Which classes of origin graphsare generated by transducers?

Mikołaj Bojańczyk, Laure Daviaud,Bruno Guillon and Vincent Penelle

Uniwersytet Warszawski – LaBRI

March 26, 2018

— Réunion Delta —

This work is part of a project lipa that has received funding from the European Research Council (erc) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreement No.683080).

1 / 18

Word transductions

DefinitionA transduction is a binary relation on words:

a subset of Σ∗ × Γ∗.

× ??? .

input alphabetoutput alphabet

aababbca aaaabbb

bbbaaaa

compute

origin information[Bojańczyk 2014]

2 / 18

Word transductions

DefinitionA transduction is a binary relation on words:

a subset of Σ∗ × Γ∗.

× ??? .

input alphabetoutput alphabet

aababbca aaaabbb

bbbaaaa

compute

origin information[Bojańczyk 2014]

2 / 18

Word transductions

DefinitionA transduction is a binary relation on words:

a subset of Σ∗ × Γ∗

.

× ??? .input alphabet

output alphabet

aababbca aaaabbb

bbbaaaa

compute

origin information[Bojańczyk 2014]

2 / 18

Origin graphs

a b c a b b a c b

a a a b b b b

input edge

output edge

origin edge

origin: a mapping from output positions into input positions

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What is the origin semanticof a transducer?featuring sst and msot

4 / 18

Streaming String Transducers (sst)

a 1-way automaton A

a finite set R of registers

including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

Streaming String Transducers (sst)

a 1-way automaton Aa finite set R of registers including a distinguished output register

e.g., R = X , Y a labelling of transitions by copyless register updates

e.g.,X ← X · a

Y ← ε,

X ← a · YY ← b · a · X ,

X ← bY ← X · a · Y ,

X ← X · aY ← X · Y .

Definition (Origin semantics for streaming string transducers)

origin of an output position: the position of the input headwhen the letter was created.

Example:w 7→ a|w |a · b|w |b

a b c a b b a c b a

register X : register Y :

A

a a a b b b bb b b b

output

a b c a b b a c b

a a a b b b b

5 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

aa bb cc aa bb bb aa cc bb

a a a

b b b b

a a a b b b b

copy 1

copy 2 6 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);

mso-Interpretationa formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

aa bb cc aa bb bb aa cc bb

a a a

b b b b

a a a b b b b

copy 1

copy 2 6 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

aa bb cc aa bb bb aa cc bb

a a a

b b b b

a a a b b b b

copy 1

copy 2 6 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

aa bb cc aa bb bb aa cc bb

a a a

b b b b

a a a b b b b

copy 1

copy 2 6 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

aa bb cc aa bb bb aa cc bb

a a a

b b b b

a a a b b b b

copy 1

copy 2 6 / 18

string-to-string mso-transduction [Courcelle 1991]

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

6 / 18

string-to-origin graph mso-transduction

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

a b c a b b a c b

b b b b

a a a

a a a b b b b

copy 1

copy 2 6 / 18

string-to-origin graph mso-transduction

Nondeterministic mso-colouring (nondeterministic case);Copy (finitely many copies of the input);mso-Interpretation

a formula for restricting the universe;a formula for each predicate of the output vocabulary.

Definition (origin semantics of mso-transduction)

origin of an output position: the input vertexof which it is a copy.

a b c a b b a c b

b b b b

a a a

a a a b b b b

copy 1

copy 2 6 / 18

What do we get from origin information?

7 / 18

Origin semantics is thinner grained

Examples

a a a a a

a a a a a

a a a a a

a a a a a

unary Identity

=6=a a a a a

a a a a a

a a a a a

a a a a a

unary Reverse

a b b a a b a

b a a6=

a b b a a b a

b a a

Subword

8 / 18

Origin semantics is thinner grained

Examples

a a a a a

a a a a a

a a a a a

a a a a a

unary Identity

=6=a a a a a

a a a a a

a a a a a

a a a a a

unary Reverse

a b b a a b a

b a a6=

a b b a a b a

b a a

Subword

8 / 18

Origin semantics is thinner grained

Examples

a a a a a

a a a a a

a a a a a

a a a a a

unary Identity

=6=a a a a a

a a a a a

a a a a a

a a a a a

unary Reverse

a b b a a b a

b a a6=

a b b a a b a

b a a

Subword

8 / 18

What is a regular transduction with origin?This is still true with origin information. [Bojańczyk 2014]

also true for closure under composition, decidability of equivalence. . .

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

sst: Streaming String Transducermsot: mso-transduction2FT : 2-way finite transducer

9 / 18

mso satisfiability on origin graphs

Theorem: The following is decidable :Input

an nsst Aan mso formula φ over the corresponding origin vocabulary

QuestionIs φ true in some origin graph in the origin semantics of A?

origin vocabulary: binary predicates , ,and labelling in Σ ∪ Γ;

Example“the origin mapping is bijective and letter-preserving.”

“the output may be split in two partssuch that the origin mapping is order-preserving on each part.”

10 / 18

mso satisfiability on origin graphs

Theorem: The following is decidable :Input

an nsst Aan mso formula φ over the corresponding origin vocabulary

QuestionIs φ true in some origin graph in the origin semantics of A?

Proof: Let A and φ be fixed.there is a string-to-origin graph mso-transduction ρ equivalentto A

we consider G = G origin graph | φ is true over Gby Backward Translation Theorem [Courcelle91],

ρ−1(G) is regular and can be tested for emptiness.

10 / 18

mso satisfiability on origin graphs

Theorem: The following is decidable :Input

an nsst Aan mso formula φ over the corresponding origin vocabulary

QuestionIs φ true in some origin graph in the origin semantics of A?

Proof: Let A and φ be fixed.there is a string-to-origin graph mso-transduction ρ equivalentto Awe consider G = G origin graph | φ is true over G

by Backward Translation Theorem [Courcelle91],ρ−1(G) is regular and can be tested for emptiness.

10 / 18

mso satisfiability on origin graphs

Theorem: The following is decidable :Input

an nsst Aan mso formula φ over the corresponding origin vocabulary

QuestionIs φ true in some origin graph in the origin semantics of A?

Proof: Let A and φ be fixed.there is a string-to-origin graph mso-transduction ρ equivalentto Awe consider G = G origin graph | φ is true over Gby Backward Translation Theorem [Courcelle91],

ρ−1(G) is regular and can be tested for emptiness.

10 / 18

Which properties of origin graphscharacterise

regular sets of origin graphs?

11 / 18

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstif and only if it is

mso-definable:an mso sentence using , , and labelling in Σ ∪ Γ;

functional:for each input word, there exists at most one origin graph;

bounded degree:each input position is the origin of at most m output positions;bounded crossing: next slide.

12 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the position

crossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

crossing:

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the position

crossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

crossing:

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the position

crossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 1

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 2

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 1

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 1

(≡ particular path decomposition of bounded width)

13 / 18

Crossing

crossing of an input positionnumber of maximal factors of the output

that originate in the input prefix ended by the positioncrossing of an origin graph: max of the crossings : 2

a b c a b b a c b

a a a b b b bleft1 right1left2right1 right2left1 right1 left2 right2left2 right2left1 right1left1

crossing: 1

(≡ particular path decomposition of bounded width)

13 / 18

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstif and only if it is

k-registers sst

mso-definable:an mso sentence using , , and labelling in Σ ∪ Γ;

functional:for each input word, there exists at most one origin graph;

bounded degree:each input position is the origin of at most m output positions;crossing bounded: previous slide

a b c a b b a c b

a a a b b b bleft1 left2right1 right2 14 / 18

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstif and only if it is

k-registers sst

mso-definable:an mso sentence using , , and labelling in Σ ∪ Γ;

functional:for each input word, there exists at most one origin graph;

bounded degree:each input position is the origin of at most m output positions;crossing bounded by k : previous slide

a b c a b b a c b

a a a b b b bleft1 left2right1 right2 14 / 18

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an

unambiguous

sstif and only if it is

k-registers sst

mso-definable:an mso sentence using , , and labelling in Σ ∪ Γ;

functional:for each input word, there exists at most one origin graph;

bounded degree:each input position is the origin of at most m output positions;crossing bounded by k : previous slide

a b c a b b a c b

a a a b b b bleft1 left2right1 right2 14 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degree

k-register =⇒ crossing bounded by knsst =⇒ string-to- mso-transduction

Proposition: we can inverse this mso-transduction=⇒ mso-definable

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by k

nsst =⇒ string-to- mso-transduction

Proposition: we can inverse this mso-transduction=⇒ mso-definable

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by knsst =⇒ string-to-origin graph mso-transduction

Proposition: we can inverse this mso-transduction=⇒ mso-definable

a b c a b b a c b

a b c a b b a c b

a a a b b b b

a b c a b b a c b

a a a b b b b

a a a b b b b

ρ

?

ρπ=

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by knsst =⇒ string-to-origin graph mso-transduction

Proposition: we can inverse this mso-transduction

=⇒ mso-definable

a b c a b b a c b

a b c a b b a c b

a a a b b b b

a b c a b b a c b

a a a b b b b

a a a b b b b

ρ

?

ρπ=

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by knsst =⇒ string-to-origin graph mso-transduction

Proposition: we can inverse this mso-transduction

=⇒ mso-definable

a b c a b b a c b

a b c a b b a c b

a a a b b b b

a b c a b b a c b

a a a b b b b

a a a b b b b

ρ

?

ρπ=

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by knsst =⇒ string-to-origin graph mso-transduction

Proposition: we can inverse this mso-transduction=⇒ mso-definable

a b c a b b a c b

a b c a b b a c b

a a a b b b b

a b c a b b a c b

a a a b b b b

a a a b b b b

ρ

?

ρπ=

15 / 18

Sketch of the proof =⇒unambiguous =⇒ functionalnsst =⇒ bounded degreek-register =⇒ crossing bounded by knsst =⇒ string-to-origin graph mso-transduction

Proposition: we can inverse this mso-transduction=⇒ mso-definable

Note: False when ε-transitions are allowed.

c

a a a b b b

n n

15 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

a b c a b b a c b

a a a b b b bleft1 left2right1 right2

a b c a b

a a b bleft1 left2

c

cc c

cc

ccc cc ccleft1left1 left2left1 left2

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.

there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transition

if functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Sketch of the proof ⇐=

Start with an mso-definable set of origin graphs G

with crossing bounded by k

we define a finite set of (partial) operations Ωk on k-BLOGs

the folding of a word w over Ω∗k is the k-BLOG αk(w)obtained from the empty graph by applying the operations.

αk can be realised by an mso-transduction.there exists a regular language L ⊆ Ω∗k such that

g ∈ G ⇐⇒ g = αk(w) for some w ∈ L

from an automaton recognising L,we build a nsst with ε-transitions realising G

if bounded degree =⇒ elimination of ε-transitionif functional =⇒ disambiguation

Definitionk-block origin graphs (k-BLOGs):An origin graph with output split in k identified blocks.

16 / 18

Equivalence

Corollary: The following is decidable:Input

Two nsst, A and B.Question

Whether they have the same origin semantics.

Proof: We show that we can check whether A ∩ B is empty.The origin semantics of B is mso-definable by a formula φ,We can check whether ¬φ is true in some origin graph in theorigin semantics of A.

17 / 18

Equivalence

Corollary: The following is decidable:Input

Two nsst, A and B.Question

Whether they have the same origin semantics.Proof: We show that we can check whether A ∩ B is empty.

The origin semantics of B is mso-definable by a formula φ,We can check whether ¬φ is true in some origin graph in theorigin semantics of A.

17 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18

Conclusion

fSST

NSST=MSOT

εNSST

2fFT=fMSOT

2NFT

2NFT with commonguess

=

REG

2NFT

2NFT with commonguess

2NFT with commonguess

MSO

functional

boun

ded deg

ree

bounded crossing

MSO

bounded crossingbounded crossing

(w , wn) | w ∈ Σ∗, n ∈ N(a, an) | n ∈ N(w , v) | v is a subword of w

(w , w 2)

(w , v 2) | v is a subword of w

(w , v 2) | v is a subword of w

∪(a, an) | n ∈ N

(w , vn) | v is a subword of w , n ∈ N

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

variants of1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

Thank you for your attention.

c

a a a b b b

n n

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

1 2 3 4 5 6 7 8 9

9 2 7 4 5 6 3 8 1

18 / 18