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Breadth-first signature of trees and rational languages Victor Marsault, joint work with Jacques Sakarovitch CNRS / Telecom-ParisTech, Paris, France Developments in Language Theory 2014, Ekateringburg, 20140830 Breadth-first
Breadth-first signature of trees and rational languages
Victor Marsault, joint work with Jacques Sakarovitch
CNRS / Telecom-ParisTech, Paris, France
Developments in Language Theory 2014, Ekateringburg, 2014–08–30
Breadth-first serialisation of languages and numeration systems: The rational case
Victor Marsault, joint work with Jacques Sakarovitch
CNRS / Telecom-ParisTech, Paris, France
Developments in Language Theory 2014, Ekateringburg, 2014–08–30
1
Outline
1 Signature of trees and of languages 2 Substitutive signatures and finite automata 3 A word on numeration system
1
We call tree a...
Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)
1
We call tree a...
Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)
1
We call tree a...
Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)
1
We call tree a...
Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)
2
Every tree has a canonical breadth-first traversal
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3
Two more features
We consider infinite trees only.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3
Two more features
We consider infinite trees only. For convenience, there is loop on the root.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s =
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2 2 1
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2 2 1 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2 2 1 2 2
4
Signature of a tree
Definition
The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
s = 2 1 2 2 1 2 1 2 2 1 2 2 1 · · ·
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6 7
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6 7 8
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6 7 8 9
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6 7 8 9 10
5
The signature is characteristic of a tree
s = ( 3 2 1 )ω
1 2 3 4 5 6 7 8 9 10 11
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Prefix-closed languages and labelled trees
Alphabets are ordered hence prefix-closed languages = labelled trees.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
Figure : Integer representations in the Fibonacci numeration system.
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Prefix-closed languages and labelled trees
Alphabets are ordered hence prefix-closed languages = labelled trees.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
1 1 1 1 1 1 1
5 = F4
Figure : Integer representations in the Fibonacci numeration system.
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Prefix-closed languages and labelled trees
Alphabets are ordered hence prefix-closed languages = labelled trees.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1
1 1 1 1 1 1
7 = 5 + 2 = F4 + F2
Figure : Integer representations in the Fibonacci numeration system.
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Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = λ =
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Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
1 1 1 1 1 1 1
s = 2 λ =01
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Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = 2 1 λ =01 0
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Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1
1
1 1 1 1 1 1
s = 2 1 2 λ =01 0 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1
1
1 1 1 1 1
s = 2 1 2 2 λ =01 0 01 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = 2 1 2 2 1 λ =01 0 01 01 0
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1
1
1 1 1 1
s = 2 1 2 2 1 2 λ =01 0 01 01 0 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = 2 1 2 2 1 2 1 λ =01 0 01 01 0 01 0
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1
1
1 1 1
s = 2 1 2 2 1 2 1 2 λ =01 0 01 01 0 01 0 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1
1
1 1
s = 2 1 2 2 1 2 1 2 2 λ =01 0 01 01 0 01 0 01 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = 2 1 2 2 1 2 1 2 2 1 λ =01 0 01 01 0 01 0 01 01 0
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1
1
1
s = 2 1 2 2 1 2 1 2 2 1 2 λ =01 0 01 01 0 01 0 01 01 0 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1
1
s = 2 1 2 2 1 2 1 2 2 1 2 2 λ =01 0 01 01 0 01 0 01 01 0 01 01
7
Serialisation of a prefix-closed language
Definition
The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 1 1 1 1 1
s = 2 1 2 2 1 2 1 2 2 1 2 2 1 · · · λ =01 0 01 01 0 01 0 01 01 0 01 01 0 · · ·
8
The pair signature/labelling is characteristic
s = (3 2 1)ω λ = (012 12 1)ω
1 2 3 4 5 6 7 8 9 10 11 1 2 1 2 1 1 2 1 2 1
8
The pair signature/labelling is characteristic
s = (3 2 1)ω λ = (012 12 1)ω
1 2 3 4 5 6 7 8 9 10 11
1
2 1
2
1 1 2 1
2
1
10 = 1×22 + 2×21 + 2×20
Figure : Non-canonical integer representations in base 2.
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton.
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A word on substitution
A substitution σ is a morphism A∗ → A∗.
Running examples
Fibonacci substitution: {a, b} → {a, b}∗ a → ab b → a
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A word on substitution
A substitution σ is a morphism A∗ → A∗.
Running examples
Fibonacci substitution: {a, b} → {a, b}∗ a → ab b → a Periodic substitution: {a, b, c} → {a, b, c}∗ a → abc b → ab c → c
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A word on substitution
A substitution σ is a morphism A∗ → A∗. σ is prolongable on a if σ(a) starts with the letter a.
Running examples
Fibonacci substitution: {a, b} → {a, b}∗ a → ab b → a Periodic substitution: {a, b, c} → {a, b, c}∗ a → abc b → ab c → c
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A word on substitution
A substitution σ is a morphism A∗ → A∗. σ is prolongable on a if σ(a) starts with the letter a. In this case, σω(a) exists and is called a purely substitutive word .
Running examples
Fibonacci substitution: {a, b} → {a, b}∗ a → ab b → a Periodic substitution: {a, b, c} → {a, b, c}∗ a → abc b → ab c → c
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Substitutive signature
σ: a substitution A∗ → A∗ prolongable on a. f : a letter-to-letter morphism f (σω(a)) is called a subtitutive word.
11
Substitutive signature
σ: a substitution A∗ → A∗ prolongable on a. f : a letter-to-letter morphism f (σω(a)) is called a subtitutive word.
Definitions
let fσ be the (letter-to-letter) morphism: A∗ → N∗ defined by ∀b, fσ(b) = |σ(b)| We call fσ(σω(a)) a subtitutive signature.
11
Substitutive signature
σ: a substitution A∗ → A∗ prolongable on a. f : a letter-to-letter morphism f (σω(a)) is called a subtitutive word.
Definitions
let fσ be the (letter-to-letter) morphism: A∗ → N∗ defined by ∀b, fσ(b) = |σ(b)| We call fσ(σω(a)) a subtitutive signature. If g is a morphism such that ∀b, |g(b)| = |σ(b)| if g(b) = c0c1 · · · ck then c0 < c1 < · · · < ck We call g(σω(a)) a substitutive labelling.
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Example 1 – the Fibonacci signature
σ(a) = ab = ⇒ fσ(a) = 2 σ(b) = a = ⇒ fσ(b) = 1 fσ(σω(a)) = 2122121221221212212122 · · · if we choose g: g(a) = 01 g(b) = 0 g(σω(a)) = 01 0 01 01 0 01 0 01 01 0 01 01 0 · · ·
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Example 1 – the Fibonacci signature
σ(a) = ab = ⇒ fσ(a) = 2 σ(b) = a = ⇒ fσ(b) = 1 fσ(σω(a)) = 2122121221221212212122 · · · if we choose g: g(a) = 01 g(b) = 0 g(σω(a)) = 01 0 01 01 0 01 0 01 01 0 01 01 0 · · · This pair signature/labelling defines the language of integer representations in the Fibonacci numeration system.
13
Example 2 – a periodic signature
σ(a) = abc (fσ(a) = 3) σ(b) = ab (fσ(b) = 2) σ(c) = c (fσ(c) = 1) σ(abc) = abc abc hence fσ(σω(a)) = (321)ω If we choose g: g(a) = 012 g(b) = 12 g(c) = 1 g(σω(a)) = (012 12 1)ω
13
Example 2 – a periodic signature
σ(a) = abc (fσ(a) = 3) σ(b) = ab (fσ(b) = 2) σ(c) = c (fσ(c) = 1) σ(abc) = abc abc hence fσ(σω(a)) = (321)ω If we choose g: g(a) = 012 g(b) = 12 g(c) = 1 g(σω(a)) = (012 12 1)ω This pair signature/labelling defines a non-canonical representation
14
Example 3 – the Thue-Morse morphism
σ(a) = ab (fσ(a) = 2) σ(b) = ba (fσ(b) = 2) fσ(σω(a)) = 2ω ∀ labelling g, the language is essentially (0 + 1)∗.
15
Forward direction of the theorem
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton.
15
Forward direction of the theorem
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton. (σ, g): a substitutive signature. (σ, g) defines a finite automaton A(σ,g). It is analogous to the prefix graph/automaton in Dumont–Thomas ’89,’91,’93
15
Forward direction of the theorem
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton. (σ, g): a substitutive signature. (σ, g) defines a finite automaton A(σ,g). It is analogous to the prefix graph/automaton in Dumont–Thomas ’89,’91,’93
Proposition
The language accepted by A(σ,g) has signature (σ, g).
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b
1
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b σ( b ) = a g( a ) = 0 1 g( b ) = 0 a b 1
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c
1
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c 1
2
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c 1 2
1
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c 1 2 1
2
16
Automaton associated with a subst. signature
σ : A∗ → A∗ prolongable on a and g : A∗ → B∗ A(σ,g) = A , B, δ , {a} , A σ( a ) = a b c σ( b ) = a b σ( c ) = c g( a ) = 0 1 2 g( b ) = 1 2 g( c ) = 1 a b c 1 2 1 2
1
17
Forward direction of the theorem
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton. (σ, g): a substitutive signature. (σ, g) defines a finite automaton A(σ,g). It is analogous to the prefix graph/automaton in Dumont Thomas ’89,’91,’93
Proposition
The language accepted by A(σ,g) has signature (σ, g).
17
Forward direction of the theorem
Theorem
L: a prefix-closed language. Signature(L) is substitutive ⇔ L is accepted by a finite automaton. (σ, g): a substitutive signature. (σ, g) defines a finite automaton A(σ,g). It is analogous to the prefix graph/automaton in Dumont Thomas ’89,’91,’93
Proposition
The language accepted by A(σ,g) has signature (σ, g). Proof: unfold the automaton A(σ,g).
18
What will be in the augmented version
Abstract Numeration System: built from an arbitrary regular language. Dumont-Thomas Numeration system: built from a substitution
18
What will be in the augmented version
Abstract Numeration System: built from an arbitrary regular language. Dumont-Thomas Numeration system: built from a substitution
Theorem (augmented version)
Two (prefix-closed) ANS built on language with same signature (but different labelling) are easily† convertible one from the other.
† Through a finite, letter-to-letter and pure sequential transducer.
18
What will be in the augmented version
Abstract Numeration System: built from an arbitrary regular language. Dumont-Thomas Numeration system: built from a substitution
Theorem (augmented version)
Two (prefix-closed) ANS built on language with same signature (but different labelling) are easily† convertible one from the other.
Theorem (augmented version)
Every DTNS is a prefix-closed ANS. Every prefix-closed ARNS is easily† convertible to a DTNS.
† Through a finite, letter-to-letter and pure sequential transducer.
19
Other works: Ultimately periodic signatures
s = u rω with r = r0 r1 r2 · · · rq−1
Definition: growth ratio
gr(s) =
r0+r1+···+rq−1 q
19
Other works: Ultimately periodic signatures
s = u rω with r = r0 r1 r2 · · · rq−1
Definition: growth ratio
gr(s) =
r0+r1+···+rq−1 q
Theorem (MS, to appear)
If gr(s) ∈ N, then s generates the language of a finite automaton. It is linked‡ to the integer base b = gr(s). If gr(s) / ∈ N, then s generates a non-context-free language. It is linked‡ to the rational base p
q = gr(s). (cf. Akiyama et al. ’08) ‡ It is a non-canonical representation of the integers (using extra digits).
20
Future works : Directed signatures
Aperiodic signature: s = s0 s1 s2 · · · Sn =
1 nΣn−1 k=0sk: partial average of s.
α : lim Sn extends the notion of growth ratio.