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

Transcription

1 A Hierarchy of Automatic ω-words having a decidable MSO Theory Vince Bárány Mathematische Grundlagen der Informatik RWTH Aachen Journées Montoises d Informatique Théorique Rennes, 2006 Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 1 / 1

2 ω-words An ω-word over Σ is a function w : N Σ. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 2 / 1

3 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 2 / 1

4 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 2 / 1

5 Automatic presentations of ω-words We associate to each word w : N Σ its word structure W w := (N, <, {w 1 (a)} a Σ ). Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 3 / 1

6 Automatic presentations of ω-words We associate to each word w : N Σ its word structure W w := (N, <, {w 1 (a)} a Σ ). An automatic presentation of a word w Σ ω comprises regular sets D and P a (a Σ), a synchronized rational binary relation over some alphabet Γ, such that (D,, {P a } a Σ ) = W w. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 3 / 1

7 Automatic presentations of ω-words We associate to each word w : N Σ its word structure W w := (N, <, {w 1 (a)} a Σ ). An automatic presentation of a word w Σ ω comprises regular sets D and P a (a Σ), a synchronized rational binary relation over some alphabet Γ, such that (D,, {P a } a Σ ) = W w. In particular, (D, ) = (N, <) is a regular weak numeration system. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 3 / 1

8 Automatic presentations of ω-words We associate to each word w : N Σ its word structure W w := (N, <, {w 1 (a)} a Σ ). An automatic presentation of a word w Σ ω comprises regular sets D and P a (a Σ), a synchronized rational binary relation over some alphabet Γ, such that (D,, {P a } a Σ ) = W w. In particular, (D, ) = (N, <) is a regular weak numeration system. General facts The FO mod theory of every automatic structure is decidable. The class of automatic structures is closed under FO mod -interpretations. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 3 / 1

9 Length-lexicographic presentations How does the choice of effect the class of words thus representable, their algorithmic properties? Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 4 / 1

10 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 4 / 1

11 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 4 / 1

12 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 4 / 1

13 Morphic words An ω-word w Σ ω 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 5 / 1

14 Morphic words An ω-word w Σ ω 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 Examples w = h(τ ω (a)) = h(a u τ(u) τ 2 (u)... τ n (u)...). 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 5 / 1

15 Morphic words An ω-word w Σ ω 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 Examples w = h(τ ω (a)) = h(a u τ(u) τ 2 (u)... τ n (u)...). 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 5 / 1

16 Morphic words An ω-word w Σ ω 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 Examples w = h(τ ω (a)) = h(a u τ(u) τ 2 (u)... τ n (u)...). 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 c 6 b... and h(τ ω (a)) is the characteristic sequence of the set of squares. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 5 / 1

17 Deciding the MSO theory of ω-words Theorem (cf. Rabinovich, Thomas 06) The MSO theory of W w is decidable iff there is a recursive factorization w = w 0 w 1... w n... f (0) f (1) f (2) f (n) f (n+1)... Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 6 / 1

18 Deciding the MSO theory of ω-words Theorem (cf. Rabinovich, Thomas 06) The MSO theory of W w is decidable iff there is a recursive factorization w = w 0 w 1... w n... 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 : w ψ f = ψ(w 0 ) ψ(w 1 )... ψ(w n )... M ω is ultimately periodic (with both period and threshold computable from ψ). Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 6 / 1

19 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 7 / 1

20 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 f τ, w ψ f τ = ψ(h(a)) ψ(h(u)) ψ(h(τ(u))) ψ(h(τ 2 (u)))... ψ(h(τ n (u)))..., Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 7 / 1

21 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 f τ, w ψ f τ = 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 7 / 1

22 Morphisms of k stacks k-stacks as parenthesized words [ [abb] [a] [ba] ] or as trees of height k a b b a b a Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 8 / 1

23 Morphisms of k stacks k-stacks as parenthesized words [ [abb] [a] [ba] ] or 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 8 / 1

24 Morphisms of k stacks k-stacks as parenthesized words [ [abb] [a] [ba] ] or 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 ) ] Application: ϕ (0) (γ (0) ) is as given, Hom (0) Γ = Γ Γ Hom (k+1) Γ = [(Hom (k) Γ ) ] (uniformity!) for ϕ (k+1) = [ϕ (k) 1... ϕ (k) s ] and γ (k+1) = [γ (k) 1... γ (k) ϕ (k+1) (γ (k+1) ) = [ϕ (k) 1 (γ(k) 1 )...ϕ(k) s (γ (k) 1 ) ϕ(k) t ] 1 (γ(k) t )...ϕ (k) s (γ (k) t )] Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 8 / 1

25 k-morphic words An word w Σ ω 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 leaves(ϕ n (γ)). n=0 Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 9 / 1

26 k-morphic words An word w Σ ω 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 leaves(ϕ n (γ)). Example n=0 Let γ = [[#]], ϕ = [ϕ 0 ϕ 1 ] with ϕ i : # i#. (Non-uniform!) # 0 # 1 # 0 0 # 0 1 # 1 0 # 1 1 #... Similarly, s = (Champernowne word) is 2-morphic. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM 06 9 / 1

27 k-length-lexicographic presentations Consider u = a 0 a 1... a tk 1 Σ tk. Its k-split is (u (1),..., u (k) ) with u (i+1) = a i a k+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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

28 k-length-lexicographic presentations Consider u = a 0 a 1... a tk 1 Σ tk. Its k-split is (u (1),..., u (k) ) with u (i+1) = a i a k+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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

29 k-length-lexicographic presentations Consider u = a 0 a 1... a tk 1 Σ tk. Its k-split is (u (1),..., u (k) ) with u (i+1) = a i a k+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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

30 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

31 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 # 0 # 1 = 0 # 0 0 = 1 # 0 1 # 1 0 # 1 1 # ε u (2) = ε ϕ 0 ϕ 1 u (1) = ϕ 0 ϕ 0 ϕ 0 ϕ 1 ϕ 1 ϕ 0 ϕ 1 ϕ 1... Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

32 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 # 0 # 1 = 0 # 0 0 = 1 # 0 1 # 1 0 # 1 1 # ε u (2) = ε ϕ 0 ϕ 1 u (1) = ϕ 0 ϕ 0 ϕ 0 ϕ 1 ϕ 1 ϕ 0 ϕ 1 ϕ 1... Notation For each k, W k is the class of k-morphic, or k-lex, words. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

33 Hierarchy theorem Clearly, W k W k+1. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

34 Hierarchy theorem Clearly, W k W k+1. Consider the following stuttering words defined for each k as s 0 = a ω s 1 = abaaba 4 ba 8 ba 16 b... s 2 = abcaabaabc(a 4 b) 4 c(a 8 b) 8 c... s 3 = abcd((a 2 b) 2 c) 2 d((a 4 b) 4 c) 4 d((a 8 b) 8 c) 8 d... s k. = n=0 ( (((a2n 0 )a 1) 2n ) ) 2n a k. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

35 Hierarchy theorem Clearly, W k W k+1. Consider the following stuttering words defined for each k as s 0 = a ω s 1 = abaaba 4 ba 8 ba 16 b... s 2 = abcaabaabc(a 4 b) 4 c(a 8 b) 8 c... s 3 = abcd((a 2 b) 2 c) 2 d((a 4 b) 4 c) 4 d((a 8 b) 8 c) 8 d... s k. = n=0 ( (((a2n 0 )a 1) 2n ) ) 2n a k. Theorem (Hierarchy Theorem) For each k N we have s k+1 W k+1 \ W k. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

36 Deciding the MSO theory of k-morphic words Theorem For all k, the MSO-theory of every k-morphic word is decidable. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

37 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 W k+1 Contraction Lemma For all w W k+1 and ψ we have w ψ = k W k effectively. By iterated contractions w ψ = i W i, in particular, w ψ = 0 is ultimately periodic. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

38 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

39 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

40 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! 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 L ϑ,ψ,a,m = {x Θ ψ(ϑ(x)(a)) = m} is regular. (effective) Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

41 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! 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 L ϑ,ψ,a,m = {x Θ ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [[#]] ϕ = [ϕ 0 ϕ 1 ] with ϕ i : # i# ψ(x) = x 1 mod 2 = 1 = 1 = 1 = # 0 1 # 1 0 # 1 1 # u (2) = u (1) = ϕ 0 ϕ 0 ϕ 0 ϕ 1 ϕ 1 ϕ 0 ϕ 1 ϕ 1 Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

42 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! 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 L ϑ,ψ,a,m = {x Θ ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [0] τ : τ 0 τ # 0 1 # 1 0 # 1 1 # 0 τ 0 τ τ 0 τ 1 τ 1 τ 0 τ 1 τ 1 Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

43 Contraction Lemma Illustration for k = 1: w generated by ϕ = [ϕ 0 ϕ 1 ] with ϕ i Hom(Σ, Σ ) and γ = [[a]]: w = a aϕ 0 aϕ 1 aϕ 0 ϕ 0 aϕ 0 ϕ 1 aϕ 1 ϕ 0 aϕ 1 ϕ 1 aϕ 0 ϕ 0 ϕ 0 aϕ 0 ϕ 0 ϕ 1... Idea: use 1-lex presentation of w ψ = 1 over {ϕ 0, ϕ 1 }! 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 L ϑ,ψ,a,m = {x Θ ψ(ϑ(x)(a)) = m} is regular. (effective) γ = [0] τ : τ 0 τ # # τ 0 τ # 1 1 # 1 0 τ 0 τ 1 τ 1 τ 0 τ 1 τ 1 For k = 0: Θ = {τ} is unary and the L ϑ,ψ,a,m are ultimately periodic. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

44 Main Results Theorem (Main Theorem) Given a k-lex. presentation of w W k and ϕ( x) MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in W w. Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

45 Main Results Theorem (Main Theorem) Given a k-lex. presentation of w W k and ϕ( x) MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in W w. Corollaries Each W k 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

46 Main Results Theorem (Main Theorem) Given a k-lex. presentation of w W k and ϕ( x) MSO having only first-order variables x free, we can compute an automaton recognizing the relation defined by ϕ in W w. Corollaries Each W k 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 w k {0, 1, #} ω obtained by concatenating all finite binary words in the k-lexicographic ordering and separated by hash marks. Theorem (Characterization) Let w Σ ω. Then w W k W w I W wk for some interpretation I = (ϕ D (x), x < y, {ϕ a (x)} a Σ ) such that = x(ϕ D (x) P # (x)). Vince Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

47 Future work and Questions To do Locate W k in the pushdown hierarchy... or 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1

48 Future work and Questions To do Locate W k in the pushdown hierarchy... or 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 Bárány (RWTH Aachen) A Hierarchy of Automatic ω-words having a decidable MSO Theory JM / 1