Binary Cellular Automata Research Articles

Emergence of Self-Replicating Hierarchical Structures in a Binary Cellular Automaton.

We have discovered a novel transition rule for binary cellular automata (CAs) that yields self-replicating structures across two spatial and temporal scales from sparse random initial conditions. Lower-level, shape-shifting clusters frequently follow a transient attractor trajectory, generating new clusters, some of which periodically self-duplicate. When the initial distribution of live cells is sufficiently sparse, these clusters coalesce into larger formations that also self-replicate. These formations may further form the boundaries of an expanding complex on an even larger scale. This rule, dubbed "Outlier," is rotationally symmetric and applies to 2-D Moore neighborhoods. It was evolved through genetic programming during an extensive search for rules that foster open-ended evolution in CAs. While self-replicating structures, both crafted and emergent, have been created in CAs with state sets intentionally designed for this purpose, the Outlier may be the first known rule to facilitate nontrivial emergent self-replication across two spatial scales in binary CAs.

Modeling gridded urban fractional change using the temporal context information in the urban cellular automata model

The cellular automata (CA) based models have been extensively used in urban sprawl modeling to support sustainable urban planning. However, in most existing urban CA models, only abrupt conversion (i.e., from non-urban to urban) was considered, whereas the difference in urbanization levels among different grids, as well as the nature of continuous urban evolution process with gradually increasing urban densities, were commonly ignored. Here, we proposed an impervious surface area (ISA) based urban CA model that can simulate the urban fractional change within each grid, using annual urban extent time series data from satellite observations. We implemented the developed ISA-based urban CA model in Beijing (China) and evaluated its performance through model comparison and scenario analyses. We found the ISA-based urban CA model can well capture the dynamics of urban sprawl with improved performance compared to the traditional urban CA model. The spatial pattern of modeled ISA is generally consistent with observed satellite data, with the root mean square error between the modeled and referred results of 0.14 across all changed pixels. Besides, results from our developed model agree well with the binary urban CA model, with notably reduced overestimation errors. The urban fractional change within each grid revealed in our model can provide more details than the traditional binary urban CA models, showing great potential in supporting sustainable urban development. Furthermore, the developed ISA-based urban CA model can be used for regional, even global scale urban sprawl modeling with urban fractional information in each grid, to support decision-making on the sustainable management and conservation of natural land resources.

The reversibility of one-dimensional cellular automata

Recently the reversible cellular automata are increasingly used to build high-performance cryptographic algorithms. The paper establishes a connection between the reversibility of homogeneous one-dimensional binary cellular automata of a finite size and the properties of a structure called binary filter with input memory and such finite automata properties as the prohibitions in automata output and loss of information. We show that finding the preimage for an arbitrary configuration of a one-dimensional cellular automaton of length L with a local transition function f is associated with reversibility of a binary filter with input memory. As a fact, the nonlinear filter with an input memory corresponding to our cellular automaton does not depend on the number of memory cells of the cellular automaton. The results obtained make it possible to reduce the complexity of solving massive enumeration problems related to the issues of reversibility of cellular automata. All the results obtained can be transferred to cellular automata with non-binary cell filling and to cellular automata of dimension greater than 1.

Formal Logic of Cellular Automata

This paper develops a formal logic, named L CA , targeting modeling of one-dimensional binary cellular automata. We first develop the syntax of L CA , then give semantics to L CA in the domain of all binary strings. Then the elementary cellular automata and four-neighborhood binary cellular automata are shown as models of the logic. These instances point out that there are other models of L CA . Finally it is proved that any one-dimensional binary cellular automaton is a model of the proposed logic.

Automatic Texture Based Classification of the Dynamics of One-Dimensional Binary Cellular Automata

Cellular automata present great variability in their temporal evolutions due to the number of rules and initial configurations. The possibility of automatically classifying its dynamic behavior would be of great value when studying properties of its dynamics. By counting on elementary cellular automata, and considering its temporal evolution as binary images, the authors created a texture descriptor of the images - based on the neighborhood configurations of the cells in temporal evolutions - so that it could be associated to each dynamic behavior class, following the scheme of Wolfram's classic classification. It was then possible to predict the class of rules of a temporal evolution of an elementary rule in a more effective way than others in the literature in terms of precision and computational cost. By applying the classifier to the larger neighborhood space containing 4 cells, accuracy decreased to just over 70%. However, the classifier is still able to provide some information about the dynamics of an unknown larger space with reduced computational cost.

Real Linear Automata with a Continuum of Periodic Solutions for Every Period

Interesting dynamical features such as periodic solutions of binary cellular automata are rare and therefore difficult to find, in general. In this paper, we illustrate an effective method in identifying fixed and periodic points of traditional one- and two-dimensional [Formula: see text]-valued cellular automaton systems, using cycle graphs. We also show that when the binary or [Formula: see text]-valued cellular states are extended to real-values and when defined as doubly-infinite vectors, there exists a continuum of periodic solutions for every period, even when the governing local rules are simply linear. In addition, we demonstrate the important effect that boundary conditions have on systems’ dynamical structures.

Block invariance and reversibility of one dimensional linear cellular automata

Consider a one-dimensional, binary cellular automaton f (the CA rule), where its n nodes are updated according to a deterministic block update (blocks that group all the nodes and such that its order is given by the order of the blocks from left to right and nodes inside a block are updated synchronously). A CA rule is block invariant over a family F of block updates if its set of periodic points does not change, whatever the block update of F is considered. In this work, we study the block invariance of linear CA rules by means of the property of reversibility of the automaton because such a property implies that every configuration has a unique predecessor, so, it is periodic. Specifically, we extend the study of reversibility done for the Wolfram elementary CA rules 90 and 150 as well as, we analyze the reversibility of linear rules with neighbourhood radius 2 by using matrix algebra techniques.

A portfolio of classification problems by one-dimensional cellular automata, over cyclic binary configurations and parallel update

Decision problems addressed by cellular automata have been historically expressed either as determining whether initial configurations would belong to a given language, or as classifying the initial configurations according to a property in them. Unlike traditional approaches in language recognition, classification problems have typically relied upon cyclic configurations and fully paralell (two-way) update of the cells, which render the action of the cellular automaton relatively less controllable and difficult to analyse. Although the notion of cyclic languages have been studied in the wider realm of formal languages, only recently a more systematic attempt has come into play in respect to cellular automata with fully parallel update. With the goal of contributing to this effort, we propose a unified definition of classification problem for one-dimensional, binary cellular automata, from which various known problems are couched in and novel ones are defined, and analyse the solvability of the new problems. Such a unified perspective aims at increasing existing knowledge about classification problems by cellular automata over cyclic configurations and parallel update.

Network Analysis Using Spatio-Temporal Patterns

Different network models have been proposed along the last years inspired by real-world topologies. The characterization of these models implies the understanding of the underlying network phenomena, which accounts structural and dynamic properties. Several mathematical tools can be employed to characterize such properties as Cellular Automata (CA), which can be defined as dynamical systems of discrete nature composed by spatially distributed units governed by deterministic rules. In this paper, we proposed a method based on the modeling of one specific CA over distinct network topologies in order to perform the classification of the network model. The proposed methodology consists in the modeling of a binary totalistic CA over a network. The transition function that governs each CA cell is based on the density of living neighbors. Secondly, the distribution of the Shannon entropy is obtained from the evolved spatio-temporal pattern of the referred CA and used as a network descriptor. The experiments were performed using a dataset composed of four different types of networks: random, small-world, scale-free and geographical. We also used cross-validation for training purposes. We evaluated the accuracy of classification as a function of the initial number of living neighbors, and, also, as a function of a threshold parameter related to the density of living neighbors. The results show high accuracy values in distinguishing among the network models which demonstrates the feasibility of the proposed method.

Facial Expression Analysis for Content-Based Video Retrieval

In this work, we propose a technique for facial expression recognition to bridge the semantic gap among the features that can be extracted in a content-based video retrieval system. The paper aims to provide accurate and reliable facial expression recognition of a dominant person in video frames using deterministic binary cellular automata (DBCA). Both geometric and appearance-based features are used. Efficient dimension reduction techniques for face detection and recognition are applied. Using the facial action coding system (FACS), one can code automatically nearly any anatomically possible facial expression, deconstructing it into what are called as action units (AUs). By employing two-dimensional deterministic binary cellular automaton systems (2D-DBCA), a scheme is developed to classify the facial expressions representing various emotions to retrieve video scenes/shots. Extensive experiments on Cohn–Kanade database, Yale database, and large movie videos show the superiority of the proposed method, in comparison with support vector machines (SVMs), hidden Markov models (HMMs), and neural network (NN) classifiers.

On solutions to evolution equations defined by lattice operators

We discuss a specific form of evolution equations defined by lattice operators. We give exact solutions for a class of those equations and evaluate the complexity of the solutions. Moreover we discuss the relationship between them and binary cellular automata, and analyze their asymptotic behavior utilizing the explicit expression of the solution.

Modeling of urban growth dynamics and its impact on surface runoff characteristics

Modeling urban growth and generating scenarios are essential for studying the impact and sustainability of an urban hydrologic system. Urban systems are regarded as complex self-organizing systems, where the dynamic transitions from one form of landuse to another occur over a period of time. Therefore, a modeling framework that captures and simulates this complex behavior is essential for generating urban growth scenarios. Cellular Automata (CA)-based models have the potential to model such discrete dynamic systems. In this study, a constraint-based binary CA model was used to predict the future urban growth scenario of the city of Roorkee (India). A hydrologic model was applied on the simulated urban catchment to study its hydrologic response. The Natural Resources Conservation Service Curve Number (NRCS-CN) method, which is suitable for ungauged urban watersheds, was adopted to determine the impact of urban growth on the quantity of storm water runoff over a period of time. The results indicate that urban growth has a linear relationship with peak discharge and time to peak for the catchment under investigation.

GEOMETRY INDUCED DELAYS OF SLIME MOULD PROPAGATION

Slime mould Physarum polycephalum propagates on nutrient substrates similarly to auto-waves in nonlinear media. In experimental laboratory studies we uncover that the width of geometrically constrained substrate affects the speed of Physarum propagation. We show that Physarum slows down when the width of the substrate increases. The slime mould propagates quicker from the vertex of a triangle to its base than from the base to the vertex. Physarum grows quicker in narrow channels than in wider channels. One can also slow down Physarum propagation by making a finite size expansion of the otherwise narrow channel. In computational experiments with a binary state cellular automaton model we demonstrate that a limitation on the slime mould's body mass production rate could be an underlying mechanism for the width-dependent slowdown of Physarum propagation.

Solving the parity problem in one-dimensional cellular automata

The parity problem is a well-known benchmark task in various areas of computer science. Here we consider its version for one-dimensional, binary cellular automata, with periodic boundary conditions: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. Since the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1), it suffices to account for odd-sized lattices only. We are interested in determining the minimal neighbourhood size that allows the problem to be solvable for any arbitrary initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can solve the parity problem, even in the simpler case of prime-sized lattices. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem; however, we review recent data against a solution in radius 3, thus providing strong empirical evidence that there may not exist a radius 3 solution even for prime-sized lattices only, contrary to a recent conjecture in the literature.

A New Adaptive Fast Cellular Automaton Neighborhood Detection and Rule Identification Algorithm

An important step in the identification of cellular automata (CA) is to detect the correct neighborhood before parameter estimation. Many authors have suggested procedures based on the removal of redundant neighbors from a very large initial neighborhood one by one to find the real model, but this often induces ill conditioning and overfitting. This is true particularly for a large initial neighborhood where there are few significant terms, and this will be demonstrated by an example in this paper. By introducing a new criteria and three new techniques, this paper proposes a new adaptive fast CA orthogonal-least-square (Adaptive-FCA-OLS) algorithm, which cannot only adaptively search for the correct neighborhood without any preset tolerance but can also considerably reduce the computational complexity and memory usage. Several numerical examples demonstrate that the Adaptive-FCA-OLS algorithm has better robustness to noise and to the size of the initial neighborhood than other recently developed neighborhood detection methods in the identification of binary CA.

Stochastic Cellular Automata Solutions to the Density Classification Problem

In the density classification problem, a binary cellular automaton should decide whether an initial configuration contains more 0s or 1s. The answer is given when all cells of the CA agree on a given state (0 or 1). This problem is known for having no exact solution in the case of binary deterministic one-dimensional CA. We investigate how randomness in CA may help us solve the problem. We analyse the behaviour of stochastic CA rules that perform the density classification task. We show that describing stochastic rules as a ''blend'' of deterministic rules allows us to derive quantitative results on the classification time and the classification time of previously studied rules. We introduce a new rule whose effect is to spread defects and to wash them out. This stochastic rule solves the problem with an arbitrary precision, that is, its quality of classification can be made arbitrarily high, though at the price of a longer time to converge. We experimentally demonstrate that this rule exhibits good scaling properties and that it attains qualities of classification never reached so far.

Simulation of dendritic growth for ternary alloys based on modified cellular automaton model

Based on the binary cellular automaton method, a modified cellular automaton model for ternary alloys is developed to simulate dendrite growth controlled by solutal effects and microsegregation in the low Peclet number regime by coupling PanEngine, which is a multicomponent thermodynamic and equilibrium calculation engine. The model can be used to calculate the interfacial equilibrium composition by considering the influence of Gibbs-Thomson effect induced curvature undercooling, and multicomponents contributed constitutional undercooling. Meanwhile, the growth velocity of interface is determined by solving the solute conservation equation simultaneously with dimensionless solute supersaturation equation for each alloying element. Moreover, equilibrium liquidus temperature and equilibrium solid concentration at the interface are derived by PanEngine. Free dendrite growth of Al-7%Si-xMg ternary alloys is simulated by the present model, which shows that the increase of solute Mg can suppress the growths of both primary and secondary dendrite arms. Meanwhile, constrained columnar dendrite growth of Al-7%Si-0.5%Mg with the increases of pulling velocity and constant thermal gradient during directional solidification is calculated. The results reveal the competitive growth of columnar dendrites, and demonstrate that the primary dendrite arm spacing would decrease as the pulling velocity increases, which accords well with the Hunt model.

COMPLEXITY INDEX OF OUTER-TOTALISTIC BINARY CELLULAR AUTOMATA WITH ARBITRARY DIMENSION AND NEIGHBORHOOD

The concept of complexity index is of key importance in the systematic analysis of the dynamics of Cellular Automata (CA); nevertheless, it has been defined only for the special case of 1D elementary CA. In this paper, we first introduce a complexity index for outer-totalistic binary CA with arbitrary dimension and neighborhood by means of a rigorous mathematical theory, and then propose a method to find it easily, given only the truth table of an outer-totalistic binary CA rule. Through our technique, we study in detail both 1D and 2D elementary CA rules, including the well-known Game of Life.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART XIV: MORE BERNOULLI στ-SHIFT RULES

Over the past eight years, we have studied one of the simplest, yet extremely interesting, dynamical systems; namely, the one-dimensional binary Cellular Automata. The most remarkable results have been presented in a series of papers which is concluded by the present article. The final stop of our odyssey is devoted to the analysis of the second half of the 30 Bernoulli στ-shift rules, which constitute the largest among the six groups in which we classified the 256 local rules. For all these 15 rules, we present the basin-tree diagrams obtained by using each bit string with L ≤ 8 as initial state, a summary of the characteristics of their ω-limit orbits, and the space-time patterns generated from the superstring. Also, in the last section we summarize the main results we obtained by means of our "nonlinear dynamics perspective".

Response Curves for Cellular Automata in One and Two Dimensions

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.