Exact results for one-dimensional cellular automata with different types of updates

We study two common types of time-noncontinuous updates for one-dimensional stochastic cellular automata with arbitrary nearest-neighbor interactions and arbitrary open boundary conditions. We first construct the stationary states using the matrix product formalism. This construction then allows to prove a general connection between the stationary states which are produced by the two different types of updates. Using this connection, we derive explicit relations between the densities and correlation functions for these different stationary states.