Turing-completeness of asynchronous non-camouflage cellular automata
Tatsuya Yamashita, Teijiro Isokawa, Ferdinand Peper, Ibuki Kawamata, Masami Hagiya · Information and Computation · 2020
Asynchronous Boolean totalistic cellular automata have recently attracted attention as promising models for implementation by reaction-diffusion systems. It is unknown, however, to what extent they are able to conduct computation. In this paper, we introduce the so-called non-camouflage property, which means that a cell's update is insensitive to neighboring states that equal its own state. This property subsumes the Boolean totalistic property, which signifies the existence of states in a cell's neighborhood, but is not concerned with how many cells are in those states. We argue that the non-camouflage property is extremely useful for the implementation of reaction-diffusion systems, and we construct an asynchronous cellular automaton with this property that is Turing-complete by directly simulating Turing machines. We also construct another asynchronous cellular automaton, but this model incorporates the so-called freezing property [1], which restricts the direction of transitions of each cell to one-way. We show that this model is Turing-complete, since it can simulate the temporal evolution of elementary cellular automata. These results indicate the feasibility of computation by reaction-diffusion systems.