On Convergence Properties of One-Dimensional Cellular Automata with Majority Cell Update Rule Predrag T. Toši´ c, Shankar N. V. Raju Department of Computer Science, University of Houston, Houston, Texas, USA ptosic@uh.edu, snvishna@mail.uh.edu Abstract— We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus on the convergence properties of a very simple kind of totalistic CA, namely, those defined on one-dimensional arrays where each cell or node updates according to the Boolean Majority function: the new state of a cell becomes 1 if and only if a simple majority of its inputs are currently in state 1, and it becomes 0 otherwise. We have observed in our prior work that such CA tend to have relatively simple asymptotic dynamics: a short transient chain followed by convergence to a “fixed point”. We now provide solid statistical evidence for these conjectures, based on our recent extensive computer simulations of Majority 1-D CA. In particular, we study the convergence properties of such CA for two communication models: one is the classical, parallel CA model with perfectly synchronous cell updates, and the other are CA whose cells update sequentially, one at a time; we consider two variants of such sequential update regimes. We simulate CA whose sizes range up to 1,000 cells, and demonstrate very fast (in particular, sublinear), and very slowly decreasing with an increase in the total number of cells, speeds of convergence. Finally, we draw conclusions based on our extensive simulations and outline some interesting questions to be considered in the future work. Keywords: models for parallel and distributed computing, cellular automata, simple threshold Boolean functions, asymptotic dynam- ics, convergence properties 1. Introduction and Motivation Cellular automata (CA) were originally introduced as an abstract mathematical model of biological systems capable of self-reproduction [13]. CA have been extensively studied in many different domains, especially in the context of mod- eling and simulation of complex physical, biological, social and socio-technical systems and their collective dynamics; see, e.g., [6], [7], [21], [25], [28], [29], [30]. However, CA have also been viewed as an abstraction of massively par- allel computers [5], [16], [23]. While most of the previous research in computer and computational sciences on CA and similar models have used these models as abstractions of parallel hardware architectures, in our prior and ongoing work we have viewed these discrete dynamical system mod- els as useful abstractions of open distributed environments at the software level [16], [17]. In particular, we view CA and related Boolean network automata as formal models of autonomously executing local processes that are reactive, persistent, and coupled to and interacting with one another. Even when the individual processes are rather simple, their mutual interaction and synergy may, in general, potentially yield a highly complex and difficult to predict long-term global behavior [17], [18], [24], [25]. This short paper has two main purposes. On the one hand, we experimentally investigate and validate several conjectures about the overall dynamics and hence possible computations of Majority CA, that were based on mainly an- alytical and conceptual considerations (but, in several cases not rigorously mathematically proven by either ourselves or, as far as we know, other researchers); see e.g. [21], [23], [24], [25]. On the other hand, our extensive simulations and statistical analysis of simulation results also provide some novel insights into the overall properties of Majority CA dynamics, with implications for various biological, social, socio-technical and computational systems and phenomena that can be modeled as such cellular automata. We have established elsewhere [20], [21] that the number of possible distinct asymptotic dynamics of Majority CA grows exponentially with the number of cells. Consequently, already for the number of cells of the order of hundreds, exhaustive simulations of all possible dynamics is com- putationally infeasible. We therefore undertake a statisti- cal experimental study, where we randomly sample initial configurations, then evolve a Majority Cellular Automaton (abbreviated as MAJ CA) from such a random initial con- figuration, then statistically analyze the obtained results with the focus on the speed of convergence to a fixed point. The rest of the paper is organized as follows. We provide formal definitions of CA models and cell update rules of interest in the next section. We briefly review the most relevant prior art. We then summarize and discuss the statistics of our simulation experiments on MAJ CA (with both parallel and sequential cell update regimes) with up to 1,000 cells, and correlate these experimental findings with our prior theoretical results and conjectures about the 308 Int'l Conf. Scientific Computing | CSC'11 |