Cellular Automata The Game of Life Matthew Willingham

Cellular Automata: The Game of Life Matthew Willingham 2/14/2008

Topics to be Covered What are Cellular Automata? Elementary Cellular Automata and Wolfram Notation Conway's the Game of Life Applications of CAs

What is Cellular Automata? Consists of a regular grid of cells, each cell being in a specific state depending on the state of its neighbors (we will clarify this) Each cell in the automaton works on the same rule set, they can be thought of as self-contained processing units. Evolves through a number of discrete time steps, we call subsequent iterations of the automaton generations

Elementary Cellular Automata and Wolfram Notation These are the simplest 1 -dimensional Cellular Automata Investigated in-depth by Stephen Wolfram in 1983 The state of the cell relies simply on the value of it's nearest neighbors

Elementary Cellular Automata and Wolfram Notation The Wolfram Notation is derived from the 8 possible neighboorhoods in reverse counting order (111, 110, 101, 100, 011, 010, 001, 000) The ruleset below is 30 in binary, hence 'rule 30' 8 There are 256 Elementary Cellular Automata (2 )

Elementary Cellular Automata and Wolfram Notation Some interesting patterns emerge from such simple rules Including randomness, in the center part of rule 30 for example (larger illustration on next page)

Elementary Cellular Automata and Wolfram Notation Rule 30, top Rule 110, bottom You can see the generation of this seashell seems to have resemblance to these patterns

The Game of Life Created by John Conway, in 1970 A more advanced form of Cellular Automata Uses a 2 d grid, and an 'infinite board' Zero player game, simply define the initial configuation and let the board evolve on its own Very simple rules, yet very interesting emergent behavior occurs

The Game of Life There are two types of neighborhoods typically applied in the game of life, we will be using r==1 Moore Neighborhood Von Neumann Neighborhood

The Game of Life Very Simple Rules 1. Any live cell with fewer than 2 live neighbors dies, as if by loneliness 2. Any live cell with more than 3 live neighbors dies, as if by overcrowding 3. Any live cell with two or 3 living neighbors lives, unchanged 4. Any dead cell with exactly 3 live neighbors comes to life

The Game of Life Can anyone see what type of problem a grid with this many neighbors might present? Illustrations of the rules in the game of life.

The Game of Life The problem with borders, 'should we leave border neighborhoods 'dead'? By using a toroidal board, aka wrapping the board around on itself, you can prevent having to deal with special edge cases

The Game of Life Some sample runs from different game of life configurations Puffer Jellyfish

The Game of Life Conway originally thought that no pattern would be able to grow indefinitely That is, for an initial configuration (group of living cells), there would be some finite upper bound on the number of living cells A team at MIT led by Bill Gosper disproved this conjecture by producing a 'glider gun' Glider Initial Configuration for Glider Gun in action

The Game of Life Many other interesting forms of emergent behavior in the game of life Turing machines have been implemented using these simple rules with very large and advanced configurations As well as simple logic gates, and 'counters' Life is a universal cellular automaton, it is effectively capable of emulating any other cellular automaton, and any turing machine, a proof of this was given by Berlekamp, Conway, and Guy (1982) and independently by Gosper (1983)

Applications of Cellular Automata Was originally thought that it might be a good biological analog, turns out to be too simple Has been used to study various emergent behavior, such as those in insects, mold growth, and traffic patterns Langton's Ant Developed by Chris Langton in 1986

Works Cited Dewdney, A. K. The New Turing Omnibus. New York: Henry Holt and Company, LLC, 1993 "Cellular Automata. " Wikipedia, The Free Encyclopedia. 28 January 2008. Wikimedia Foundation, Inc. 10 Feb. 2008. <http: //en. wikipedia. org/wiki/Cellular_automata>. “Cellular Automaton. ” Wolfram Mathworld. 10 February 2008. Wolfram Research Inc. 10 Feb. 2008. <http: //mathworld. wolfram. com/Cellular. Automaton. html>

Homework Questions What are the two possible neighborhoods used in the game of life? Write out the ruleset for the 'rule 110' elementary cellular automata, in wolfram notation.