Von Neumann Neighborhood Research Articles

A SYMMETRIC IDENTITY-RULE-VARIATION-BASED METHOD FOR ENUMERATING AND BUILDING RADIUS-1 TWO-STATE NCCA RULES

The class of cellular automata that preserve quantities, referred to as Number-Conserving Cellular Automata (NCCA), serves as a crucial tool for modeling various complex systems that exhibit the preservation of specific physical properties. In this paper, we first present some well-known necessary and/or sufficient conditions that must satisfy any NCCA rule. These conditions can be used to find NCCA rules using a brute-force method. However, the process of examining the set of all rules becomes impractical for complex cases with larger neighborhoods, dimensions, or number of CA states. To address this challenge, we propose a new approach to constructing and writing radius-1 two-state NCCA rules. The main idea of our contribution is the use of symmetric variations injected into the CA identity rule, which allows us to efficiently find and write NCCA rules. The proposed method has successfully reproduced the well-known 1D- and 2D-NCCA with the von Neumann neighborhood. Moreover, it has also been able to give the codes of the seventeen 2D-NCCA conservative rules with the Moore neighborhood. We believe that our approach could be generalized for higher dimensions and larger neighborhood radius.

Isotropic Growth on Cartesian Voxel Grids with von Neumann Neighborhoods for Rapid Generation of Synthetic Microstructures

Randomized instantiations of synthetic microstructures are needed to assess the statistical significance of microstructure variability. Algorithms that use Voronoi tessellations of random points produce synthetic polycrystalline microstructures with a less realistic appearance than random packing of spheroidal particles followed by growth-until-impingement. Additionally, Voronoi tessellations offer limited control of morphological parameters and are challenging to implement when the desired volume fraction is less than unity. However, unless additional physics-based constraints are applied, growth on a Cartesian voxel grid utilizing a von Neumann neighborhood element results in anisotropic growth that is generally not observable along the primary axes. The present work describes both analytical and empirically optimized corrections for directional growth rates and a framework for their inclusion in a synthetic microstructure generation algorithm. The presented algorithm can produce synthetic microstructures similar to those produced by random seed Monte Carlo techniques in a fraction of the computational time.

Conformity and Mass Media Influence in the Sznajd Model on Regular Lattices.

The polarization of opinions and difficulties in reaching a consensus are central problems of many modern societies. Understanding the dynamics governing those processes is, therefore, one of the main aims of sociophysics. In this work, the Sznajd model of opinion dynamics is investigated with Monte Carlo simulations performed on four different regular lattices: triangular, honeycomb, and square with von Neumann or Moore neighborhood. The main objective is to discuss the interplay of the probability of convincing (conformity) and mass media (external) influence and to provide the details of the possible phase transitions. The results indicate that, while stronger bonds and openness to discussion and argumentation may help in reaching a consensus, external influence becomes destructive at different levels depending on the lattice.

Heterogeneous contributions can jeopardize cooperation in the public goods game.

When studying social dilemma games, a crucial question arises regarding the impact of general heterogeneity on cooperation, which has been shown to have positive effects in numerous studies. Here, we demonstrate that heterogeneity in the contribution value for the focal public goods game can jeopardize cooperation. We show that there is an optimal contribution value in the homogeneous case that most benefits cooperation depending on the lattice. In a heterogeneous scenario, where strategy and contribution coevolve, cooperators making contributions higher than the optimal value end up harming those who contribute less. This effect is notably detrimental to cooperation in the square lattice with von Neumann neighborhood, while it can have no impact in other lattices. Furthermore, in parameter regions where a higher-contributing cooperator cannot normally survive alone, the exploitation of lower-value contribution cooperators allows their survival, resembling a parasitic behavior. To obtain these results, we examined the effect of various distributions for the contribution values in the initial condition and we conducted Monte Carlo simulations.

2D Linear CA with Mixing Boundary Conditions and Reversibility

In this paper, we consider two-dimensional cellular automata (CA) with the von Neumann neighborhood. We study the characterization of 2D linear cellular automata defined by the von Neumann neighborhood with new type of boundary conditions over the field [Formula: see text]. Furthermore, we investigate the rule matrices of 2D von Neumann CA by applying the group of permutations [Formula: see text]. Moreover, the algorithm for computing the rank of rule matrices is given. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for considered two-dimensional cellular automata are obtained.

Irreversibility of 2D Linear CA and Garden of Eden

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field [Formula: see text] (the set of integers modulo [Formula: see text]) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.

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.

Cooperation in regular lattices

In the context of Evolutionary Game Theory, one of the most noteworthy mechanisms to support cooperation is spatial reciprocity, usually accomplished by distributing players in a spatial structure allowing cooperators to cluster together and avoid exploitation. This raises an important question: how is the survival of cooperation affected by different topologies? Here, to address this question, we explore the Focal Public Goods (FPGG) and classic Public Goods Games (PGG), and the Prisoner’s Dilemma (PD) on several regular lattices: honeycomb, square (with von Neumann and Moore neighborhoods), kagome, triangular, cubic, and 4D hypercubic lattices using both analytical methods and agent-based Monte Carlo simulations. We found that for both Public Goods Games, a consistent trend appears on all two-dimensional lattices: as the number of first neighbors increases, cooperation is enhanced. Besides this, clustered topologies, i.e., those that allow two connected players to share neighbors, are the most beneficial to cooperation for the FPGG. The same is not always true for the classic PGG, where having shared neighbors between connected players may or may not benefit cooperation. We also provide a reinterpretation of the classic PGG as a focal game by representing the lattice structure of this category of games as a single interaction game with longer-ranged, weighted neighborhoods, an approach valid for any regular lattice topology. Finally, we show that depending on the payoff parametrization of the PD, there can be an equivalency between the PD and the FPGG; when the mapping between the two games is imperfect, the definition of an effective synergy parameter can still be useful to show their similarities.

Cellular automata rules solving the wireless sensor network coverage problem

The problem of an optimal coverage of a wireless sensor network area is considered. To solve this problem, a Cellular Automata (CA) approach is proposed. More specifically, the objective is to find CA rules which are able to cover the 2D space by a minimum number of so–called “Sensor Tiles”. A sensor tile consists of a von Neumann neighborhood of range 2 centered at sensor “point” and surrounded by 12 sensing “pixels”. Two probabilistic CA rules were designed that can perform this task. Results of an experimental study show that the first rule evolves very fast stable sub-optimal coverings, starting from a random configuration. The second rule finds optimal coverings, however it needs much more time for their evolution. The results are supported by a theoretical study on von Neumann neighborhoods and borrowing either from heuristics or from the spectral theory of circulant graphs.

Cooperation enhanced by the interaction diversity for the spatial public goods game on regular lattices

We investigate the evolution of cooperative behaviors of the spatial public goods game on lattices where the number of interactive neighbors are not identical for all players in the population but they can randomly select the interactive objects from the immediate neighbors in the von Neumann or Moore neighborhood. Extensive numerical simulations indicate that the cooperation level is greatly enhanced by the interaction diversity compared with the traditional model in which the social scope is homogeneous. The mechanism that explains this improvement is based on the inhomogeneous interactive domain in which cooperators can maintain their strategies easily and stabilize the cooperative neighborhood, resulting in the emergence of close cooperation clusters. Moreover, increasing uncertainty in strategy adoption process can resolve the trapped condition with a strong synergy factor and help the system to achieve the full cooperation state when the diversity mechanism is considered.

Influence of the neighborhood on cyclic models of biodiversity

This work deals with the influence of the neighborhood in simple rock–paper–scissors models of biodiversity. We consider the case of three distinct species which evolve under the standard rules of mobility, reproduction and competition. The rule of competition follows the guidance of the rock–paper–scissors game, with the prey being annihilated, leaving an empty site in accordance with the May-Leonard proposal for the predator and prey competition. We use the von Neumann neighborhood, but we consider mobility under the presence of the nearest, next nearest and next to next nearest neighbors in three distinct environments, one with equal probability and the others with probability following power law and exponential profiles. The results are different, but they all show that increasing the neighborhood increases the characteristic length of the system in an important way. We have studied other models, in particular the case where one modifies the manner a specific species competes, breaking the cyclic evolution and unveiling the interesting result in which the strongest individuals may constitute the less abundant population.

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

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.

The resolution of a mathematically bad-defined problem in the approximation of a cellular automation by the example of the description of human states

A mathematical model describing the mutual influence of bad-defined various human characteristics is constructed. This model is described by a system of differential equations that reflect the “rate” of change in a characteristic as a function of the frequency of interaction with other characteristics. The transition from differential equations to equations in finite differences and the introduction of the von Neumann neighborhood on the resulting square space of the frequency of interaction of various human characteristics allows us to introduce a cellular automaton. The sequential execution of iterations in the cellular automaton allows to track how each of the entered characteristics depends on the behavior of other characteristics.

Cluster solutions in networks of weakly coupled oscillators on a 2D square torus


 
 
 We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.
 
 

Two-dimensional rotation-symmetric number-conserving cellular automata

We present a novel method to study two-dimensional rotation-symmetric number-conserving multi-state cellular automata with the von Neumann neighborhood with radius one. This method enables a succinct and easy enumeration in all cases examined so far in literature, i.e., cellular automata with at most five states. Moreover, it allows to find all such cellular automata with six and seven states, while so far, even enumerating six-state rules was beyond the reach of computing machines. Such enumeration allows us to revisit some unresolved questions in the field. Furthermore, we give some rough estimates of the asymptotic growth of the number of such cellular automata with n states, as n tends to infinity. The results are obtained for finite square grids with periodic boundary conditions, but they are also valid in the case of the infinite square grid.

What Makes a UI Simple? Difficulty and Complexity in Tasks Engaging Visual-Spatial Working Memory

Tasks that imply engagement of visual-spatial working memory (VSWM) are common in interaction with two-dimensional graphical user interfaces. In our paper, we consider two groups of factors as predictors of user performance in such tasks: (1) the metrics based on compression algorithms (RLE and Deflate) plus the Hick’s law, which are known to be characteristic of visual complexity, and (2) metrics based on Gestalt groping principle of proximity, operationalized as von Neumann and Moore range 1 neighborhoods from the cellular automata theory. We involved 88 subjects who performed about 5000 VSWM-engaging tasks and 78 participants who assessed the complexity of the tasks’ configurations. We found that the Gestalt-based predictors had a notable advantage over the visual complexity-based ones, as the memorized chunks best corresponded to von Neumann neighborhood groping. The latter was further used in the formulation of index of difficulty and throughput for VSWM-engaging tasks, which we proposed by analogy with the infamous Fitts’ law. In our experimental study, throughput amounted to 3.75 bit/s, and we believe that it can be utilized for comparing and assessing UI designs.

Wasserstein metric-based Boltzmann entropy of a landscape mosaic: a clarification, correction, and evaluation of thermodynamic consistency

Boltzmann entropy, also called thermodynamic entropy, has long been suggested and recently reemphasized as a basis for achieving a profound understanding of landscape dynamics with thermodynamic insights. The difficulty in practically applying this entropy lies in its computation with landscapes, and many solutions have attempted to address this. The latest solution for landscape mosaics is the Wasserstein metric-based method. The first objective is to provide a clarification of and a correction to the Wasserstein metric-based method. The second is to evaluate the method in terms of thermodynamic consistency using different implementations. Two implementation methods, namely the von Neumann and the Moore neighborhood, were used, which led to two different Wasserstein metric-based entropies. Thermodynamic consistency, the fundamental property of entropy, was used as the evaluation principle. Three criteria (validity, reliability, and ability) were designed in terms of thermodynamic consistency, and corresponding indicators were proposed. Boltzmann entropies computed using all existing methods were used as benchmarks. The three indicators of the five Boltzmann entropies (including two based on the Wasserstein metric and three using existing methods) against 100,000 landscapes were computed and investigated. The reasons for both the good and poor performance of the Wasserstein metric-based entropies were identified. The Wasserstein metric-based method can be safely used with the von Neumann neighborhood. Compared with the entropies produced by existing methods, the Wasserstein metric-based entropy has worse reliability but better ability (i.e., working range). The most reliable entropy was computed using the total edge-based method.

Freezing sandpiles and Boolean threshold networks: Equivalence and complexity

The NC versus P-hard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second, we can highlight some necessary and sufficient elements to the dynamical complexity of sandpiles, with a surprisingly crucial role of cells with two grains.

EVOLUTION OF TWO-DIMENSIONAL CELLULAR AUTOMATA. NEW FORMS OF PRESENTATION

The paper considers cellular automata and forms of reflection of their evolution. Forms of evolution of elementary cellular automata are known and widely used, which allowed specialists to model different dynamic processes and behavior of systems in different directions. In the context of the easy construction of the form of evolution of elementary cellular automata, difficulties arise in representing the form of evolution of two-dimensional cellular automata, both synchronous and asynchronous. The evolution of two-dimensional cellular automata is represented by a set of states of two-dimensional forms of cellular automata, which complicates the perception and determination of the dynamics of state change. The aim of this work is to solve the problem of a fixed mapping of the evolution of a two-dimensional cellular automaton in the form of a three-dimensional representation, which is displayed in different colors on a two-dimensional image The paper proposes the evolution of two-dimensional cellular automata in the form of arrays of binary codes for each cell of the field. Each time step of the state change is determined by the state of the logical "1" or "0". Moreover, each subsequent state is determined by increasing the binary digit by one. The resulting binary code identifies the color code that is assigned to the corresponding cell at each step of the evolution iteration. As a result of such coding, a two-dimensional color matrix (color image) is formed, which in its color structure indicates the evolution of a two-dimensional cellular automaton. To represent evolution, Wolfram coding was used, which increases the number of rules for a two-dimensional cellular automaton. The rules were used for the von Neumann neighborhood without taking into account the own state of the analyzed cell. In accordance with the obtained two-dimensional array of codes, a discrete color image is formed. The color of each pixel of such an image is encoded by the obtained evolution code of the corresponding cell of the two-dimensional cellular automaton with the same coordinates. The bitness of the code depends on the number of time steps of evolution. The proposed approach allows us to trace the behavior of the cellular automaton in time depending on its initial states. Experimental analysis of various rules for the von Neumann neighborhood made it possible to determine various rules that allow the shift of an image in different directions, as well as various affine transformations over images. Using this approach, it is possible to describe various dynamic processes and natural phenomena.

Transitional Channel Flow: A Minimal Stochastic Model.

In line with Pomeau’s conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes observed in channel flow before its return to laminar flow. Numerical simulations show that a regime with bands obliquely drifting in two stream-wise symmetrical directions bifurcates into an asymmetrical regime, before ultimately decaying to laminar flow. The model is expressed in terms of a probabilistic cellular automaton of evolving von Neumann neighborhoods with probabilities educed from a close examination of simulation results. It implements band propagation and the two main local processes: longitudinal splitting involving bands with the same orientation, and transversal splitting giving birth to a daughter band with an orientation opposite to that of its mother. The ultimate decay stage observed to display one-dimensional DP properties in a two-dimensional geometry is interpreted as resulting from the irrelevance of lateral spreading in the single-orientation regime. The model also reproduces the bifurcation restoring the symmetry upon variation of the probability attached to transversal splitting, which opens the way to a study of the critical properties of that bifurcation, in analogy with thermodynamic phase transitions.