Finite Automata and Randomness Ludwig Staiger - PowerPoint PPT Presentation

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Finite Automata and Randomness Ludwig Staiger Martin-Luther-Universität Halle-Wittenberg Jewels of Automata: from Mathematics to Applications Leipzig, May, 2015

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Outline 1 Notation and Preliminaries Notation Algorithmic Randomness 2 Automata and Measure Automata on ω -words Subword complexity 3 Unpredictability Gambling Strategies for Automata Finite-state dimension Other concepts 4 Incompressibility Sequential compression Finite-state complexity

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Notation: Strings and Languages Finite Alphabet X = { 0 ,..., r − 1 } , cardinality | X | = r Finite strings (words) w = x 1 ··· x n ∈ { 0 , 1 } ∗ , x i ∈ { 0 , 1 } Length | w | = n W ⊆ X ∗ Languages Infinite strings ( ω -words) ξ = x 1 ··· x n ··· ∈ X ω Prefixes of infinite strings ξ [ 0 .. n ] ∈ X ∗ , � = n � � ξ [ 0 .. n ] � ω -Languages F ⊆ X ω

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility X ω as C ANTOR space Metric: ρ ( η , ξ ) : = inf { r −| w | : w ∈ pref ( η ) ∩ pref ( ξ ) } Balls: w · X ω = { η : w ∈ pref ( η ) } = { η : w ⊏ η } Diameter: diam w · X ω = r −| w | diam F = inf { r −| w | : F ⊆ w · X ω } Open sets: W · X ω = � w ∈ W w · X ω Closure: (Smallest closed set containing F ) C ( F ) = { ξ : pref ( ξ ) ⊆ pref ( F ) } Fact F ⊆ X ω is closed if and only if pref ( ξ ) ⊆ pref ( F ) implies ξ ∈ F.

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Algorithmic Randomness measure-theoretic paradigm An ω -word is random if and only if it is not contained in a constructive null-set. unpredictability paradigm An ω -word is random if and only if no constructive predicting strategy can win against it. incompressibility (complexity-theoretic) paradigm An ω -word is random if and only if one cannot constructively compress infinitely many of its prefixes.

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Measure Measure on base sets: µ ( w · X ω ) : = r −| w | V n · X ω , � Constructive null-sets: Unions of ω -languages of the form n ∈ N where � ( v , n ) : v ∈ X ∗ ∧ n ∈ N � is constructive, V ⊆ V n : = { v : ( v , n ) ∈ V } and µ ( V n · X ω ) ≤ r − n . Definition (Randomness) ξ ∈ X ω is random if and only if no constructive null-set contains ξ .

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Predicting strategy: Gambling Our model: • Playing against an ω -word ξ ∈ X ω . • Gambling strategy Γ : X ∗ × X → [ 0 , 1 ] (bet on outcome x ∈ X ) x ∈ X Γ ( w , x ) ≤ 1 for w ∈ X ∗ � • yields a (super-)martingale V Γ : X ∗ → R + • V Γ ( ξ [ 0 .. n ]) is the capital after the n th round, that is, V Γ ( ξ [ 0 .. n ]) = r · Γ ( ξ [ 0 .. n ] , x ) · V Γ ( ξ [ 0 .. n − 1 ]) , for ξ ( n ) = x Fact (super-martingale property) V Γ ( w ) ≥ 1 � x ∈ X V Γ ( wx ) r · Definition (Randomness) ξ ∈ X ω is random if and only if no constructive gambling strategy Γ can win against ξ , that is, limsup n →∞ V Γ ( ξ [ 0 .. n ]) < ∞ .

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Gambling strategies: martingale V V ( e ) ✟ ❍❍❍❍❍❍❍ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ V ( 0 ) V ( 1 ) � ❅ ❅ � ❅ � ❅ ❅ � ❅ � ❅ ❅ � ❅ � ❅ ❅ � ❅ V ( 00 ) V ( 01 ) V ( 10 ) V ( 11 ) ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆❆ ✁ ❆❆ ✁ ❆❆ ✁ ❆❆ ✁ ✁ ✁ ✁ V ( 111 ) ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ❈❈ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Compression: The Principle of Lossless Compression description ( or program) π ∈ X ∗ text w ∈ X ∗ f ✲ ϕ X ∗ X ∗ ✛ space of texts space of descriptions f is injective and ϕ ( f ( w )) = w for all w ∈ X ∗ Complexity of w w.r.t. ϕ : C ϕ ( w ) : = inf { | π | : ϕ ( π ) = w } Definition (Randomness = Incompressibility) ξ ∈ X ω is random if and only if all constructive decompression functions ϕ satisfy ∃ c ∀ n ( C ϕ ( ξ [ 0 .. n ])) ≥ n − c , that is, prefixes of ξ cannot be compressed.

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility References: Algorithmic Randomness • Calude, C.S.: Information and Randomness. An Algorithmic Perspective , 2nd ed., Springer, Berlin (2002). • Downey, R., Hirschfeldt D.: Algorithmic Randomness and Complexity , Springer, Heidelberg (2010). • Li M., Vitányi: An Introduction to Kolmogorov Complexity and Its Applications , Springer, Berlin (1993). • Nies, A.: Computability and Randomness , Oxford Univ. Press, Oxford (2009).

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Automata on ω -words: Büchi-automata Automaton: A = ( X , Q , ∆ , q 0 , Q fin ) with ∆ ⊆ Q × X × Q , q 0 ∈ Q , Q fin ⊆ Q Run on ξ : ( q i ) i ∈ N with ∀ i ≥ 0 : ( q i , ξ ( i + 1 ) , q i + 1 ) ∈ ∆ q 0 q 1 q 2 q i − 1 q i ց ↑ ց ↑ ··· ↑ ց ↑ ց ··· ξ ( 1 ) ξ ( 2 ) ξ ( i − 1 ) ξ ( i ) A accepts ξ : ∀ i ≥ 0 : ( q i , ξ ( i + 1 ) , q i + 1 ) ∈ ∆ ∃ ( q i ) i ∈ N ∧ ∃ ∞ k : q k ∈ Q fin A accepts F : F = � ξ : A accepts ξ �

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Regular ω -languages Definition (Regular ω -language) An ω -language F ⊆ X ω is called regular if and only if F is accepted by a finite automaton Theorem (B ÜCHI 1962) 1 An ω -language F ⊆ X ω is regular if and only if F = � n i = 1 W i · V ω for i some n ∈ N and regular languages W i , V i ⊆ X ∗ . 2 The set of regular ω -languages over X is closed under Boolean operations.

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Regular null-sets Theorem ( St’76,St’98 ) Let F be a regular ω -language. 1 If F is closed then µ ( F ) = 0 if and only if there is word w ∈ X ∗ such that F ⊆ X ω \ X ∗ · w · X ω . 2 µ ( F ) = 0 if and only if w ∈ X ∗ X ω \ X ∗ · w · X ω . F ⊆ � Remark This theorem holds for a much larger class of finite measures on X ω . Definition (Randomness = Disjunctivity) An ω -word ξ ∈ X ω is called disjunctive (or rich or saturated ) if and only if it contains every word w ∈ X ∗ as subword (infix) [ infix ( ξ ) = X ∗ ].

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Partial randomness: Subword complexity Definition (Asymptotic subword complexity) log r | infix ( ξ ) ∩ X n | τ ( ξ ) : = limsup n →∞ n infix ( ξ ) ∩ X n + m ⊆ ( infix ( ξ ) ∩ X n ) · ( infix ( ξ ) ∩ X m ) Fact � log r | infix ( ξ ) ∩ X n | � The limit exists and equals τ ( ξ ) = inf : n ∈ N . n Proposition 0 ≤ τ ( ξ ) ≤ 1 and an ω -word ξ ∈ X ω is disjunctive if and only if τ ( ξ ) = 1 .

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Hausdorff dimension I r − α ·| v | : F ⊆ � � v · X ω ∧ min � L α ( F ) : = lim n →∞ inf � v ∈ V | v | ≥ n v ∈ V v ∈ V L α ( F ) ✻ ∞ L α 0 ( F ) r α ✲ 0 0 1 α 0 = dim F dim F : = inf { α : L α ( F ) = 0 } = sup { α : L α ( F ) = ∞ }

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Hausdorff dimension II Fact 1 dim � F i = sup � dim F i : i ∈ N � and dim { ξ } = 0 i ∈ N 2 If µ ( F ) > 0 then dim F = 1 . 3 If F is regular then dim F = 1 implies µ ( F ) > 0 . Fact � dim F : F is a regular ω -language � Q ⊂

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility Partial randomness: The hierarchy Lemma If F ⊆ X ω is a regular ω -language and ξ ∈ F then τ ( ξ ) ≤ dim F. Theorem � � 1 τ ( ξ ) = inf dim F : ξ ∈ F ∧ F is a regular ω -language 2 If α = dim F for some regular ω -language then there is a ξ such that τ ( ξ ) = α . 3 For all α , γ , 0 ≤ α < γ ≤ 1 , the level sets F ( τ ) : = { ξ : τ ( ξ ) ≤ α } α satisfy F ( τ ) ⊂ F ( τ ) . α γ Open question Does there, for every α , 0 ≤ α ≤ 1 , exist a ξ with τ ( ξ ) = α .

Notation and Preliminaries Automata and Measure Unpredictability Incompressibility References: Automata and Measure • Staiger, L.: Reguläre Nullmengen, Elektron. Informationsverarb. Kybernet. EIK 12: 307–311 (1976). • Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. , 103(2):159–194, (1993). • Staiger, L.: Rich ω -words and monadic second-order arithmetic. In Mogens Nielsen and Wolfgang Thomas, editors, Computer Science Logic (Aarhus, 1997) , LNCS 1414, Springer, 478–490 (1998). • Staiger, L.: Asymptotic Subword Complexity, In Languages Alive 2012 , LNCS 7300, Springer, 236–245 (2012).