Probabilistic cellular automata: between deterministic Wolfram's rules 23, 77, 178 and 232

We study one-dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighbourhood states, the central site takes either the majority state or the opposite and (ii) if the neighbourhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters p,r∈[0,1], and we analytically solve small-size cases by using a Markov process formulation.