Deterministic phase transitions and self-organization in logistic cellular automata

We present a simple extension in which a single parameter tunes the dynamics of cellular automata (CA) by consequently expanding their discrete state space into a Cantor set. Such an implementation serves as a potent platform for further investigation of several emergent phenomena, including deterministic phase transitions, pattern formation, autocatalysis, and self-organization. We first apply this approach to Conway's Game of Life and observe sudden changes in the asymptotic dynamics of the system accompanied by the emergence of complex propagators. Incorporation of the new state space with system features is used to explain the transitions and formulate the tuning parameter range where the propagators adaptively survive by investigating their autocatalytic local interactions. Similar behavior is present when the same recipe is applied to Rule 90, an outer totalistic elementary one-dimensional cellular automaton. In addition, the latter case shows that deterministic transitions between classes of CA can be achieved by tuning a single parameter continuously.