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

▶
regular sets of trees and probability
SMART_READER_LITE
LIVE PREVIEW

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

Jan 25, 2024 132 likes •1.12k views

Regular Sets of Trees and Probability Matteo Mio CNRS & ENSLyon 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

slide-1
SLIDE 1

Regular Sets of Trees and Probability

Matteo Mio CNRS & ENS–Lyon

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

slide-2
SLIDE 2

Some quick background

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

slide-3
SLIDE 3

Some quick background

Automata Theory is used to prove decidability

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

slide-4
SLIDE 4

Some quick background

Automata Theory is used to prove decidability

◮ Presburger arithmetic FO(N, <, +), ◮ Linear arithmetic FO(R, <, +), ◮ Some other theories FO

  • Π0

1(R), ∪, ∩

  • Matteo Mio

Workshop on Wadge Theory and Automata II, Torino, 2018

slide-5
SLIDE 5

Some quick background

Automata Theory is used to prove decidability

◮ Presburger arithmetic FO(N, <, +), ◮ Linear arithmetic FO(R, <, +), ◮ Some other theories FO

  • Π0

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

slide-6
SLIDE 6

Classical Temporal Logics

◮ Example: Computation Tree Logic (CTL) ◮ Models: Labeled Trees

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

slide-7
SLIDE 7

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

slide-8
SLIDE 8

. . . . . . . . . . . . . . . . . . . . . . . . a b a b a a a b b a b a a b b

1 2 1 2 1 3 2 3 1 4 3 4

p 1 − p

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

slide-9
SLIDE 9

. . . . . . . . . . . . . . . . . . . . . . . . a b a b a a a b b a b a a b b

1 2 1 2 1 3 2 3 1 4 3 4

p 1 − p A pCTL formula: µ

  • π | π has infinitely many a
  • ≥ 1

3

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

slide-10
SLIDE 10

. . . . . . . . . . . . . . . . . . . . . . . . a b a b a a a b b a b a a b b

1 2 1 2 1 3 2 3 1 4 3 4

p 1 − p A pCTL formula: µ

  • π | π has infinitely many a
  • ≥ 1

3

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

slide-11
SLIDE 11

. . . . . . . . . . . . . . . . . . . . . . . . a b a b a a a b b a b a a b b

1 2 1 2 1 3 2 3 1 4 3 4

p 1 − p A pCTL formula: µ

  • π | π has infinitely many a
  • ≥ 1

3

Open Problem (Lehmann–Shelah, 82). Given a formula φ ∃M.(M | = φ)?

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

slide-12
SLIDE 12

Regular Sets of Trees and Probability

Matteo Mio CNRS – ENS Lyon, France

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

slide-13
SLIDE 13

Σ = a finite alphabet

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

slide-14
SLIDE 14

Σ = a finite alphabet TΣ = Σ-labeled binary trees: t : {L, R}∗ →Σ

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

slide-15
SLIDE 15

Σ = a finite alphabet TΣ = Σ-labeled binary trees: t : {L, R}∗ →Σ Example: Σ = {0, 1} . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1

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

slide-16
SLIDE 16

Σ = a finite alphabet TΣ = Σ-labeled binary trees: t : {L, R}∗ →Σ Example: Σ = {0, 1} . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 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

slide-17
SLIDE 17

Random Generation of Σ-labeled trees

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

slide-18
SLIDE 18

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

slide-19
SLIDE 19

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

slide-20
SLIDE 20

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

slide-21
SLIDE 21

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

slide-22
SLIDE 22

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

slide-23
SLIDE 23

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1

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

slide-24
SLIDE 24

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1

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

slide-25
SLIDE 25

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1

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

slide-26
SLIDE 26

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1

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

slide-27
SLIDE 27

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1 1

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

slide-28
SLIDE 28

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1 1 1

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

slide-29
SLIDE 29

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1 1 1

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

slide-30
SLIDE 30

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1 1 1

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

slide-31
SLIDE 31

Random Generation of Σ-labeled trees

Intuition: generate a Σ-labeled tree by fair-coin tosses. Example: Σ = {0, 1} 1 1 1 1 1 1 1 1

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

slide-32
SLIDE 32

Formally: Probability measure µ on the (Cantor) space TΣ.

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

slide-33
SLIDE 33

Formally: Probability measure µ on the (Cantor) space TΣ.

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

slide-34
SLIDE 34

Formally: Probability measure µ on the (Cantor) space TΣ. Question 1: Given a regular set L⊆TΣ what is the value of µ(L) ?

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

slide-35
SLIDE 35

Formally: Probability measure µ on the (Cantor) space TΣ. Question 1: Given a regular set L⊆TΣ 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

slide-36
SLIDE 36

Measurability of Regular Sets

Question 0: Are regular sets L⊆TΣ measurable?

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

slide-37
SLIDE 37

Measurability of Regular Sets

Question 0: Are regular sets L⊆TΣ measurable?

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

slide-38
SLIDE 38

Measurability of Regular Sets

Question: Are regular sets L⊆TΣ measurable?

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

slide-39
SLIDE 39

Measurability of Regular Sets

Question: Are regular sets L⊆TΣ measurable? Spoiler: answer is yes.

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

slide-40
SLIDE 40

Measurability of Regular Sets

Question: Are regular sets L⊆TΣ 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

slide-41
SLIDE 41

Kolmogorov’s R-sets

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

slide-42
SLIDE 42

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

slide-43
SLIDE 43

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

slide-44
SLIDE 44

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) =

σ

  • Open, A, ¬
  • Matteo Mio

Workshop on Wadge Theory and Automata II, Torino, 2018

slide-45
SLIDE 45

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) =

σ

  • Open, A, ¬
  • Idea: define operator (transform) R acting on operations.

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

slide-46
SLIDE 46

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) =

σ

  • Open, A, ¬
  • Idea: define operator (transform) R acting on operations.

◮ R( n) = A.

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

slide-47
SLIDE 47

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) =

σ

  • Open, A, ¬
  • Idea: define operator (transform) R acting on operations.

◮ R( n) = A. ◮ R(A) a new and more expressive operation on sets.

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

slide-48
SLIDE 48

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) =

σ

  • Open, A, ¬
  • Idea: define operator (transform) R acting on operations.

◮ R( n) = A. ◮ R(A) a new and more expressive operation on sets. ◮ RR(A) . . .

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

slide-49
SLIDE 49

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) =

σ

  • Open, A, ¬
  • Idea: define operator (transform) R acting on operations.

◮ R( n) = A. ◮ R(A) a new and more expressive operation on sets. ◮ RR(A) . . .

Kolmogorov’s σ-algebra of R-sets: σ

  • Open, {Rn}n, ¬
  • Matteo Mio

Workshop on Wadge Theory and Automata II, Torino, 2018

slide-50
SLIDE 50

Theorem (Kolmogorov, 1928): Every R-set is measurable.

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

slide-51
SLIDE 51

Theorem (Kolmogorov, 1928): Every R-set is measurable.

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

slide-52
SLIDE 52

Theorem (Kolmogorov, 1928): Every R-set is measurable. Theorem: (Gogacz, Michalewski, Mio, Skrzypczak, 2017)

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

slide-53
SLIDE 53

Theorem (Kolmogorov, 1928): Every R-set is measurable. Theorem: (Gogacz, Michalewski, Mio, Skrzypczak, 2017)

◮ Game languages Wi,k are complete for the finite levels of the

hierarchy of R-sets.

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

slide-54
SLIDE 54

Question 1: Given a regular set L⊆TΣ, what is the value µ(L)?

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

slide-55
SLIDE 55

Question 1: Given a regular set L⊆TΣ, what is the value µ(L)? Algorithmic version: Given a presentation of L (e.g., tree automaton) and numbers a, b ∈ [0, 1] ∩ Q decide if a < µ(L) < b

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

slide-56
SLIDE 56

Question 1: Given a regular set L⊆TΣ, what is the value µ(L)? Algorithmic version: Given a presentation of L (e.g., tree automaton) and numbers a, b ∈ [0, 1] ∩ Q decide if a < µ(L) < b Open problem

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

slide-57
SLIDE 57

A partial solution. Theorem: (2015, Michalewski, Mio) µ(L) is computable if L is definable by so-called game automata.

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

slide-58
SLIDE 58

A partial solution. Theorem: (2015, Michalewski, Mio) µ(L) is computable if L is definable by so-called game automata. Our algorithm is quite involved:

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

slide-59
SLIDE 59

A partial solution. Theorem: (2015, Michalewski, Mio) µ(L) is computable if L is definable by so-called game automata. Our algorithm is quite involved:

  • 1. Reduction to a systems of nested

(co)inductive polynomial equations,

  • 2. Solve these equations using Tarski’s decision procedure for

FO(R, +, ×, 0, 1).

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

slide-60
SLIDE 60

L1 ⊆T{a,b,c}

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

slide-61
SLIDE 61

L1 ⊆T{a,b,c} Algorithm calculates: µ(L1) = 1

2

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

slide-62
SLIDE 62

L2 ⊆T{a,b,c}

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

slide-63
SLIDE 63

L2 ⊆T{a,b,c} Algorithm calculates: µ(L2) = 1

4(3 −

√ 7) ≈ 0.088

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

slide-64
SLIDE 64

One application Model Checking of Markov Branching Processes

R 0.1 D R 0.89 (R, R) R 0.01 M M 0.9 D M 0.1 (R, M, M, M, M) D

1

D

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

slide-65
SLIDE 65

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

slide-66
SLIDE 66

Other curiosities

Lemma (Measure = Baire Category)

There are regular sets L such that: µ(L) = 0 and L is comeager (second category)

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

slide-67
SLIDE 67

Other curiosities

Lemma (Measure = Baire Category)

There are regular sets L such that: µ(L) = 0 and L is comeager (second category) Example: {t | every path has ∞ many a}

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

slide-68
SLIDE 68

Other curiosities

Lemma (Measure = Baire Category)

There are regular sets L such that: µ(L) = 0 and L is comeager (second category) Example: {t | every path has ∞ many a}

Lemma (A Zero-One law)

The language Wi,k ⊆ TΣ with Σ = {∀, ∃} × {i, . . . , k} µ(Wi,k) = 1 if k is even µ(Wi,k) = 0 if k is odd.

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

slide-69
SLIDE 69

Some Questions (possibly easy)

Theorem (Regularity)

For every measurable A ⊆ TΣ, there is a Gδ set B ⊇ A such that µ(A) = µ(B).

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

slide-70
SLIDE 70

Some Questions (possibly easy)

Theorem (Regularity)

For every measurable A ⊆ TΣ, there is a Gδ set B ⊇ A such that µ(A) = µ(B). Question: Given A ⊆ TΣ regular, can we find B regular?

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

slide-71
SLIDE 71

Theorem

Let E ⊆ [0, 1] a Borel set. Then there exists a basic interval (a, b) such that µ

  • E ∩ (a, b)
  • ∈ {0, b − a}

(i.e., E has full or null measure in (a, b)).

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

slide-72
SLIDE 72

Theorem

Let E ⊆ [0, 1] a Borel set. Then there exists a basic interval (a, b) such that µ

  • E ∩ (a, b)
  • ∈ {0, b − a}

(i.e., E has full or null measure in (a, b)). Source: Facebook Group

Mathematical theorems you had no idea existed, cause they are false.

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

slide-73
SLIDE 73

Theorem

Let E ⊆ [0, 1] a Borel set. Then there exists a basic interval (a, b) such that µ

  • E ∩ (a, b)
  • ∈ {0, b − a}

(i.e., E has full or null measure in (a, b)). Source: Facebook Group

Mathematical theorems you had no idea existed, cause they are false.

Question: Is the theorem true for E ⊆ TΣ regular?

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

slide-74
SLIDE 74

Generalised Quantifiers

a b a . . . . . . a . . . . . . b a . . . . . . b . . . . . .

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

slide-75
SLIDE 75

Generalised Quantifiers

a b a . . . . . . a . . . . . . b a . . . . . . b . . . . . . Properties we can express in MSO:

◮ ∃X.

  • “X is a branch” ∧ φ(X)
  • Matteo Mio

Workshop on Wadge Theory and Automata II, Torino, 2018

slide-76
SLIDE 76

Generalised Quantifiers

a b a . . . . . . a . . . . . . b a . . . . . . b . . . . . . Properties we can express in MSO:

◮ ∃X.

  • “X is a branch” ∧ φ(X)
  • Properties we wish to express in probabilistic logics:

◮

X | “X is a branch” ∧ φ(X)

  • has probability = 1.

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

slide-77
SLIDE 77

Definition (Syntax of MSO+∀=1

π )

φ ::= ¬φ | φ1 ∨ φ2 | ∀x.φ | ∀X.φ | x ∈ X

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

slide-78
SLIDE 78

Definition (Syntax of MSO+∀=1

π )

φ ::= ¬φ | φ1 ∨ φ2 | ∀x.φ | ∀X.φ | x ∈ X | ∀=1

π X.φ

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

slide-79
SLIDE 79

Definition (Syntax of MSO+∀=1

π )

φ ::= ¬φ | φ1 ∨ φ2 | ∀x.φ | ∀X.φ | x ∈ X | ∀=1

π X.φ

∀=1

π X. φ(X) holds

⇔ {X | X is a branch and φ(X) holds } has coinflipping measure 1.

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

slide-80
SLIDE 80

Definition (Syntax of MSO+∀=1

π )

φ ::= ¬φ | φ1 ∨ φ2 | ∀x.φ | ∀X.φ | x ∈ X | ∀=1

π X.φ

∀=1

π X. φ(X) holds

⇔ {X | X is a branch and φ(X) holds } has coinflipping measure 1. Question: Is MSO+∀=1

π

decidable?

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

slide-81
SLIDE 81

∀=1

π X. φ(X) holds

⇔ {X | X is a branch and φ(X) holds } has coinflipping measure 1. ∀=1X. φ(X) holds ⇔ {X | φ(X) holds } has coinflipping measure 1.

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

slide-82
SLIDE 82

A formula φ(X, Y ) defines a set in the product space: TΣ × T n

Σ

# » Y X

φ(X, # » Y ) Matteo Mio Workshop on Wadge Theory and Automata II, Torino, 2018

slide-83
SLIDE 83

Interpretation of the ∀ quantifier:

# » Y X

φ(X, # » Y ) ∀X.φ(X, # » Y )

Takes full sections.

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

slide-84
SLIDE 84

Interpretation of the ∀=1 quantifier: Takes full large sections.

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

slide-85
SLIDE 85

Interpretation of the ∀=1 quantifier: Takes full large sections. Quantifier ∀=1X.φ(X) first studied by H. Friedman in 1979.

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

slide-86
SLIDE 86

Interpretation of the ∀∗ quantifier: taking large (= comeager) sections.

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

slide-87
SLIDE 87

Interpretation of the ∀∗ quantifier: taking large (= comeager) sections.

Theorem (Friedman, on Borel structures)

The theories of FO + ∀=1 and FO + ∀∗ coincide.

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

slide-88
SLIDE 88

Interpretation of the ∀∗ quantifier: taking large (= comeager) sections.

Theorem (Friedman, on Borel structures)

The theories of FO + ∀=1 and FO + ∀∗ coincide.

Theorem (Staiger, 97)

A regular L ⊆ Σω is comeager iff it has measure 1.

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

slide-89
SLIDE 89

Some Results

Mio, Skrzypczak, Michalewski: Monadic Second Order Logic with Measure and Category Quantifiers in LMCS, vol 14(2), 2018.

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

slide-90
SLIDE 90

Some Results

Mio, Skrzypczak, Michalewski: Monadic Second Order Logic with Measure and Category Quantifiers in LMCS, vol 14(2), 2018.

Theorem (on ω-words)

MSO + ∀∗ = MSO and therefore decidable.

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

slide-91
SLIDE 91

Some Results

Mio, Skrzypczak, Michalewski: Monadic Second Order Logic with Measure and Category Quantifiers in LMCS, vol 14(2), 2018.

Theorem (on ω-words)

MSO + ∀∗ = MSO and therefore decidable. MSO + ∀=1 MSO and undecidable.

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

slide-92
SLIDE 92

Theorem (on trees)

MSO + ∀=1 MSO and undecidable.

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

slide-93
SLIDE 93

Theorem (on trees)

MSO + ∀=1 MSO and undecidable. MSO + ∀∗ ? = MSO is an open problem.

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

slide-94
SLIDE 94

Theorem (on trees)

MSO + ∀=1 MSO and undecidable. MSO + ∀∗ ? = MSO is an open problem.

◮ Equality claimed in ICALP 2015, but wrong proof :-(

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

slide-95
SLIDE 95

Theorem (on trees)

MSO + ∀=1 MSO and undecidable. MSO + ∀∗ ? = MSO is an open problem.

◮ Equality claimed in ICALP 2015, but wrong proof :-( ◮ if φ(X,

Y ) is game–automata–definable then ∀∗X. φ(X, Y ) is MSO–definable (without ∀∗).

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

slide-96
SLIDE 96

Theorem (on trees)

MSO + ∀=1 MSO and undecidable. MSO + ∀∗ ? = MSO is an open problem.

◮ Equality claimed in ICALP 2015, but wrong proof :-( ◮ if φ(X,

Y ) is game–automata–definable then ∀∗X. φ(X, Y ) is MSO–definable (without ∀∗). MSO + ∀=1

π

MSO but (un)decidability is open.

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

slide-97
SLIDE 97

Theorem (on trees)

MSO + ∀=1 MSO and undecidable. MSO + ∀∗ ? = MSO is an open problem.

◮ Equality claimed in ICALP 2015, but wrong proof :-( ◮ if φ(X,

Y ) is game–automata–definable then ∀∗X. φ(X, Y ) is MSO–definable (without ∀∗). MSO + ∀=1

π

MSO but (un)decidability is open.

◮ (Bojanczyk) WMSO + ∀=1 π

is decidable.

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

slide-98
SLIDE 98

My Conclusion: Tree–Automata Theory + Probability = Easy

THANKS

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