Spatial Complexity Measure for Characterising Cellular Automata Generated 2D Patterns

Cellular automata (CA) are known for their capacity to generate complex patterns through the local interaction of rules. Often the generated patterns, especially with multi-state two-dimensional CA, can exhibit interesting emergent behaviour. This paper addresses quantitative evaluation of spatial characteristics of CA generated patterns. It is suggested that the structural characteristics of two-dimensional (2D) CA patterns can be measured using mean information gain. This information-theoretic quantity, also known as conditional entropy, takes into account conditional and joint probabilities of cell states in a 2D plane. The effectiveness of the measure is shown in a series of experiments for multi-state 2D patterns generated by CA. The results of the experiments show that the measure is capable of distinguishing the structural characteristics including symmetry and randomness of 2D CA patterns.