Which classes of origin graphsare generated by transducers?

joint work by:Mikołaj Bojańczyk, Laure Daviaud, Bruno Guillon and Vincent Penelle

Uniwersytet Warszawski – Università degli Studi di Milano

London, July 10, 2017

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 / 7

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 / 7

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 / 7

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 / 7

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 / 7

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 / 7

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

3 / 7

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

3 / 7

Origin graphs

example: square function (w 7→ w · w)

l o n d o n

l o n d o n l o n d o n

origin: a mapping from output positions into input positions

3 / 7

What is a regular transduction?

Functional caseTheorem: the following models are equivalent:

2-way automata with outputsMonadic Second Order TransductionsStreaming String Transducers

[Engelfriet & Hooogeboom 2001], [Alur & Černy 2010]

Theorem: closed under composition.[Chytil & Jákl 1977], [DFJL 2017]

Remark: still true with origin information. [Bojańczyk 2014]

4 / 7

What is a regular transduction?

Functional caseTheorem: the following models are equivalent:

2-way automata with outputsMonadic Second Order TransductionsStreaming String Transducers

[Engelfriet & Hooogeboom 2001], [Alur & Černy 2010]

Theorem: closed under composition.[Chytil & Jákl 1977], [DFJL 2017]

Remark: still true with origin information. [Bojańczyk 2014]

4 / 7

Which properties of origin graphscharacterise

regular sets of origin graphs?

5 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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:

a particular path decomposition of bounded width.

6 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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:

a particular path decomposition of bounded width.

6 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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:

a particular path decomposition of bounded width.

6 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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 crossing:

a particular path decomposition of bounded width.

6 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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 crossing:a particular path decomposition of bounded width.

6 / 7

Which sets of origin graphs are generated by transducers?

Theorem:A set of origin graphs is realised by an unambiguous sstk-registers 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;

crossing bounded by k :a particular path decomposition of bounded width.

6 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7

Conclusion

uSST

NSST=MSOT

εNSST

2DFT

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

7 / 7