Almost sure theory for first order logic on Galton- Watson trees - - PowerPoint PPT Presentation

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Almost sure theory for first order logic on Galton-Watson trees and probabilities of local neighbourhoods of the root

Moumanti Podder Joint work with Joel Spencer

Courant Institute of Mathematical Sciences New York University

14th Annual Northeast Probability Seminar November 20, 2015

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The First Order (F.O.) World

T random Galton-Watson tree, Poisson(λ) offspring distribution.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The First Order (F.O.) World

T random Galton-Watson tree, Poisson(λ) offspring distribution. Constant Symbol: root Equality: x = y, Parent: π(y) = x (x is parent of y, binary predicate), Variable Symbols x, y, z . . ., Boolean ∨, ∧, ¬, →, ↔, etc, Quantification ∀x, ∃y over vertices only.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The First Order (F.O.) World

T random Galton-Watson tree, Poisson(λ) offspring distribution. Constant Symbol: root Equality: x = y, Parent: π(y) = x (x is parent of y, binary predicate), Variable Symbols x, y, z . . ., Boolean ∨, ∧, ¬, →, ↔, etc, Quantification ∀x, ∃y over vertices only. Example ∃ a node with exactly one child and one grandchild.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

3

(xi, yi) ∈ T1 × T2, 1 ≤ i ≤ k, pairs of nodes selected.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

3

(xi, yi) ∈ T1 × T2, 1 ≤ i ≤ k, pairs of nodes selected.

4

Duplicator wins if

xi = R1 ⇐ ⇒ yi = R2,

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

3

(xi, yi) ∈ T1 × T2, 1 ≤ i ≤ k, pairs of nodes selected.

4

Duplicator wins if

xi = R1 ⇐ ⇒ yi = R2, π(xj) = xi ⇐ ⇒ π(yj) = yi,

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

3

(xi, yi) ∈ T1 × T2, 1 ≤ i ≤ k, pairs of nodes selected.

4

Duplicator wins if

xi = R1 ⇐ ⇒ yi = R2, π(xj) = xi ⇐ ⇒ π(yj) = yi, xi = xj ⇐ ⇒ yi = yj.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games

Definition

1

Trees T1, T2, roots R1, R2, # moves = k.

2

Spoiler picks any one tree and a node from it. Duplicator chooses a node from the other tree.

3

(xi, yi) ∈ T1 × T2, 1 ≤ i ≤ k, pairs of nodes selected.

4

Duplicator wins if

xi = R1 ⇐ ⇒ yi = R2, π(xj) = xi ⇐ ⇒ π(yj) = yi, xi = xj ⇐ ⇒ yi = yj.

Theorem If Duplicator wins EHR[T1, T2, k] then T1 | = A ⇐ ⇒ T2 | = A for F .O. A of depth k.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht value

Definition T1 ≡k T2 if Duplicator wins EHR[T1, T2, k].

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht value

Definition T1 ≡k T2 if Duplicator wins EHR[T1, T2, k]. Theorem Fix k. Only finitely many equivalence classes.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht value

Definition T1 ≡k T2 if Duplicator wins EHR[T1, T2, k]. Theorem Fix k. Only finitely many equivalence classes. Definition Equivalence class of T its Ehrenfeucht value.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Our results on almost sure theory for F.O.

Theorem Fix k ∈ N. Fix a finite tree T0. A[T0] := {∃ a subtree ∼ = T0 in T}. Conditioned on the tree being infinte, A is almost surely true. Schema A = {A[T0] : ∀ T0 finite tree} gives almost sure theory for infinite trees.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Consequence of previous result

Corollary Fix k ∈ N. Condition on T being infinite. Ehrenfeucht value of T depends on the local neighbourhood of the root, of radius ≈ 3k+2. For all A = A[T0], P[A] = P[A∗] where A∗ only depends on the local neighbourhood of the root.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness

1

Only concerned with Γ1 = {0, 1, 2, . . . k − 1, ω}, ω indicates ≥ k.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness

1

Only concerned with Γ1 = {0, 1, 2, . . . k − 1, ω}, ω indicates ≥ k.

2

Ai = {R has i children}, i = 1, 2, . . . k − 1, ω.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness

1

Only concerned with Γ1 = {0, 1, 2, . . . k − 1, ω}, ω indicates ≥ k.

2

Ai = {R has i children}, i = 1, 2, . . . k − 1, ω.

3

B = {T is finite}, P[B] = p.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness

1

Only concerned with Γ1 = {0, 1, 2, . . . k − 1, ω}, ω indicates ≥ k.

2

Ai = {R has i children}, i = 1, 2, . . . k − 1, ω.

3

B = {T is finite}, P[B] = p.

4

For i ∈ {0, 1, . . . k − 1} P[Ai ∩ Bc] =P[Ai] − P[Ai ∩ B] =e−λ · λi i! (1 − pi).

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness

1

Only concerned with Γ1 = {0, 1, 2, . . . k − 1, ω}, ω indicates ≥ k.

2

Ai = {R has i children}, i = 1, 2, . . . k − 1, ω.

3

B = {T is finite}, P[B] = p.

4

For i ∈ {0, 1, . . . k − 1} P[Ai ∩ Bc] =P[Ai] − P[Ai ∩ B] =e−λ · λi i! (1 − pi).

5

For ω children: P[Aω ∩ Bc] =P[Aω] − P[Aω ∩ B] =

∞

• j=k

e−λ · λj j! [1 − pj].

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Some definitions

Definition

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Some definitions

Definition

1

For 0 ≤ i ≤ k − 1, Pi(x) = P[Poi(x) = i] = e−x xi

i! .

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Some definitions

Definition

1

For 0 ≤ i ≤ k − 1, Pi(x) = P[Poi(x) = i] = e−x xi

i! .

2

For ≥ k, Pω(x) = P[Poi(x) ≥ k] = e−x ∞

j=k xj j! .

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Some definitions

Definition

1

For 0 ≤ i ≤ k − 1, Pi(x) = P[Poi(x) = i] = e−x xi

i! .

2

For ≥ k, Pω(x) = P[Poi(x) ≥ k] = e−x ∞

j=k xj j! .

3

(i + 1)-generation neighbourhood Γi+1 = {g : Γi → Γ1}.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Some definitions

Definition

1

For 0 ≤ i ≤ k − 1, Pi(x) = P[Poi(x) = i] = e−x xi

i! .

2

For ≥ k, Pω(x) = P[Poi(x) ≥ k] = e−x ∞

j=k xj j! .

3

(i + 1)-generation neighbourhood Γi+1 = {g : Γi → Γ1}.

4

For τ ∈ Γi, Pτ(x) = P[i−generation neighbourhood ∈ τ for Poi(x)].

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Example + Illustration

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Example + Illustration

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Example + Illustration

k = 3, g(0) = 0, g(1) = ω, g(2) = 1, g(ω) = 1.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Example + Illustration

k = 3, g(0) = 0, g(1) = ω, g(2) = 1, g(ω) = 1. Pσ(x) =

2

• i=0

Pg(i)(xPi(x)) · Pg(ω)(xPω(x)) =e−x ·   

∞

• j=3

(xP1(x))j j!    · (xP2(x)) · (xPω(x))

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Recursive computation of probabilities

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Recursive computation of probabilities

Theorem Pσ(x) =

• τ∈Γi

Pg(τ)(xPτ(x)) ∀ σ = g ∈ Γi+1.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Recursive computation of probabilities

Theorem Pσ(x) =

• τ∈Γi

Pg(τ)(xPτ(x)) ∀ σ = g ∈ Γi+1. Theorem B = {T is finite}, P[B] = p.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Recursive computation of probabilities

Theorem Pσ(x) =

• τ∈Γi

Pg(τ)(xPτ(x)) ∀ σ = g ∈ Γi+1. Theorem B = {T is finite}, P[B] = p. By duality for Galton-Watson trees, P[{i − generation neighbourhood = σ}|B] = Pσ(pλ).

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Recursive computation of probabilities

Theorem Pσ(x) =

• τ∈Γi

Pg(τ)(xPτ(x)) ∀ σ = g ∈ Γi+1. Theorem B = {T is finite}, P[B] = p. By duality for Galton-Watson trees, P[{i − generation neighbourhood = σ}|B] = Pσ(pλ). If P∗[σ] = P[{i − generation neighbourhood = σ}|Bc], then P∗[σ] = Pσ(λ) − p · Pσ(pλ) 1 − p .

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The probabilities are nice functions

Remark For all i and σ ∈ Γi, Pσ(x) nice function. Consists of polynomials in p, x, e−x, and base e exponentiation.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The probabilities are nice functions

Remark For all i and σ ∈ Γi, Pσ(x) nice function. Consists of polynomials in p, x, e−x, and base e exponentiation. Example

1

A := {Root has no child with no child}, k ≥ 1.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The probabilities are nice functions

Remark For all i and σ ∈ Γi, Pσ(x) nice function. Consists of polynomials in p, x, e−x, and base e exponentiation. Example

1

A := {Root has no child with no child}, k ≥ 1.

2

A = {g(0) = 0}.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The probabilities are nice functions

Remark For all i and σ ∈ Γi, Pσ(x) nice function. Consists of polynomials in p, x, e−x, and base e exponentiation. Example

1

A := {Root has no child with no child}, k ≥ 1.

2

A = {g(0) = 0}.

3

P[A] = P0[λP0(λ)] = e−λP0(λ) = e−λe−λ.

Almost sure theory for first

• rder logic on

Galton- Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Thank you.