Evolving multiplication as emergent behavior in cellular automata using conditionally matching rules

In this paper a special representation technique called conditionally matching rules will be applied in order to design computational processes in uniform cellular automata. The goal is to verify abilities of this approach in combination with genetic algorithm on the problem of disigning various cellular automata that exhibit a given computational process. The principle of a computational process in a cellular automaton is to interpret some cells as input bits and some (possibly other) cells as output bits (i.e. the result of the computation). The genetic algorithm is applied to find a suitable transition function of a cellular automaton according to which the given computation could be observed during its development for all the possible binary combinations stored in the input cells. Both the input values and the result is represented by state values of cells. The input of the computation will be represented by the initial state of the cellular automaton. After a finite number of development steps the cells representing the output bits are expected to contain the result of the computation. A set of experiments will be performed considering various setups of the evolutionary system and arrangements of the target computation. It will be shown that non-trivial computations can be realized in a uniform two-dimensional cellular array.