Elementary Cellular Automaton Rule Research Articles

Elementary cellular automata realized by stateful three-memristor logic operations

Cellular automata (CA) are computational systems that exhibit complex global behavior arising from simple local rules, making them a fascinating candidate for various research areas. However, challenges such as limited flexibility and efficiency on conventional hardware platforms still exist. In this study, we propose a memristor-based circuit for implementing elementary cellular automata (ECA) by extending the stateful three-memristor logic operations derived from material implication (IMP) logic gates. By leveraging the inherent physical properties of memristors, this approach offers simplicity, minimal operational steps, and high flexibility in implementing ECA rules by adjusting the circuit parameters. The mathematical principles governing circuit parameters are analyzed, and the evolution of multiple ECA rules is successfully demonstrated, showcasing the robustness in handling the stochastic nature of memristors. This approach provides a hardware solution for ECA implementation and opens up new research opportunities in the hardware implementation of CA.

Cellular automata imbedded memristor-based recirculated logic in-memory computing

Memristor-based circuits offer low hardware costs and in-memory computing, but full-memristive circuit integration for different algorithm remains limited. Cellular automata (CA) has been noticed for its well-known parallel, bio-inspired, computational characteristics. Running CA on conventional chips suffers from low parallelism and high hardware costs. Establishing dedicated hardware for CA remains elusive. We propose a recirculated logic operation scheme (RLOS) using memristive hardware and 2D transistors for CA evolution, significantly reducing hardware complexity. RLOS’s versatility supports multiple CA algorithms on a single circuit, including elementary CA rules and more complex majority classification and edge detection algorithms. Results demonstrate up to a 79-fold reduction in hardware costs compared to FPGA-based approaches. RLOS-based reservoir computing is proposed for edge computing development, boasting the lowest hardware cost (6 components/per cell) among existing implementations. This work advances efficient, low-cost CA hardware and encourages edge computing hardware exploration.

Operator Representation and Class Transitions in Elementary Cellular Automata

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.

Synchronous solution of the parity problem on cyclic configurations, with elementary cellular automaton rule 150, over a family of directed, non-circulant, regular graphs

The parity problem is one of the classical benchmarks for probing the computational ability of cellular automata. It refers to conceiving a rule to allow deciding whether the number of 1s in an arbitrary binary sequence is an odd or even number, without resorting to globally accessing the sequence. In its most traditional formulation, it refers to cyclic configurations of odd length, which, under the action of a cellular automaton, should converge to a fixed point of all cells in the 0-state, if the configuration initially has an even number of 1s, or to a fixed point of all cells in the 1-state, otherwise. It has been shown that the problem can be solved in this formulation, in particular by a rule alone. Here, we provide a synchronous solution to the problem, by relying upon elementary cellular automaton rule 150, the elementary local parity rule, over a connection pattern among the cells defined by a family of directed, non-circulant, regular graphs. This represents the simplest synchronous solution currently known for cyclic configurations and, quite interestingly, being an instance of the solution of a non-trivial global problem by means of its local counterpart.

Temporally Stochastic Cellular Automata: Classes and Dynamics

This article introduces temporally stochastic cellular automata where a cell of the cellular system can be updated using one of the two elementary cellular automata rules, say [Formula: see text] and [Formula: see text]. The [Formula: see text] is the default rule and [Formula: see text] is temporally applied to the overall system with some probability that acts as noise to the system. As the mathematical analysis of such system is a difficult task, we use qualitative and quantitative simulations to study the dynamics of these temporally stochastic cellular automata. We are specially interested to exploring if it is possible that two periodic (resp. chaotic) rules together depict chaotic (resp. periodic) dynamics. To answer these questions, we fully classify temporally stochastic cellular automata. The study further observes phase transition and various types of class transition in the temporally stochastic cellular automata.

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

<p style='text-indent:20px;'>This paper presents a singular function on the unit interval <inline-formula><tex-math id="M1">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula> derived from the dynamics of one-dimensional elementary cellular automaton Rule <inline-formula><tex-math id="M2">\begin{document}$ 150 $\end{document}</tex-math></inline-formula>. We describe the properties of the resulting function, which is strictly increasing, uniformly continuous, and differentiable almost everywhere, and show that it is not differentiable at dyadic rational points. We also derive functional equations that this function satisfies and show that this function is the only solution of the functional equations.</p>

Phase transitions in random mixtures of elementary cellular automata

We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

Fungal Automata

We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungal mycelium. Such a filament is divided in compartments (here also called cells) by septa. These septa are invaginations of the cell wall and their pores allow for the flow of cytoplasm between compartments and hyphae. The septal pores of the fungal phylum of the Ascomycota can be closed by organelles called Woronin bodies. Septal closure is increased when the septa become older and when exposed to stress conditions. Thus, Woronin bodies act as informational flow valves. The one-dimensional fungal automaton is a binary-state ternary neighborhood CA, where every compartment follows one of the elementary cellular automaton (ECA) rules if its pores are open and either remains in state 0 (first species of fungal automata) or its previous state (second species of fungal automata) if its pores are closed. The Woronin bodies closing the pores are also governed by ECA rules. We analyze a structure of the composition space of cell-state transition and pore-state transition rules and the complexity of fungal automata with just a few Woronin bodies, and exemplify several important local events in the automaton dynamics.

Comments on Statistical Properties for Cellular Automaton Models with Probabilistic Global and Asymmetric Local Rules

We focus on several stochastic cellular automaton (CA) models which consist of the elementary CA rule 254 and probabilistic global rules in common. Statistical properties of spatiotemporal patterns...

Maximum sensitivity to update schedules of elementary cellular automata over periodic configurations

This work is a thoughtful extension of the ideas sketched in [Montalva et al. 2017], aiming at classifying elementary cellular automata (ECA) according to their maximal one-step sensitivity to changes in the schedule of cells update. It provides a complete classification of the ECA rule space for all period sizes n > 9 and, together with the classification for all period sizes n ≤ 9 presented in [Montalva et al. 2018], closes this problem and opens further questionings. Most of the 256 ECA rule's sensitivity is proved or disproved to be maximum thanks to an automatic application of basic methods. We formalize meticulous case disjunctions that lead to the results, and patch failing cases for some rules with simple arguments. This gives new insights on the dynamics of ECA rules depending on the proof method employed, as for the last rules 45 and 105 requiring (0011) * induction patterns.

Chaotic properties of elementary cellular automata with majority memory

In this paper, a practical framework of symbolic vector space is applied to uncover the time-asymptotic evolutionary behaviors of cellular automata with majority memory. This work focuses on elementary cellular automata rules with majority memory (ECAMs) and Bernoulli-shift parameters σ=1,τ=2. The concepts of forward time−τ map and characteristic function are exploited to display the Bernoulli-shift features and modes. Particularly, it is rigorously verified that ECAMs rule 10 actually defines a Bernoulli-measure global attractor in the bi-infinite symbolic vector space. It is furthermore identified that ECAMs rule 10 possesses complicated symbolic dynamics; namely, it is endowed with temporal chaotic features as positive topological entropy and topologically mixing. Therefore, ECAMs rule 10 is chaotic on its global attractor according to definitions of both Li-York and Devaney. To this end, it should be underlined that the procedure proposed in this study is applied to other ECAMs rules with the same shifting mode, and the corresponding results are exhibited.

Flexible toggles and symmetric invertible asynchronous elementary cellular automata

A sequential dynamical system (SDS) consists of a graph G with vertices v1,…,vn, a state set A, a collection of “vertex functions” {fvi}i=1n, and a permutation π∈Sn that specifies how to compose these functions to yield the SDS map [G,{fvi}i=1n,π]:An→An. In this paper, we study symmetric invertible SDS defined over the cycle graph Cn using the set of states F2. These are, in other words, asynchronous elementary cellular automata (ECA) defined using ECA rules 150 and 105. Each of these SDS defines a group action on the set F2n of n-bit binary vectors. Because the SDS maps are products of involutions, this relates to generalized toggle groups, which Striker recently introduced. In this paper, we further generalize the notion of a generalized toggle group to that of a flexible toggle group; the SDS maps we consider are examples of Coxeter elements of flexible toggle groups.Our main result is the complete classification of the dynamics of symmetric invertible SDS defined over cycle graphs using the set of states F2 and the identity update order π=123⋯n. More precisely, if T denotes the SDS map of such an SDS, then we obtain an explicit formula for |Perr(T)|, the number of periodic points of T of period r, for every positive integer r. It turns out that if we fix r and vary n and T, then |Perr(T)| only takes at most three nonzero values.

Inferring the Limit Behavior of Some Elementary Cellular Automata

Cellular automata locally define dynamical systems, discrete in space, time and in the state variables, capable of displaying arbitrarily complex global emergent behavior. One core question in the study of cellular automata refers to their limit behavior, that is, to the global dynamical features in an infinite time evolution. Previous works have shown that for finite time evolutions, the dynamics of one-dimensional cellular automata can be described by regular languages and, therefore, by finite automata. Such studies have shown the existence of growth patterns in the evolution of such finite automata for some elementary cellular automata rules and also inferred the limit behavior of such rules based upon the growth patterns; however, the results on the limit behavior were obtained manually, by direct inspection of the structures that arise during the time evolution. Here we present the formalization of an automatic method to compute such structures. Based on this, the rules of the elementary cellular automata space were classified according to the existence of a growth pattern in their finite automata. Also, we present a method to infer the limit graph of some elementary cellular automata rules, derived from the analysis of the regular expressions that describe their behavior in finite time. Finally, we analyze some attractors of two rules for which we could not compute the whole limit set.

A Language for Particle Interactions in Rule 54 and Other Cellular Automata

This is a study of localised structures in one-dimensional cellular automata, with the elementary cellular automaton Rule 54 as a guiding example. A formalism for particles on a periodic background is derived, applicable to all one-dimensional cellular automata. One can compute which particles collide and in how many ways. One can also compute the fate of a particle after an unlimited number of collisions - whether they only produce other particles, or the result is a growing structure that destroys the background pattern. For Rule 54, formulas for the four most common particles are given and all two-particle collisions are found. We show that no other particles arise, which particles are stable and which can be created, provided that only two particles interact at a time. More complex behaviour of Rule 54 requires therefore multi-particle collisions.

Computing cellular automata spectra under fixed boundary conditions via limit graphs

Cellular automata are fully discrete complex systems with parallel and homogeneous behavior studied both from the theoretical and modeling viewpoints. The limit behaviors of such systems are of particular interest, as they give insight into their emerging properties. One possible approach to investigate such limit behaviors is the analysis of the growth of graphs describing the finite time behavior of a rule in order to infer its limit behavior. Another possibility is to study the Fourier spectrum describing the average limit configurations obtained by a rule. While the former approach gives the characterization of the limit configurations of a rule, the latter yields a qualitative and quantitative characterisation of how often particular blocks of states are present in these limit configurations. Since both approaches are closely related, it is tempting to use one to obtain information about the other. Here, limit graphs are automatically adjusted by configurations directly generated by their respective rules, and use the graphs to compute the spectra of their rules. We rely on a set of elementary cellular automata rules, on lattices with fixed boundary condition, and show that our approach is a more reliable alternative to a previously described method from the literature.

Glider Collisions in Hybrid Cellular Automaton Rules

Elementary cellular automaton (ECA) rules 9 and 74, members of Chua’s Bernoulli shift rules and Wolfram’s class 2, can generate a host of gliders and complicated glider collisions by introducing the hybrid mechanism. These gliders and collisions are more plentiful than those generated by ECA rule 110. This paper presents a discussion of the symbolic dynamics of the gliders in hybrid cellular automaton (HCA) rules�9 and 74. Moreover, with regard to HCA rules that are composed of more than a couple of ECA rules, three other HCA rules endowed with gliders and glider collisions are discussed.

Dynamics of universal computation and [formula omitted] noise in elementary cellular automata

Elementary (one-dimensional two-state three-neighbor) cellular automaton rule 110 is capable of supporting universal computation and exhibits 1/f noise. However, the power spectra of rule 110 do not always exhibit 1/f-type spectrum because of periodic background particular to rule 110. Since periodic background in rule 110 does not play any role in supporting universal computation, we can expect that the removal of periodic background does not change the behavior essential to supporting universal computation. As a result of removing periodic background, we found that the power spectra fit better to 1/f noise. This result suggests 1/f noise is an intrinsic property of computational universality in cellular automata.

Spatial pattern of discrete and ultradiscrete Gray-Scott model

Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton.In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations.In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters.The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters.The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern.A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.

Ulam's cellular automaton and Rule 150

In this paper we study Ulam's cellular automaton, a nonlinear almost equicontinuous two-dimensional cell-model of crystalline growths. We prove that Ulam's automaton contains a linear chaotic elementary cellular automaton (Rule 150) as a subsystem. We also study the application of the inverse ultradiscretization, a method for deriving partial differential equations from a given cellular automaton, to Ulam's automaton. It is shown that the partial differential equation obtained by the inverse ultradiscretization preserves the self-organizing pattern of Ulam's automaton.

Expressing the Entropy of Lattice Systems as Sums of Conditional Entropies

Whether a system is to be considered complex or not depends on how one searches for correlations. We propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively taken into account. Compared to the traditional approach for estimating the entropy density, in which successive approximations builds on step-wise extensions of blocks of symbols, we show that one can take larger steps when collecting the statistics necessary to calculate the entropy density of the system. In one dimension this means that, instead of a single sweep over the system in which states are read sequentially, one take several sweeps with larger steps so that eventually the whole lattice is covered. This means that the information in correlations is captured in a different way, and in some situations this will lead to a considerably much faster convergence of the entropy density estimate as a function of the size of the configurations used in the estimate. The formalism is exemplified with both an example of a free energy minimisation scheme for the two-dimensional Ising model, and an example of increasingly complex spatial correlations generated by the time evolution of elementary cellular automaton rule 60.