Von Neumann Neighborhood Research Articles

Some Investigations About Synchronization and Density Classification Tasks in One-dimensional and Two-dimensional Cellular Automata Rule Spaces

The study of computational aspects of cellular automata (CA) is a recurrent theme being that the investigation of specific tasks to be solved by CA rules a common and widely-known approach. We investigated two of the most-studied computational tasks: synchronization (ST) and density classification (DCT). Different specifications of CA rule space were analyzed for both tasks: one-dimensional rules with radius 1 and 2, and two-dimensional rules with von Neumann and Moore neighborhoods. We also analyzed different lattice sizes when trying to execute these tasks. Several evolutionary experiments were performed to characterize ST and DCT on these different scenarios. Some interesting results have been occurred from these experiments as the adequacy of the tasks to be solved in two-dimensional spaces instead of 1D even using rules with the same length and the dependency to the parity of the lattice size related to good rules for DCT in 1D and 2D spaces.

Boundaries, kinetic properties, and final domain structure of plane discrete uniform Poisson-Voronoi tessellations with von Neumann neighborhoods

Discrete random tessellations appear not infrequently in describing nucleation and growth transformations. Generally, several non-Euclidean metrics are possible in this case. Previously [A. Korobov, Phys. Rev. B 76, 085430 (2007)] continual analogs of such tessellations have been studied. Here one of the simplest discrete varieties of the Kolmogorov-Johnson-Mehl-Avrami model, namely, the model with von Neumann neighborhoods, has been examined per se, i.e., without continualization. The tessellation is uniform in the sense that domain boundaries consist of tiles. Similarities and distinctions between discrete and continual models are discussed.

Geometrical tile design for complex neighborhoods.

Recent research has showed that tile systems are one of the most suitable theoretical frameworks for the spatial study and modeling of self-assembly processes, such as the formation of DNA and protein oligomeric structures. A Wang tile is a unit square, with glues on its edges, attaching to other tiles and forming larger and larger structures. Although quite intuitive, the idea of glues placed on the edges of a tile is not always natural for simulating the interactions occurring in some real systems. For example, when considering protein self-assembly, the shape of a protein is the main determinant of its functions and its interactions with other proteins. Our goal is to use geometric tiles, i.e., square tiles with geometrical protrusions on their edges, for simulating tiled paths (zippers) with complex neighborhoods, by ribbons of geometric tiles with simple, local neighborhoods. This paper is a step toward solving the general case of an arbitrary neighborhood, by proposing geometric tile designs that solve the case of a “tall” von Neumann neighborhood, the case of the f-shaped neighborhood, and the case of a 3 × 5 “filled” rectangular neighborhood. The techniques can be combined and generalized to solve the problem in the case of any neighborhood, centered at the tile of reference, and included in a 3 × (2k + 1) rectangle.

Topology matters: Network topology affects outcomes from community ecology neutral models

Topology affects outcomes of processes in planar networks. Hexagon tessellations have different performance than, and are often superior to, square tessellations in applications such as fluid dynamics, percolation theory, self-avoiding walks, survey sample design, and quantization. Hexagon tessellations and square tessellations using both von Neumann and Moore neighborhoods were examined as a network topology for neutral ecology community models, following the simulation approach of Graham Bell. These models, which assume identical life history and movement properties for each individual of each species, produce collective properties of communities, such as abundance distributions, range distributions, spatial variation in abundance, species–area curves, and spatial variation in species composition, that match many empirical patterns. The simulations in this study varied the dispersal rate but kept birth, death, and immigration rates constant. For these experiments, ending community populations, species richness, Shannon diversity, and Simpson diversity were clearly different for the different topologies, but the relationship between the topologies varied as the dispersal rate changed. Empirical distributions of the performance measures also showed clear differences among topologies. The interaction of topology with dispersal, spatial boundary effects, and other parameters of these models appears to be quite complex and warrants further research.

A simple 2-dimensional model of cardiac tissue conduction

Objectives: The purpose of this study is to simulate cardiac tissue conduction using a 2-dimensional cellular automata (CA) model. Methods: Using a 2-dimensional CA model, we represented ventricular myocytes by cells arranged in a 33-by-33 grid to simulate a single ventricle. The neighborhood of each cell is a von Neumann neighborhood composed of 4 directly adjoining cells. The grid was composed in the shape of a torus to permit cells across the opposite face of the grid to interact. The design allows simulations of sustainable waves of electrical activity. The NetLogo 4.01 Beta software platform (Northwestern Illinois) was used to implement the CA model. Each tick of the simulation clock is equivalent to 1 millisecond. The simulations were run over millions of iterations. The pacemaker was set at 60 beats per minute. Results: Given normal physiologic parameters of heart rate, length of Na/Ca channel opening times, and QTc, the CA model simulates the normal conduction of the heart. The waves of depolarization and repolarization spread through the cardiac tissue without formation of pathologic dysrhythmias over hundreds of thousands of heartbeats. The virtual rhythm strip created by a virtual electrocardiogram lead shows a narrow complex rhythm at the rate set by the user-defined pacemaker parameter (Fig. 1A). Given the 2-dimensional nature of the model, the T wave is in the opposite direction to the QRS because there is no epicardium or endocardium. The plot of the electric potential at a time t + 1 vs time t shows a stable system that occupies a stable orbit (Fig. 1B). Once the system is perturbed by increasing the QTc, the CA model enters into an unstable tachycardia. The rate of ventricle depolarization ranges from 120 to 180 beats per minute depending on the exact setting of the QTc. The resultant rhythm strip (Fig. 1C) shows the tachyarrhythmia with a pacemaker spike occurring at regular intervals that is independent of the ventricular rhythm (ie, AV dissociation). The tachycardia is narrow complex because of the small number of cells in the virtual cardiac tissue. The state diagram shows the system trapped in a tachyarrhythmia from which it is unable to escape (Fig. 1D). Conclusions: The initial results of this article indicate that the cardiac conduction system can be readily simulated using CA. Cellular automata simulations offer a major advantage in their ability to monitor the electric potential of each myocyte as depolarization and repolarization proceeds. This implies that the effect of a large number of factors that influence conduction, for example, electrolyte concentrations, congenital and acquired alterations in voltage-gated channels, and ischemia-induced changes in conduction, can be readily simulated. These CA simulations are expected to provide information on the overall progress of both normal and abnormal electric activity as well as on individual myocyte function over time. Thus, CA modeling of cardiac tissue affords the novel possibility of creating simulations of the effect of antiarrhythmic medications and ischemia at the cardiac myocyte and tissue level.

Reliable Self-Replicating Machines in Asynchronous Cellular Automata

We propose a self-replicating machine that is embedded in a two-dimensional asynchronous cellular automaton with von Neumann neighborhood. The machine dynamically encodes its shape into description signals, and despite the randomness of cell updating, it is able to successfully construct copies of itself according to the description signals. Self-replication on asynchronously updated cellular automata may find application in nanocomputers, where reconfigurability is an essential property, since it allows avoidance of defective parts and simplifies programming of such computers.

Construction universality in purely asynchronous cellular automata

Universality in cellular automata (CAs), first studied by von Neumann, has attracted much research efforts over the years, especially for CA employing synchronous timing. This paper proposes a computation- and construction-universal CA with a von Neumann neighborhood that is updated in a purely asynchronous way, rather than by the conventional but less efficient way of simulating synchronous CAs on asynchronous CAs. The proposed asynchronous CA is capable of implementing self-reproducing machines. Our model employs strongly symmetric cells with 15 states.

REALIZATION OF BOOLEAN FUNCTIONS VIA CNN WITH VON NEUMANN NEIGHBORHOODS

Recently, an effective method for realizing linearly separable Boolean functions via Cellular Neural Networks (CNN), called the threshold bifurcation method, was introduced, with a CNN gene bank of four variables established [Chen & Chen, 2005]. Based on this success, the present paper is to further explore the realization of all linearly separable Boolean functions of five variables via CNN with von Neumann neighborhoods. This paper provides: (i) important and essential relations among the genes (or templates) and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set, (ii) a neat truth table of uncoupled CNN with five input variables, (iii) 94572 linearly separable Boolean functions (LSBF) in the family of 225 = 4.294967296 × 109 Boolean functions of five variables, realizable by a single CNN, and (iv) all 94572 CNN linearly separable Boolean genes (LSBG), which can be determined to form the CNN gene bank of five variables.

Delay-insensitive computation in asynchronous cellular automata

Asynchronous cellular automata (ACA) are cellular automata that allow cells to update their states independently at random times. Because of the unpredictability of the order of update, computing on ACA is usually done by simulating a timing mechanism to force all cells into synchronicity after which well-established synchronous methods of computation can be used. In this paper, we present a more effective method of computation based upon a 4-state two-dimensional ACA with von Neumann neighborhood that is based on the construction in the cellular space of delay-insensitive circuits, a special type of asynchronous circuits, whose operations are robust to arbitrary delays in operators or interconnection lines. We show that this novel ACA model can be used to construct a universal Turing machine, which suffices to prove its computational universality.

A CA-based edge operator and its performance evaluation

This paper presents a new edge detector based on a cellular automata model. A uniform cellular automaton rule using a von Neumann neighborhood is proposed for carrying out edge detection on binary and gray-scaled images. A computational model and characterization of the state space of the rule are analyzed using a finite state machine. The work shows that a cellular automata-based model often provides an optimum edge map on binary images, and on average is better than the compared edge operators for gray-scaled images.

Self-reproduction in a reversible cellular space

We investigate a problem whether self-reproduction is possible in a two-dimensional “reversible” cellular space, and give an affirmative answer. A reversible (or injective) cellular automaton (RCA) is a CA such that every configuration has at most one predecessor. In order to design an RCA we use a framework of partitioned cellular automaton (PCA). A PCA with von Neumann neighborhood is a special type of CA whose cell is divided into five parts. We designed here a reversible PCA SR 8 having 8 states in each part (thus one cell has 8 5 states). In this cellular space, encoding the shape of an object into a “gene” represented by a command sequence, copying the gene, and interpreting the gene to create an object, are all performed reversibly. We show that, by using these operations, various objects can reproduce themselves in a very simple manner.

Computation-universality of one-dimensional one-way reversible cellular automata

A reversible (or injective) cellular automaton (RCA) is a “backward deterministic” CA, i.e., every configuration of it has at most one predecessor. Reversible systems have been studied to investigate the limitation of inevitable power dissipation in computing processes, and many interesting results have been shown (see e.g. [2-41X As for RCA, Toffoli [9] showed that every (irreversible) k-dimensional CA can be simulated by a k + l-dimensional RCA and thus two-dimensional RCA is computation-universal. In the two-dimensional case, Margolus [5] gave a very simple computation-universal model of 2-state RCA with so-called Margolus neighborhood. On the other hand, Morita and Ueno [8] showed two universal models of two-dimensional 16-state RCA with usual von Neumann neighborhood. In the one-dimensional case, it has been shown by Morita and Harao [6] that a three-neighbor RCA is computation-universal. In this paper, we investigate the computing ability of a one-dimensional two-neighbor (i.e., one-way) RCA. We first investigate a one-dimensional partitioned cellular automaton (PCA), a special subclass of a CA, and show that a three-neighbor RPCA can be simu-

Symmetrical one-dimensional cellular spaces

It is shown that every one-dimensional cellular space with von Neumann neighborhood (each cell connected to its two immediate neighbors) can be simulated in real-time by a cellular space with a symmetrical local transition function with the same neighborhood, and starting with the same configuration. Symmetrical means that the individual cells cannot distinguish between left and right.

Completeness problem of multidimensional tessellation automata

The completeness problem for tessellation automata is the question of whether or not, for a given tessellation automaton, there exists a finite pattern that cannot evolve from a given primitive pattern no matter what sequence of environmental input transformations are applied. The question has been solved for one-dimensional tessellation automata. In this paper we prove that, for any d ⩾ 1 and q ⩾ 2, a d -dimentional q -state tessellation automaton with von Neumann neighborhood index is complete.

A note on cellular automata simulations

During the past several years, a number of papers have appeared on the subject of simulation in cellular automata systems. This paper describes a simulation of a two-dimensional system using a square neighborhood of n × n cells by one, using the generalized von Neumann neighborhood, which costs significantly less in terms of cell complexity than existing simulations. In addition, the equivalences between certain neighborhoods are demonstrated.

Machine Complexity Versus Interconnection Complexity in Iterative Arrays

A procedure is given for the construction of a von Neumann neighborhood iterative array to simulate a Moore neighborhood array in real time for the cases of up to three-dimensional arrays. The increase in the typical machine complexity accompanying this neighborhood complexity reduction is significantly less than that found in the general n-dimensional study of Cole [1].