Regular Sets of Trees and Probability Matteo Mio CNRS & - PowerPoint PPT Presentation

Regular Sets of Trees and Probability Matteo Mio CNRS & ENS–Lyon Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Some quick background Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Some quick background Automata Theory is used to prove decidability Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Some quick background Automata Theory is used to prove decidability ◮ Presburger arithmetic FO( N , <, +), ◮ Linear arithmetic FO( R , <, +), ◮ Some other theories FO � Π 0 � 1 ( R ) , ∪ , ∩ Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Some quick background Automata Theory is used to prove decidability ◮ Presburger arithmetic FO( N , <, +), ◮ Linear arithmetic FO( R , <, +), ◮ Some other theories FO � Π 0 � 1 ( R ) , ∪ , ∩ and ◮ (temporal) logics in computer science: ◮ MSO(words), LTL, ◮ MSO(trees), CTL, CTL ∗ , µ -calculus, . . . Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Classical Temporal Logics ◮ Example: Computation Tree Logic (CTL) ◮ Models: Labeled Trees Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Classical Temporal Logics ◮ Example: Computation Tree Logic (CTL) ◮ Models: Labeled Trees Probabilistic Temporal Logics ◮ Example: Probabilistic Computation Tree Logic (pCTL) ◮ Models: Labeled Markov Chains (= trees with probabilities) Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

. . . . . . . . . . . . . . . . . . . . . . . . a a a b b b b b p 1 − p a a a b 1 2 1 3 3 3 4 4 a b 1 a 1 2 2 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

. . . . . . . . . . . . . . . . . . . . . . . . a a a b b b b b p 1 − p a a a b 1 2 1 3 3 3 4 4 a b 1 a 1 2 2 ≥ 1 � � A pCTL formula: µ π | π has infinitely many a 3 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

. . . . . . . . . . . . . . . . . . . . . . . . a a a b b b b b p 1 − p a a a b 1 2 1 3 3 3 4 4 a b 1 a 1 2 2 ≥ 1 � � A pCTL formula: µ π | π has infinitely many a 3 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

. . . . . . . . . . . . . . . . . . . . . . . . a a a b b b b b p 1 − p a a a b 1 2 1 3 3 3 4 4 a b 1 a 1 2 2 ≥ 1 � � A pCTL formula: µ π | π has infinitely many a 3 Open Problem (Lehmann–Shelah, 82). Given a formula φ ∃ M . ( M | = φ )? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Regular Sets of Trees and Probability Matteo Mio CNRS – ENS Lyon, France Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Σ = a finite alphabet Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Σ = a finite alphabet t : { L , R } ∗ → Σ T Σ = Σ-labeled binary trees: Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Σ = a finite alphabet t : { L , R } ∗ → Σ T Σ = Σ-labeled binary trees: Example: Σ = { 0 , 1 } . . . . . . . . . . . . . . . . . . . . . . . . 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Σ = a finite alphabet t : { L , R } ∗ → Σ T Σ = Σ-labeled binary trees: Example: Σ = { 0 , 1 } . . . . . . . . . . . . . . . . . . . . . . . . 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 Definition : A set L ⊆T { 0 , 1 } is regular if it is definable by a S2S formula φ ( X ). Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 1 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 1 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 1 0 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 1 0 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Random Generation of Σ-labeled trees Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = { 0 , 1 } 0 1 1 0 1 1 0 1 0 1 1 0 1 0 0 Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Formally: Probability measure µ on the (Cantor) space T Σ . Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Formally: Probability measure µ on the (Cantor) space T Σ . Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Formally: Probability measure µ on the (Cantor) space T Σ . Question 1: Given a regular set L ⊆T Σ what is the value of µ ( L ) ? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Formally: Probability measure µ on the (Cantor) space T Σ . Question 1: Given a regular set L ⊆T Σ what is the value of µ ( L ) ? Question 0: are all regular sets µ -measurable? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Measurability of Regular Sets Question 0: Are regular sets L ⊆T Σ measurable? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Measurability of Regular Sets Question 0: Are regular sets L ⊆T Σ measurable? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Measurability of Regular Sets Question: Are regular sets L ⊆T Σ measurable? Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Measurability of Regular Sets Question: Are regular sets L ⊆T Σ measurable? Spoiler: answer is yes. Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Measurability of Regular Sets Question: Are regular sets L ⊆T Σ measurable? Spoiler: answer is yes. ◮ Using a rather advanced theorem (proved using forcing ) from set-theory. J. Fenstad and D. Normann, On absolutely measurable sets , Fundamenta Mathematicae, 1974. Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Kolmogorov’s R -sets Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Kolmogorov’s R -sets Goal (1928) : Find a large σ -algebra of definable measurable sets. Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Kolmogorov’s R -sets Goal (1928) : Find a large σ -algebra of definable measurable sets. � � ◮ Borel Sets = σ Open , � n , ¬ Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

Kolmogorov’s R -sets Goal (1928) : Find a large σ -algebra of definable measurable sets. � � ◮ Borel Sets = σ Open , � n , ¬ ◮ σ -algebra generated by Suslin operation (1918) = � � Open , A , ¬ σ Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018