Effect of random fluctuations on minimizing the complexity of universal asynchronous cellular automata

Abstract

Asynchronous cellular automata (ACAs) are cellular automata (CAs) that allow cells to undergo state transitions independently and randomly. Notwithstanding the unpredictable update timings of each cell, universal computation can be carried out on ACAs via the embedding of logic circuits into cell spaces, like the ACA with von Neumann neighborhood that uses 5 cell states and 55 transition rules (Lee et al. (2003)). Recently, the complexity in terms of the numbers of cell states and rules has been reduced via the use of local configurations representing signals that may fluctuate between moving forward and backward in the cell space. This forms the basis for the implementation of simple circuit elements in ACAs that can exploit signal fluctuations. Nevertheless, universal ACA still appears much more complex than the best synchronous CAs found to date, e.g., the 2-state and 3-rule CA (Banks 1970). This paper aims to propel the fluctuation-based scheme further, by proposing a novel ACA with von Neumann neighborhood which requires merely 3 cell states and 10 rules. The universality of our ACA is shown by implementing a new set of circuit elements that are able to exploit fluctuations, as well as a special circuitry to conduct the crossing of signals in the two-dimensional cell space. The further reduction in the numbers of both cell states and rules demonstrates that random fluctuations may play an important role in pursuing the minimal complexity of universal ACAs.