Introduction to the Modeling and Analysis of Complex Systems

Chapter 12Cellular Automata II: Analysis12.1 Sizes of Rule Space and Phase SpaceOne of the unique features of typical CA models is that time, space, and states of cellsare all discrete. Because of such discreteness, the number of all possible state-transitionfunctions is finite, i.e., there are only a finite number of “universes” possible in a given CAsetting. Moreover, if the space is finite, all possible configurations of the entire system arealso enumerable. This means that, for reasonably small CA settings, one can conduct anexhaustive search of the entire rule space or phase space to study the properties of allthe “parallel universes.” Stephen Wolfram did such an exhaustive search for a binary CArule space to illustrate the possible dynamics of CA [34, 46].Let’s calculate how large a rule space/phase space of a given CA setting can be. Herewe assume the following:• Dimension of space: D• Length of space in each dimension: L• Radius of neighborhood: r• Number of states for each cell: kTo calculate the number of possible rules, we first need to know the size of the neighborhoodof each cell. For each dimension, the length of each side of a neighborhood isgiven by 2r + 1, including the cell itself. In a D-dimensional space, this is raised to thepower of D, assuming that the neighborhood is a D-dimensional (hyper)cube. So, thesize (volume) of the neighborhood is given byn = (2r + 1) D . (12.1)209