Cellular Automata with Sparse Communication

Abstract

Abstract. 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 Zustandsänderungskom-plexität von Zellularautomaten. In: Beiträge zur Theorie der Polyautomaten-zweite Folge, Braunschweig (1982) 139-151 (in German)].

Hierarchies and undecidability results for iterative arrays with sparse communication

“…We start by showing that realtime-SC(n)-IA are less powerful than realtime-SC(n 2 )-IA. We remark that a similar result is known between realtime-SC(n)-CA and realtime-SC(n 2 )-CA (see, e.g., [11]). Theorem 6.…”

“…Second, by adapting the proof given in [11] showing that { wcw R | w ∈ {a, b} + } does not belong to L rt (SC(f )-CA) if f ∈ o(n 2 / log(n)), we obtain that L does not belong to L rt (SC(n • √ n)-CA) as well which gives the desired contradiction. Now, let L be accepted by some realtime-SC(n)-IA M .…”

“…To show that L is not accepted by any realtime-MC(const)-IA we combine two techniques which have successfully been applied for SC-CA [11] and for IA with a bounded constant number of different messages to be communicated [13]. First, we derive an upper bound for the number of different communications that the communication cell can perform while processing an input of length n and performing communications.…”

“…To measure the communication in cellular automata in a quantitative way we count the number of uses of the links between cells and we consider, on the one hand, bounds on the sum of all communications of an accepting computation and, on the other hand, bounds on the maximum number of communications per cell that may appear in accepting computations. Many results on this quantitative measure have been obtained for cellular automata in [11,12], and cellular automata that are restricted with respect to the qualitative and the quantitative measure are investigated in [10,12] as well. As main results we would like to mention hierarchy results and the undecidability of almost all commonly studied decidability questions such as emptiness, finiteness, equivalence, inclusion, regularity, and context-freeness.…”

Remembering Chandra Kintala

“…For example, limited nondeterminism was considered in [4,6,7] and time hierarchies were refined in [5,30]. Recently, lim-itations on the number of cells [59,61] as well as on the amount of communication between the cells were studied in [50,51].…”