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A Hierarchy of Automatic ω-Words having a decidable MSO Theory

Vince B´ ar´ any

Mathematische Grundlagen der Informatik RWTH Aachen

Journ´ ees Montoises d’Informatique Th´ eorique Rennes, 2006

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 1 / 1

ω-Words

An ω-word over Σ is a function w : N → Σ.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 2 / 1

ω-Words

An ω-word over Σ is a function w : N → Σ. We are interested in ω-words having

◮ finite descriptions, ◮ favourable logical/algorithmic properties.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 2 / 1

ω-Words

An ω-word over Σ is a function w : N → Σ. We are interested in ω-words having

◮ finite descriptions, ◮ favourable logical/algorithmic properties.

In this talk: finite descriptions using automata.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 2 / 1

Automatic presentations of ω-Words

We associate to each word w : N → Σ its word structure Ww := (N, <, {w−1(a)}a∈Σ).

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 3 / 1

Automatic presentations of ω-Words

We associate to each word w : N → Σ its word structure Ww := (N, <, {w−1(a)}a∈Σ). An automatic presentation of a word w ∈ Σω comprises regular sets D and Pa (a ∈ Σ), a synchronized rational binary relation ≺

• ver some alphabet Γ, such that (D, ≺, {Pa}a∈Σ) ∼

= Ww.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 3 / 1

Automatic presentations of ω-Words

We associate to each word w : N → Σ its word structure Ww := (N, <, {w−1(a)}a∈Σ). An automatic presentation of a word w ∈ Σω comprises regular sets D and Pa (a ∈ Σ), a synchronized rational binary relation ≺

• ver some alphabet Γ, such that (D, ≺, {Pa}a∈Σ) ∼

= Ww. In particular, (D, ≺) ∼ = (N, <) is a regular weak numeration system.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 3 / 1

Automatic presentations of ω-Words

We associate to each word w : N → Σ its word structure Ww := (N, <, {w−1(a)}a∈Σ). An automatic presentation of a word w ∈ Σω comprises regular sets D and Pa (a ∈ Σ), a synchronized rational binary relation ≺

• ver some alphabet Γ, such that (D, ≺, {Pa}a∈Σ) ∼

= Ww. In particular, (D, ≺) ∼ = (N, <) is a regular weak numeration system. General facts

◮ The FOmod theory of every automatic structure is decidable. ◮ The class of automatic structures is closed under

FOmod-interpretations.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 3 / 1

Length-lexicographic presentations

How does the choice of ≺ effect

◮ the class of words thus representable, ◮ their algorithmic properties?

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 4 / 1

Length-lexicographic presentations

How does the choice of ≺ effect

◮ the class of words thus representable, ◮ their algorithmic properties?

In the unary numeration system, when ≺ compares length only, precisely the ultimately periodic words are representable.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 4 / 1

Length-lexicographic presentations

How does the choice of ≺ effect

◮ the class of words thus representable, ◮ their algorithmic properties?

In the unary numeration system, when ≺ compares length only, precisely the ultimately periodic words are representable. In (generalized) numeration systems the usual (greedy) choice for ≺ is the length-lexicographic ordering x <llex y ⇐ ⇒ |x| < |y| or |x| = |y| and x <lex y

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 4 / 1

Length-lexicographic presentations

How does the choice of ≺ effect

◮ the class of words thus representable, ◮ their algorithmic properties?

In the unary numeration system, when ≺ compares length only, precisely the ultimately periodic words are representable. In (generalized) numeration systems the usual (greedy) choice for ≺ is the length-lexicographic ordering x <llex y ⇐ ⇒ |x| < |y| or |x| = |y| and x <lex y

Proposition (Rigo,Maes ’02)

An ω-word is morphic iff it is automatically presentable using <llex.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 4 / 1

Morphic words

An ω-word w ∈ Σω is morphic if there is a morphism τ : Γ∗ → Γ∗ with τ(a) = au for some a ∈ Γ and a morphism h : Γ∗ → Σ∗ such that w = h(τ ω(a)) = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 5 / 1

Morphic words

An ω-word w ∈ Σω is morphic if there is a morphism τ : Γ∗ → Γ∗ with τ(a) = au for some a ∈ Γ and a morphism h : Γ∗ → Σ∗ such that w = h(τ ω(a)) = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) . Examples

◮ The fixed point of τ : a → ab, b → ba is the Prouhet-Thue-Morse

sequence t = τ ω(a) = a · b · ba · baab · baababba · . . .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 5 / 1

Morphic words

An ω-word w ∈ Σω is morphic if there is a morphism τ : Γ∗ → Γ∗ with τ(a) = au for some a ∈ Γ and a morphism h : Γ∗ → Σ∗ such that w = h(τ ω(a)) = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) . Examples

◮ The fixed point of τ : a → ab, b → ba is the Prouhet-Thue-Morse

sequence t = τ ω(a) = a · b · ba · baab · baababba · . . .

◮ The fixed point of φ : a → ab, b → a is the Fibonacci word

f = a · b · a · ab · aba · abaab · abaababa · . . .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 5 / 1

Morphic words

An ω-word w ∈ Σω is morphic if there is a morphism τ : Γ∗ → Γ∗ with τ(a) = au for some a ∈ Γ and a morphism h : Γ∗ → Σ∗ such that w = h(τ ω(a)) = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) . Examples

◮ The fixed point of τ : a → ab, b → ba is the Prouhet-Thue-Morse

sequence t = τ ω(a) = a · b · ba · baab · baababba · . . .

◮ The fixed point of φ : a → ab, b → a is the Fibonacci word

f = a · b · a · ab · aba · abaab · abaababa · . . .

◮ Consider τ : a → ab, b → ccb, c → c and h : a, b → 1, c → 0. Then

τ ω(a) = a · b · ccb · ccccb · c6b · . . . and h(τ ω(a)) is the characteristic sequence of the set of squares.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 5 / 1

Deciding the MSO theory of ω-words

Theorem (cf. Rabinovich, Thomas ’06)

The MSO theory of Ww is decidable iff there is a recursive factorization w = w0 · w1 · . . . · wn · . . .

f (0) f (1) f (2) f (n) f (n+1) ...

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 6 / 1

Deciding the MSO theory of ω-words

Theorem (cf. Rabinovich, Thomas ’06)

The MSO theory of Ww is decidable iff there is a recursive factorization w = w0 · w1 · . . . · wn · . . .

f (0) f (1) f (2) f (n) f (n+1) ...

such that for every morphism ψ into a finite monoid M the contraction of w wrt. ψ and f : wψ

f = ψ(w0) · ψ(w1) · . . . · ψ(wn) · . . . ∈ Mω

is ultimately periodic (with both period and threshold computable from ψ).

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 6 / 1

Deciding the MSO theory of morphic words [Carton,Thomas ’02]

Consider w = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) and a morphism ψ into a finite monoid M.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 7 / 1

Deciding the MSO theory of morphic words [Carton,Thomas ’02]

Consider w = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) and a morphism ψ into a finite monoid M. The contraction of w wrt. ψ and fτ, wψ

fτ = ψ(h(a)) · ψ(h(u)) · ψ(h(τ(u))) · ψ(h(τ 2(u))) · . . . · ψ(h(τ n(u))) · . . . ,

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 7 / 1

Deciding the MSO theory of morphic words [Carton,Thomas ’02]

Consider w = h(a · u · τ(u) · τ 2(u) · . . . · τ n(u) · . . .) and a morphism ψ into a finite monoid M. The contraction of w wrt. ψ and fτ, wψ

fτ = ψ(h(a)) · ψ(h(u)) · ψ(h(τ(u))) · ψ(h(τ 2(u))) · . . . · ψ(h(τ n(u))) · . . . ,

is ultimately periodic, since there are (computable) N and p such that ψ ◦ h ◦ τ n+p = ψ ◦ h ◦ τ n (n > N)

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 7 / 1

Morphisms of k stacks

k-stacks as parenthesized words [ [abb] [a] [ba] ]

• r

as trees of height k a b b a b a

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 8 / 1

Morphisms of k stacks

k-stacks as parenthesized words [ [abb] [a] [ba] ]

• r

as trees of height k a b b a b a Morphisms of k-stacks ≈ k-stack of morphisms: Stack(0)

Γ

= Γ Stack(k+1)

Γ

= [(Stack(k)

Γ )∗]

Hom(0)

Γ

= Γ → Γ Hom(k+1)

Γ

= [(Hom(k)

Γ )∗] (uniformity!)

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 8 / 1

Morphisms of k stacks

k-stacks as parenthesized words [ [abb] [a] [ba] ]

• r

as trees of height k a b b a b a Morphisms of k-stacks ≈ k-stack of morphisms: Stack(0)

Γ

= Γ Stack(k+1)

Γ

= [(Stack(k)

Γ )∗]

Hom(0)

Γ

= Γ → Γ Hom(k+1)

Γ

= [(Hom(k)

Γ )∗] (uniformity!)

Application:

◮ ϕ(0)(γ(0)) is as given, ◮ for ϕ(k+1) = [ϕ(k) 1

. . . ϕ(k)

s ] and γ(k+1) = [γ(k) 1

. . . γ(k)

t

] ϕ(k+1)(γ(k+1)) = [ϕ(k)

1 (γ(k) 1 )...ϕ(k) s (γ(k) 1 ) · · · ϕ(k) 1 (γ(k) t

)...ϕ(k)

s (γ(k) t

)]

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 8 / 1

k-Morphic words

An word w ∈ Σω is k-morphic if there is a morphism ϕ ∈ Hom(k)

Γ , a

k-stack γ ∈ Stack(k)

Γ , and a homomorphism h : Γ∗ → Σ∗ such that

w = h ∞

• n=0

leaves(ϕn(γ))

• .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 9 / 1

k-Morphic words

An word w ∈ Σω is k-morphic if there is a morphism ϕ ∈ Hom(k)

Γ , a

k-stack γ ∈ Stack(k)

Γ , and a homomorphism h : Γ∗ → Σ∗ such that

w = h ∞

• n=0

leaves(ϕn(γ))

• .

Example

Let γ = [[#]], ϕ = [ϕ0ϕ1] with ϕi :

• →

1 → 1 # → i# . (Non-uniform!)

# 0 # 1 # 0 0 # 0 1 # 1 0 # 1 1 # . . . Similarly, s = 12345678910111213 . . . (Champernowne word) is 2-morphic.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 9 / 1

k-length-lexicographic presentations

Consider u = a0a1 . . . atk−1 ∈ Σtk. Its k-split is (u(1), . . . , u(k)) with u(i+1) = aiak+i . . . a(t−1)k+i f.a. i < k. Additionally, let u(0) = 1|u|. Conversely, u = ⊗k(u(1), . . . , u(k)) is the k-shuffle of the u(i)-s.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 10 / 1

k-length-lexicographic presentations

Consider u = a0a1 . . . atk−1 ∈ Σtk. Its k-split is (u(1), . . . , u(k)) with u(i+1) = aiak+i . . . a(t−1)k+i f.a. i < k. Additionally, let u(0) = 1|u|. Conversely, u = ⊗k(u(1), . . . , u(k)) is the k-shuffle of the u(i)-s. For 0 ≤ i < k we define the equivalence u =i v

def

⇐ ⇒ ∀j ≤ i u(j) = v(j) (implying |u| = |v|).

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 10 / 1

k-length-lexicographic presentations

Consider u = a0a1 . . . atk−1 ∈ Σtk. Its k-split is (u(1), . . . , u(k)) with u(i+1) = aiak+i . . . a(t−1)k+i f.a. i < k. Additionally, let u(0) = 1|u|. Conversely, u = ⊗k(u(1), . . . , u(k)) is the k-shuffle of the u(i)-s. For 0 ≤ i < k we define the equivalence u =i v

def

⇐ ⇒ ∀j ≤ i u(j) = v(j) (implying |u| = |v|). Consider some lin. ord. < of Σ with induced <lex. The induced k-length-lexicographic ordering <k-llex is defined as u <k-llex v

def

⇐ ⇒ |u| < |v| ∨ ∃i < k : u =i v ∧ u(i+1) <lex v(i+1) .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 10 / 1

k-Morphic = k-lexicographically presentable

Theorem

For all k, an ω-word is k-morphic iff it has an aut. pres. using <k-llex .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 11 / 1

k-Morphic = k-lexicographically presentable

Theorem

For all k, an ω-word is k-morphic iff it has an aut. pres. using <k-llex . Illustration #

ε ε

#

1 ϕ0

1 #

1 ϕ1

=0

u(2) = 00 10

#

11 u(1) = ϕ0ϕ0 00

1

10

#

11 ϕ0ϕ1

1

00 10

#

11 ϕ1ϕ0

1

00

1

10

#

11 ϕ1ϕ1

=1 . . .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 11 / 1

k-Morphic = k-lexicographically presentable

Theorem

For all k, an ω-word is k-morphic iff it has an aut. pres. using <k-llex . Illustration #

ε ε

#

1 ϕ0

1 #

1 ϕ1

=0

u(2) = 00 10

#

11 u(1) = ϕ0ϕ0 00

1

10

#

11 ϕ0ϕ1

1

00 10

#

11 ϕ1ϕ0

1

00

1

10

#

11 ϕ1ϕ1

=1 . . .

Notation

For each k, Wk is the class of k-morphic, or k-lex, words.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 11 / 1

Hierarchy theorem

Clearly, Wk ⊆ Wk+1.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 12 / 1

Hierarchy theorem

Clearly, Wk ⊆ Wk+1. Consider the following stuttering words defined for each k as s0 = aω s1 = abaaba4ba8ba16b . . . s2 = abcaabaabc(a4b)4c(a8b)8c . . . s3 = abcd((a2b)2c)2d((a4b)4c)4d((a8b)8c)8d . . . . . . sk = ∞

n=0(· · · (((a2n 0 )a1)2n) · · · )2nak

. . .

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 12 / 1

Hierarchy theorem

Clearly, Wk ⊆ Wk+1. Consider the following stuttering words defined for each k as s0 = aω s1 = abaaba4ba8ba16b . . . s2 = abcaabaabc(a4b)4c(a8b)8c . . . s3 = abcd((a2b)2c)2d((a4b)4c)4d((a8b)8c)8d . . . . . . sk = ∞

n=0(· · · (((a2n 0 )a1)2n) · · · )2nak

. . .

Theorem (Hierarchy Theorem)

For each k ∈ N we have sk+1 ∈ Wk+1 \ Wk.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 12 / 1

Deciding the MSO theory of k-morphic words

Theorem

For all k, the MSO-theory of every k-morphic word is decidable.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 13 / 1

Deciding the MSO theory of k-morphic words

Theorem

For all k, the MSO-theory of every k-morphic word is decidable. Proof plan (=0, ..., =k) provide a “built in” factorization of depth k of each w ∈ Wk+1 Contraction Lemma For all w ∈ Wk+1 and ψ we have wψ

=k ∈ Wk effectively.

By iterated contractions wψ

=i ∈ Wi, in particular, wψ =0 is ultimately

periodic.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 13 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Higher-Order Regularity Lemma Consider Θ a finite set of morphisms on Σ⋆, ϑ : Θ⋆ → Hom(Σ⋆, Σ⋆) s.t. ϑ(x · y) = ϑ(y) ◦ ϑ(x). Let ψ ∈ Hom(Σ⋆, M), a ∈ Σ, m ∈ M.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Higher-Order Regularity Lemma Consider Θ a finite set of morphisms on Σ⋆, ϑ : Θ⋆ → Hom(Σ⋆, Σ⋆) s.t. ϑ(x · y) = ϑ(y) ◦ ϑ(x). Let ψ ∈ Hom(Σ⋆, M), a ∈ Σ, m ∈ M. Then Lϑ,ψ,a,m = {x ∈ Θ⋆ | ψ(ϑ(x)(a)) = m} is regular. (effective)

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Higher-Order Regularity Lemma Consider Θ a finite set of morphisms on Σ⋆, ϑ : Θ⋆ → Hom(Σ⋆, Σ⋆) s.t. ϑ(x · y) = ϑ(y) ◦ ϑ(x). Let ψ ∈ Hom(Σ⋆, M), a ∈ Σ, m ∈ M. Then Lϑ,ψ,a,m = {x ∈ Θ⋆ | ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [[#]] ϕ = [ϕ0ϕ1] with ϕi :

• →

1 → 1 # → i#

ψ(x) = |x|1 mod 2

u(2) = 00 10

#

11 u(1) = ϕ0ϕ0 00

1

10

#

11 ϕ0ϕ1

1

00 10

#

11 ϕ1ϕ0

1

00

1

10

#

11 ϕ1ϕ1

=1 =1 =1 =1

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Higher-Order Regularity Lemma Consider Θ a finite set of morphisms on Σ⋆, ϑ : Θ⋆ → Hom(Σ⋆, Σ⋆) s.t. ϑ(x · y) = ϑ(y) ◦ ϑ(x). Let ψ ∈ Hom(Σ⋆, M), a ∈ Σ, m ∈ M. Then Lϑ,ψ,a,m = {x ∈ Θ⋆ | ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [0] τ :

• τ0

τ1 → 1 1 → 1

#

τ0τ0

1 1 #

τ0τ1

1 1 #

τ1τ0

1 1 #

τ1τ1

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Contraction Lemma

Illustration for k = 1: w generated by ϕ = [ϕ0ϕ1] with ϕi ∈ Hom(Σ⋆, Σ⋆) and γ = [[a]]: w = a · aϕ0 aϕ1 · aϕ0ϕ0 aϕ0ϕ1 aϕ1ϕ0 aϕ1ϕ1 · aϕ0ϕ0ϕ0 aϕ0ϕ0ϕ1 . . . Idea: use 1-lex presentation of wψ

=1 over {ϕ0, ϕ1}!

Higher-Order Regularity Lemma Consider Θ a finite set of morphisms on Σ⋆, ϑ : Θ⋆ → Hom(Σ⋆, Σ⋆) s.t. ϑ(x · y) = ϑ(y) ◦ ϑ(x). Let ψ ∈ Hom(Σ⋆, M), a ∈ Σ, m ∈ M. Then Lϑ,ψ,a,m = {x ∈ Θ⋆ | ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [0] τ :

• τ0

τ1 → 1 1 → 1

#

τ0τ0

1 1 #

τ0τ1

1 1 #

τ1τ0

1 1 #

τ1τ1

For k = 0: Θ = {τ} is unary and the Lϑ,ψ,a,m are ultimately periodic.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 14 / 1

Main Results

Theorem (Main Theorem)

Given a k-lex. presentation of w ∈ Wk and ϕ( x) ∈ MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in Ww.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 15 / 1

Main Results

Theorem (Main Theorem)

Given a k-lex. presentation of w ∈ Wk and ϕ( x) ∈ MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in Ww. Corollaries

◮ Each Wk is closed under MSO-definable recolorings. ◮ If a structure is MSO-interpretable in a k-lexicographic word by

formulas ϕ( x) as in the theorem, then it is automatic.

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 15 / 1

Main Results

Theorem (Main Theorem)

Given a k-lex. presentation of w ∈ Wk and ϕ( x) ∈ MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in Ww. Corollaries

◮ Each Wk is closed under MSO-definable recolorings. ◮ If a structure is MSO-interpretable in a k-lexicographic word by

formulas ϕ( x) as in the theorem, then it is automatic. For each k consider wk ∈ {0, 1, #}ω obtained by concatenating all finite binary words in the k-lexicographic ordering and separated by hash marks.

Theorem (Characterization)

Let w ∈ Σω. Then w ∈ Wk ⇐ ⇒ Ww ≤I Wwk for some interpretation I = (ϕD(x), x < y, {ϕa(x)}a∈Σ) such that | = ∀x(ϕD(x) → P#(x)).

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 15 / 1

Future work and Questions

To do

◮ Locate Wk in the pushdown hierarchy...

• r generate them from simply-typed schemes.

◮ Extend results to other (all?) automatic presentations of (N, <)...

to other linear orderings.... ∗∗ Is isomorphism of k-lexicographic words decidable? ∗∗ Let k > k′. Is it decidable whether a k-morphic word is k′-lexicographic? In particular, is eventual periodicity of k-morphic words decidable? (Cf. same problems for ω-words generated by HD0L systems, i.e. k = 1)

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 16 / 1

Future work and Questions

To do

◮ Locate Wk in the pushdown hierarchy...

• r generate them from simply-typed schemes.

◮ Extend results to other (all?) automatic presentations of (N, <)...

to other linear orderings.... ∗∗ Is isomorphism of k-lexicographic words decidable? ∗∗ Let k > k′. Is it decidable whether a k-morphic word is k′-lexicographic? In particular, is eventual periodicity of k-morphic words decidable? (Cf. same problems for ω-words generated by HD0L systems, i.e. k = 1)

THANK YOU!

Vince B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω-Words having a decidable MSO Theory JM’06 16 / 1