Investigation of Replicating Tiles in Cellular Automata Designed by Evolution Using Conditionally Matching Rules

In this paper we investigate the evolutionary design of replicating tiles in cellular automata. In particular, various sizes of the tiles will be considered whose replication ought to be performed by satisfying a given arrangement of the tiles with respect to each other. The goal is to determine the abilities of the genetic algorithm in combination with conditionally matching rules used for representing the transition functions of cellular automata to find solutions for tiles consisting of up to a hundred of cells. A comparative study will be presented considering the success rate, computational effort and complexity of the obtained solutions as the main values of interest. It will be shown that, in addition to the tile size and the number of states of the cellular automaton, the probability of finding a correct solution is also substantially influenced by the arrangement style. The results show that the tile arrangement that may be considered as the simplest one does not have to necessarily be easily realisable by the genetic algorithm as a transition function for a cellular automaton.