Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be...
Transcript of Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be...
![Page 1: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/1.jpg)
Cellular Automata, Lecture 2
Math Circles 2018University of Waterloo
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Part I
Multistate and Totalistic Automata
![Page 3: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/3.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 4: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/4.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).
These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 5: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/5.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 6: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/6.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 7: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/7.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 8: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/8.jpg)
Multistate automata
Last time: only allowed cells to be ALIVE or DEAD (1 or 0).These are called the elementary cellular automata, and there are256 of them.
What about three states?
![Page 9: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/9.jpg)
Multistate automata
Some interesting 3-color automata:
I Rule 679458
I Rule 3333333
I Rule 21111
CellularAutomaton[{679458,3}]
How many 3-color rules are there?
How many k-color rules are there?
![Page 10: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/10.jpg)
Multistate automata
Some interesting 3-color automata:
I Rule 679458
I Rule 3333333
I Rule 21111
CellularAutomaton[{679458,3}]
How many 3-color rules are there?
How many k-color rules are there?
![Page 11: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/11.jpg)
Multistate automata
Some interesting 3-color automata:
I Rule 679458
I Rule 3333333
I Rule 21111
CellularAutomaton[{679458,3}]
How many 3-color rules are there?
How many k-color rules are there?
![Page 12: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/12.jpg)
Multistate automata
Some interesting 3-color automata:
I Rule 679458
I Rule 3333333
I Rule 21111
CellularAutomaton[{679458,3}]
How many 3-color rules are there?
How many k-color rules are there?
![Page 13: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/13.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 14: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/14.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 15: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/15.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 16: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/16.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 17: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/17.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 18: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/18.jpg)
Totalistic automata
Totalistic rules: the state of a cell only depends on the total ofits neighboring cells.
Total 3 2 1 0
State 0 1 1 0
It’s Rule 126! The friendship rule!
CellularAutomaton[{6, {2, 1}}]
How many 2-color totalistic rules are there?
![Page 19: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/19.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 20: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/20.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 21: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/21.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 22: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/22.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 23: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/23.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 24: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/24.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 25: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/25.jpg)
Totalistic automata
What about for 3-color automata?
CellularAutomaton[{2049, {3, 1}}]
Total 6 5 4 3 2 1 0
State 2 2 1 0 2 2 0
Notice 2049 is just 2210220 in base-3.
How many 3-color totalistic rules are there?
How many k-color totalistic rules are there?
![Page 26: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/26.jpg)
Totalistic automata: A Code Challenge
Exercise: Using Wolfram Language, write code that generates a4-color totalistic rule with initial state 102321.
CellularAutomaton[{30,{4,1}},{{1,0,2,3,2,1},0},100]
Change 30 to another rule. See if you can find one that looks cool!Then use RulePlot to print out its transition rules.
![Page 27: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/27.jpg)
Part II
The Mathematical Formalism of CA’s
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Mathematical definition?
In order to make precise statements about cellular automata, weneed to form a rigorous foundation.
We should be able to provide a rigorous explanation for each of thefollowing:
I What is a state?
I What is a neighborhood?
I What do 1D and 2D automata have in common?
![Page 29: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/29.jpg)
Mathematical definition?
In order to make precise statements about cellular automata, weneed to form a rigorous foundation.
We should be able to provide a rigorous explanation for each of thefollowing:
I What is a state?
I What is a neighborhood?
I What do 1D and 2D automata have in common?
![Page 30: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/30.jpg)
Mathematical definition?
In order to make precise statements about cellular automata, weneed to form a rigorous foundation.
We should be able to provide a rigorous explanation for each of thefollowing:
I What is a state?
I What is a neighborhood?
I What do 1D and 2D automata have in common?
![Page 31: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/31.jpg)
Mathematical definition?
In order to make precise statements about cellular automata, weneed to form a rigorous foundation.
We should be able to provide a rigorous explanation for each of thefollowing:
I What is a state?
I What is a neighborhood?
I What do 1D and 2D automata have in common?
![Page 32: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/32.jpg)
Mathematical definition?
In order to make precise statements about cellular automata, weneed to form a rigorous foundation.
We should be able to provide a rigorous explanation for each of thefollowing:
I What is a state?
I What is a neighborhood?
I What do 1D and 2D automata have in common?
![Page 33: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/33.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0,
f (x0), f (f (x0)), f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 34: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/34.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0),
f (f (x0)), f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 35: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/35.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0), f (f (x0)),
f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 36: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/36.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0), f (f (x0)), f (f (f (x0))),
. . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 37: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/37.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0), f (f (x0)), f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 38: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/38.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0), f (f (x0)), f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 39: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/39.jpg)
Iterated function systems
Suppose X is a set, f : X → X is a function, and x0 ∈ X is somepoint.
Then you can form a sequence
x0, f (x0), f (f (x0)), f (f (f (x0))), . . . , f n(x0), . . .
In general, the nth term is f n(x0) which is obtained by iterating fon the initial value x0.
This sequence is called the orbit of x0.
![Page 40: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/40.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17,
52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 41: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/41.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52,
26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 42: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/42.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26,
13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 43: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/43.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13,
40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 44: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/44.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40,
20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 45: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/45.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20,
10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 46: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/46.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10,
5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 47: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/47.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5,
16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 48: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/48.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16,
8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 49: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/49.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8,
4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 50: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/50.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4,
2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 51: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/51.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2,
1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 52: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/52.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1,
4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 53: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/53.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4,
2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 54: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/54.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2,
1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 55: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/55.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 56: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/56.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 57: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/57.jpg)
Example: The Collatz Sequence
Consider the function f : N→ N given as follows.
f (n) :=
{3n + 1 if n is odd,
n/2 if n is even.
Starting at the initial value n = 17, you get the sequence
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .
This is called a Collatz sequence.
Exercise: Find the Collatz sequence starting at 9.
![Page 58: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/58.jpg)
Example: The Collatz Sequence
Millenium Problem: Prove that every Collatz sequence reachesthe 4, 2, 1 loop, no matter what the initial value.
You’ll win $1,000,000!
![Page 59: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/59.jpg)
Example: The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
It is defined as follows: a0 = a1 = 1, and
an+2 = an+1 + an.
Here’s another way to define it!
f : R2 → R2
f (a, b) = (a + b, a).
Find the orbit of (1, 1) under this function f .
![Page 60: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/60.jpg)
Example: The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
It is defined as follows: a0 = a1 = 1, and
an+2 = an+1 + an.
Here’s another way to define it!
f : R2 → R2
f (a, b) = (a + b, a).
Find the orbit of (1, 1) under this function f .
![Page 61: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/61.jpg)
Example: The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
It is defined as follows: a0 = a1 = 1, and
an+2 = an+1 + an.
Here’s another way to define it!
f : R2 → R2
f (a, b) = (a + b, a).
Find the orbit of (1, 1) under this function f .
![Page 62: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/62.jpg)
Example: The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
It is defined as follows: a0 = a1 = 1, and
an+2 = an+1 + an.
Here’s another way to define it!
f : R2 → R2
f (a, b) = (a + b, a).
Find the orbit of (1, 1) under this function f .
![Page 63: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/63.jpg)
Example: The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
It is defined as follows: a0 = a1 = 1, and
an+2 = an+1 + an.
Here’s another way to define it!
f : R2 → R2
f (a, b) = (a + b, a).
Find the orbit of (1, 1) under this function f .
![Page 64: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/64.jpg)
Cellular automata as iterated function systems
A cellular automaton starts with an initial value, and then we drawthe orbit after iterating the rule.
I What is the “space” X?
I What is the function f : X → X?
![Page 65: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/65.jpg)
Cellular automata as iterated function systems
A cellular automaton starts with an initial value, and then we drawthe orbit after iterating the rule.
I What is the “space” X?
I What is the function f : X → X?
![Page 66: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/66.jpg)
Cellular automata as iterated function systems
A cellular automaton starts with an initial value, and then we drawthe orbit after iterating the rule.
I What is the “space” X?
I What is the function f : X → X?
![Page 67: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/67.jpg)
What should a “state” be? (1D)
A state should assign 0 or 1 to each “cell”. The cells should beindexed by integers.
A state is just a function X : Z→ {0, 1}.
(For k colors, a state is a function X : Z→ {1, 2, . . . , k}.)
![Page 68: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/68.jpg)
What should a “state” be? (1D)
A state should assign 0 or 1 to each “cell”. The cells should beindexed by integers.
A state is just a function X : Z→ {0, 1}.
(For k colors, a state is a function X : Z→ {1, 2, . . . , k}.)
![Page 69: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/69.jpg)
What should a “state” be? (1D)
A state should assign 0 or 1 to each “cell”. The cells should beindexed by integers.
A state is just a function X : Z→ {0, 1}.
(For k colors, a state is a function X : Z→ {1, 2, . . . , k}.)
![Page 70: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/70.jpg)
What should a “neighborhood” be? (1D)
The neighborhood of a cell consists of itself, and the two cells toits left and right.
So the neighborhood of cell n is defined to be
{n − 1, n, n + 1}.
![Page 71: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/71.jpg)
What should a “neighborhood” be? (1D)
The neighborhood of a cell consists of itself, and the two cells toits left and right.
So the neighborhood of cell n is defined to be
{n − 1, n, n + 1}.
![Page 72: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/72.jpg)
How should we define a 1D cellular automaton?
A cellular automaton should take a state x and produce a newstate y .
Common notation: for sets A,B, the set of functions from A toB is denoted
BA := {f : A→ B}.
So the set of states is {0, 1}Z.
Thus, a cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
x 7→ τ(x).
This is a function whose inputs and outputs are ALSO functions!
![Page 73: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/73.jpg)
How should we define a 1D cellular automaton?
A cellular automaton should take a state x and produce a newstate y .
Common notation: for sets A,B, the set of functions from A toB is denoted
BA := {f : A→ B}.
So the set of states is {0, 1}Z.
Thus, a cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
x 7→ τ(x).
This is a function whose inputs and outputs are ALSO functions!
![Page 74: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/74.jpg)
How should we define a 1D cellular automaton?
A cellular automaton should take a state x and produce a newstate y .
Common notation: for sets A,B, the set of functions from A toB is denoted
BA := {f : A→ B}.
So the set of states is {0, 1}Z.
Thus, a cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
x 7→ τ(x).
This is a function whose inputs and outputs are ALSO functions!
![Page 75: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/75.jpg)
How should we define a 1D cellular automaton?
A cellular automaton should take a state x and produce a newstate y .
Common notation: for sets A,B, the set of functions from A toB is denoted
BA := {f : A→ B}.
So the set of states is {0, 1}Z.
Thus, a cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
x 7→ τ(x).
This is a function whose inputs and outputs are ALSO functions!
![Page 76: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/76.jpg)
How should we define a 1D cellular automaton?
A cellular automaton should take a state x and produce a newstate y .
Common notation: for sets A,B, the set of functions from A toB is denoted
BA := {f : A→ B}.
So the set of states is {0, 1}Z.
Thus, a cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
x 7→ τ(x).
This is a function whose inputs and outputs are ALSO functions!
![Page 77: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/77.jpg)
Example: Rule 126
Here are the transition rules for Rule 126.
Let’s write this as a function τ : {0, 1}Z → {0, 1}Z.
I Since Rule 126 is totalistic, we can define its memory rule by
µ(a, b, c) :=
{1 if a + b + c ∈ {0, 3}0 if a + b + c ∈ {1, 2}
I For a state x : Z→ {0, 1}, define τx to be the state
τ(x)n := µ(xn−1, xn, xn+1).
![Page 78: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/78.jpg)
Example: Rule 126
Here are the transition rules for Rule 126.
Let’s write this as a function τ : {0, 1}Z → {0, 1}Z.
I Since Rule 126 is totalistic, we can define its memory rule by
µ(a, b, c) :=
{1 if a + b + c ∈ {0, 3}0 if a + b + c ∈ {1, 2}
I For a state x : Z→ {0, 1}, define τx to be the state
τ(x)n := µ(xn−1, xn, xn+1).
![Page 79: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/79.jpg)
Example: Rule 126
Here are the transition rules for Rule 126.
Let’s write this as a function τ : {0, 1}Z → {0, 1}Z.
I Since Rule 126 is totalistic, we can define its memory rule by
µ(a, b, c) :=
{1 if a + b + c ∈ {0, 3}
0 if a + b + c ∈ {1, 2}
I For a state x : Z→ {0, 1}, define τx to be the state
τ(x)n := µ(xn−1, xn, xn+1).
![Page 80: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/80.jpg)
Example: Rule 126
Here are the transition rules for Rule 126.
Let’s write this as a function τ : {0, 1}Z → {0, 1}Z.
I Since Rule 126 is totalistic, we can define its memory rule by
µ(a, b, c) :=
{1 if a + b + c ∈ {0, 3}0 if a + b + c ∈ {1, 2}
I For a state x : Z→ {0, 1}, define τx to be the state
τ(x)n := µ(xn−1, xn, xn+1).
![Page 81: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/81.jpg)
Example: Rule 126
Here are the transition rules for Rule 126.
Let’s write this as a function τ : {0, 1}Z → {0, 1}Z.
I Since Rule 126 is totalistic, we can define its memory rule by
µ(a, b, c) :=
{1 if a + b + c ∈ {0, 3}0 if a + b + c ∈ {1, 2}
I For a state x : Z→ {0, 1}, define τx to be the state
τ(x)n := µ(xn−1, xn, xn+1).
![Page 82: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/82.jpg)
Exercise For You: Rule 90
Here are the transition rules for Rule 90 (the XOR automaton).
Find the memory function µ for Rule 90, and write the cellularautomaton τ in terms of µ.
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Exercise For You, 2: Rule 30
Here are the transition rules for Rule 30.
Find the memory function µ for Rule 30, and write the cellularautomaton τ in terms of µ.
(Hint: the memory function involves XOR.)
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I lied about the definition of “cellular automaton”
Now we’ve seen that a cellular automaton τ should have a memoryfunction.
So the REAL definition of cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
which is determined by a memory rule µ : {0, 1}3 → {0, 1}:
τ(x)n = µ(xn−1, xn, xn+1).
(If you want k colors, replace {0, 1} by {1, . . . , k}.)
![Page 85: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/85.jpg)
I lied about the definition of “cellular automaton”
Now we’ve seen that a cellular automaton τ should have a memoryfunction.
So the REAL definition of cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
which is determined by a memory rule µ : {0, 1}3 → {0, 1}:
τ(x)n = µ(xn−1, xn, xn+1).
(If you want k colors, replace {0, 1} by {1, . . . , k}.)
![Page 86: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/86.jpg)
I lied about the definition of “cellular automaton”
Now we’ve seen that a cellular automaton τ should have a memoryfunction.
So the REAL definition of cellular automaton is a function
τ : {0, 1}Z → {0, 1}Z
which is determined by a memory rule µ : {0, 1}3 → {0, 1}:
τ(x)n = µ(xn−1, xn, xn+1).
(If you want k colors, replace {0, 1} by {1, . . . , k}.)
![Page 87: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/87.jpg)
Part III
(A little bit on) 2-Dimensional Automata
![Page 88: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/88.jpg)
Return to Planet Friendship
Zoink is back!
Zoink has a new power: he can grow friends in all eightdirections around them!
This is called 2-dimensional friendification.
![Page 89: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/89.jpg)
Return to Planet Friendship
Zoink is back!
Zoink has a new power: he can grow friends in all eightdirections around them!
This is called 2-dimensional friendification.
![Page 90: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/90.jpg)
Return to Planet Friendship
Zoink is back!
Zoink has a new power: he can grow friends in all eightdirections around them!
This is called 2-dimensional friendification.
![Page 91: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/91.jpg)
Return to Planet Friendship
Zoink is back!
Zoink has a new power: he can grow friends in all eightdirections around them!
This is called 2-dimensional friendification.
![Page 92: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/92.jpg)
Rules for 2-Dimensional Friendification
1. Loneliness: If an alien has ≤ 1 live neighbors, he dies :(
2. Survival: If an alien has two or three live neighbors, hesurvives :D
3. Overpopulation: If an alien has ≥ 4 live neighbors, he dies :(
4. Birth: If a cell has exactly three live neighbors, an alien willbe born there!
![Page 93: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/93.jpg)
Rules for 2-Dimensional Friendification
1. Loneliness: If an alien has ≤ 1 live neighbors, he dies :(
2. Survival: If an alien has two or three live neighbors, hesurvives :D
3. Overpopulation: If an alien has ≥ 4 live neighbors, he dies :(
4. Birth: If a cell has exactly three live neighbors, an alien willbe born there!
![Page 94: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/94.jpg)
Rules for 2-Dimensional Friendification
1. Loneliness: If an alien has ≤ 1 live neighbors, he dies :(
2. Survival: If an alien has two or three live neighbors, hesurvives :D
3. Overpopulation: If an alien has ≥ 4 live neighbors, he dies :(
4. Birth: If a cell has exactly three live neighbors, an alien willbe born there!
![Page 95: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/95.jpg)
Rules for 2-Dimensional Friendification
1. Loneliness: If an alien has ≤ 1 live neighbors, he dies :(
2. Survival: If an alien has two or three live neighbors, hesurvives :D
3. Overpopulation: If an alien has ≥ 4 live neighbors, he dies :(
4. Birth: If a cell has exactly three live neighbors, an alien willbe born there!
![Page 96: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/96.jpg)
Rules for 2-Dimensional Friendification
1. Loneliness: If an alien has ≤ 1 live neighbors, he dies :(
2. Survival: If an alien has two or three live neighbors, hesurvives :D
3. Overpopulation: If an alien has ≥ 4 live neighbors, he dies :(
4. Birth: If a cell has exactly three live neighbors, an alien willbe born there!
![Page 97: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/97.jpg)
Exercise
Suppose the aliens stand in the following arrangement. Figure outwho will live, die, or survive over the next two days.
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Exercise
Suppose the aliens stand in the following arrangement. Figure outwho will live, die, or survive over the next two days.
![Page 99: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/99.jpg)
More exercises
(a) For each of these two states, see what happens over thecourse of seven days.
(b) Find an example of a state that never changes.
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More exercises
(a) For each of these two states, see what happens over thecourse of seven days.
(b) Find an example of a state that never changes.
![Page 103: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/103.jpg)
2D Cellular Automata
Each cell has eight neighbors, which can be either alive or dead(black or white).
I How many 2D cellular automata are there?
I If there are k colors, how many 2D cellular automata arethere?
I How many 2D totalistic (two-state) automata are there?
![Page 104: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/104.jpg)
2D Cellular Automata
Each cell has eight neighbors, which can be either alive or dead(black or white).
I How many 2D cellular automata are there?
I If there are k colors, how many 2D cellular automata arethere?
I How many 2D totalistic (two-state) automata are there?
![Page 105: Cellular Automata, Lecture 2 - CEMC · Multistate automata Last time: only allowed cells to be ALIVE or DEAD (1 or 0). These are called the elementary cellular automata, and there](https://reader036.fdocuments.in/reader036/viewer/2022070805/5f03af6a7e708231d40a44ad/html5/thumbnails/105.jpg)
Have a nice holiday!