Introduction to the Modeling and Analysis of Complex Systems

210 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSISEach of the n cells in a neighborhood can take one of k states. Therefore, the total numberof local situations possible is given bym = k n = k (2r+1)D . (12.2)Now we can calculate the number of all possible state-transition functions. A function hasto map each of the m situations to one of the k states. Therefore, the number of possiblemappings is given byR = k m = k kn = k k(2r+1)D . (12.3)For example, a one-dimensional (D = 1) binary (k = 2) CA model with radius 1 (r = 1)has 2 (2×1+1)1 = 2 3 = 8 different possible situations, and thus there are 2 8 = 256 statetransitionfunctions possible in this CA universe 1 . This seems reasonable, but be careful—this number quickly explodes to astronomical scales for larger k, r, or D.Exercise 12.1 Calculate the number of possible state-transition functions for atwo-dimensional CA model with two states and Moore neighborhoods (i.e., r = 1).Exercise 12.2 Calculate the number of possible state-transition functions for athree-dimensional CA model with three states and 3-D Moore neighborhoods (i.e.,r = 1).You must have faced some violently large numbers in these exercises. Various symmetryassumptions (e.g., rotational symmetries, reflectional symmetries, totalistic rules,etc.) are often adopted to reduce the size of the rule spaces of CA models.How about the size of the phase space? It can actually be much more manageablecompared to the size of rule spaces, depending on how big L is. The total number of cellsin the space (i.e., the volume of the space) is given byV = L D . (12.4)Each cell in this volume takes one of the k states, so the total number of possible configurations(i.e., the size of the phase space) is simply given byS = k V = k LD . (12.5)1 There is a well-known rule-numbering scheme for this particular setting proposed by Wolfram, but wedon’t discuss it in detail in this textbook.