Random expansion method for the generation of complex cellular automata

The complex behaviors in cellular automata have been widely developed in recent years to generate and analyze automata that produce space-moving patterns or gliders that interact in a periodic background. This type of automata has been frequently found either by conducting an exhaustive search or through a meticulous construction of the evolution rule. In this study, the specification of cellular automata with complex behaviors was obtained by utilizing randomly generated specimens. In particular, it was proposed that a cellular automaton of n states should be specified at random and then extended to another automaton with a higher number of states so that the original automaton operates as a periodic background where the additional states serve to define the gliders. Moreover, this study presents an explanation of this method. Furthermore, the random way of defining complex cellular automata was studied by using mean-field approximations for various states and local entropy measures. This specification was refined with a genetic algorithm to obtain specimens of a higher degree of complexity. By adopting this methodology, it was possible to generate complex automata with hundreds of states, demonstrating the fact that randomly defined local interactions with multiple states can construct complexity.