Introduction Transducers Logic Algebra New logic Summary
Automata, Logic and Algebra for(Finite) Word Transductions
Emmanuel Filiot
Universite libre de Bruxelles & FNRS
ACTS 2017, Chennai
1 / 31
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
2 / 31
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
2 / 31
Introduction Transducers Logic Algebra New logic Summary
Objective of the talk
Automata Logic
Algebra
Transductions
f : Σ∗ → Σ∗
? ?
?
3 / 31
Introduction Transducers Logic Algebra New logic Summary
Automata models for transductions
4 / 31
Introduction Transducers Logic Algebra New logic Summary
Automata for transductions: transducers
fdel :
b:ε b:εa:a
a:a
aabaa 7→ aaaa
aaba 7→ undefined
dom(fdel) = ’even number of a’
5 / 31
Introduction Transducers Logic Algebra New logic Summary
Automata for transductions: transducers
fdel :
b:ε b:εa:a
a:a
aabaa 7→ aaaa
aaba 7→ undefined
dom(fdel) = ’even number of a’
5 / 31
Introduction Transducers Logic Algebra New logic Summary
Automata for transductions: transducers
fdel :
b:ε b:εa:a
a:a
aabaa 7→ aaaa
aaba 7→ undefined
dom(fdel) = ’even number of a’
5 / 31
Introduction Transducers Logic Algebra New logic Summary
Automata for transductions: transducers
fdel :
b:ε b:εa:a
a:a
aabaa 7→ aaaa
aaba 7→ undefined
dom(fdel) = ’even number of a’
5 / 31
Introduction Transducers Logic Algebra New logic Summary
Non-determinism
In general, transducers define binary relations in Σ∗ × Σ∗
σ:ε
σ:σ
realizes {(u, v) | v is a subword of u}
6 / 31
Introduction Transducers Logic Algebra New logic Summary
Sequential vs Non-deterministic functionalNon-deterministic transducers may define functions:
fsw :
qa
qb
for all σ ∈ Σ
σ:σ
σ:σ
σ:aσ
σ:bσ
a:ε
b:ε
babaa 7→ ababa
uσ 7→ σu |u| ≥ 1
input-determinism (aka sequential) < non-determinism ∩ functions
7 / 31
Introduction Transducers Logic Algebra New logic Summary
Sequential vs Non-deterministic functionalNon-deterministic transducers may define functions:
fsw :
qa
qb
for all σ ∈ Σ
σ:σ
σ:σ
σ:aσ
σ:bσ
a:ε
b:ε
babaa 7→ ababa
uσ 7→ σu |u| ≥ 1
input-determinism (aka sequential) < non-determinism ∩ functions
7 / 31
Introduction Transducers Logic Algebra New logic Summary
Sequential vs Non-deterministic functionalNon-deterministic transducers may define functions:
fsw :
qa
qb
for all σ ∈ Σ
σ:σ
σ:σ
σ:aσ
σ:bσ
a:ε
b:ε
babaa 7→ ababa
uσ 7→ σu |u| ≥ 1
input-determinism (aka sequential) < non-determinism ∩ functions7 / 31
Introduction Transducers Logic Algebra New logic Summary
Determinizability
= white space
0 12
a:a :ε:
a:a
:ε
:ε
aa a 7→ aa a
Is non-determinism needed ? No.
3 4
a:a :ε:ε
a: a
8 / 31
Introduction Transducers Logic Algebra New logic Summary
Determinizability
= white space
0 12
a:a :ε:
a:a
:ε
:ε
aa a 7→ aa a
Is non-determinism needed ? No.
3 4
a:a :ε:ε
a: a
8 / 31
Introduction Transducers Logic Algebra New logic Summary
Determinizability
= white space
0 12
a:a :ε:
a:a
:ε
:ε
aa a 7→ aa a
Is non-determinism needed ?
No.
3 4
a:a :ε:ε
a: a
8 / 31
Introduction Transducers Logic Algebra New logic Summary
Determinizability
= white space
0 12
a:a :ε:
a:a
:ε
:ε
aa a 7→ aa a
Is non-determinism needed ? No.
3 4
a:a :ε:ε
a: a
8 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
`
s
s t
t
r
r
e
e
s
s
s
s
e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s t
t r
r
e
e
s
s
s
s
e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t r
r e
e
s
s
s
s
e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r e
e s
s
s
s
e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e s
s s
s
e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s s
s e
e
d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s
s
s e
e d
d
a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s
s
s
e
e d
d a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s
s
s
e
e
d
d a
a
dd ee ss ss ee rr tt s
1
1 2
2
3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s
s
s
e
e d
d a
a
dd ee ss ss ee rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s
s
s e
e d
d
a
a
d
d ee ss ss ee rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e
s
s s
s e
e
d
d
a
a
d
d e
e ss ss ee rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r
e
e s
s s
s
e
e
d
d
a
a
d
d
e
e s
s ss ee rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t
r
r e
e s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s s
s ee rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s
t
t r
r e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s e
e rr tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
` s
s t
t r
r
e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s
e
e r
r tt s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
`
s
s t
t
r
r
e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s
e
e
r
r t
t s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
`
s
s
t
t
r
r
e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s
e
e
r
r
t
t s
1
1 2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
`
`
s
s
t
t
r
r
e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s
e
e
r
r
t
t s
1
1
2
2 3
3
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Two-way transducers
input
output
` s
s
t
t
r
r
e
e
s
s
s
s
e
e
d
d
a
a
d
d
e
e
s
s
s
s
e
e
r
r
t
t s
1
1
2
2
33
σ:ε,→
a:ε,←
σ:σ,←
`:ε
one-way < two-way
© decidable equivalence problem (Culik, Karhumaki, 87).
© closed under composition ◦ (Chytil, Jakl, 77)
9 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classes
valu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classes
valu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classes
valu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classesvalu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classesvalu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classesvalu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classesvalu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Landscape of Transducer Classesvalu
edn
ess
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
Figure: A landscape of transducers of finite words.
10 / 31
Introduction Transducers Logic Algebra New logic Summary
Other recent results
Transducers with registersX
σ | X := σX
mirror
Y X
σ
∣∣∣∣∣∣ X := σX
Y := Y σ
id.mirrorI deterministic one-way
I equivalent to 2DFT if copyless updates (Alur, Cerny, 10)
I decidable equivalence problem (F., Reynier) ∼ HDT0L
I regular expressions to register transducer, implemented inDReX (Alur, D’Antoni, Raghothaman, 2015)
I register minimization for a subclass (Baschenis, Gauwin,Muscholl, Puppis, 16)
Two-way to one-way transducers
I decidable, but non-elementary complexity in (FGRS13)
I elementary complexity first obtained for subclasses(sweeping) by (BGMP15)
I recently for the full class (BGMP17)
11 / 31
Introduction Transducers Logic Algebra New logic Summary
Other recent results
Transducers with registersX
σ | X := σX
mirror
Y X
σ
∣∣∣∣∣∣ X := σX
Y := Y σ
id.mirrorI deterministic one-way
I equivalent to 2DFT if copyless updates (Alur, Cerny, 10)
I decidable equivalence problem (F., Reynier) ∼ HDT0L
I regular expressions to register transducer, implemented inDReX (Alur, D’Antoni, Raghothaman, 2015)
I register minimization for a subclass (Baschenis, Gauwin,Muscholl, Puppis, 16)
Two-way to one-way transducers
I decidable, but non-elementary complexity in (FGRS13)
I elementary complexity first obtained for subclasses(sweeping) by (BGMP15)
I recently for the full class (BGMP17)
11 / 31
Introduction Transducers Logic Algebra New logic Summary
Other recent results
Transducers with registersX
σ | X := σX
mirror
Y X
σ
∣∣∣∣∣∣ X := σX
Y := Y σ
id.mirrorI deterministic one-way
I equivalent to 2DFT if copyless updates (Alur, Cerny, 10)
I decidable equivalence problem (F., Reynier) ∼ HDT0L
I regular expressions to register transducer, implemented inDReX (Alur, D’Antoni, Raghothaman, 2015)
I register minimization for a subclass (Baschenis, Gauwin,Muscholl, Puppis, 16)
Two-way to one-way transducers
I decidable, but non-elementary complexity in (FGRS13)
I elementary complexity first obtained for subclasses(sweeping) by (BGMP15)
I recently for the full class (BGMP17) 11 / 31
Introduction Transducers Logic Algebra New logic Summary
Logic for transductions
12 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
(Courcelle) MSO Transformations“interpreting the output structure in the input structure”
I output predicates defined by MSO[S] formulas interpretedover the input structure
s t r e s s e d
S S S S S S S
SSSSSSS
φS(x, y) ≡ S(y, x)
φσ(x) ≡ σ(x)
I input structure can be copied a fixed number of times:u 7→ uu, or u 7→ u.mirror(u).
13 / 31
Introduction Transducers Logic Algebra New logic Summary
Buchi 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,jS (x, y) |= x � y.
14 / 31
Introduction Transducers Logic Algebra New logic Summary
Buchi 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,jS (x, y) |= x � y.
14 / 31
Introduction Transducers Logic Algebra New logic Summary
Buchi 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,jS (x, y) |= x � y.
14 / 31
Introduction Transducers Logic Algebra New logic Summary
First-order transductions
Replace MSO by FO formulas.
Results
I equivalent to aperiodic transducers with registers (F.,Trivedi, Krishna S., 14)
I and to aperiodic 2DFT (Carton, Dartois, 15) (Dartois,Jecker, Reynier, 16)
15 / 31
Introduction Transducers Logic Algebra New logic Summary
Algebraic characterizations oftransductions
16 / 31
Introduction Transducers Logic Algebra New logic Summary
Myhill-Nerode congruence for L ⊆ Σ∗
I u ∼L v if: for all w ∈ Σ∗, uw ∈ L iff vw ∈ LI u and v have the same “effect” on continuations w
I Myhill-Nerode’s Thm: L is regular iff Σ∗/∼L is finite
I canonical (and minimal) deterministic automaton for L,Σ∗/∼L as set of states
GoalExtend Myhill-Nerode’s theorem to classes of transductions
17 / 31
Introduction Transducers Logic Algebra New logic Summary
Myhill-Nerode congruence for L ⊆ Σ∗
I u ∼L v if: for all w ∈ Σ∗, uw ∈ L iff vw ∈ LI u and v have the same “effect” on continuations w
I Myhill-Nerode’s Thm: L is regular iff Σ∗/∼L is finite
I canonical (and minimal) deterministic automaton for L,Σ∗/∼L as set of states
GoalExtend Myhill-Nerode’s theorem to classes of transductions
17 / 31
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) v2.2 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. domainmembership 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]
18 / 31
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) v2.2 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. domainmembership 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]
18 / 31
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) v2.2 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. domainmembership 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] 18 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
b
b
b
b
b
b
b
b
a
ε
b
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
b
b
b
b
b
b
b
b
aεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
b
b
b
b
b
b
bbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
b
b
b
b
bbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
b
b
bbbbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
b
a
b
b
b
b
b
a
b
a
b
a
b
a
b
bbbbbbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
babbbbbababababbbbbbbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
babbbbbababababbbbbbbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Rational transductions are almost sequential
I fsw : uσ 7→ σu is not sequential
I but sequential modulo look-ahead information I = {a, b, ε}.
babbbbbababababbbbbbbbbaεb
aσ:aσ
bσ:bσ
aσ:σ
bσ:σ
εa:ε
εb:ε
I look-ahead information: L : Σ∗ → II f [L]: f with input words extended with look-ahead
information
19 / 31
Introduction Transducers Logic Algebra New logic Summary
Results
Theorem (Elgot, Mezei, 65)
f is rational iff f [L] is sequential, for some finite look-aheadinformation L computable by a right sequential transducer.
Original statement: RAT = SEQ ◦RightSEQ.
Reutenauer, Schutzenberger, 91
I canonical look-ahead given by a congruence ≡fI identify suffixes with a ’bounded’ effect on the transduction
of prefixes
I characterization of rational transductionsI f is rationalI ≡f has finite index and f [≡f ] is sequentialI ≡f and ∼f [≡f ] have finite index.
20 / 31
Introduction Transducers Logic Algebra New logic Summary
Results
Theorem (Elgot, Mezei, 65)
f is rational iff f [L] is sequential, for some finite look-aheadinformation L computable by a right sequential transducer.
Original statement: RAT = SEQ ◦RightSEQ.
Reutenauer, Schutzenberger, 91
I canonical look-ahead given by a congruence ≡fI identify suffixes with a ’bounded’ effect on the transduction
of prefixes
I characterization of rational transductionsI f is rationalI ≡f has finite index and f [≡f ] is sequentialI ≡f and ∼f [≡f ] have finite index.
20 / 31
Introduction Transducers Logic Algebra New logic Summary
Definability problems
Rational TransductionsGiven f defined by T , is it definable by some C-transducer ?
I sufficient conditions on C to get decidability (F., Gauwin,Lhote, LICS’16)
I includes aperiodic congruences: decidable FO-definability
I even PSpace-c (F., Gauwin, Lhote, FSTTCS’16)
Regular Transductions
I existence of a canonical transducer if origin is taken intoaccount (Bojanczyk, ICALP’14)
a aa aa aa 7→ 6=a aa aa aa 7→
I decidable FO-definability with origin, open without
21 / 31
Introduction Transducers Logic Algebra New logic Summary
A new logic for transductionsjoint with Luc Dartois and Nathan Lhote
22 / 31
Introduction Transducers Logic Algebra New logic Summary
23 / 31
Introduction Transducers Logic Algebra New logic Summary
24 / 31
Introduction Transducers Logic Algebra New logic Summary
25 / 31
Introduction Transducers Logic Algebra New logic Summary
26 / 31
Introduction Transducers Logic Algebra New logic Summary
27 / 31
Introduction Transducers Logic Algebra New logic Summary
28 / 31
Introduction Transducers Logic Algebra New logic Summary
Summary: sequential transductions
Automata Logic
Algebra
Sequential
Transductions
input-deterministic
one-way transducersprefix-independent MSO, ?
finiteness of ∼f (Choffrut)
29 / 31
Introduction Transducers Logic Algebra New logic Summary
Summary: rational transductions
Automata Logic
Algebra
Rational
Transductions
one-way transducersorder-preserving MSO
finiteness of ≡f and ∼f [≡f ] (Reutenauer, Schutzenberger)
29 / 31
Introduction Transducers Logic Algebra New logic Summary
Summary: regular transductions
Automata Logic
Algebra
Regular
Transductions
deterministic two-way transducers(Courcelle) MSO
??
29 / 31
Introduction Transducers Logic Algebra New logic Summary
Other works
I AC0 transductions (Cadilhac,Krebs,Ludwig,Paperman,15)
I variants of two-way transducers (Guillon, Choffrut,14,15,16), (Carton, 12) (McKenzie, Schwentick, Therien,Vollmer, 06)
I model-checking and synthesis problems for rationaltransductions with “similar origins” (F., Jecker, Loding,Winter, 16)
I non-determinism
I infinite words, nested words, trees
30 / 31
Introduction Transducers Logic Algebra New logic Summary
SIGLOG News 9th
Thank You.
31 / 31
Introduction Transducers Logic Algebra New logic Summary
SIGLOG News 9th
Thank You.
31 / 31
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