On Completeness and Decidability of Phase Space Invertible Asynchronous Cellular Automata

While for synchronous deterministic cellular automata there is an accepted definition of reversibility, this is not the case for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata. We will show that for each Turing machine there is a phase space invertible purely asynchronous cellular automaton simulating it and that it is decidable whether an arbitrary-dimensional purely and a one-dimensional fully asynchronous cellular automaton is phase space invertible.