Non-uniform Cellular Automata Research Articles

On the Garden of Eden theorem for non-uniform cellular automata

We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.

On linear non-uniform cellular automata: Duality and dynamics

For linear non-uniform cellular automata (NUCA) which are global perturbations of CA over an arbitrary universe, we introduce and investigate their dual linear NUCA, which are also endomorphisms over a generally infinite dimensional vector space. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Surprisingly, we show that linear NUCA with finite memory satisfy the shadowing property also known as the pseudo-orbit tracing property. Thus, linear NUCA can be useful for fault-tolerant computation. Applications on the dual surjunctivity are also obtained.

Development of a non-uniform cellular automata framework for sizing, topology and layout optimization of truss structures

This article presents a bi-level non-uniform cellular automata (CA) algorithm for the solution of sizing, topology and layout optimization of truss structures. The non-uniform CA was successfully used in a previous study to solve the weight optimization problem of truss structures for topology and sizing (El Bouzouiki, Sedaghati, and Stiharu 2021. Computers & Structures 242: 106394). In this article, an extended version of the non-uniform CA algorithm is proposed, based on the fully stressed design approach and the distribution of strain energy within the structure, to find the optimal position of the cell’s (joint’s) coordinates. The proposed non-uniform CA algorithm can solve the minimum weight optimization problem of truss structures subjected to both stress and displacement constraints. Several benchmark problems are presented to demonstrate the efficiency and accuracy of the proposed methodology.

DICOM Medical Image Security with DNA- Non-Uniform Cellular Automata and JSMP Map Based Encryption Technique

DICOM Medical Image Security with DNA- Non-Uniform Cellular Automata and JSMP Map Based Encryption Technique

On invertible and stably reversible non-uniform cellular automata

For non-uniform cellular automata (NUCA) over an arbitrary universe with multiple local transition rules, we introduce and investigate fundamental dynamical properties such as stable injectivity, stable reversibility, and stable post-surjectivity. Over infinite amenable group universes (e.g. infinite abelian groups), we show that it is impossible to obtain injective NUCA by disturbing the local transition rules of a finite number of cells of non-injective cellular automata. Moreover, we establish the equivalence between reversibility, stable reversibility, and stable injectivity for NUCA. The surjectivity and invertibility of several classes of injective and stably injective NUCA are also obtained.

A Non-Uniform Continuous Cellular Automata for Analyzing and Predicting the Spreading Patterns of COVID-19

During the COVID-19 outbreak, modeling the spread of infectious diseases became a challenging research topic due to its rapid spread and high mortality rate. The main objective of a standard epidemiological model is to estimate the number of infected, suspected, and recovered from the illness by mathematical modeling. This model does not capture how the disease transmits between neighboring regions through interaction. A more general framework such as Cellular Automata (CA) is required to accommodate a more complex spatial interaction within the epidemiological model. The critical issue of modeling in the spread of diseases is how to reduce the prediction error. This research aims to formulate the influence of the interaction of a neighborhood on the spreading pattern of COVID-19 using a neighborhood frame model in a Cellular-Automata (CA) approach and obtain a predictive model for the COVID-19 spread with the error reduction to improve the model. We propose a non-uniform continuous CA (N-CCA) as our contribution to demonstrate the influence of interactions on the spread of COVID-19. The model has succeeded in demonstrating the influence of the interaction between regions on the COVID-19 spread, as represented by the coefficients obtained. These coefficients result from multiple regression models. The coefficient obtained represents the population’s behavior interacting with its neighborhood in a cell and influences the number of cases that occur the next day. The evaluation of the N-CCA model is conducted by root mean square error (RMSE) for the difference in the number of cases between prediction and real cases per cell in each region. This study demonstrates that this approach improves the prediction of accuracy for 14 days in the future using data points from the past 42 days, compared to a baseline model.

Nilpotency and periodic points in non-uniform cellular automata

Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

Simple method of selecting totalistic rules for pseudorandom number generator based on nonuniform cellular automaton

This paper is devoted to selecting rules for one-dimensional (1D) totalistic cellular automaton (TCA). These rules are used for the generation of pseudorandom sequences, which could be useful in cryptography. The power of pseudorandom number generator (PRNG) based on nonuniform TCA can be improved using not only one rule but a large set of rules. For this purpose, each subset of rules should be analyzed with its assignation to cellular automaton (CA) cells should be analyzed. We examine each of the subsets of totalistic rules, consisting of rules with neighborhood radius equal to 1 and 2. The entropy of bitstreams generated by the nonuniform TCA points out the best set of rules appropriate for the TCA-based generator. The paper also presents the method of simple selection of CA rules based on a cryptographic criterion known as a balance. The proposed method selects a maximal size of the set of available CA rules for a given neighborhood radius and suitable for PRNG. The method guarantees to avoid conflicting assignments of rules resulting in the creation of unwanted stable bit sequences, and provides high-quality pseudorandom sequences. This technique is used to verify the subsets of rules selected experimentally. Verified rules are proposed for 1D TCA-based PRNG as a new subset of best nonuniform TCA rules. New picked, examined, and verified subset of rules could be used in TCA-based PRNG and provide cryptographically strong bit sequences and huge keyspace.

Design of New One-Time Pad Encryption Using Non-Uniform Cellular Automata With Parity Bit

This paper attempts to propose a one-time pad block encryption using non-uniform cellular automata for the public cloud which satisfies avalanche diffusion and confusion properties. It is found that by combining non-uniform and reversible cellular automata mechanisms, a robust master key in the public cloud could be generated. The simulation results show that the proposed method has large key space and protects the traditional brute force and chosen plaintext attacks in addition to having better execution time for accessing cloud data. The theoretical analysis confirmed that the proposed cellular automata–based random key generation solution was marked by acceptable security.

Reachability problem in non-uniform cellular automata

This paper deals with the CREP (Configuration REachability Problem) for non-uniform cellular automata (CAs). The cells of non-uniform CAs, we have considered here, can use different Wolfram’s rules to generate their next states. We report an algorithm which decides whether or not a configuration of a given (non-uniform) cellular automaton is reachable from another configuration. We develop a characterization tool, named Reachability tree, is used to develop theories and the decision algorithm for the CREP. Though the worst case complexity of the algorithm is exponential in time and space, but the average performance is very good.

Characteristics of fractal cellular automata constructed from linear rules

Cellular automata (CAs) have played a significant role in the study of complex systems. Recently, the recursive estimation of neighbors (REN) algorithm was proposed to extend a CA rule with a unit rule radius to rules with larger radii. This framework enables the construction of non-uniform CAs comprising cells that follow different CA rules. A non-uniform CA, referred to as fractal CA (F-CA), which comprises fractally arranged cells, inherits certain characteristics of basic CAs, including pattern replicability and time-reversibility of linear rules. In this paper, F-CAs based on linear rules, particularly the elementary CA #90 and #150, and life-like CA B1357S1357 and B1357S02468 are investigated. Cells in the F-CAs of #90 and B1357S1357 are separated into groups by their rule radius and each group has an independent lifetime. The explicitly constructed inverse rule of F-CA of #150 is more complex than that of F-CA. The complexity of the F-CA of B1357S02468 and its inverse CA is demonstrated by image scrambling. The F-CAs can be applied to encoding and decoding processes for encryption systems.

Deterministic Computing Techniques for Perfect Density Classification

The aim of this paper is to solve the density classification task (DCT), an extensively studied classical problem, using one-dimensional nonuniform Cellular Automata (CA) rules. A perfect solution of DCT requires searching for CA rules for binary strings of all possible lengths. But the generic problem is still open though the solution exists only for a specific fixed length CA. This paper provides two fundamental ideas to solve this problem in a better way. The first technique solves this problem using deterministic Turing machines which ultimately leads to generation of different CA rules under periodic boundary conditions. In the second technique, the existence of DCT solution by analyzing the state transition diagrams (STDs) of number conserving CA rules is investigated. The possibility of finding the exact solutions of DCT using Turing machine, STD and number conservation property of CA rules can be viewed as an improvement over the approximate solutions obtained by evolutionary techniques.

A Study of Chaos in Non-uniform Cellular Automata

This paper studies chaotic behavior of 1-dimensional non-uniform cellular automata (CAs). The chaotic system is unpredictable in the long run. To understand the source of unpredictability, we add an information to a cell and check how it flows to its neighbors. That is, once the state of a cell is changed, the probability to which the change affects its neighbors, is calculated. The probabilistic value decides whether one cell communicates with another cell. Depending on the communication among cells, a binary relation over the set of cells is defined in this paper which is known as communication relation. It is shown that the relation should be an equivalence relation to make a cellular automaton (CA) chaotic. The equivalence relation among the cells form an equivalence class which is named as communication class. Another important property of a chaotic CA is transitivity of the CA. Transitivity of a CA does not allow disjoint subsets of a configuration space. Based on the communication class and transitivity property of a CA, we newly define the chaos in CA. Sometimes one cell may not be able to communicate with another cell due to blocking word. The presence of blocking word in a CA makes the CA non-transitive. This paper shows a method to find out such blocking words. Finally, a parametrization technique is developed in this work based on the transitivity, communication class and blocking words. Depending on the parameter value of a non-uniform CA, one can predict whether the behavior of the CA is chaotic.

On the dynamical behaviour of linear higher-order cellular automata and its decidability

Higher-order cellular automata (HOCA) are a variant of cellular automata (CA) used in many applications (ranging, for instance, from the design of secret sharing schemes to data compression and image processing), and in which the global state of the system at time t depends not only on the state at time t−1, as in the original model, but also on the states at time t−2, …,t−n, where n is the memory size of the HOCA. We provide decidable characterizations of two important dynamical properties, namely, sensitivity to the initial conditions and equicontinuity, for linear HOCA over the alphabet Zm. These characterizations have an impact in applications since the involved linear HOCA are usually required to exhibit a chaotic or stable behaviour. Moreover, they extend the ones shown in [28] for linear CA (LCA) over the alphabet Zmn in the case n=1. We also show that linear HOCA of memory size n over Zm form a class that is indistinguishable from a specific subclass of LCA over Zmn. This enables to decide injectivity and surjectivity for linear HOCA of memory size n over Zm using the decidable characterization provided in [2] and [25] for injectivity and surjectivity of LCA over Zmn. Finally, we prove an equivalence between LCA over Zmn and an important class of non-uniform CA, another variant of CA used in many applications.

A survey of cellular automata: types, dynamics, non-uniformity and applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modeling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behavior of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.

Cellular automata in fractal arrangement

Cellular automata (CAs) have been successfully used to investigate complex phenomena across a broad range of research fields. Standard CAs can be extended by applying a novel algorithm known as the recursive estimation of neighbors. This process allows the construction of non-uniform CAs that are composed of cells with different perception areas parameterized by an extra radius. This paper proposes a non-uniform CA called Fractal CA (F-CA), which is composed of cells with self-similar fractal arrangement. By focusing on the extension of 1D elementary CAs, certain characteristics of standard CAs are carried over into F-CAs, including time reversibility of the linear rules. F-CAs based on 2D outer-totalistic CAs are also mentioned briefly.

Design of New Pseudo-Random Number Generator Based on Non-Uniform Cellular Automata

Design of New Pseudo-Random Number Generator Based on Non-Uniform Cellular Automata

Color image encryption based on hybrid hyper-chaotic system and cellular automata

This paper proposes an image encryption scheme based on Cellular Automata (CA). CA is a self-organizing structure with a set of cells in which each cell is updated by certain rules that are dependent on a limited number of neighboring cells. The major disadvantages of cellular automata in cryptography include limited number of reversal rules and inability to produce long sequences of states by these rules. In this paper, a non-uniform cellular automata framework is proposed to solve this problem. This proposed scheme consists of confusion and diffusion steps. In confusion step, the positions of the original image pixels are replaced by chaos mapping. Key image is created using non-uniform cellular automata and then the hyper-chaotic mapping is used to select random numbers from the image key for encryption. The main contribution of the paper is the application of hyper chaotic functions and non-uniform CA for robust key image generation. Security analysis and experimental results show that the proposed method has a very large key space and is resistive against noise and attacks. The correlation between adjacent pixels in the encrypted image is reduced and the amount of entropy is equal to 7.9991 which is very close to 8 which is ideal.

Three research directions in non-uniform cellular automata

The paper deals with recent developments about non-uniform cellular automata. After reviewing known results about structural stability we complete them by showing that also sensitivity to initial conditions is not structurally stable. The second part of the paper reports the complexity results about the main dynamical properties. Some proofs are shortened and clarified. The third part is completely new and starts the exploration of the fixed points set of non-uniform cellular automata.

Non-uniform Cellular Automata based on Open-ended Rule Evolution

Cellular automata (CAs) are mathematical models of spatially and temporally discrete mathematical systems. Non-uniform CAs are the cellular automata in which each cell may contain a different transition rule and change it with time, while all cells share the same transition rule in regular CAs. Little is still known about the dynamics of open-ended evolution of rules in non-uniform CAs. The purpose of our study is to construct and investigate a model of non-uniform CAs capable of open-ended rule evolution exhibiting a wide variety of behavior across all Wolfram's classes. For this purpose, we construct 1-dimensional 2-state 3 neighborhood non-uniform CAs with evolving transition rules. In the model, we found an interesting dynamics that Class II (periodical behavior) and III (chaotic behavior) patterns emerged alternately, between which Class IV patterns sometimes emerged.