Differential Equations for Creating Complex Cellular Automaton Patterns

This paper explores the concept of isomorphism in cellular automata (CAs), focusing on identifying and understanding isomorphic relationships between distinct CAs. A cellular automaton (CA) is said to be isomorphic to another CA if the configuration transition diagrams of the two CAs are identical. To illustrate how isomorphism entails the preservation of structural properties while allowing for variations in state representations, we exhaustively explore the isomorphic reversible rules in the first degree CAs (FDCAs), that is, the simple three-neighborhood CAs represented by eight parameters. The experimental study includes reversible FDCAs for states two to 10 with cell lengths of four to seven under the null boundary condition and states two to seven with cell lengths of four to seven under the periodic boundary condition. Given a CA, we also provide two synthesis methods for identifying the set of CAs that are isomorphic to it. The first method operates by permuting the states of a group of CA cell locations, whereas the second method involves exchanging cells at various points. We demonstrate that, depending on the range of exchanged cell positions, the isomorphic CAs produced in this way may have a bigger neighborhood than the input CA. Finally, we provide an exhaustive classification of the elementary CAs (ECAs) using the first synthesis method and compare it with existing results.

Evolution-in-Materio (EIM) is a method of using artificial evolution to exploit physical properties of materials for computation. It has previously been successfully used to evolve a multitude of different computational devices implemented in physical materials. One of the main problems has been the stability of solutions due to physical changes in the substrate materials, e.g. topological or electrical, leading to devices sensitive to the evolutionary history, i.e. instability. This paper presents stable linearly and non-linearly separable logic gates, and elementary cellular automata (CA) transition functions, successfully evolved in single-walled carbon nanotube (SWCNT) and polymer composite materials. The results herein show the feasibility of building stand-alone in-materio devices and outline a novel computational abstraction for EIM based on CA. This work is done within the European Project NASCENCE.

Cellular Automata (CA) have a long history being employed as pseudo-random number generators (PRNG), especially for cryptographic applications such as keystream generation in stream ciphers. Initially starting from the study of rule 30 of elementary CA, multiple rules where the objects of investigation and were shown to be able to pass most of the rigorous statistical tests used to assess the quality of PRNG. In all cases, the CA employed where of the classical, synchronous kind. This assumes a global clock regulating all CA updates which can be a weakness if an attacker is able to tamper it. Here we study how much asynchrony is necessary to make a CA-based PRNG ineffective. We have found that elementary CA are subdivided into three class: (1) there is a “state transition” where, after a certain level of asynchrony, the CA loses the ability to generate strong random sequences, (2) the randomness of the sequences increases with a limited level of asynchrony, or (3) CA normally unable to be used as PRNG exhibit a much stronger ability to generate random sequences when asynchrony is introduced.

Random Boolean networks (RBN) and Cellular Automata (CA) operate in a very similar way. They update their state with simple deterministic functions called Boolean function or Transition Table (TT), both being essentially the same mechanism under different names. This paper applies a concept most known from CA called Minimum Equivalence (ME). ME is applied to RBN and shows how to calculate the number of unique computations for a given number of neighbours. Crucially, it is shown how RBN rules are even more equivalent than in CA, how the set can be reduced into even fewer unique rules, and how the concept becomes more relevant with larger neighbourhoods. For example, switching transformation alone reduces the number of unique rules in RBN with 4 neighbours from 65 536 to only 3 984 (6.1%) rules. Additionally, this paper examines the ME and transformations in substrates beyond Elementary CA (ECA), such as CA with additional spatial dimensions and number of states.

HiFlow3 Version 2.0 continues the path as a multi-purpose finite element software, which provides powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differen- tial equations (PDEs). New features and functionalities, which allow to run numerical simulations with more advanced solution algorithms and discretizations in comparison to previous releases of HiFlow3, have been implemented. These comprise Uncertainty Quantification (UQ), energy-efficient multigrid techniques, Schur complement solvers for saddle-point problems, extended third-party library support, and adaptive local mesh refinement in a parallel computing environment. Furthermore, HiFlow3 has been successfully integrated into advanced and state-of-the-art simulation environments by means of the Medical Simulation Markup Language (MSML), for example.The presented new algorithms and features as well as general under-the-hood improvements have enabled excellent and relevant research activities in the fields of both medical engineering and meteorol- ogy and environmental sciences. The described show cases demonstrate the potential and advantages, which HiFlow3 can offer in performing a numerical simulation by means of finite element methods. Especially, the high performance computing capabilities of HiFlow3 – not only in the mentioned fields of applications, but also in general – have been significantly improved in Version 2.0.

Image scrambling is the process of converting an image to an unintelligible format, mainly for security reasons. The scrambling is considered as a pre-process or a post-process of security related applications such as watermarking, information hiding, fingerprinting, and encryption. Cellular automata are parallel models of computation that prove an interesting concept where a simple configuration can lead to a complex behavior. Since there are a lot of parameters to configure, cellular automata have many types and these types differ in terms of complexity and behavior. Cellular automata were previously used in scrambling different types of multimedia, but only complex two-dimensional automata were explored. We propose a scheme where the simplest type of cellular automata is used that is the elementary type. We test the scrambling degree for different cellular automata rules that belong to classes three and four of Wolfram's classification which correspond to complex and chaotic behavior; we also check the effect of other parameters such as the number of generations and the boundary condition. Experimental results show that our proposed scheme outperforms other schemes based on cellular automata in terms of scrambling degree.

Elementary cellular automata (ECA) rule 184 can be used as mathematical models of traffic flows. The slow-to-start model is obtained from the ECA rule 184 model by taking time lag for restart into consideration and is represented by 3-state 3-neighbor cellular automata (CA). In the present paper, we propose a traffic model where vehicles following the slow-to-start rule and those not following the slow-to-start rule are mixed. This model, called the mixed slow-to-start model, is represented by 4-state 3-neighbor CA. Further, we introduce the vector representation of CA with the slow-to-start rule and with the mixed slow-to-start rule, and then get their corresponding fuzzy CA. These fuzzy CA provide continuous-valued traffic models with the slow-to-start effect. Comparing the fundamental diagrams of the the slow-to-start model, the mixed slow-to-start model, and their fuzzy counterparts, we investigate the influence of the density and the mixing ratio of vehicles on traffic jams.

This paper proposes an unique image encryption scheme in the hybrid domain based on Logistic Map, Reversible Integer Wavelet Transform, and Elementary Cellular Automata (CA). In the proposed work, position permutation is done using Logistic Map equation in the spatial domain and pixel value transformation is done using CA rule and randomly generated matrix created using Logistic map equation in the Transfer Domain with the help of Reversible Integer Wavelet Transform. This paper handouts a pristine application of Logistic map functions and CA in the transfer domain and in spatial domain. Analysis shows that the relationship among the transfer domain coefficients is greatly reduced and the randomness metric is equal to 7.999 much closer to the ideal value of 8.

This work investigates elementary cellular automata comparing their dynamic evolution on two topologies: the usual regular lattice and a small-world network constructed as a directed graph. The number of iterations for any initial perturbation to propagate over all cells of the automaton corresponds to how quickly an initial configuration reaches a high entropy state, and the temporal rate of propagation of the perturbation can be evaluated from a chaotic set of elementary cellular automata rules using an entropy measure. Results indicate that the average entropy can be nearly tripled in the small-world network topology, suggesting much faster applications (e.g. in Cryptography) from alterations in the topological arrangement of usual cellular automata regular lattices.

The elementary cellular automata (ECAs) under the chaotic rule possess long periodicity and are widely used in pseudo-random number generators. However, their period is limited, related to the rule and the number of cells. Meanwhile, the Boolean functions of some ECAs are linear and vulnerable to linear analysis. Thus, the ECA cannot be directly implemented in the stream cipher. In this paper, a hybrid ECA (HECA) with dynamic mask (HECA-M) is designed. The HECA-M consists of two parts: the driving and mask parts. The driving part based on a HECA is used in generating the keystream, and the mask part based on a chaotic ECA is utilized to determine the iterative rule of the driving part. Subsequently, a stream cipher based on the HECA-M and SHA-512 is proposed. The statistic and secure analyses indicate that the proposed stream cipher possesses good randomness and can resist stream cipher analyses, such as exhaustive search, Berlekamp–Massey synthesis, guess and determine attack, time–memory–data tradeoff attack, etc. Hence, the proposed scheme can meet security requirements. Moreover, the time and space consumption of the proposed stream cipher is qualified.

One of the main goals of sentiment analysis is to analyze human perception to continuously adapt to each person’s demands. The information gathered is structured to understand the mood or emotional tone of the reviews and comments. In every aspect of society, there is always an opportunity for opinions and suggestions from the public. So, the quality of the services provided by various resources can considerably increase if we can determine whether those thoughts are positive, negative or neutral. In this paper, we employ cellular automata (CAs) to categorize the reviews as positive or negative. The fundamental advantage of the suggested technique is that there is less need to be concerned about the linguistic characteristics of human languages. The proposed method performs better by leveraging a variety of parameters that are extracted by analyzing cycle length (CL) graphs generated using CL signals produced by elementary CAs (ECAs) without employing any form of pre-trained language model. We evaluated the effectiveness of the proposed cellular automaton–based machine learning (CAML) classifier by assessing certain parameters such as recall, precision, F1 score and accuracy. Our method achieves a maximum accuracy of 92.44% for the sentiment tweet dataset. We compared the proposed technique with all existing machine learning models, as well as recent methodologies employed in the field. The suggested method performs well in predicting the positive and negative classes of sentiments using CAs.

Non-linear oscillators with Kuramoto-like local coupling: Complexity analysis and spatiotemporal pattern generation

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.

We approach the general problem of the emergence of complexity in natural, artificial or formal systems, by studying the particular problems that arise during the investigation of elementary cellular automata (ECA). Our objective is to investigate the origin of complex behaviours in the universe of ECA by using a formalism, named “vector logic”, that represents the logical functions in terms of vectors and matrices. This algebraic representation reveals the rich inner structure of the logical functions and provides a new point of view to analyse the laws that govern ECA and their interactions. We show that in this class of automata, the emergence of complexity from completely simple elements does not seem possible, and that complex behaviours result from the coexistence of intrinsic complexities in the elementary laws, together with appropriate initial conditions. “The meaning of words and of expressions, is no longer grasped. An isolated word, or a detail of a design can be understood. But the meaning of the whole escapes. Once we know the number one, we believe that we know the number two because one plus one equals two. We forget that first we must know the meaning of plus”. Comments emitted by computer Alpha 60, in “Alphaville” by Jean-Luc Godard (1965)

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.