Fast Cellular Automata with Restricted Inter-Cell Communication: Computational Capacity

A d-dimensional cellular automaton with sequential input mode is a d-dimensional grid of interconnecte d interacting finite au­ tomata. The distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequen­ tially. Often in the literature this model is referred to as iterative ar­ ray. We investigate d-dimensional iterative arrays and one-dimensional cellular automata operating in real and linear time, whose inter-cell communication is restricted to some constant number of bits indepen­ dent of the number of states. It is known that even one-dimensional one-bit iterative arrays accept rather complicated languages such as {a^ I p prim} or {a^ | n G N} (16). We show that there is an infinite strict double dimension-bit hierarchy. The computational capacity of the one-dimensional devices in question is compared with the power of communication-restricted two-way cellular automata. It turns out that the relations are quite different from the relations in the unrestricted case. On passing, we obtain an infinite strict bit hierarchy for real-time two-way cellular automata and, moreover, a very dense time hierarchy for every fc-bit cellular automata, i.e., just one more time step leads to a proper superfamily of accepted languages.