Dualities for Multi-State Probabilistic Cellular Automata
  In this paper a new form of duality for probabilistic cellular automata (PCA) is introduced. Using this duality, an ergodicity result for processes having a dual is proved. Also, conditions on the probabilities defining the evolution of the processes for the existence of a dual are provided. The results are applied to wide classes of PCA which include multi-opinion voter models, competition models and the Domany-Kinzel model. 
F.J. Lopez, G. Sanz, and M. Sobottka