A Probabilistic Approach To Conway\u27s Game of Life (And Related Cellular Automata)

Conway\u27s Game of Life is the most well-known instance of a class of computational structures known as cellular automata. Conway\u27s Game of Life, and other life-like cellular automata, are played on an infinite/arbitrarily large square lattice. At the beginning of the game every square (known as a cell ) is either alive or dead , and at each stage of the game ( generation ), every cell changes its state or not based on the number of alive cells around it. Locally, the game is very simple, but globally, surpisingly complex and chaotic patterns emerge from these rules. The central question in our investigation is: given a starting density , what can we say about the eventual or long-term behavior of our board? In particular, what can we say about the long-term density? We attempted to tackle this question theoretically and experimentally. We collected large amounts of data running finite boards in Mathematica that seems to converge to likely behavior for the idealized infinite case. In addition to that we used a combination of algebraic, probabilistic and combinatorial reasoning to develop a polynomial that both accurately predicts short-term behavior and, in a looser way, reflects the behavior of various automata rules