Cellular Automata

History John von Neumann and Ulam (1940’s) Crystal structure Self reproducing machines John Conway – Game of Life (1970) Student interest Stephen Wolfram Cellular Automata New Kind of Science

Properties Discrete, local, synchronous Grid of cells – often 2D Symbols, often binary Neighborhood Moore 8 surrounding cells Von Neumann 4 cells (orthogonally) Rules for cell state: f(neighborhood)

Conway’s Game of life Any live cell with fewer than two live neighbors dies, as if caused by underpopulation. Any live cell with more than three live neighbors dies, as if by overcrowding. Any live cell with two or three live neighbors lives on to the next generation. Any dead cell with exactly three live neighbors becomes a live cell.

R-pentomonio

Still Lifes Ship Loaf Beehive Boat

Ocsillators Blinker Period 2 Toad Period 2 Pulsar Period 3

Gliders Glider – cycle length 6 Gosper’s Glider Gun

Computation Gliders for data streams Structures that act as gates Turing Computatable

Wolfram Numeric description Detailed analysis: ‘Cellular Automata and Complexity”

Wolfram’s results Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears. Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures. Some of the randomness in the initial pattern may filtered out, but some remains. Local changes to the initial pattern tend to remain local. Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely. Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways. Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps required to reach this state may be very large, even when the initial pattern is relatively simple. Local changes to the initial pattern may spread indefinitely. Wolfram has conjectured that many, if not all class 4 cellular automata are capable of universal computation. This has been proved for Rule 110 and Conway's game of life.

New Kind of Science Replace differential equations with cellular structures CA become a standard fabric for constructions

MCell By Mirek Wojtowciz Windows app and now Java based versions Large set of rules in families 1D systems General binary Larger than Life