A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson - - PowerPoint PPT Presentation

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a first investigation of sturmian trees

A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson - - PowerPoint PPT Presentation

Mar 31, 2024 762 likes •1.18k views

A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle Fagnot 2 1 LIAFA, CNRS Universit e Paris 7 2 IGM, CNRS Universit e de Marne-la-Vall ee STACS2007, Aachen Olivier Carton (LIAFA)

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A First Investigation of Sturmian Trees

Jean Berstel2, Luc Boasson1 Olivier Carton1, Isabelle Fagnot2

1LIAFA, CNRS

Universit´ e Paris 7

2IGM, CNRS

Universit´ e de Marne-la-Vall´ ee

STACS’2007, Aachen

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 1 / 14

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Outline

1 Sturmian words 2 Sturmian trees 3 Slow automata 4 Rank and degree 5 Results

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 2 / 14

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Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 4

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 5

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 6

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 7

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 8

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 9

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 10

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 11

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 12

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aab aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 13

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba aba baa bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 14

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab bab 4 4 aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 15

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 16

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 17

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba aaba abaa abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 18

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab abab baab baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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SLIDE 19

Factors of an infinite word

A factor w of a word x is a finite word that occurs in x, that is, there are words u and y such that x = uwy.

Example (Fibonacci word)

x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n Factors # 1 a b 2 2 aa ab ba 3 3 aab aba baa bab 4 4 aaba abaa abab baab baba baba 5 5 aabaa aabab abaab ababa baaba babaa 6

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

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Sturmian words

Proposition (Hedlund & Morse)

An infinite word x ultimately periodic iff there is an integer n such that x has at most n distinct factors of length n. An infinite word x is Sturmian if the number of its factors of length n is n + 1 for each n. Sturmian words are non ultimately periodic words with the smallest complexity.

Example (Fibonacci word: fn+2 = fn+1fn)

f0 = a f1 = ab f2 = aba f3 = abaab fω = abaababa · · ·

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 4 / 14

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Characterization: cutting sequences

y = βx + ρ x = a b b a b ab b a b ba b

Theorem

A infinite word is Sturmian iff it is the cutting sequence of a straight line y = βx + ρ with an irrational slope β.

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 5 / 14

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Factor of a tree

A factor of height h of a tree t is a subtree of height h that occurs in t. Factor

f height 3

Factor

f height 2
Olivier Carton

(LIAFA) Sturmian Trees STACS’2007, Aachen 6 / 14

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Sturmian tree

Proposition (Carpi et al)

A complete tree t is rational if there is some integer h such that t has at most h distinct factors of height h. A tree is Sturmian if the number of its factors of height h is h + 1 for each h.

Example (Easy one: uniform tree)

An Sturmian word x = abaaba . . . is repeated on each branch.

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 7 / 14

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Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree D∗

2 = {ε, 01, 0101, 0011, . . .}

1 1 Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 27

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 28

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 29

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 31

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 32

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 33

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 34

Example (Unexpected one: Dyck tree)

A node is • if it is a Dyck word over the alphabet {0, 1}. The Dyck tree Its factors

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 8 / 14

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SLIDE 35

Slow automata

Let A = (Q, A, ·, q0, F) be a (infinite) deterministic automaton. Define the Moore equivalence ∼n by induction. q ∼1 q′ ⇐ ⇒ (q ∈ F ⇔ q′ ∈ F) q ∼n+1 q′ ⇐ ⇒ (q ∼n q′) and (∀a ∈ A q · a ∼n q′ · a) The relation q ∼n q′ does not hold if there is a word w of length n such that q · w ∈ F and q′ · w / ∈ F (or q · w / ∈ F and q′ · w ∈ F) An infinite automaton is slow iff each equivalence ∼n has n + 1 classes.

Proposition

A tree t is Sturmian iff the minimal automaton of t−1(a) is slow.

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 9 / 14

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SLIDE 36

Application to the Dyck tree

The minimal automaton of the Dyck language is the following. ∞ 1 2 3 . . . 0, 1 1 1 1 1 1 The Moore equivalences of this automaton ∼1: 0 | 1, 2, 3, 4, . . . ∞ ∼2: 0 | 1 | 234 . . . ∞ ∼3: 0 | 1 | 2 | 3, 4, . . . ∞ ∼4: 0 | 1 | 2 | 3 | 4, . . . ∞

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 10 / 14

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SLIDE 37

Rank and degree

A node is called irrational if the infinite subtree rooted in this node is not rational. The rank is the number of distinct rational subtrees. The degree is the number of branches of irrational nodes.

Examples

The uniform tree has rank 0 and degree ∞. The Dyck tree has rank 1 and degree ∞.

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 11 / 14

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SLIDE 38

Results

rank degree finite infinite 1 characterized example later ≥ 2, finite proved to be empty example in full paper infinite example of Dyck tree example in full paper

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 12 / 14

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Characterization

Example (Indicator tree)

Take any Sturmian word (e.g. 01001010 · · · ) and distinguish the branch labeled by this word. Rank 1 Degree 1

Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 13 / 14

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Yet another example

Example (Rank ∞ and degree 1)

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24· · · 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24· · · 1 1 1 1 1 1 1 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 14 / 14