Continuity &RationalFunctions

Cadilhac, Carton,Paperman

Continuity & Rational FunctionsA Theory of Composability

Michaël Cadilhac, Olivier Carton, Charles Paperman

1 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if

(∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)

[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈

{J ,R,L,DA,A,AB,Gsol,G, COM}I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,

AB,Gsol,G, COM}I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G,

COM}I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

One definition, and contributions

V a class of languages; notion of “functions preserving V”

Definition (V-continuity)

τ : A∗ → A∗ is V-continuous if (∀L ∈ V)[τ−1(L) ∈ V]

Contributions

I Decidable given transducer whether continuous for:V ∈ {J ,R,L,DA,A,AB,Gsol,G, COM}

I Study when structure and continuity are related

I Study when continuity propagated to sub/superclass

2 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Contents

The Many Faces of Continuity

Structure vs Continuity

Conclusion

3 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Outline

The Many Faces of Continuity

Structure vs Continuity

Conclusion

3 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:

I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}

I F is closed under compositionis the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 1: A big box of functions with no new χL

I Languages are transductions: χL : A∗ → {0, 1}

I Closure of {χL | L ∈ V} under composition uninteresting

Theorem

The largest class F of transductions s.t.:I F ∩ {0, 1}A∗ = {χL | L ∈ V}I F is closed under composition

is the class of V-continuous functions

4 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L

if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L

if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)

[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)

[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from L

impliesτ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 2: Functions that do not help separation

K is V-separable from L if (∃S ∈ V)[K ⊆ S ∧ L ∩ S = ∅]

Theorem

τ is V-continuous iff

∀K , L ⊆ A∗

K not V-separable from Limplies

τ(K ) not V-separable from τ(L)

5 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V),

(∀s, t),

if n big enough then[

s·

wn

· t

∈ L⇔

s·

wn+1

· t

∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:

(∀L ∈ V),

(∀s, t),

if n big enough then[

s·

wn

· t

∈ L⇔

s·

wn+1

· t

∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V),

(∀s, t),

if n big enough then

[

s·

wn

· t

∈ L⇔

s·

wn+1

· t

∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V),

(∀s, t),

if n big enough then[

s·

wn

· t

∈ L⇔

s·

wn+1

· t

∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”

I ~w =V ~w′ iff (w1,w

′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”

I ~w =V ~w′ iff (w1,w

′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REG

I G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REG

I G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REG

I G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REG

I G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REG

I G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Examples

I For x word, xω = (x1!, x2!, x3!, . . .) fools REGI G: Largest class verifying xω =G (ε, ε, . . .)

I AB: Largest class verifying xω =AB (ε, ε, . . .) and(ab, ab, . . .) =AB (ba, ba, . . .)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Lemma (Preservation)

τ is V-continuous iff

~w =V ~w′ ⇒ τ(

s ·

~w

· t

) =V τ(

s ·

~w ′

· t

)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Lemma (Preservation)

τ is V-continuous iff~w =V ~w

′ ⇒

τ(

s ·

~w

· t

) =V τ(

s ·

~w ′

· t

)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Lemma (Preservation)

τ is V-continuous iff~w =V ~w

′ ⇒ τ(

s ·

~w

· t

) =V τ(

s ·

~w ′

· t

)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Lemma (Preservation)

τ is V-continuous iff~w =V ~w

′ ⇒ τ(s · ~w · t) =V τ(s · ~w ′ · t)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Face 3: Preserving indistinguishability of behaviors

Definition (Profinite approach)

I Sequence of words (w1,w2, . . .) fools V:(∀L ∈ V), (∀s, t), if n big enough then

[s·wn · t ∈ L⇔ s·wn+1 · t ∈ L]

I Sequences that fool REG = “profinite words”I ~w =V ~w

′ iff (w1,w′1,w2,w

′2, . . .) fools V

Lemma (Preservation)

τ is V-continuous iff~w =V ~w

′ ⇒ τ(s · ~w · t) =V τ(s · ~w ′ · t)

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Outline

The Many Faces of Continuity

Structure vs Continuity

Conclusion

6 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:

I Transducer is structurally V:“Computing it is like recognizing a V language”

I Transducer is continuous for V:“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”

I Transducer is continuous for V:“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languages

I For any ~w =V ~w′, if n big enough then

q.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languages

I For any ~w =V ~w′, if n big enough then

q.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′,

if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough then

q.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Transducer structure

Previous studies (Filiot et al., 2016) focus on structure:I Transducer is structurally V:

“Computing it is like recognizing a V language”I Transducer is continuous for V:

“The function is compatible with V languages”

Definition (τ is structurally V)

I Transition monoid of τ recognizes only V-languagesI For any ~w =V ~w

′, if n big enough thenq.wn = q.w ′n for all states q

7 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does structure imply continuity?“If τ is computed like a V-language, is it compatible with V?”

YES, for A,Gsol,G

NO, for J ,L,R,DA,AB,Gnil, COM

8 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Does continuity imply structure?“If τ is compatible with V, is it computed like a V-language?”

YES, for group varieties containing Gnil

NO, for J ,L,R,DA,A,AB, COM

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Outline

The Many Faces of Continuity

Structure vs Continuity

Conclusion

9 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with V

I Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structure

I Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classes

I When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?

I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked at

I Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Conclusion

We saw:I Robust notion of “functional compatibility” with VI Semantical notion unrelated with transducer structureI Techniques rooted in the “profinite approach”

We did not see (but you can!):I Decidability of continuity for a wealth of classesI When continuity propagates to sub/superclass

Foreseeable future:I Can we uniformize decidability proofs?I Could not crack all classes V we looked atI Study “(∀L ∈ V)[τ−1(L) ∈ W]”

10 / 11

Continuity &RationalFunctions

Cadilhac, Carton,Paperman

V-continuity∀L ∈ V,τ−1(L) ∈ V

~w fools V∀L ∈ V,ultimately alls ·wn ·t in/out L

~w =V ~w′

(w1,w′1,w2,w

′2,

. . .) fools V

ω-termxω = (xn!)n>0

Preservationτ V-cont. iff~w =V ~w′ ⇒τ(~w)=V τ(~w

′)

Thank you, merci!

The Many Faces of ContinuityContinuity = No new char func

= No help for separation= Preserving equations

Structure vs ContinuityTransducer structure & continuity unrelated in general

Conclusion

Cadilhac, Krebs, Ludwig, and Paperman (2015).A circuit complexity approach to transductions.In MFCS, pages 141–153.

Filiot, Gauwin, and Lhote (2016).First-order definability of rational transductions: An algebraic approach.In LICS, pages 387–396. ACM.

Pin and Silva (2017).On uniformly continuous functions for some profinite topologies.Theoretical Computer Science, 658, Part A:246 – 262.Formal Languages and Automata: Models, Methods and Application In honour of the70th birthday of Antonio Restivo.

Reutenaeur and Schützenberger (1995).Variétés et fonctions rationnelles.Theoretical Computer Science, 145(1–2):229–240.

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