Three states hybrid cellular automata with periodic boundary condition

Even though cellular automata (CA) is a discrete model, the behaviors at many iterative times can be a close a continuous system. After modelling the CA structure, one of the important tasks is able to move forwards and backwards on CA to understand their behaviors. This happens if CA is to be a reversible one. In this paper, it is studied main theoretical views of 2D hybrid-linear periodic CA over the ternary field, i.e. Z3 or three states case. Here, we set up a relation between reversibility of CA and characterization of 2D hybrid linear CA with this special bound- ary conditions by using of the matrix algebra theory. For given special transition (information) rule matrices, it is proved that which hybrid 2D CAs are reversible or not. In other words, the reversibility problem of 2D hybrid, linear CA with periodic boundary is resolved completely over ternary field. It is believed that these type of CAs could be found many different real life applications in special case situation e.g. computability theory, mathematics, theoretical biology, image processing area, textile design, video processing, DNA research and microstructure modelling, etc. in near future