Automata, Logic and Algebra for (Finite) Word Transductions - - PowerPoint PPT Presentation

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Automata, Logic and Algebra for (Finite) Word Transductions - - PowerPoint PPT Presentation

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Introduction Transducers Logic Algebra New logic Summary Automata, Logic and Algebra for (Finite) Word Transductions Emmanuel Filiot Universit e libre de Bruxelles & FNRS ACTS 2017, Chennai 1 / 31 Introduction Transducers Logic

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Introduction Transducers Logic Algebra New logic Summary

Automata, Logic and Algebra for (Finite) Word Transductions

Emmanuel Filiot

Universit´ e libre de Bruxelles & FNRS

ACTS 2017, Chennai

1 / 31

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Introduction Transducers Logic Algebra New logic Summary

Trinity for Regular Languages

Automata Logic Algebra

Regular languages L ⊆ Σ∗

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Introduction Transducers Logic Algebra New logic Summary

Trinity for Regular Languages

Automata Logic Algebra

Regular languages L ⊆ Σ∗ DFA=NFA=2DFA=2NFA MSO[S] Finite monoids

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Introduction Transducers Logic Algebra New logic Summary

Objective of the talk

Automata Logic Algebra

Transductions f : Σ∗ → Σ∗

? ? ?

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Introduction Transducers Logic Algebra New logic Summary

Automata models for transductions

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Introduction Transducers Logic Algebra New logic Summary

Automata for transductions: transducers

fdel : b:ǫ b:ǫ a:a a:a

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Introduction Transducers Logic Algebra New logic Summary

Automata for transductions: transducers

fdel : b:ǫ b:ǫ a:a a:a aabaa → aaaa

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Introduction Transducers Logic Algebra New logic Summary

Automata for transductions: transducers

fdel : b:ǫ b:ǫ a:a a:a aabaa → aaaa aaba → undefined

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Introduction Transducers Logic Algebra New logic Summary

Automata for transductions: transducers

fdel : b:ǫ b:ǫ a:a a:a aabaa → aaaa aaba → undefined dom(fdel) = ’even number of a’

5 / 31

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Introduction Transducers Logic Algebra New logic Summary

Non-determinism

In general, transducers define binary relations in Σ∗ × Σ∗ σ:ǫ σ:σ realizes {(u, v) | v is a subword of u}

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Introduction Transducers Logic Algebra New logic Summary

Sequential vs Non-deterministic functional

Non-deterministic transducers may define functions: fsw : qa qb for all σ ∈ Σ σ:σ σ:σ σ:aσ σ:bσ a:ǫ b:ǫ

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SLIDE 12

Introduction Transducers Logic Algebra New logic Summary

Sequential vs Non-deterministic functional

Non-deterministic transducers may define functions: fsw : qa qb for all σ ∈ Σ σ:σ σ:σ σ:aσ σ:bσ a:ǫ b:ǫ babaa → ababa

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SLIDE 13

Introduction Transducers Logic Algebra New logic Summary

Sequential vs Non-deterministic functional

Non-deterministic transducers may define functions: fsw : qa qb for all σ ∈ Σ σ:σ σ:σ σ:aσ σ:bσ a:ǫ b:ǫ babaa → ababa uσ → σu |u| ≥ 1 input-determinism (aka sequential) < non-determinism ∩ functions

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Introduction Transducers Logic Algebra New logic Summary

Determinizability

= white space 1 2 a:a :ǫ : a:a :ǫ :ǫ

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Introduction Transducers Logic Algebra New logic Summary

Determinizability

= white space 1 2 a:a :ǫ : a:a :ǫ :ǫ aa a → aa a

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Introduction Transducers Logic Algebra New logic Summary

Determinizability

= white space 1 2 a:a :ǫ : a:a :ǫ :ǫ aa a → aa a Is non-determinism needed ?

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Introduction Transducers Logic Algebra New logic Summary

Determinizability

= white space 1 2 a:a :ǫ : a:a :ǫ :ǫ aa a → aa a Is non-determinism needed ? No. 3 4 a:a :ǫ :ǫ a: a

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Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

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SLIDE 19

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 20

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

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SLIDE 21

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

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SLIDE 22

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 23

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 24

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 25

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 26

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 27

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 28

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 29

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

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SLIDE 30

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 31

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 32

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s e 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 33

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s e r 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 34

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s e r t 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 35

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s e r t s 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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SLIDE 36

Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

⊢ s t r e s s e d ⊣ d e s s e r t s 1 2 3 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

9 / 31

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Introduction Transducers Logic Algebra New logic Summary

Two-way transducers

input

  • utput

s t r e s s e d ⊣ d e s s e r t s 1 2 σ:ǫ, → ⊣:ǫ, ← σ:σ, ← ⊢:ǫ

  • ne-way < two-way

decidable equivalence problem (Culik, Karhumaki, 87). closed under composition ◦ (Chytil, Jakl, 77)

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

expressiveness

SFTs FT 2DFT=2FT ⊂ ⊂

sequential transductions rational transductions regular transductions

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

expressiveness

SFTs FT 2DFT=2FT ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

expressiveness

SFTs FT 2DFT=2FT ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

valuedness expressiveness

SFTs FT 2DFT=2FT NFT 2NFT ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

valuedness expressiveness

SFTs FT 2DFT=2FT NFT 2NFT ⊂ ⊂ ⊂ ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

valuedness expressiveness

SFTs FT 2DFT=2FT NFT 2NFT ⊂ ⊂ ⊂ ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 PTime Schutzenberger75 GurIba83,BealCartonPS03 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

valuedness expressiveness

SFTs FT 2DFT=2FT NFT 2NFT ⊂ ⊂ ⊂ ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 PTime Schutzenberger75 GurIba83,BealCartonPS03 decidable CulKar87 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Landscape of Transducer Classes

valuedness expressiveness

SFTs FT 2DFT=2FT NFT 2NFT ⊂ ⊂ ⊂ ⊂ ⊂

sequential transductions rational transductions regular transductions PTime Chof77 WK95 BealCartonPS03 PTime Schutzenberger75 GurIba83,BealCartonPS03 decidable CulKar87 undecidable BaschenisGauwinMuschollPuppis15 decidable FGRS13

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Introduction Transducers Logic Algebra New logic Summary

Other recent results

Transducers with registers

X σ | X := σX mirror Y X σ

  • X := σX

Y := Y σ id.mirror ◮ deterministic one-way ◮ equivalent to 2DFT if copyless updates (Alur, Cerny, 10) ◮ decidable equivalence problem (F., Reynier) ∼ HDT0L

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Introduction Transducers Logic Algebra New logic Summary

Other recent results

Transducers with registers

X σ | X := σX mirror Y X σ

  • X := σX

Y := Y σ id.mirror ◮ deterministic one-way ◮ equivalent to 2DFT if copyless updates (Alur, Cerny, 10) ◮ decidable equivalence problem (F., Reynier) ∼ HDT0L ◮ regular expressions to register transducer, implemented in

DReX (Alur, D’Antoni, Raghothaman, 2015)

◮ register minimization for a subclass (Baschenis, Gauwin,

Muscholl, Puppis, 16)

11 / 31

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Introduction Transducers Logic Algebra New logic Summary

Other recent results

Transducers with registers

X σ | X := σX mirror Y X σ

  • X := σX

Y := Y σ id.mirror ◮ deterministic one-way ◮ equivalent to 2DFT if copyless updates (Alur, Cerny, 10) ◮ decidable equivalence problem (F., Reynier) ∼ HDT0L ◮ regular expressions to register transducer, implemented in

DReX (Alur, D’Antoni, Raghothaman, 2015)

◮ register minimization for a subclass (Baschenis, Gauwin,

Muscholl, Puppis, 16)

Two-way to one-way transducers

◮ decidable, but non-elementary complexity in (FGRS13) ◮ elementary complexity first obtained for subclasses

(sweeping) by (BGMP15)

◮ recently for the full class (BGMP17)

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Introduction Transducers Logic Algebra New logic Summary

Logic for transductions

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Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

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Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

s t r e s s e d S S S S S S S

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Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

s t r e s s e d S S S S S S S φS(x, y) ≡ S(y, x) φσ(x) ≡ σ(x)

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Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

s t r e s s e d S S S S S S S S S S S S S S φS(x, y) ≡ S(y, x) φσ(x) ≡ σ(x)

13 / 31

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SLIDE 54

Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

s t r e s s e d S S S S S S S φS(x, y) ≡ S(y, x) φσ(x) ≡ σ(x)

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SLIDE 55

Introduction Transducers Logic Algebra New logic Summary

(Courcelle) MSO Transformations

“interpreting the output structure in the input structure”

◮ output predicates defined by MSO[S] formulas interpreted

  • ver the input structure

s t r e s s e d S S S S S S S φS(x, y) ≡ S(y, x) φσ(x) ≡ σ(x)

◮ input structure can be copied a fixed number of times:

u → uu, or u → u.mirror(u).

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Introduction Transducers Logic Algebra New logic Summary

B¨ uchi Theorems for Word Transductions

Let f : Σ∗ → Σ∗.

Theorem (Engelfriet, Hoogeboom, 01)

f is 2FT-definable iff f is MSO-definable.

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Introduction Transducers Logic Algebra New logic Summary

B¨ uchi Theorems for Word Transductions

Let f : Σ∗ → Σ∗.

Theorem (Engelfriet, Hoogeboom, 01)

f is 2FT-definable iff f is MSO-definable. Consequence Equivalence is decidable for MSO-transducers.

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Introduction Transducers Logic Algebra New logic Summary

B¨ uchi Theorems for Word Transductions

Let f : Σ∗ → Σ∗.

Theorem (Engelfriet, Hoogeboom, 01)

f is 2FT-definable iff f is MSO-definable. Consequence Equivalence is decidable for MSO-transducers.

Theorem (Bojanczyk 14, F. 15)

f is (1)FT-definable iff f is order-preserving MSO-definable. Order-preserving MSO: φi,j

S (x, y) |

= x y.

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Introduction Transducers Logic Algebra New logic Summary

First-order transductions

Replace MSO by FO formulas.

Results

◮ equivalent to aperiodic transducers with registers (F.,

Trivedi, Krishna S., 14)

◮ and to aperiodic 2DFT (Carton, Dartois, 15) (Dartois,

Jecker, Reynier, 16)

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Introduction Transducers Logic Algebra New logic Summary

Algebraic characterizations of transductions

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Introduction Transducers Logic Algebra New logic Summary

Myhill-Nerode congruence for L ⊆ Σ∗

◮ u ∼L v if: for all w ∈ Σ∗, uw ∈ L iff vw ∈ L ◮ u and v have the same “effect” on continuations w ◮ Myhill-Nerode’s Thm: L is regular iff Σ∗/∼L is finite ◮ canonical (and minimal) deterministic automaton for L,

Σ∗/∼L as set of states

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Introduction Transducers Logic Algebra New logic Summary

Myhill-Nerode congruence for L ⊆ Σ∗

◮ u ∼L v if: for all w ∈ Σ∗, uw ∈ L iff vw ∈ L ◮ u and v have the same “effect” on continuations w ◮ Myhill-Nerode’s Thm: L is regular iff Σ∗/∼L is finite ◮ canonical (and minimal) deterministic automaton for L,

Σ∗/∼L as set of states

Goal

Extend Myhill-Nerode’s theorem to classes of transductions

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Introduction Transducers Logic Algebra New logic Summary

Sequential transductions (Choffrut)

Refinement of the MN congruence.

Two ideas

  • 1. produce asap: F(u) = LCP{f(uw) | uw ∈ dom(f)}

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SLIDE 64

Introduction Transducers Logic Algebra New logic Summary

Sequential transductions (Choffrut)

Refinement of the MN congruence.

Two ideas

  • 1. produce asap: F(u) = LCP{f(uw) | uw ∈ dom(f)}
  • 2. u ∼f v if

2.1 u ∼dom(f) v 2.2 F(u)−1f(uw) = F(v)−1f(vw) ∀w ∈ u−1dom(f)

“u and v have the same effect on continuations w w.r.t. domain membership and produced outputs”

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SLIDE 65

Introduction Transducers Logic Algebra New logic Summary

Sequential transductions (Choffrut)

Refinement of the MN congruence.

Two ideas

  • 1. produce asap: F(u) = LCP{f(uw) | uw ∈ dom(f)}
  • 2. u ∼f v if

2.1 u ∼dom(f) v 2.2 F(u)−1f(uw) = F(v)−1f(vw) ∀w ∈ u−1dom(f)

“u and v have the same effect on continuations w w.r.t. domain membership and produced outputs”

Theorem (Choffrut)

f is sequential iff ∼f has finite index ∼f is a right congruence canonical and minimal transducer ! Transitions: [u]

σ|F(u)−1F(uσ)

− − − − − − − − − → [uσ]

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Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b b b b a b

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SLIDE 67

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b b b b a

ǫ

b

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SLIDE 68

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b b b b

b

a

ǫ

b

19 / 31

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SLIDE 69

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b b b

b

b

b

a

ǫ

b

19 / 31

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SLIDE 70

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b b

b

b

b

b

b

a

ǫ

b

19 / 31

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SLIDE 71

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}.

a b b a a a a b

b

b

b

b

b

b

b

a

ǫ

b

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SLIDE 72

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}. b

a

b

b

b

b

b

a

b

a

b

a

b

a

b

b

b

b

b

b

b

b

b

a

ǫ

b

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SLIDE 73

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}. b

a

b

b

b

b

b

a

b

a

b

a

b

a

b

b

b

b

b

b

b

b

b

a

ǫ

b

a

σ:aσ

b

σ:bσ

a

σ:σ

b

σ:σ

ǫ

a:ǫ

ǫ

b:ǫ

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SLIDE 74

Introduction Transducers Logic Algebra New logic Summary

Rational transductions are almost sequential

◮ fsw : uσ → σu is not sequential ◮ but sequential modulo look-ahead information I = {a, b, ǫ}. b

a

b

b

b

b

b

a

b

a

b

a

b

a

b

b

b

b

b

b

b

b

b

a

ǫ

b

a

σ:aσ

b

σ:bσ

a

σ:σ

b

σ:σ

ǫ

a:ǫ

ǫ

b:ǫ

◮ look-ahead information: L : Σ∗ → I ◮ f[L]: f with input words extended with look-ahead

information

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Introduction Transducers Logic Algebra New logic Summary

Results

Theorem (Elgot, Mezei, 65)

f is rational iff f[L] is sequential, for some finite look-ahead information L computable by a right sequential transducer.

Original statement: RAT = SEQ ◦ RightSEQ.

20 / 31

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Introduction Transducers Logic Algebra New logic Summary

Results

Theorem (Elgot, Mezei, 65)

f is rational iff f[L] is sequential, for some finite look-ahead information L computable by a right sequential transducer.

Original statement: RAT = SEQ ◦ RightSEQ.

Reutenauer, Sch¨ utzenberger, 91

◮ canonical look-ahead given by a congruence ≡f ◮ identify suffixes with a ’bounded’ effect on the transduction

  • f prefixes

◮ characterization of rational transductions

◮ f is rational ◮ ≡f has finite index and f[≡f] is sequential ◮ ≡f and ∼f[≡f ] have finite index. 20 / 31

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Introduction Transducers Logic Algebra New logic Summary

Definability problems

Rational Transductions

Given f defined by T, is it definable by some C-transducer ?

◮ sufficient conditions on C to get decidability (F., Gauwin,

Lhote, LICS’16)

◮ includes aperiodic congruences: decidable FO-definability ◮ even PSpace-c (F., Gauwin, Lhote, FSTTCS’16)

Regular Transductions

◮ existence of a canonical transducer if origin is taken into

account (Bojanczyk, ICALP’14) a a a a a a a → = a a a a a a a →

◮ decidable FO-definability with origin, open without

21 / 31

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Introduction Transducers Logic Algebra New logic Summary

A new logic for transductions

joint with Luc Dartois and Nathan Lhote

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Introduction Transducers Logic Algebra New logic Summary

Summary: sequential transductions

Automata Logic Algebra

Sequential Transductions

input-deterministic

  • ne-way transducers

prefix-independent MSO, ? finiteness of ∼f (Choffrut)

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Introduction Transducers Logic Algebra New logic Summary

Summary: rational transductions

Automata Logic Algebra

Rational Transductions

  • ne-way transducers
  • rder-preserving MSO

finiteness of ≡f and ∼f[≡f] (Reutenauer, Sch¨ utzenberger)

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Introduction Transducers Logic Algebra New logic Summary

Summary: regular transductions

Automata Logic Algebra

Regular Transductions

deterministic two-way transducers (Courcelle) MSO ??

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Introduction Transducers Logic Algebra New logic Summary

Other works

◮ AC0 transductions (Cadilhac,Krebs,Ludwig,Paperman,15) ◮ variants of two-way transducers (Guillon, Choffrut,

14,15,16), (Carton, 12) (McKenzie, Schwentick, Th´ erien, Vollmer, 06)

◮ model-checking and synthesis problems for rational

transductions with “similar origins” (F., Jecker, L¨

  • ding,

Winter, 16)

◮ non-determinism ◮ infinite words, nested words, trees

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Introduction Transducers Logic Algebra New logic Summary

SIGLOG News 9th

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Introduction Transducers Logic Algebra New logic Summary

SIGLOG News 9th

Thank You.

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