Probability of any given neighbourhood of root, conditioned on the - PowerPoint PPT Presentation

Probability of any given neighbour- hood of the Probability of any given neighbourhood of root, conditioned on the root, conditioned on the tree being the tree being infinite infinite Moumanti Podder Moumanti Podder Courant Institute of Mathematical Sciences New York University Logic and Random Graphs Lorentz Center, Leiden August 31, 2015

Motivation Probability of any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

Motivation Probability of any given neighbour- For every given finite subtree T 0 , a random infinite tree hood of the T will almost surely contain v such that T ( v ) ∼ root, = T 0 . conditioned on the tree being infinite Moumanti Podder

Motivation Probability of any given neighbour- For every given finite subtree T 0 , a random infinite tree hood of the T will almost surely contain v such that T ( v ) ∼ root, = T 0 . conditioned on the tree being infinite Almost surely there exists v ∈ T far from root R such Moumanti that T ( v ) ∼ = UNIV ( k ) . Podder

Motivation Probability of any given neighbour- For every given finite subtree T 0 , a random infinite tree hood of the T will almost surely contain v such that T ( v ) ∼ root, = T 0 . conditioned on the tree being infinite Almost surely there exists v ∈ T far from root R such Moumanti that T ( v ) ∼ = UNIV ( k ) . Podder k -move Ehrenfeucht value of T thus determined by neighbourhood of R (of radius ≈ 4 k ). Need to compute probabilities of neighbourhoods conditioned on T infinite.

Motivation Probability of any given neighbour- For every given finite subtree T 0 , a random infinite tree hood of the T will almost surely contain v such that T ( v ) ∼ root, = T 0 . conditioned on the tree being infinite Almost surely there exists v ∈ T far from root R such Moumanti that T ( v ) ∼ = UNIV ( k ) . Podder k -move Ehrenfeucht value of T thus determined by neighbourhood of R (of radius ≈ 4 k ). Need to compute probabilities of neighbourhoods conditioned on T infinite. Remark In fact, able to compute P [ A | T is infinite] for any first order statement A.

First generation Probability of any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, conditioned on the tree being infinite Moumanti Podder

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, P [ T is finite ] = p ∈ ( 0 , 1 ) . 3 conditioned on the tree being infinite Moumanti Podder

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, P [ T is finite ] = p ∈ ( 0 , 1 ) . 3 conditioned on P [ A i ∩ B ] = e − λ · λ i the tree being i ! · p i , i ∈ { 1 , . . . k − 1 } . 4 infinite Moumanti Podder

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, P [ T is finite ] = p ∈ ( 0 , 1 ) . 3 conditioned on P [ A i ∩ B ] = e − λ · λ i the tree being i ! · p i , i ∈ { 1 , . . . k − 1 } . 4 infinite 5 Moumanti Podder P [ A i ∩ B c ] = P [ A i ] − P [ A i ∩ B ] = e − λ · λ i i ! ( 1 − p i ) , i ∈ { 1 , . . . k − 1 } .

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, P [ T is finite ] = p ∈ ( 0 , 1 ) . 3 conditioned on P [ A i ∩ B ] = e − λ · λ i the tree being i ! · p i , i ∈ { 1 , . . . k − 1 } . 4 infinite 5 Moumanti Podder P [ A i ∩ B c ] = P [ A i ] − P [ A i ∩ B ] = e − λ · λ i i ! ( 1 − p i ) , i ∈ { 1 , . . . k − 1 } . j = k e − λ · λ j P [ A many ∩ B ] = � ∞ j ! · p j . 6

First generation Probability of A i = { R has i children } , i = 1 , 2 , . . . k − 1 , many . 1 any given neighbour- B = { T is finite } . 2 hood of the root, P [ T is finite ] = p ∈ ( 0 , 1 ) . 3 conditioned on P [ A i ∩ B ] = e − λ · λ i the tree being i ! · p i , i ∈ { 1 , . . . k − 1 } . 4 infinite 5 Moumanti Podder P [ A i ∩ B c ] = P [ A i ] − P [ A i ∩ B ] = e − λ · λ i i ! ( 1 − p i ) , i ∈ { 1 , . . . k − 1 } . j = k e − λ · λ j P [ A many ∩ B ] = � ∞ j ! · p j . 6 7 P [ A many ∩ B c ] = P [ A many ] − P [ A many ∩ B ] ∞ e − λ · λ j � j ! ( 1 − p j ) . = j = k

Second generation Probability of any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

Second generation Probability of any given neighbour- hood of the root, conditioned on We will do this in two parts: the tree being infinite Moumanti Podder

Second generation Probability of any given neighbour- hood of the root, conditioned on We will do this in two parts: the tree being infinite First find P [ A ∩ B ] where Moumanti 1 Podder

Second generation Probability of any given neighbour- hood of the root, conditioned on We will do this in two parts: the tree being infinite First find P [ A ∩ B ] where Moumanti 1 Podder A is the event that the root R has a given 2-generation neighbourhood;

Second generation Probability of any given neighbour- hood of the root, conditioned on We will do this in two parts: the tree being infinite First find P [ A ∩ B ] where Moumanti 1 Podder A is the event that the root R has a given 2-generation neighbourhood; B is the event that the tree is finite.

Second generation Probability of any given neighbour- hood of the root, conditioned on We will do this in two parts: the tree being infinite First find P [ A ∩ B ] where Moumanti 1 Podder A is the event that the root R has a given 2-generation neighbourhood; B is the event that the tree is finite. Then find P [ A ] . Finally find P [ A ∩ B c ] = P [ A ] − P [ A ∩ B ] . 2

First step Probability of any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

First step Probability of Definition any given neighbour- A child v of the root R is of type: hood of the root, conditioned on α i , i ∈ { 0 , 1 , . . . k − 1 , many } , if v has i children and 1 the tree being infinite T ( v ) finite; Moumanti O otherwise. Podder 2

First step Probability of Definition any given neighbour- A child v of the root R is of type: hood of the root, conditioned on α i , i ∈ { 0 , 1 , . . . k − 1 , many } , if v has i children and 1 the tree being infinite T ( v ) finite; Moumanti O otherwise. Podder 2 X i , i ∈ { 0 , 1 , . . . k − 1 , many } , number of children of R of type α i ;

First step Probability of Definition any given neighbour- A child v of the root R is of type: hood of the root, conditioned on α i , i ∈ { 0 , 1 , . . . k − 1 , many } , if v has i children and 1 the tree being infinite T ( v ) finite; Moumanti O otherwise. Podder 2 X i , i ∈ { 0 , 1 , . . . k − 1 , many } , number of children of R of type α i ; Y number of children of R of type O .

First step Probability of Definition any given neighbour- A child v of the root R is of type: hood of the root, conditioned on α i , i ∈ { 0 , 1 , . . . k − 1 , many } , if v has i children and 1 the tree being infinite T ( v ) finite; Moumanti O otherwise. Podder 2 X i , i ∈ { 0 , 1 , . . . k − 1 , many } , number of children of R of type α i ; Y number of children of R of type O . Lemma X i ∼ Poi ( λ p i ) , i ∈ { 0 , . . . many } , Y ∼ Poi ( λ ( 1 − p 0 − . . . p many )) and all mutually independent.

What are these p i ’s? Probability of any given neighbour- hood of the root, conditioned on the tree being infinite Moumanti Podder

What are these p i ’s? Probability of Given R has n children, any given neighbour- hood of the ( X 0 , . . . X many , Y ) ∼ Mult ( n , p 0 , . . . p many , 1 − p 0 − . . . p many ) . root, conditioned on the tree being infinite Moumanti Podder

What are these p i ’s? Probability of Given R has n children, any given neighbour- hood of the ( X 0 , . . . X many , Y ) ∼ Mult ( n , p 0 , . . . p many , 1 − p 0 − . . . p many ) . root, conditioned on the tree being infinite Moumanti For i ∈ { 0 , 1 , . . . k − 1 } , Podder

What are these p i ’s? Probability of Given R has n children, any given neighbour- hood of the ( X 0 , . . . X many , Y ) ∼ Mult ( n , p 0 , . . . p many , 1 − p 0 − . . . p many ) . root, conditioned on the tree being infinite Moumanti For i ∈ { 0 , 1 , . . . k − 1 } , Podder p i = e − λ · λ i i ! · p i , and