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# Population Dynamics in Conway’s Game of Life and its Variants

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The presentation for the project of high school students Yonatan Biel and David Hua made in the Students and Teachers As Research Scientists (STARS) program at the Missouri Estimation of Distribution Algorithms Laboratory (MEDAL). To see animations, please download the powerpoint presentation.

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### Population Dynamics in Conway’s Game of Life and its Variants

1. 1. Population Dynamics in Conway’sGame of Life and its VariantsDavid Hua and Yoni Biel
2. 2. Background and Motivation Cellular automata (CA) as computational models  Can simulate any algorithm (computation).  Useful in computational theory, biology, physics, mathematics, artificial intelligence.  Used to study emergence of complex behavior, self- organization, self-replication, and other aspects.  Conway’s game of life is one of simplest yet powerful CA. The purpose of this project  Study Conway’s game of life and its variants.  Focus on population dynamics in terms of the rule set used and initialization of the simulation.  Learned programming in C++.
3. 3. Outline1. Cellular automata.2. Conway’s game of life and its variants.3. Population dynamics of studied CA variants.4. Summary, conclusions, and future work.
4. 4. What are Cellular Automata? Computational models arranged on a grid of cells. Each cell is in a state. Grid changes over a number of discrete time steps. Change of cell state determined by its current Cellular automaton for state, states of its neighbors, simulating diffusion/aggregation and the set of rules. [http://www.hermetic.ch/pca/da.htm]
5. 5. Differences from Most Other Models Three interesting features of CA  Parallelism: Every cell is updated at the same time.  Localism: Every cell is updated based upon its neighbors.  Homogeneity: Every cell is updated using the same rules.
6. 6. Why Cellular Automata? Cellular automata can simulate any algorithm via implementing universal Turing machine. Cellular automata can demonstrate and model emergence of complex global behavior from simple local rules, self-organization, self-replication, population dynamics. Cellular automata useful in computational theory, biology, physics, artificial intelligence…
7. 7. 2D Cellular Automata Cells arranged in a two-dimensional grid. Each cell has 8 neighbors• Opposite sides may connect so that the grid wraps around (for top/bottom row and left/right column).
8. 8. Conway’s Game of Life Conway’s game of life is 2D cellular automaton. Two possible states for each cell  Alive  Dead States can change  Living cell can die (death).  Dead cell can become alive (birth). Simple set of rules specifying  Death (overcrowding, underpopulation).  Birth (reproduction).
9. 9. Basic Rules of Conway’s Game of Life1. Living cells die if they have 2. Living cells die if they have fewer than 2 neighbors more than 3 neighbors (underpopulation/loneliness) (overpopulation)3. Dead cells that have 3 neighbors 4. Otherwise, there is no change become alive (whether cell is alive or dead) (reproduction)
10. 10. Game of Life - Behaviors  Wide range of common structure types:  Mobile groups, spaceships  Oscillators  Static structures  …  Structures and their interaction crucial for simulating computations or processes.
11. 11. Summarizing Rules; Game of Life Variants Rules can be summarized in a simple statement defining the rules for survival and birth (also else is just dead). Examples:  B3 / S23 (Conway’s original rules)  Born if 3 neighbors are alive.  Survives if 2 or 3 neighbors are alive.  B36 / S23 (high life)  Born if 3 or 6 neighbors alive.  Survives if 2 or 3 neighbors alive.  B2 / S (seeds)  Born if 2 neighbors are alive.  Never survives.
12. 12. Rule Sets Used for 2-State Game of Life Rule sets:  Game of life (B3 / S23)  Reversed GOL (B23 / S3)  Evens (B02468 / S02468)  Day and night (B3678 / S34678)  Maze (B3 / S12345)  Walled cities (B45678 / S2345)
13. 13. Our Research• For each rule set, set a few important inputs as controls for each simulation • World size – 20x20 cells • Number of time steps – 100 steps • Number of runs of each simulation – 100 runs• Independent variable • Initialization percentages of living cells – 10%-90%• Analyze behavior of various rule sets.• For each rule set, analyze influence of controllable variables on 1. Percentages of living cell populations. 2. Percentages of changed cell states per time step (rate of change).
14. 14. Game of Life - Dynamics Population Level Over Time 70  Convergence upon 60 20.00% common asymptote.% Living Cells 50 30.00% 40 40.00% 30 50.00% 20 10 60.00%  Initial population 0 Time Interval decline.  Limited range of Rate of Change Over Time initializations% Change From Previous Time 60 50 20.00% achieve this. 40 30.00% 40.00% 30 20 50.00% 60.00%  Restrictive survival 10 0 rule. Time Interval
15. 15. Reversed GOL (B23/S3) Example Begins by expanding very quickly Seems to change in waves Not very many cells stay alive from time step to time step
16. 16. Reversed GOL (B23/S3) Population Dynamics % Living % Changed 90 90 80 80 70 70 Percentage of Cells ChangedPercentage of Living Cells 60 60 50 50 40 40 30 30 20 20 10 10 0 0 1 11 21 31 41 51 61 71 81 91 101 1 11 21 31 41 51 61 71 81 91 Number of Time Steps Number of Time Steps 20% initialization 40% initialization 20% Initialization 40% Initialization 60% initialization 80% initialization 60% Initialization 80% Initialization • Populations very stable • Maybe many live & dead cells • Overcrowding still kills switch places
17. 17. Evens (B02468/S02468) Example  No recognizable patterns  All regions seem to change constantly  All movement is chaotic
18. 18. Evens (B02468/S02468) Population Dynamics % Living % Changed 90 90 80 80 70 70 Percentage of Cells ChangedPercent of Living Cells 60 60 50 50 40 40 30 30 20 20 10 10 0 0 1 11 21 31 41 51 61 71 81 91 101 1 11 21 31 41 51 61 71 81 91 Number of Time Steps Number of Time Steps 20% Initialization 40% Initialization 20% Initialization 40% Initialization 60% Initialization 80% Initialization 60% Initialization 80% Initialization • Populations very stable • Initial population size doesn’t matter • Half changes and half is static
19. 19. Day and Night (S34678/B3678) Example  Life and death are symmetrical – living and dead cells behave the same way  Large regions of living/dead cells  Regions have similar activity, chaotic boundaries
20. 20. Day and Night (S34678/B3678) Population Dynamics Population Level Over Time No convergence in 120 10.00% 20.00% population level or 30.00% % Living Cells 100 80 40.00% rate of change. 60 50.00% 60.00% 40 70.00% 20 80.00% Relatively stable; 0 90.00% no significant Time Interval initial population % Change From Previous Time Rate of Change Over Time decline. 40 10.00% 35 20.00% 30.00% Rule set – living 30 25 40.00% 50.00% and dead are 20 15 60.00% 70.00% treated 10 5 80.00% 90.00% equally, less 0 Time Interval survival pressures.
21. 21. Maze (S12345/B3) Example  Static rule set: stops changing after pattern is complete.  Consistent maze pattern for all initializations.
22. 22. Maze (S12345/B3) Population Dynamics Population Level Over Time Convergence of a range of 100 initializations. 90 80 Rate of change 70 10.00% 20.00% quickly drops to 30.00% % Living Cells 60 40.00% 50.00% zero. 50 60.00% 70.00% 40 80.00% Stable, expanding 30 90.00% population – 20 tolerant survival 10 rule. 0 Time Interval
23. 23. Walled Cities (S2345/B45678) Example Polygonal cities filled with chaotic activity. Activity continues only within cities after they are built.
24. 24. Walled Cities (S2345/B45678) Population Dynamics Initial population Population Level Over Time drop. 100 90 Limited range of 80 10.00% initializations 70 20.00% 30.00% converge despite lack % Living Cells 60 40.00% 50.00% of interaction between 50 60.00% 70.00% cities. 40 80.00% 90.00% 30 Restrictive survival 20 10 rule similar to game of 0 life. Time Interval
25. 25. 3-State Game of Life Example An additional living cell type represents a second species or group. Rules used are the game of life with identity of new species determined by dominant neighboring cell type. Cells coalesce into homogeneous mobile masses. Each region becomes overtaken by one cell type.
26. 26. 3-State Game of Life Population Dynamics Initializations are Population Level Over Time Population Level Over Time based on difference between initial Initial Difference: 16% Initial Difference: 12% populations, with % Living Cells % Living Cells 20 20 2.00% 4.00% total initial 15 10 18.00% 15 10 16.00% population of 20%. 5 5 0 0 Rates of change are Time Interval Time Interval same as game of life. Little correlation Population Level Over Time Population Level Over Time between populations Initial Difference: 8% Initial Difference: 4% of each species. % Living Cells % Living Cells 15 15 6.00% 8.00% 10 14.00% 10 12.00% Higher initializations 5 5 had higher rates of 0 0 decline. Time Interval Time Interval
27. 27. Summary Presented basics of CA. Presented basics of Conway’s game of life (simple CA). Explored population dynamics for several variants of the game of life as well as concrete examples. Considered both 2-state and 3-state variants.
28. 28. Conclusions Simple rule sets for CA can yield complex behavior. Small change to rule set can yield completely different results. Changes in the initialization of cell populations can sometimes yield similar dynamics, but sometimes the dynamics are dramatically affected (depends on rules). Rule sets can be categorized by population dynamics, which appear to be affected by the survival rule  Convergence upon optimum population levels/rates of change.  Initial behavior and time for stabilization.  Limited range of initializations achieving an optimum state. Additional states introduce new possibilities for simulating competition and species-specific pressures.
29. 29. Future Work Simulations of biological and ecological systems  Example: Spreading of forest fires  3 colors for live plants, fire, & empty space  Rules  Fire consumes all plant neighbors  Fire can’t pass over empty spaces  Plants survives with any neighbors until fire reaches it  Space stays as space  This can simulate a very important phenomenon rather easily. Simulate ecosystems, evolutionary systems, social systems, …
30. 30. Acknowledgments STARS-2012 sponsors  Pfizer Inc.  LMI Aerospace Inc. / D3Technologies  St. Louis Symphony Orchestra  Solae  University of Missouri in St. Louis  Washington University in St. Louis Martin Pelikan (mentor) supported from NSF under grants ECS-0547013 and IIS-1115352. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.