Cellular Automata with Sparse Communication

We investigate cellular automata whose internal inter-cell communication is bounded. The communication is quantitatively measured by the number of uses of the links between cells. Bounds on the sum of all communications of a computation as well as bounds on the maximal number of communications that may appear between each two cells are considered. It is shown that even the weakest non-trivial device in question, that is, one-way cellular automata where each two neighboring cells may communicate constantly often only, accept rather complicated languages. We investigate the computational capacity of the devices in question and prove an infinite strict hierarchy depending on the bound on the total number of communications during a computation. Despite their sparse communication even for the weakest devices, by reduction of Hilbert’s tenth problem undecidability of several problems is derived. Finally, the question whether a given real-time one-way cellular automaton belongs to the weakest class is shown to be undecidable. This result can be adapted to answer an open question posed in [Vollmar, R.: Zur Zustandsänderungskomplexität von Zellularautomaten. In: Beiträge zur Theorie der Polyautomaten—zweite Folge, Braunschweig (1982) 139–151 (in German)]. F.1.1 [Computation by Abstract Devices]: Models of Computation—Cellular automata; F.1.2 [Computation by Abstract Devices]: Modes of Computation—Parallelism and concurrency; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes—Complexity hierarchies; F.4.3 [Mathematical Logic and Formal Languages]: Formal Languages—Decision problems Additional