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

Almost sure theory for first order logic on Almost sure theory for first order logic on Galton- Watson trees and Galton-Watson trees and probabilities of probabilities of local neigh- bourhoods of local neighbourhoods of the root the root Moumanti Podder Joint work with Joel Moumanti Podder Spencer Joint work with Joel Spencer Courant Institute of Mathematical Sciences New York University 14th Annual Northeast Probability Seminar November 20, 2015

The First Order (F.O.) World Almost sure theory for first order logic on T random Galton-Watson tree, Poisson ( λ ) offspring Galton- Watson trees distribution. and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

The First Order (F.O.) World Almost sure theory for first order logic on T random Galton-Watson tree, Poisson ( λ ) offspring Galton- Watson trees distribution. and probabilities of Constant Symbol: root local neigh- bourhoods of the root Equality: x = y , Moumanti Parent: π ( y ) = x ( x is parent of y , binary predicate), Podder Joint work with Joel Variable Symbols x , y , z . . . , Spencer Boolean ∨ , ∧ , ¬ , → , ↔ , etc, Quantification ∀ x , ∃ y over vertices only .

The First Order (F.O.) World Almost sure theory for first order logic on T random Galton-Watson tree, Poisson ( λ ) offspring Galton- Watson trees distribution. and probabilities of Constant Symbol: root local neigh- bourhoods of the root Equality: x = y , Moumanti Parent: π ( y ) = x ( x is parent of y , binary predicate), Podder Joint work with Joel Variable Symbols x , y , z . . . , Spencer Boolean ∨ , ∧ , ¬ , → , ↔ , etc, Quantification ∀ x , ∃ y over vertices only . Example ∃ a node with exactly one child and one grandchild.

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root ( x i , y i ) ∈ T 1 × T 2 , 1 ≤ i ≤ k, pairs of nodes selected. 3 Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root ( x i , y i ) ∈ T 1 × T 2 , 1 ≤ i ≤ k, pairs of nodes selected. 3 Moumanti Podder Duplicator wins if 4 Joint work with Joel x i = R 1 ⇐ ⇒ y i = R 2 , Spencer

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root ( x i , y i ) ∈ T 1 × T 2 , 1 ≤ i ≤ k, pairs of nodes selected. 3 Moumanti Podder Duplicator wins if 4 Joint work with Joel x i = R 1 ⇐ ⇒ y i = R 2 , Spencer π ( x j ) = x i ⇐ ⇒ π ( y j ) = y i ,

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root ( x i , y i ) ∈ T 1 × T 2 , 1 ≤ i ≤ k, pairs of nodes selected. 3 Moumanti Podder Duplicator wins if 4 Joint work with Joel x i = R 1 ⇐ ⇒ y i = R 2 , Spencer π ( x j ) = x i ⇐ ⇒ π ( y j ) = y i , x i = x j ⇐ ⇒ y i = y j .

Ehrenfeucht games Almost sure Definition theory for first order logic on Galton- Trees T 1 , T 2 , roots R 1 , R 2 , # moves = k. 1 Watson trees and Spoiler picks any one tree and a node from it. probabilities of 2 local neigh- Duplicator chooses a node from the other tree. bourhoods of the root ( x i , y i ) ∈ T 1 × T 2 , 1 ≤ i ≤ k, pairs of nodes selected. 3 Moumanti Podder Duplicator wins if 4 Joint work with Joel x i = R 1 ⇐ ⇒ y i = R 2 , Spencer π ( x j ) = x i ⇐ ⇒ π ( y j ) = y i , x i = x j ⇐ ⇒ y i = y j . Theorem If Duplicator wins EHR [ T 1 , T 2 , k ] then T 1 | = A ⇐ ⇒ T 2 | = A for F .O. A of depth k .

Ehrenfeucht value Almost sure theory for first order logic on Galton- Watson trees and Definition probabilities of local neigh- T 1 ≡ k T 2 if Duplicator wins EHR [ T 1 , T 2 , k ] . bourhoods of the root Moumanti Podder Joint work with Joel Spencer

Ehrenfeucht value Almost sure theory for first order logic on Galton- Watson trees and Definition probabilities of local neigh- T 1 ≡ k T 2 if Duplicator wins EHR [ T 1 , T 2 , k ] . bourhoods of the root Moumanti Podder Theorem Joint work with Joel Fix k. Only finitely many equivalence classes. Spencer

Ehrenfeucht value Almost sure theory for first order logic on Galton- Watson trees and Definition probabilities of local neigh- T 1 ≡ k T 2 if Duplicator wins EHR [ T 1 , T 2 , k ] . bourhoods of the root Moumanti Podder Theorem Joint work with Joel Fix k. Only finitely many equivalence classes. Spencer Definition Equivalence class of T its Ehrenfeucht value.

Our results on almost sure theory for F.O. Almost sure theory for first order logic on Galton- Watson trees and Theorem probabilities of local neigh- Fix k ∈ N . bourhoods of the root Fix a finite tree T 0 . A [ T 0 ] := {∃ a subtree ∼ Moumanti = T 0 in T } . Podder Joint work with Joel Conditioned on the tree being infinte, A is almost surely Spencer true. Schema A = { A [ T 0 ] : ∀ T 0 finite tree } gives almost sure theory for infinite trees.

Consequence of previous result Almost sure theory for first order logic on Galton- Watson trees and probabilities of Corollary local neigh- bourhoods of Fix k ∈ N . Condition on T being infinite. the root Moumanti Podder Ehrenfeucht value of T depends on the local Joint work with Joel neighbourhood of the root, of radius ≈ 3 k + 2 . Spencer P [ A ] = P [ A ∗ ] where A ∗ only For all A = A [ T 0 ] , depends on the local neighbourhood of the root.

First generation probability conditioned on infiniteness Almost sure Only concerned with Γ 1 = { 0 , 1 , 2 , . . . k − 1 , ω } , ω 1 theory for first order logic on indicates ≥ k . 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 Almost sure Only concerned with Γ 1 = { 0 , 1 , 2 , . . . k − 1 , ω } , ω 1 theory for first order logic on indicates ≥ k . Galton- Watson trees A i = { R has i children } , i = 1 , 2 , . . . k − 1 , ω. 2 and probabilities of local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness Almost sure Only concerned with Γ 1 = { 0 , 1 , 2 , . . . k − 1 , ω } , ω 1 theory for first order logic on indicates ≥ k . Galton- Watson trees A i = { R has i children } , i = 1 , 2 , . . . k − 1 , ω. 2 and probabilities of B = { T is finite } , P [ B ] = p . 3 local neigh- bourhoods of the root Moumanti Podder Joint work with Joel Spencer

First generation probability conditioned on infiniteness Almost sure Only concerned with Γ 1 = { 0 , 1 , 2 , . . . k − 1 , ω } , ω 1 theory for first order logic on indicates ≥ k . Galton- Watson trees A i = { R has i children } , i = 1 , 2 , . . . k − 1 , ω. 2 and probabilities of B = { T is finite } , P [ B ] = p . 3 local neigh- bourhoods of For i ∈ { 0 , 1 , . . . k − 1 } 4 the root Moumanti P [ A i ∩ B c ] = P [ A i ] − P [ A i ∩ B ] Podder Joint work = e − λ · λ i with Joel i ! ( 1 − p i ) . Spencer

First generation probability conditioned on infiniteness Almost sure Only concerned with Γ 1 = { 0 , 1 , 2 , . . . k − 1 , ω } , ω 1 theory for first order logic on indicates ≥ k . Galton- Watson trees A i = { R has i children } , i = 1 , 2 , . . . k − 1 , ω. 2 and probabilities of B = { T is finite } , P [ B ] = p . 3 local neigh- bourhoods of For i ∈ { 0 , 1 , . . . k − 1 } 4 the root Moumanti P [ A i ∩ B c ] = P [ A i ] − P [ A i ∩ B ] Podder Joint work = e − λ · λ i with Joel i ! ( 1 − p i ) . Spencer For ω children: 5 P [ A ω ∩ B c ] = P [ A ω ] − P [ A ω ∩ B ] ∞ e − λ · λ j � j ! [ 1 − p j ] . = j = k