Multi-state Cellular Automata Research Articles

Lyapunov exponents of multi-state cellular automata.

In order to describe the sensitivity of a cellular automaton (CA) to a small change in its initial configuration, one can attempt to extend the notion of Lyapunov exponents as defined for continuous dynamical systems to a CA. So far, such attempts have been limited to a CA with two states. This poses a significant limitation on their applicability, as many CA-based models rely on three or more states. In this paper, we generalize the existing approach to an arbitrary N-dimensional k-state CA with either a deterministic or probabilistic update rule. Our proposed extension establishes a distinction between different kinds of defects that can propagate, as well as the direction in which they propagate. Furthermore, in order to arrive at a comprehensive insight into CA's stability, we introduce additional concepts, such as the average Lyapunov exponent and the correlation coefficient of the difference pattern growth. We illustrate our approach for some interesting three-state and four-state rules, as well as a CA-based forest-fire model. In addition to making the existing methods generally applicable, our extension makes it possible to identify some behavioral features that allow us to distinguish a Class IV CA from a Class III CA (according to Wolfram's classification), which has been proven to be difficult.

Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one.

We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two properties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of the neighborhood by 90^{∘}) and number conservation (i.e., the sum of all the cell states is conserved upon every update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0,1,...,k-1}. In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate cellular automata should not only focus on those that have {0,1,...,k-1} as a state set.

Analysis of information gain and Kolmogorov complexity for structural evaluation of cellular automata configurations

ABSTRACTShannon entropy fails to discriminate structurally different patterns in two-dimensional images. We have adapted information gain measure and Kolmogorov complexity to overcome the shortcomings of entropy as a measure of image structure. The measures are customised to robustly quantify the complexity of images resulting from multi-state cellular automata (CA). Experiments with a two-dimensional multi-state cellular automaton demonstrate that these measures are able to predict some of the structural characteristics, symmetry and orientation of CA generated patterns.

Modeling traffic flow at a single-lane urban roundabout

Abstract In this paper, we propose a new model to study traffic flow at a single-lane urban roundabout, using a multi-state cellular automata (CA) ring under the offside-priority rule (by which a vehicle entering gives way to one already on the roundabout). Each vehicle entering the roundabout is randomly characterized by a predetermined exit with specified probability. Driver behavior at the roundabout entrance is randomly grouped into four categories based on space required to enter the roundabout. Three aspects of roundabout performance in particular have been studied. The first looks at overall throughput (the number of vehicles that navigate the roundabout in a given time). This is considered for different geometries, turning and arrival rates (vehicles arrive at random with a Poisson distribution, with parameter λ ⩽0.5 in general for free flow). The second investigates changes in queue length, delay time and vehicle density (ratio of the number vehicles to the number of cells) for an individual road. The third considers the impact of driver choices on throughput and operation of the roundabout. We find that throughput is influenced by the topology of the roundabout and turning rates, but only incidentally by size. Throughput reaches a maximum for critical arrival rate on one or more roads. Driver behavior has considerable impact on overall performance, with rapid congestion resulting from reckless choices. Vehicles drive on the left in Ireland, but rules are generally applicable.

Collision-free path planning for a diamond-shaped robot using two-dimensional cellular automata

This paper presents a new parallel algorithm for collision-free path planning of a diamond-shaped robot among arbitrarily shaped obstacles, which are represented as a discrete image, and its implementation in VLSI. The proposed algorithm is based on a retraction of free space onto the Voronoi diagram, which is constructed through the time evolution of cellular automata, after an initial phase during which the boundaries of obstacles are identified and coded with respect to their orientation. The proposed algorithm is both space and time efficient, since it does not require the modeling of objects or distance and intersection calculations. Additionally, the proposed twodimensional multistate cellular automaton architecture achieves high frequency of operation and it is particularly suited for VLSI implementation due to its inherent parallelism, structural locality, regularity, and modularity.

Multistate cellular automaton for reaction-diffusion processes.

A multistate cellular automaton model for computational simulation of reaction-diffusion processes at the microscopic level is presented. Diffusion is introduced through a standard stochastic scheme. On the other hand, the reaction dynamics, which applies to generic chemical reactions, consists of a deterministic algorithm driven by a discrete mapping. By means of a proper average on the states of the cellular automaton, the microscopic dynamics is fully identified with the corresponding macroscopic reaction-diffusion laws. As an illustration of the capability of the model for describing the behavior of reaction-diffusion systems, some results on pattern formation for various chemical systems are presented.

STABLE, MULTI-STATE, TIME-REVERSIBLE CELLULAR AUTOMATA WITH RICH PARTICLE CONTENT

Abstract We discuss a new class of stable multi-state Cellular Automata (CA). The CA significantly extends the two-state, time-irreversible CA proposed by Park, Steiglitz, and Thurston by providing for arbitrary numbers of states as well as time-reversibility. These new CA are dynamical systems possessing an enormous array of coherent particle-like structures. The particle interaction picture is surprisingly rich and includes particle production as well.

A model of wave propagation in an inhomogeneous excitable meduim

Wave propagation in an inhomogeneous excitable medium is modeled by a two-dimensional, multi-state cellular automaton with absorbing boundary conditions. The inhomogeneity of the meduim affects the normal pattern of wave propagation and various phenomena like delayed propagation and circular wave formation are observed. The different types of wave propagation observed in our computer simulations are summarized in a phase diagram and the predictions of the stochastic models for two of the wave propagation phenomena are presented. This system has been suggested as a simple model of the atrioventricular (AV) node.