w, ww) | w ∈ Σ∗}

⊲ a b a c c a b ⊳ a b a c c a b a b a c c a b ◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word

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A simple example: SQUARE = {(w, ww) | w ∈ Σ∗}

⊲ a b a c c a b ⊳ a b a c c a b a b a c c a b ◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a

copy the input word

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a

copy the input word rewind the input tape

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a a a a a

copy the input word rewind the input tape

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a a a a a

copy the input word rewind the input tape

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a a a a a a a a a

copy the input word rewind the input tape

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a a a a a a a a a

copy the input word rewind the input tape accept and halt with nondeterminism

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Another example: UnaryMult =

• (an, akn) | k, n ∈ N
• ⊲

a a a a ⊳ a a a a a a a a a a a a

copy the input word rewind the input tape accept and halt with nondeterminism

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Rational operations

◮ Union

R1 ∪ R2

◮ Componentwise concatenation

R1 · R2 = {(u1u2, v1v2) | (u1, v1) ∈ R1 and (u2, v2) ∈ R2}

◮ Kleene star

R∗ = {(u1u2 · · · uk, v1v2 · · · vk) | ∀i (ui, vi) ∈ R}

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Rational operations

◮ Union

R1 ∪ R2

◮ Componentwise concatenation

R1 · R2 = {(u1u2, v1v2) | (u1, v1) ∈ R1 and (u2, v2) ∈ R2}

◮ Kleene star

R∗ = {(u1u2 · · · uk, v1v2 · · · vk) | ∀i (ui, vi) ∈ R}

Definition (Rat(Σ∗ × Γ∗))

The class of rational relations is the smallest class:

◮ that contains finite relations ◮ and which is closed under rational operations

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Rational operations

◮ Union

R1 ∪ R2

◮ Componentwise concatenation

R1 · R2 = {(u1u2, v1v2) | (u1, v1) ∈ R1 and (u2, v2) ∈ R2}

◮ Kleene star

R∗ = {(u1u2 · · · uk, v1v2 · · · vk) | ∀i (ui, vi) ∈ R}

Definition (Rat(Σ∗ × Γ∗))

The class of rational relations is the smallest class:

◮ that contains finite relations ◮ and which is closed under rational operations

Theorem (Elgot, Mezei - 1965)

1-way transducers = = the class of rational relations.

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Hadamard operations

◮ H-product

R1 H R2 = {(u, v1v2) | (u, v1) ∈ R1 and (u, v2) ∈ R2}

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Hadamard operations

◮ H-product

R1 H R2 = {(u, v1v2) | (u, v1) ∈ R1 and (u, v2) ∈ R2}

Example: SQUARE = {(w, ww) | w ∈ Σ∗} = Identity H Identity

⊲ a b a c c a b ⊳ a b a c c a b a b a c c a b

◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word

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Hadamard operations

◮ H-product

R1 H R2 = {(u, v1v2) | (u, v1) ∈ R1 and (u, v2) ∈ R2}

◮ H-star

RH⋆ = {(u, v1v2 · · · vk) | ∀i (u, vi) ∈ R}

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Hadamard operations

◮ H-product

R1 H R2 = {(u, v1v2) | (u, v1) ∈ R1 and (u, v2) ∈ R2}

◮ H-star

RH⋆ = {(u, v1v2 · · · vk) | ∀i (u, vi) ∈ R}

Example: UnaryMult =

• (an, akn) | k, n ∈ N
• = Identity H⋆

⊲ a a a a ⊳ a a a a a a a a a a a a

copy the input word rewind the input tape accept and halt with nondeterminism

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H-Rat relations

Definition

A relation R is in H-Rat(Σ∗ × Γ∗) if R =

• 0≤i≤n

Ai H BH⋆

i

where for each i, Ai and Bi are rational relations.

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Main result

When Σ = {a} and Γ = {a}:

Theorem (Elgot, Mezei - 1965)

1-way transducers = = the class of rational relations .

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Main result

When Σ = {a} and Γ = {a}:

Theorem (Elgot, Mezei - 1965)

1-way transducers = = the class of rational relations . T h i s t a l k 2

• w

a y t r a n s d u c e r s H

• R

a t r e l a t i

• n

s

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Main result

When Σ = {a} and Γ = {a}:

Theorem (Elgot, Mezei - 1965)

1-way transducers = = the class of rational relations . T h i s t a l k 2

• w

a y t r a n s d u c e r s H

• R

a t r e l a t i

• n

s

Proof

◮ ⊇: easy ◮ ⊆: difficult part

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Known results

◮ 2-way functional =

= MSO definable functions [Engelfriet, Hoogeboom - 2001]

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Known results

◮ 2-way functional =

= MSO definable functions [Engelfriet, Hoogeboom - 2001]

◮ 2-way general incomparable MSO definable relations

[Engelfriet, Hoogeboom - 2001]

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Known results

◮ 2-way functional =

= MSO definable functions [Engelfriet, Hoogeboom - 2001]

◮ 2-way general incomparable MSO definable relations

[Engelfriet, Hoogeboom - 2001]

◮ 1-way simulation of 2-way functional transducer:

decidable and constructible [Filiot et al. - 2013]

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Known results

◮ 2-way functional =

= MSO definable functions [Engelfriet, Hoogeboom - 2001]

◮ 2-way general incomparable MSO definable relations

[Engelfriet, Hoogeboom - 2001]

◮ 1-way simulation of 2-way functional transducer:

decidable and constructible [Filiot et al. - 2013] When Γ = {a}:

◮ 2-way unambiguous −

→ 1-way [Anselmo - 1990]

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Known results

◮ 2-way functional =

= MSO definable functions [Engelfriet, Hoogeboom - 2001]

◮ 2-way general incomparable MSO definable relations

[Engelfriet, Hoogeboom - 2001]

◮ 1-way simulation of 2-way functional transducer:

decidable and constructible [Filiot et al. - 2013] When Γ = {a}:

◮ 2-way unambiguous −

→ 1-way [Anselmo - 1990]

◮ 2-way unambiguous =

= 2-way deterministic [Carnino, Lombardy - 2014]

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From H-Rat to 2-way transducers (unary case)

Property

The family of relations accepted by 2-way transducers is closed under ∪ ,

H and H⋆ .

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From H-Rat to 2-way transducers (unary case)

Property

The family of relations accepted by 2-way transducers is closed under ∪ ,

H and H⋆ .

Proof.

◮ R1 ∪ R2:

◮ simulate T1 or T2 10 / 15

From H-Rat to 2-way transducers (unary case)

Property

The family of relations accepted by 2-way transducers is closed under ∪ ,

H and H⋆ .

Proof.

◮ R1 ∪ R2:

◮ simulate T1 or T2

◮ R1 H R2:

◮ simulate T1 ◮ rewind the input tape ◮ simulate T2 10 / 15

From H-Rat to 2-way transducers (unary case)

Property

The family of relations accepted by 2-way transducers is closed under ∪ ,

H and H⋆ .

Proof.

◮ R1 ∪ R2:

◮ simulate T1 or T2

◮ R1 H R2:

◮ simulate T1 ◮ rewind the input tape ◮ simulate T2

◮ RH⋆:

◮ repeat an arbitrary

number of times:

◮ simulate T ◮ rewind the input tape ◮ reach the right endmarker

and accept

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From H-Rat to 2-way transducers (unary case)

Property

The family of relations accepted by 2-way transducers is closed under ∪ ,

H and H⋆ .

Corollary

H-Rat ⊆ accepted by 2-way transducers

 

0≤i≤n

Ai H BH⋆

i

 

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From 2-way transducers to H-Rat (unary case)

A first ingredient, a preliminary result:

Lemma

With arbitrary Σ and Γ = {a}: H-Rat is closed under ∪ ,

H and H⋆ .

Proof.

Tedious formal computations. . .

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From 2-way transducers to H-Rat (unary case)

We fix a transducer T .

◮ Consider border to border run segments;

⊲

u

⊳ q1 q2 q3 q4

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From 2-way transducers to H-Rat (unary case)

We fix a transducer T .

◮ Consider border to border run segments;

⊲

u

⊳ q1 q2 q3 q4 R1 = {(u, v1)} R2 = {(u, v2)} R3 = {(u, v3)}

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From 2-way transducers to H-Rat (unary case)

We fix a transducer T .

◮ Consider border to border run segments; ◮ Compose border to border segments;

⊲

u

⊳ q1 q2 q3 q4 R1 = {(u, v1)} R2 = {(u, v2)} R3 = {(u, v3)}

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From 2-way transducers to H-Rat (unary case)

We fix a transducer T .

◮ Consider border to border run segments; ◮ Compose border to border segments;

⊲

u

⊳ q1 q2 q3 q4 R1 = {(u, v1)} R2 = {(u, v2)} R3 = {(u, v3)} R1 H R2 H R3 = {(u, v1v2v3)}

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From 2-way transducers to H-Rat (unary case)

We fix a transducer T .

◮ Consider border to border run segments; ◮ Compose border to border segments; ◮ Conclude using the closure properties of H-Rat.

⊲

u

⊳ q1 q2 q3 q4 R1 = {(u, v1)} R2 = {(u, v2)} R3 = {(u, v3)} R1 H R2 H R3 = {(u, v1v2v3)}

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From 2-way transducers to H-Rat (unary case)

q1 q2 ⊲

u

⊳ define a relation R bi , bj

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From 2-way transducers to H-Rat (unary case)

q1 q2 ⊲

u

⊳ define a relation R bi , bj

Q × {⊲, ⊳}

∈ ∈

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From 2-way transducers to H-Rat (unary case)

q1 q2 ⊲

u

⊳ define a relation R bi , bj

Q × {⊲, ⊳}

∈ ∈ R0,0 R0,1 · · · R0,2 R1,0 R1,1 · · · R1,2 · · · Ri,j · · · Rk,0 Rk,1 · · · Rk,k

                         

HIT =

2 |Q| 2 |Q|

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ .

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ).

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ). ⊲ a a a a a a a a a a ⊳

q q

• utput

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ). ⊲ a a a a a a a a a a ⊳ q rational output

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ). By closure property:

Corollary

Each entry of HIT H⋆ is in H-Rat.

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ). By closure property:

Corollary

Each entry of HIT H⋆ is in H-Rat.

Remark

The relation accepted by T is a union of entries of HIT H⋆.

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From 2-way transducers to H-Rat (unary case)

Second ingredient: The behavior of T is given by the matrix HIT H⋆ . Third ingredient:

Lemma

Each entry Rb1,b2 of the matrix HIT is rational ( constructible ). By closure property:

Corollary

Each entry of HIT H⋆ is in H-Rat.

Remark

The relation accepted by T is a union of entries of HIT H⋆.

Corollary

accepted by 2-way transducers ⊆ H-Rat

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations.

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows:

◮ 2-way transducers can be made sweeping .

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows:

◮ 2-way transducers can be made sweeping .

With only Γ = {a}:

◮ 2-way

    

deterministic unambiguous functional

    

accept rational relations.

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows:

◮ 2-way transducers can be made sweeping .

With only Γ = {a}:

◮ 2-way

    

deterministic unambiguous functional

    

accept rational relations.

◮ 2-way transducers are uniformizable by 1-way transducers.

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows:

◮ 2-way transducers can be made sweeping .

With only Γ = {a}:

◮ 2-way

    

deterministic unambiguous functional

    

accept rational relations.

◮ 2-way transducers are uniformizable by 1-way transducers.

Every thing is constructible .

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Conclusion

Theorem

When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows:

◮ 2-way transducers can be made sweeping .

With only Γ = {a}:

◮ 2-way

    

deterministic unambiguous functional

    

accept rational relations.

◮ 2-way transducers are uniformizable by 1-way transducers.

Every thing is constructible . Thank you for your attention. 15 / 15