Limit dynamics of elementary cellular automaton 18

Complex dynamics of the type used in random number generators may emerge in elementary cellular automata with properly designed structures and cells. This chapter reviews recent results in quantifying the complexity of the dynamics in cellular automata with an emphasis on the recently discovered phenomenon called binary synchronization. It allows that two cellular automata systems with the same structure will synchronize (the receiver will duplicate the n-dimensional state vector of the transmitter) receiving only a single bit stream, produced by the output of a single cell of the transmitter cellular automaton. The decoding of this stream is possible only when the structure of the cellular automata (encryption key) is known. It is shown how the key space may be increased using various methods (e.g. using hybrid models or perturbing the cellular network model into a small-worlds model). Applications in cryptography, spread spectrum communications, and compressed sensing are reviewed. Some particularities for the implementation of such cellular automata systems in FPGA technologies are provided.

Elementary cellular automata (ECA) are linear arrays of finite-state machines (cells) which take binary states, and update their states simultaneously depending on states of their closest neighbours. We design and study ECA with memory (ECAM), where every cell remembers its states during some fixed period of evolution. We characterize complexity of ECAM in a case study of rule 126, and then provide detailed behavioural classification of ECAM. We show that by enriching ECA with memory we can achieve transitions between the classes of behavioural complexity. We also show that memory helps to ‘discover’ hidden information and behaviour on trivial (uniform, periodic), and non-trivial (chaotic, complex) dynamical systems.

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts S_beta . We show that any reversible CA F:S_beta rightarrow S_beta has an almost equicontinuous direction whenever S_beta is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations Xtimes Y of two nontrivial subshifts X and Y. We also make some preliminary observations on direct topological factorizations of beta-shifts that are subshifts of finite type.

Cellular automata are discrete dynamical systems having the ability to generate highly complex behaviour starting from a simple initial configuration and set of update rules. The discovery of rules exhibiting a high degree of global self-organization is of major importance in the study and understanding of complex systems. This task is not easily achieved since coordinated global information processing must rise from the interactions of simple components with local information and communication. In this paper, a fast supporting heuristic of linear complexity is proposed to encourage the development of rules characterized by increased dynamics with regard to cell state changes. This heuristic is integrated in an evolutionary approach to the density classification task. Computational experiments emphasize the ability of the proposed approach to facilitate an efficient exploration of the search space leading to the discovery of complex rules situated beyond the simple block-expanding rules.

We exploit the mirror and complementary symmetries of elementary cellular automata (ECAs) to rewrite their rules in terms of logical operators. The operator representation based on these fundamental symmetries enables us to construct a periodic table of ECAs that maps all unique rules in clusters of similar asymptotic behavior. We also expand the elementary cellular automaton (ECA) dynamics by introducing a parameter that scales the pace with which operators iterate the system. While tuning this parameter continuously, further emergent behavior in ECAs is unveiled as several rules undergo multiple phase transitions between periodic, chaotic and complex (class 4) behavior. This extension provides an environment for studying class transitions and complex behavior in ECAs. Moreover, the emergence of class 4 structures can potentially enlarge the capacity of many ECA rules for universal computation.

Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.

The rapid growth of online information transmission has made the secure transmission and storage of sensitive visual information crucial, particularly in light of the continuous increase in cyber threats. Traditional encryption algorithms are not very effective on images due to their very high pixel correlation, larger file sizes, and intrinsic redundancy. This paper presents a new framework of secure image encryption that will make use of advanced chaotic maps and cellular automata dynamics. It creates deterministic randomness, high sensitivity to initial conditions, and unpredictability through chaotic maps. CA introduces a series of dynamic, rule-based transformations that ensure robust properties for diffusion and confusion. During encryption, chaotic maps create pseudo-random sequences that control both the CA-based pixel scrambling and value transformation. This provides a very secure encryption method with many layers. It ensures high entropy in key generation, resistance against brute-force attacks, and high sensitivity to initial parameters. The scheme works because it can withstand differential, statistical, and noise-based attacks and show performance metrics like entropy analysis, correlation coefficients, histogram uniformity, and robustness. The suggested method is characterized by large-scale use, adaptability, and safety from modern cryptographic threats.

The two-dimensional (2-D) dynamic cellular automata (CA) model for photoresist etching simulation has been originally extended to simulate the negative chemical amplification process to further investigate its possibility to simulate monolithic simulation of lithography processes. Simulation profiles of the 2-D dynamic CA model show a good agreement with experiment profiles, and the computation time of negative chemical amplification process using the 2-D dynamic CA model is greatly reduced compared with that of the 2-D static CA model. The results indicate that the 2-D dynamic CA model is accurate, fast, and capable of being integrated into monolithic lithography process simulation. This is identified to be greatly useful to increasingly needed monolithic simulation of various lithography processes for integrated circuits (IC), Microelectromechanical Systems (MEMS), and even future Nanoelectromechanical Systems (NEMS).

For the three-dimensional (3-D) numerical study of photoresist etching processes, the 2-D dynamic cellular automata (CA) model has been successfully extended to a 3-D dynamic CA model. Only the boundary cells will be processed in the 3-D dynamic CA model and the structure of “if-else” description in the simulation program is avoided to speed up the simulation. The 3-D dynamic CA model has found to be stable, fast and accurate for the numerical study of photoresist etching processes. The exposure simulation, post-exposure bake (PEB) simulation and etching simulation are integrated together to further investigate the performances of the CA model. Simulation results have been compared with the available experimental results and the simulations show good agreement with the available experiments.

A novel three-dimensional (3-D) dynamic cellular automata (CA) model is presented for a photoresist-etching process simulation (photoresist-dissolution simulation and development simulation). In the 3-D dynamic CA model, the Moore neighborhood is adopted, and the boundary cells are only processed by using a boundary cell array, a corresponding linked list of pointers to the boundary cells, and a state flag to indicate the relations between the cells and the etching boundary. A time-compensation method is also introduced to speed up the photoresist-etching simulation. Therefore, the simulation speed is greatly increased compared with that of the static 3-D CA model, and the preferential etch in different directions reported in cell-removal models is significantly reduced. The 3-D dynamic CA model was successfully tested using some well-known etch-rate distribution test functions and has been shown to be stable, accurate, and fast. Exposure simulation, post-exposure bake simulation, and photoresist-etching simulation have been successfully integrated together to further study the effectiveness of the 3-D dynamic CA model. Simulation results show an agreement with available experimental results

Gliders, or localized propagating structures, play a role of autonomous signals in cellular-automata models of collision-based computing devices. A method is described for automatically classifying cellular automata rules for a spectrum of ordered, complex and chaotic dynamics, and thus identify rules that support interacting gliders. This is achieved by a measure of the variance of input-entropy over time. The distribution of rule classes in rule-space is discovered. The method also allows automatic “filtering” of cellular automata space-time patterns to show up gliders and related emergent configurations more clearly. Cellular automata dynamics is shown to exhibit some approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of in-degree sizes in their branching structure, and to the rule parameter Z. The research is based on computer experiments using the software Discrete Dynamics Lab (DDLab) [26]

Nara et al. (1999) proposed a method of describing digital sound data by means of rule dynamics in two states and three neighbors cellular automats. In this paper, we report the results of evaluation of the reproducibility and the performance of describing original sounds by circulating distortion rate, power spectra and return map of signal amplitude. A typical result in our numerical evaluations gives 31.49 dB as averaged distortion rate. Comparison between the two return maps taken from the original signals and the amplitude data generated by the two states and three neighbors cellular automata shows some considerable differences, which indicates that the present method of rule extraction is not yet optimized in describing sound data. In order to improve the fidelity of describing the original data, we have tried to apply all the possible rule sequences within four rules to a certain initial bit pattern. In the best case, the trial can reproduce all the possible 16 bit binary patterns. These results suggest that the our method has strong possibility of describing digital data with high fidelity and with considerable compression.

The dynamics of cellular automata that are homogeneous and symmetric with respect to up-down symmetry is expressed by the probability of the appearance of different neighbourhoods on a lattice. The distribution function found in computer simulations is used to specify the differences in the set of cellular automata. The intrinsic structure of a rule has been proposed to explain the results obtained. The problem of whether or not automata are stable, the length of time needed to reach the stabilization and the type of stabilization, are also discussed.

Signal machines form an abstract and idealized model of collision computing. Based on dimensionless signals moving on the real line, they model particle/signal dynamics in Cellular Automata. Each particle, or signal , moves at constant speed in continuous time and space. When signals meet, they get replaced by other signals. A signal machine defines the types of available signals, their speeds, and the rules for replacement in collision. A signal machine A simulates another one B if all the space-time diagrams of B can be generated from space-time diagrams of A by removing some signals and renaming other signals according to local information. Given any finite set of speeds S we construct a signal machine that is able to simulate any signal machine whose speeds belong to S . Each signal is simulated by a macro-signal , a ray of parallel signals. Each macro-signal has a main signal located exactly where the simulated signal would be, as well as auxiliary signals that encode its id and the collision rules of the simulated machine. The simulation of a collision, a macro-collision , consists of two phases. In the first phase, macro-signals are shrunk, and then the macro-signals involved in the collision are identified and it is ensured that no other macro-signal comes too close. If some do, the process is aborted and the macro-signals are shrunk, so that the correct macro-collision will eventually be restarted and successfully initiated. Otherwise, the second phase starts: the appropriate collision rule is found and new macro-signals are generated accordingly. Considering all finite sets of speeds S and their corresponding simulators provides an intrinsically universal family of signal machines.

Binary Synchronization of Complex Dynamics in Cellular Automata and its Applications in Compressed Sensing and Cryptography