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 W w := ( 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 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 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 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 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 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 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 ∈ 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 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 ∈ 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 ) · . . . ) . 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 ∈ 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 ) · . . . ) . 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 ∈ 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 ) · . . . ) . 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 · c 6 b · . . . 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 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 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 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 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 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 B´ ar´ any (RWTH Aachen) A Hierarchy of Automatic ω -Words having a decidable MSO Theory JM’06 7 / 1
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