������������� � � � � � � � � � � � � � Asymptotic Approximation � � � � � � � � by Regular Languages ���������������������� � 9/27/2020 ��l�_l�g�.��g Ryoma Sin’ya Akita University YR-OWLS 30 Sep 2020 �le:///U�e��/����a/De�k���/��l�_l�g�.��g 1/1
This talk is based on [S1] Ryoma Sin’ya. Asymptotic Approximation by Regular Languages, SOFSEM2021 (to appear), draft is available at http://www.math.akita-u.ac.jp/~ryoma
Outline 1. Motivation of this work 2. Set of natural numbers and measure density 3. Density of regular languages and REG-measurability 4. REG-(im)measurability of several languages 5. Open problems
The Primitive Words Conjecture [Dömösi-Horvath-Ito 1991] • is said to be primitive if it can not be represented as a A non-empty word w w = u n ⇒ u = w ( and n = 1) power of shorter words, i.e., denotes the set of all primitive words over . 𝖱 A A • The case is trivial ( ). Here after we only consider the case #( A ) = 1 𝖱 A = A for , and simply write . 𝖱 A 𝖱 A = { a , b } ababab = ( ab ) 3 ∉ 𝖱 Example ： ababa ∈ 𝖱 Conjecture: is not context-free. 𝖱
Why is “primitivity” important? • Primitive words are like prime numbers. Fact: For every non-empty word , there exists a unique primitive word w v w = v k such that for some . k ≥ 1 • u − 1 wu = vu by For a word , we denote its conjugate (by ) . w = uv u vu u − 1 wu If and are non-empty, is called a proper conjugate. u v ⇔ w ≠ u − 1 wu Fact: is primitive for every proper conjugate. w Note: if we regard a conjugation as a (partial) morphism on words, “ is primitive” means w “ has no non-trivial automorphism” (cf. rigid graphs, rigid models in model theory) . w • Primitive words and its special class called Lyndon words play a central role in algebraic coding theory and combinatorics on words, also in text compression (cf. Lyndon factorisation, Burrows–Wheeler transformation).
The Primitive Words Conjecture [Dömösi-Horvath-Ito 1991] On the Connection between Formal Languages and Primitive Words Masami Ito Pál Dömösi [Dömösi-Ito 2014]
The Primitive Words Conjecture [Dömösi-Horvath-Ito 1991] On the Connection between Formal Languages and Primitive Words Szilárd Fazekas Masami Ito Pál Dömösi
My motivating intuition (Intuition 1) is “very large” while there is no “good approximation” 𝖱 by regular languages. (Intuition 2) Every “very large” context-free language has some “good approximation” by regular languages. My (naive) idea: if we can formalise the above intuition and prove it, then the primitive words conjecture is true! → I proved that (the formal statement) of Intuition 1 is true, but Intuition 2 is false.
Approximation of languages We adopt and extend Buck’s measure density to formalise “approximation by regular languages”. Measure density [Buck 1946] Rough set approximation [P ă un-Polkowski-Skowron 1996] Minimal cover-automata [Câmpeanu-Sânten-Yu 1999] Minimal regular cover [Domaratzki-Shallit-Yu 2001] Convergent-reliability / Slender-reliability [Kappes-Kintala 2004] Bounded- ε -approximation [Eisman-Ravikumar 2005] Degree of approximation [Cordy-Salomaa 2007]
Outline 1. Motivation of this work 2. Set of natural numbers and measure density 3. Density of regular languages and REG-measurability 4. REG-(im)measurability of several languages 5. Open problems
Natural density of a subset of ℕ ( ∋ 0) • For an arithmetic progression S = { cn + d ∣ n ∈ ℕ } we define its natural density as δ ( S ) ・ if (i.e., ) then c = 0 S = { d } δ ( S ) = 0 δ ( S ) = 1 ・ if (i.e., is infinite) then c ≠ 0 S c Intuitively, represents the “largeness” of . More formally, it represents δ ( S ) S the probability that a randomly chosen natural number is in . n S
Measure density of a subset of ℕ [Buck 1946] "The measure theoretic approach to density” • For a set of numbers , its outer measure of is defined as S ⊆ ℕ μ *( S ) S μ *( S ) = inf { ∑ δ ( X i ) ∣ S ⊆ X , X is a disjoint union of finitely many arithmetic progressions X 1 , …, X k } i Theorem (Buck) ： • If a set satisfies the condition S ⊆ ℕ 0 ⊊ μ ( ☆ ) μ *( S ) + μ *( S ) = 1 then we call the measure density of , and we say that “ is measurable ”. μ *( S ) S S satisfying ( ☆ ) is the Carathéodory extension of • The class of all subsets of μ ℕ 0 = { X ⊆ ℕ ∣ X is a disjoint union of finitely many arithmetic progresssions }
Observation 0 = { X ⊆ ℕ ∣ X is a finitely many disjoint union of arithemtic progressions } • can be seen as the class of regular languages over a unary alphabet REG A : A = { a } 0 = {{ | w | ∣ w ∈ L } ∣ L ∈ REG A } The set of lengths of words in a regular language (i.e., the Parikh image of ) L L is a finite union of arithmetic progressions (i.e., ultimately periodic set ). If we can define a “density” notion on for an arbitrary alphabet , we can REG A A naturally extend Buck’s measure density to formal languages!
Outline 1. Motivation of this work 2. Set of natural numbers and measure density 3. Density of regular languages and REG-measurability 4. REG-(im)measurability of several languages 5. Open problems
Density of formal languages • The asymptotic density of a Fact: if converges then δ A ( L ) δ A ( L ) language over is defined as L A also converges, and δ * A ( L ) moreover . #( L ∩ A n ) δ A ( L ) = δ * A ( L ) δ A ( L ) = lim #( A n ) n →∞ • The density is defined as But the converse is not true! δ * A ( L ) n − 1 #( L ∩ A i ) 1 trivial example: L = ( AA )* ∑ δ * A ( L ) = lim (diverges) but #( A i ) n δ A ( L ) = ⊥ n →∞ i =0 δ * A ( L ) = 1/2
Density of formal languages • The asymptotic density of a Fact1 (cf. [Salomaa-Soittla 1978]): for any δ A ( L ) language over is defined as regular language over , converges L A L A δ * A ( L ) to a rational number. #( L ∩ A n ) δ A ( L ) = lim Fact2 (cf. [S2]): A regular language is not L #( A n ) n →∞ null (i.e., ) if and only if is dense δ * A ( L ) ≠ 0 L • The density is defined as δ * A ( L ) (i.e., L ∩ A * wA * ≠ ∅ for any w ∈ A * ). n − 1 #( L ∩ A i ) 1 ∑ δ * A ( L ) = lim Not null: measure theoretic “largeness” #( A i ) n n →∞ Dense: topological “largeness” i =0 Note: “ is not null is dense” is true for any language , but L ⇒ L L “ is dense is not null” is false for general non-regular languages. L ⇒ L
Density of formal languages Note: “ is not null is dense” is true for any language , but L ⇒ L L “ is dense is not null” is false for general non-regular languages. L ⇒ L Infinite Monkey Theorem (cf. [Borel 1913]): δ A ( A * wA *) = 1 for any w ∈ A * . is not dense means that there exists such that L w L ∩ A * wA * = ∅ (such word is called a forbidden word of ), L thus by the infinite monkey theorem. δ A ( L ) ≤ 1 − δ A ( A * wA *) = 0 The semi-Dyck language over 𝖤 = { ε , (), (()), ()(), ((())), …} A = {(, )} is dense, but actually null. ( )(()( ))
Density of formal languages • The asymptotic density of a Fact1 (cf. [Salomaa-Soittla 1978]): for any δ A ( L ) language over is defined as regular language over , converges L A L A δ * A ( L ) to a rational number. #( L ∩ A n ) δ A ( L ) = lim #( A n ) n →∞ • The density is defined as Fact2 (cf. [S2]): A regular language is not δ * A ( L ) L n − 1 #( L ∩ A i ) null (i.e., ) if and only if is dense 1 δ * A ( L ) ≠ 0 L ∑ δ * A ( L ) = lim (i.e., ). #( A i ) ∀ w ∈ A * L ∩ A * wA * ≠ ∅ n n →∞ i =0
Measure density of languages • We now consider the Carathéodory extension of the class of regular languages: For , its outer measure is defined as L ⊆ A * . μ REG ( L ) = inf{ δ * A ( R ) ∣ L ⊆ R ∈ REG A } We say that is REG-measurable if holds. L μ REG ( L ) + μ REG ( L ) = 1 Lemma: the followings are equivalent (1) is REG-measurable L (2) REG ( L ) = sup{ δ * A ( R ) ∣ L ⊇ R ∈ REG A } μ REG ( L ) = μ the inner measure of L Note: always holds (if is defined). REG ( L ) ≤ δ * A ( L ) ≤ μ REG ( L ) μ δ * A ( L )
・ ・ Measure density of languages A * K 1 K 2 ・ ・ ・ L ・ M 2 M 1 is REG-measurable if we can take an infinite sequence of pairs or regular languages L such that . A ( K n ∖ M n ) = 0 ( M n ⊆ L ⊆ K n ) n n →∞ δ * lim
Outline 1. Motivation of this work 2. Set of natural numbers and measure density 3. Density of regular languages and REG-measurability 4. REG-(im)measurability of several languages 5. Open problems
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