One-dimensional Cellular Automata Research Articles

Order-disorder on cluster dynamics in the Q2R-Potts cellular automaton

Cellular automata (CA) are mathematical models that allow the study of emergent behavior from a bottom-up point of view, while the Potts model is renowned for its rich dynamics capable of developing both first and second order phase transitions. Here, we study the Q2R-Potts cellular automaton, which is a model that merges cellular automata and the Potts model through a microcanonical ensemble characterized by a conservative energy-like function. Our study, conducted via numerical simulations, focuses on the one-dimensional Q2R-Potts CA with three (q = 3) states, examining its dynamics through both macroscopic and microscopic quantities. We discover that the model exhibits a first-order phase transition from order to disorder around of a critical energy density, characterized by a discontinuity in a phase diagram and self-organizing clusters that follow a power-law behavior.

The algebraic entropy of one-dimensional finitary linear cellular automata

Abstract The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on Z m \mathbb{Z}_{m} from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual S ̂ \hat{S} of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on Z m \mathbb{Z}_{m} ; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of 𝑆, which coincides with the topological entropy of T = S ̂ T=\hat{S} by the so-called Bridge Theorem. In order to better understand and describe entropy, we introduce the degree deg ⁡ ( S ) \deg(S) and deg ⁡ ( T ) \deg(T) of 𝑆 and 𝑇.

Reversibility of linear cellular automata with intermediate boundary condition

<abstract><p>This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over $ \mathbb{Z}_2 $.</p></abstract>

One-Dimensional Cellular Automaton Transitions and Integral Value Transformations Representing Deoxyribonucleic Acid Sequence Evolutions

The cellular automaton (CA) and an integral value transformation (IVT) evolving in discrete time steps are two mathematical models that are well established. Theoretically, it can be suggested that a CA possesses the capacity to produce varieties of evolutionary patterns. However, computing a CA in higher dimensions or computing a nonlinear CA may be complex. In such cases, an IVT can be conveniently used. This paper presents the relation between the transition functions of a one-dimensional CA and an IVT. It also highlights the algebraic structures on the basis of binary operations for a set of transition functions of a one-dimensional CA and for a set of IVTs. The suitability of using an IVT over a CA is discussed. Also, we present the evolutionary models of two deoxyribonucleic acid (DNA) sequences through IVTs and their spacetime diagrams. This can eventually bring out some characteristic features of the evolutionary sequences.

Non-uniform number-conserving elementary cellular automata

In this paper, we investigate non-uniform elementary cellular automata (i.e., one-dimensional cellular automata whose cells can use different Wolfram rules to update their states) in the context of number conservation. As a result, we obtain an exhaustive characterization of such number-conserving cellular automata on all finite grids both with periodic and null boundary conditions. The characterization obtained allows, inter alia, to enumerate all number-conserving non-uniform elementary cellular automata, in particular those that are reversible. Surprisingly, the numbers obtained are closely related to the Fibonacci sequence.

Coupling from the past for exponentially ergodic one-dimensional probabilistic cellular automata

Coupling from the past for exponentially ergodic one-dimensional probabilistic cellular automata

МЕТОД ОБЕСПЕЧЕНИЯ ЗАЩИТЫ ПЕРЕДАВАЕМЫХ СООБЩЕНИЙ В СИСТЕМЕ ADS-B С ИСПОЛЬЗОВАНИЕМ АППАРАТА КЛЕТОЧНЫХ АВТОМАТОВ

The purpose of the study is to develop a method for encoding of transmitted ADS-B messagesbetween aircraft. The open format 1090ES of transmitted data is critical in terms of carryingout various types of attacks that can lead to a violation of the safety of aircraft operations.The work is aimed at using means of encoding and decoding messages with a private key. Researchmethods are based on the application and development of streaming data encryption usingone-dimensional cellular automata. They operate as a generator of pseudo-random sequences thattransform the elementary states of a cell of a one-dimensional cellular automaton based on typicalhardware-oriented operations. The processes of encoding and decoding data fields are based onan analytical expression using typical logical operations (or, xor). This property allows parallelprocessing of message data fields. The result is the created method for ensuring the protection oftransmitted data, additionally encoding on transmission and decoding on message reception.A distinctive feature of the method is the preservation of the protocol forma. The method uses aone-dimensional cellular automaton that encodes and decodes the target fields (coordinates, heading,etc.) using a pseudo-random number generator. The developed method belongs to the class ofhardware-oriented methods. Critical for encoding and decoding properties of periodicity of datafields and key length are eliminated by choosing an initial irrational value and organizing the“streaming” work of the encoder. If the encoding automaton is running in streaming mode, thecurrent value depends on the history of some depth, determining the length of the "automatic key"from the ADS-B message will be algorithmically impossible due to data loss. The linear complexityof the method allows you to perform transformations at the data rate. Conclusion: the developmentof hardware-oriented methods of data encoding makes it possible to increase the efficiency ofusing the ADS-B system by counteracting various types of destructive actions.

Characterization of Single Length Cycle Two-Attractor Cellular Automata Using Next-State Rule Minterm Transition Diagram

Cellular automata (CAs) are simple mathematical models that are effectively being used to analyze and understand the behavior of complex systems. Researchers from a wide range of fields are interested in CAs due to their potential for representing a variety of physical, natural and real-world phenomena. Three-neighborhood one-dimensional CAs, a special class of CAs, have been utilized to develop various applications in the field of very large-scale integration (VLSI) design, error-correcting codes, test pattern generation, cryptography and others. A thorough analysis of a three-neighborhood cellular automaton (CA) with two states per cell is presented in this paper. A graph-based tool called the next-state rule minterm transition diagram (NSRTD) is presented for analyzing the state transition behavior of CAs with fixed points. A linear time mechanism has been proposed for synthesizing a special class of irreversible CAs referred to as single length cycle two-attractor CAs (TACAs), having only two fixed points.

Cellular automata that generate symmetrical patterns give singular functions

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem’s singular functions. We also discuss two nonlinear elementary CAs, Rule 22 and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.

Analytical Solution of the Mixed Traffic Flow Cellular Automaton FI Model with the Next-Nearest-Neighbor Interaction

Based on a one-dimensional (1D) traffic flow cellular automaton (CA) FI model, a deterministic next-nearest-neighbor interaction FI model (NIFI model) is proposed. Using the mean-field analysis, the analytical solution of the NIFI model in one-dimensional traffic flow is derived under periodic boundary conditions. For the mixed traffic flow, the occupancy and the mixing ratio are introduced to describe the mixing effect. Similarly, using the mean-field method, the exact solution of the mixed traffic flow is derived from the long-time evolution to reach the steady state. The numerical simulations are carried out for the mixed traffic flow with different vehicle lengths, maximum velocities, and mixing ratios to verify the analytical solutions. The results show that the numerical simulation results agree well with the analytical solution.

On hardest languages for one-dimensional cellular automata

The existence of “hardest languages” in the sense of Greibach (“The hardest context-free language”, 1973) is investigated for two families of cellular automata. For one-way real-time cellular automata, also known as trellis automata, it is shown that there is no hardest language under reductions by deterministic finite transducers. For linear-time cellular automata, a hardest language under reductions by homomorphisms is constructed.

КЛЕТОЧНЫЕ АВТОМАТЫ И ИХ ОБОБЩЕНИЯ В ЗАДАЧАХ КРИПТОГРАФИИ. ЧАСТЬ 2

The purpose of the article is an analytical review of the application of cellular automata and their generalizations in cryptography. Research method: an analysis of scientific publications on the topic of the article. Results: The review article analyzes the literature devoted to the use of cellular automata and their generalizations for the construction of cryptographic algorithms. The article consists of two parts. The first part was devoted to classical cellular automata and symmetric cryptographic algorithms based on them. It briefly discussed the history of the theory of cellular automata and its application in various scientific fields. A review of the works of a number of authors who proposed symmetric cryptographic algorithms and pseudorandom sequence generators based on one-dimensional cellular automata was presented. The security of such cryptographic algorithms turned out to be insufficient. The following was a review of articles devoted to the use of two-dimensional cellular automata for constructing ciphers (this approach gave the best results). Multidimensional cellular automata were also mentioned. This second part of the article is devoted to a review of works devoted to the use of generalized cellular automata in cryptography – on the basis of such automata, it is possible to create symmetric encryption algorithms and cryptographic hash functions that provide a high level of security and high performance in hardware implementation (for example, on FPGA), as well as having fairly low requirements for hardware resources. In addition, an attention is paid to interesting connections of generalized cellular automata, in the context of their use in cryptography, with the theory of expander graphs. Attention is also paid to the security of cryptographic algorithms based on generalized cellular automata. The works devoted to the implementation of various cryptographic algorithms based on generalized cellular automata on FPGA and GPU are mentioned. In addition, an overview of asymmetric cryptoalgorithms based on cellular automata is given. The questions about the belonging of some problems on cellular automata and their generalizations to the class of NP-complete problems, as well as to some other complexity classes, are also considered.

Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules


 
 
 We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.
 
 

Maximal Temporal Period of a Periodic Solution Generated by a One-Dimensional Cellular Automaton

One-dimensional cellular automata evolutions with both temporal and spatial periodicity are studied. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period σ and number of states n. When σ = 2, 3, 4 or 6, and the rules are restricted to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. Non-additive rules are also constructed with temporal period on the same order as the trivial upper bound n σ . Experimental results, open problems and possible extensions of the results are also discussed.

Synthesis of Scalable Single Length Cycle, Single Attractor Cellular Automata in Linear Time

This paper proposes the synthesis of single length cycle, single attractor cellular automata (SACAs) for arbitrary length. The n-cell single length cycle, single attractor cellular automaton (SACA), synthesized in linear time O(n), generates a pattern and finally settles to a point state called the single length cycle attractor state. An analytical framework is developed around the graph-based tool referred to as the next state transition diagram to explore the properties of SACA rules for three-neighborhood, one-dimensional cellular automata. This enables synthesis of an (n+1)-cell SACA from the available configuration of an n-cell SACA in constant time and an (n+m)-cell SACA from the available configuration of n-cell and m-cell SACAs also in constant time.

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.

Nonlocal quantum computing theory and Poincare cycle in spherical states

Four new fundamental nonlocal quantum computing diagonal operator-state relations are derived which model the interaction between two adjacent atoms of an entangled atomic chain. Each atom possesses four eigen-states. These relations lead to four momentum-space cyclic transformations and are used as the computation states in one-dimensional cellular automaton. Four interacting half-observable periodic planar states appear with the same Poincare cycle. Due to the space-time rotational symmetry of these operator-state relations, a new type of periodic spherical state can be constructed consisting of eight finite space-time quadrants as the special quantum computing result.

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes [Formula: see text] and (uniform) [Formula: see text]. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

The cellular automata model of sound propagations and its application in structural noise calculations

Based on the cellular automata (CA) method and the principle of wave superposition, this study examines a computational technique to analyze the sound radiation characteristics of arbitrary shape structures. The local evolution rule of the one-dimensional (1D) CA model of Y-function was derived from the acoustic wave equation in spherical coordinate, and the relationship between Y-function and sound pressure was used to calculate the sound pressure value. On this basis, the 1D CA model of sound propagation for pulsating sphere source was established, and the superimposed sound field of multiple sphere sources in two-dimensional (2D) space was analyzed. Furthermore, combined with the principle of the wave superposition, the acoustic radiation characteristics of a rectangular piston in three-dimensional (3D) space were discussed, and the feasibility of the proposed method was verified by experiments. Finally, by introducing the finite element method, the calculation method of noise radiated by the 3D structure under external loads was presented, and the radiated sound field of a simply supported beam under loads was analyzed. The results shown that the proposed method could avoid anisotropic updating of cell state variables in the traditional 2D CA model. Moreover, the modeling was simpler, the calculation accuracy and efficiency were also higher. The results calculated by the proposed method matched well with the analytical solutions, experimental results, and numerical results, which indicated the veracity of the calculation method. The proposed method can provide a novel way for noise calculation of engineering structures.

Scattering and Perturbation Theory for Discrete-Time Dynamics.

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasienergy which is defined only modulo 2π. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative quantum field theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyze the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.