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Life, the Game
In the field of complexity theory, perhaps the most well-known examples come from the mathematical study of cellular automata. Cellular automata are grid-based systems that vaguely resemble game boards. At any moment, a grid square can either be occupied or unoccupied. A set of rules dictates how the cells behave and how their states change over time. The name "cellular automata" comes from the grid of squares, called "cells," and the fact that once the system is set into motion, the rules move the system forward so that it runs "autonomously," without further input from the outside. Not all cellular automata systems automatically create emergence. It all depends on the rules that govern the behavior of the cells. For example, if we wanted to replicate a blank TV screen in a cellular automata grid, we could create the following rules:
To begin, all of the cells are unoccupied.
The state of the cells does not change.
These rules would result in the fixed system of a blank, empty field. If we wanted to replicate a cellular automata system that would result in pure chaos, we could create the following rules:
To begin, half of the cells are occupied and half are unoccupied, determined randomly.
Every thirtieth of a second, a new set of half of the cells is occupied, randomly determined.
These rules would result in the ever-changing but patternless sheen of dots that make up television static. Cellular automata were systems originally developed by the mathematicians John Von Neumann and Stanislaw Ulam in the 1930s and 1940s. But the most famous application comes from John Conway. His cellular automata system, The Game of Life, or more simply, Life, was popularized when Martin Gardener published it as a puzzle in Scientific American in 1970. There are many fascinating ways to look at The Game of Life. What interests us about Life is not that it is an early form of artificial life, or even that it is called a game (it isn't a game by our definition), but instead that it is a wonderful example of the principles of emergence and complexity. Life takes place on an infinite grid; the cells of the grid are governed by a simple set of rules, summarized here by Steven Levy in Artificial Life:
Life occurs on a virtual checkerboard. The squares are called cells. They are in one of two states: alive or dead. Each cell has eight possible neighbors, the cells which touch its sides or its corners. If a cell on the checkerboard is alive, it will survive in the next time step (or generation) if there are either two or three neighbors also alive. It will die of overcrowding if there are more than three live neighbors, and it will die of exposure if there are fewer than two. If a cell on the checkerboard is dead, it will remain dead in the next generation unless exactly three of its eight neighbors are alive. In that case, the cell will be "born" in the next generation.[6]
Although Conway's Life was not originally created on a computer, it is commonly manifest today as computer software. Life always begins with some of the cells full and some empty, either in predetermined or random patterns. When the program is run, the grid is updated in steps, each step iterating the rules according to the current configuration on the grid. Visually, Life manifests as a set of dynamic, shifting geometric patterns as grid cells die and come to life.
The Game of Life: The Glider
Some triplet histories in the Game of Life
The startling thing about Life is that from these extremely simple rules, strikingly unexpected patterns result. Some shapes die out quickly and disappear from the grid. Others reach stable and fixed patterns, or oscillate forever between two states. Still others move steadily across the grid (like the famous "glider" shape), and yet others exhibit even more complex behavior (such as the "glider gun," which produces a new glider every 30 generations). By combining these different patterns and unexpected behaviors, fans of Life have made everything from rich visual patterns to functional calculators and other "virtual machines" that operate using the cells on the grid. Life is a complex system that exhibits emergent behavior from a simple set of rules. Using Life as a case study, it becomes clear why emergence is formed from coupled and context-dependent interactions between the elements of a system: Coupled: In Life, the interaction between any two objects is not an isolated relationship. Instead, the relationships depend on a set of coupled relationships. Whether or not a cell is alive or dead in Life is dependent on the positions of the adjacent cells; the life and death of those cells are also dependent on all of their adjacent cells. These linked relationships create interlocking forms that multiply across the space as a whole, giving rise to surprising patterns.
Context-dependent: Over time, as the context of an object in a system changes, its relationships to other objects change, giving rise to global transformations in the system. In the case of Life, the context of any given cell is its eight neighbors. At each time step, this context shifts, propagating dynamic behavior across the system. Just as coupled interactions reverberate across the space of the grid, context-dependent interactions ripple out over time as the space of the system transforms and creates new contexts for the system objects. [6]Langton, Artificial Life, p. 52.
