Introducing Encoder-Decoder Relation Between Cellular Automata to Uncover Their Computational Structure

Cellular automata are fascinating models of parallel computation, yet it remains a challenge to program them efficiently to solve complex tasks. In this paper, we propose a novel method of analyzing their computational capacity. It is based on measuring the richness of the automaton's dynamics by assessing how many other automata it can simulate. Using this method, we can identify automata that can perform many computations efficiently. Moreover, there is no need to choose an arbitrary set of computational tasks as they are self-defined by the cellular automata space itself. The novelty of our method is based on a new definition of automata simulation that generalizes previous approaches. Thus, we obtain a rich structure of relations that we demonstrate on the class of elementary cellular automata. We show that the number of elementary automata with unique dynamics can be reduced from 88 to 52. We further show that some automata, such as 14, 43, and 142, have in fact equivalent computational capacity. We believe that using similar approaches, the number of unique automata can be dramatically reduced in the future.