Cellular automata universality revisited

We propose an original arithmetization of cellular automata independent of Turing machines and present a new classification of universal cellular automata according to the way universality is achieved. Indeed, there are many possibilities to get the universality of cellular automata all based on simulations. Consider, for instance the simulation of any other cellular automaton or the simulation of a given universal Turing machine. The two simulations are quite different but both lead to the construction of a universal cellular automaton. We will distinguish three different simulations. The three kinds of corresponding universal machines are defined as simulation-universal, hereditary-universal and construction-universal. As an illustration, we propose an alternative definition of Kolmogorov complexity. We also recall an undecidability result and a well-known complexity result. The last two results hold as soon as we have a hereditary-universal machine.