Cellular automata are one-dimensional arrays of interconnected interacting finite automata, also called cells. Here, we investigate one of the weakest class of cellular automata, namely the class of real-time one-way cellular automata. Additionally, we impose two restrictions on the inter-cell communication. First, the number of allowed uses of the links between cells is bounded. Moreover, the amount of information to be communicated in one time step is bounded by a constant being independent from the given automaton. In the weakest case, we consider cellular automata with one-way information flow where each cell receives one bit of information exactly once. In this case, we obtain a characterization of the regular languages for unary alphabets and the acceptance of non-context-free languages for non-unary alphabets. Next, a proper language hierarchy can be derived when increasing the number of allowed uses of the links between cells step by step. Finally, decidability problems of these restricted cellular automata are studied and undecidability of almost all problems can be proven even in the most restricted case of one-way one-bit communication.
