Deterministic Cellular Automata Research Articles

Self-Reproduction and Evolution in Cellular Automata: 25 Years After Evoloops.

The year 2024 marks the 25th anniversary of the publication of evoloops, an evolutionary variant of Chris Langton's self-reproducing loops, which proved constructively that Darwinian evolution of self-reproducing organisms by variation and natural selection is possible within deterministic cellular automata. Over the last few decades, this line of Artificial Life research has since undergone several important developments. Although it experienced a relative dormancy of activity for a while, the recent rise of interest in open-ended evolution and the success of continuous cellular automata models have brought researchers' attention back to how to make spatiotemporal patterns self-reproduce and evolve within spatially distributed computational media. This article provides a review of the relevant literature on this topic over the past 25 years and highlights the major accomplishments made so far, the challenges being faced, and promising future research directions.

MemCA: All-Memristor Design for Deterministic and Probabilistic Cellular Automata Hardware Realization

Inspired by the behavior of natural systems, Cellular Automata (CA) tackle the demanding long-distance information transfer of conventional computers by the massive parallel computation performed by a set of locally-coupled dynamical nodes. Although CA are envisioned as powerful deterministic computers, their intrinsic capabilities are expanded after the memristor’s probabilistic switching is introduced into CA cells, resulting in new hybrid deterministic and probabilistic memristor-based CA (MemCA). In the proposed MemCA hardware realization, memristor devices are incorporated in both the cell and rule modules, composing the very first all-memristor CA hardware, designed with mixed CMOS/Memristor circuits. The proposed implementation accomplishes high operating speed and reduced area requirements, exploiting also memristor as an entropy source in every CA cell. MemCA’s functioning is showcased in deterministic and probabilistic operation, which can be externally modified by the selection of programming voltage amplitude, without changing the design. Also, the proposed MemCA system includes a reconfigurable rule module implementation that allows for spatial and temporal rule inhomogeneity.

Universality for two‐dimensional critical cellular automata

We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or ‘infected') by certain configurations of nearby infected sites. These models have close connections to statistical physics, and several specific examples have been extensively studied in recent years by both mathematicians and physicists. This general setting was first studied only recently, however, by Bollobás, Smith and Uzzell, who showed that the family of all such ‘bootstrap percolation’ models on Z 2 $\mathbb {Z}^2$ can be naturally partitioned into three classes, which they termed subcritical, critical and supercritical. In this paper we determine the order of the threshold for percolation (complete occupation) for every critical bootstrap percolation model in two dimensions. This ‘universality’ theorem includes as special cases results of Aizenman and Lebowitz, Gravner and Griffeath, Mountford and van Enter and Hulshof; significantly strengthens bounds of Bollobás, Smith and Uzzell; and complements recent work of Balister, Bollobás, Przykucki and Smith on subcritical models.

Heider balance of a chain of actors as dependent on the interaction range and a thermal noise

The Heider balance is investigated in a chain of actors, with periodic boundary conditions and the neighborhood of range r, with r as a parameter. Two model dynamics are applied: a deterministic cellular automaton [Malarz et al., Physica D 411 (2020) 132556] and the heat-bath algorithm, with the density of unbalanced–balanced triads in the role of energy. The outcome is a spectrum of energy in stationary and blinking states and a balanced–unbalanced network transition driven by thermal noise. The critical point Tc increases with the range r and it does not depend on the system size.

Hardness of Approximation for Langton’s Ant on a Twisted Torus

Langton’s ant is a deterministic cellular automaton studied in many fields, artificial life, computational complexity, cryptography, emergent dynamics, Lorents lattice gas, and so forth, motivated by the hardness of predicting the ant’s macroscopic behavior from an initial microscopic configuration. Gajardo, Moreira, and Goles (2002) proved that Langton’s ant is PTIME -hard for reachability. On a twisted torus, we demonstrate that it is PSPACE hard to determine whether the ant will ever visit almost all vertices or nearly none of them.

Multi-scale modeling of liquid-metal cooling directional solidification and solidification behavior of nickel-based superalloy casting

Liquid-metal cooling (LMC) process can offer refinement of microstructure and reduce defects due to the increased cooling rate from enhanced heat extraction, and thus an understanding of solidification behavior in nickel-based superalloy casting during LMC process is essential for improving mechanical performance of single crystal (SC) castings. In this effort, an integrated heat transfer model coupling meso grain structure and micro dendrite is developed to predict the temperature distribution and microstructure evolution in LMC process. An interpolation algorithm is used to deal with the macro-micro grids coupling issues. The algorithm of cells capture is also modified, and a deterministic cellular automaton (DCA) model is proposed to describe neighborhood cell tracking. In addition, solute distribution is also considered to describe the dendrite growth. Temperature measuring, EBSD, OM and SEM experiments are implemented to verify the proposed model, and the experiment results agree well with the simulation results. Several simulations are performed with a range of withdrawal rates, and the results indicate that 12 mm·min-1 is suitable for LMC process in this work, which can result in a fairly narrow and flat mushy zone and correspondingly exhibited fairly straight grains. The mushy zone length is about 4.8 mm in the steady state and the average deviation angle of grains is about 13.9° at the height 90 mm from the casting base under 12 mm·min-1 withdrawal process. The competitive phenomenon of dendrites at different withdrawal rates is also observed, which has a great relevant to the temperature fluctuation.

Modeling of Lizard Skin Pattern by Cellular Automaton

Properties of lizard skin pattern (LSP) comprised of light and dark scales are characterized statistically and compared with the corresponding properties of a random binary field (RBF). The similarity function of these fields exhibits an outstanding peak that indicates their stochastic character. Stochastic properties are still more generally indicated by the probability distribution of scales in hexagonal cells comprised of a center and ring. It shows that similar scales are grouped together in LSP, but not in RBF. This difference is characterized by the conditional probability that reveals why LSP appears more striped than RBF. For generation of fields resembling LSP the cellular automaton (CA) is adapted to LSP by the non-parametric regression. Its deterministic performance is demonstrated by the operation on RBF. By adding a random number generator to this model the deterministic CA is generalized to a probabilistic one. Its actions cause more expressive changing of the input field as the actions of the deterministic CA.

Ore and Chvátal‐type degree conditions for bootstrap percolation from small sets

AbstractBootstrap percolation is a deterministic cellular automaton in which vertices of a graph begin in one of two states, “dormant” or “active.” Given a fixed positive integer , a dormant vertex becomes active if at any stage it has at least active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices , we say that G ‐percolates (from ) if every vertex in becomes active after some number of steps. Let denote the minimum size of a set such that G ‐percolates from . Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure . In particular, we give an Ore‐type degree sum result that states that if a graph satisfies , then either or is in one of a small number of classes of exceptional graphs. (Here, is the minimum sum of degrees of two nonadjacent vertices in .) We also give a Chvátal‐type degree condition: If is a graph with degree sequence such that or for all , then or falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore‐type result in a paper by Freund et al.

Determination of parameters for the prognosis of rock mass deformation with the use of the cellular automaton method

The article describes the basics of construction and operation of a simple computational model built on the basis of the cellular automata theory designed to simulate rock mass subsidence caused by underground mining exploitation. The construction of the computational model as a deterministic finite cellular automaton has never been described in the literature in the context of the choice of method parameters. In the context of the construction of the model, an exemplary methodology for the selection of model parameters was presented on the example of the results of the geodetic observation of the consequences of underground mining exploitation.

Ergodicity of some classes of cellular automata subject to noise

Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise. We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.

A novel recurrence-transience transition and Tracy-Widom growth in a cellular automaton with quenched noise

We study the growing patterns formed by a deterministic cellular automaton, the rotor-router model, in the presence of quenched noise. By the detailed study of two cases, we show that: a) the boundary of the pattern displays KPZ fluctuations with a Tracy-Widom distribution, b) as one increases the amount of randomness, the rotor-router path undergoes a transition from a recurrent to a transient walk. This transition is analysed here for the first time, and it is shown that it falls in the 3D anisotropic directed percolation universality class.

Time, the Arrow of Time, and Quantum Mechanics

It is brought forward that viable theories of the physical world that have no variable at all that can play the role of time, do not exist; some notion of time is one of the very first ingredients a candidate theory should possess. Almost by definition, time has an arrow. In contrast, time reversibility, or even the possibility to run the equations of motion backwards in time, is not at all a primary requirement. This means that the direction of the arrow of time may well be uniquely defined in the theory, even locally. It is explained that a rigorous definition of time, as well as a formulation of the causality and locality concepts, can only be given when one has a model for the physical phenomena described. The only viable causality condition is one that is symmetric under time reversal. We explain these statements in terms of the author’s favoured deterministic cellular automaton interpretation of quantum mechanics, also to be referred to as 'vector space analysis', and expand on these ideas. It is also summarised how our more rigorous causality condition affects Bell’s theorems. What distinguishes quantum systems from classical ones is our fundamental inability to control the microscopic details of the initial state when phenomena are studied in the light of some theoretical model.

Land use and cover change modelling and scenarios in the Upper Uruguay Basin (Brazil)

The objective of this work was to implement a spatial dynamics model to simulate land use and cover change (LUCC) in the period between 1986, 2002 and 2008 and to generate future scenarios until 2030 in an area characterized by multiple, competitive and complex LUCC processes. The model structure follows the majority of the existing LUCC models, being composed by six consecutive procedures: (a) calculation of transition demands, (b) definition and mapping of LUCC processes, (c) generation of transition probabilities maps, (d) spatial allocation of transitions, (e) calibration and validation of the simulation results, (f) generation of scenarios. The present model which is called LanDSCAM differs from other LUCC models reported in the literature in that: (a) LanDSCAM provision for the aggregation of multiple land use and cover transitions, simplifying LUCC modelling of study areas with multiple and complex land use and cover transitions; (b) a mandatory application of user-defined regions for transition demands calculation, transitions spatial allocation and model calibration, (c) introduction of specific procedures for spatial allocation of transitions divided on a deterministic cellular automata and a stochastic patch generation functions. The use of a theoretical approach for systems dynamics modelling allowed an overview on the different models of development by society. The LUCC model developed in this study provided contributions to the generation of simulations and scenarios in situations related to multiple, competitive and complex LUCC processes.

Numerical simulation of dendrite growth in nickel-based superalloy and validated by in-situ observation using high temperature confocal laser scanning microscopy

Dendritic structures are the predominant microstructural constituents of nickel-based superalloys, an understanding of the dendrite growth is required in order to obtain the desirable microstructure and improve the performance of castings. For this reason, numerical simulation method and an in-situ observation technology by employing high temperature confocal laser scanning microscopy (HT-CLSM) were used to investigate dendrite growth during solidification process. A combined cellular automaton-finite difference (CA-FD) model allowing for the prediction of dendrite growth of binary alloys was developed. The algorithm of cells capture was modified, and a deterministic cellular automaton (DCA) model was proposed to describe neighborhood tracking. The dendrite and detail morphology, especially hundreds of dendrites distribution at a large scale and three-dimensional (3-D) polycrystalline growth, were successfully simulated based on this model. The dendritic morphologies of samples before and after HT-CLSM were both observed by optical microscope (OM) and scanning electron microscope (SEM). The experimental observations presented a reasonable agreement with the simulation results. It was also found that primary or secondary dendrite arm spacing, and segregation pattern were significantly influenced by dendrite growth. Furthermore, the directional solidification (DS) dendritic evolution behavior and detail morphology were also simulated based on the proposed model, and the simulation results also agree well with experimental results.

Exact matrix product decay modes of a boundary driven cellular automaton

We study integrability properties of a reversible deterministic cellular automaton (Rule 54 of (Bobenko et al 1993 Commun. Math. Phys. 158 127)) and present a bulk algebraic relation and its inhomogeneous extension which allow for an explicit construction of Liouvillian decay modes for two distinct families of stochastic boundary driving. The spectrum of the many-body stochastic matrix defining the time propagation is found to separate into sets, which we call orbitals, and the eigenvalues in each orbital are found to obey a distinct set of Bethe-like equations. We construct the decay modes in the first orbital (containing the leading decay mode) in terms of an exact inhomogeneous matrix product ansatz, study the thermodynamic properties of the spectrum and the scaling of its gap, and provide a conjecture for the Bethe-like equations for all the orbitals and their degeneracy.

Quantum models as classical cellular automata

A synopsis is offered of the properties of discrete and integer-valued, hence “natural”, cellular automata (CA). A particular class comprises the “Hamiltonian CA” with discrete updating rules that resemble Hamilton’s equations. The resulting dynamics is linear like the unitary evolution described by the Schrödinger equation. Employing Shannon’s Sampling Theorem, we construct an invertible map between such CA and continuous quantum mechanical models which incorporate a fundamental discreteness scale l. Consequently, there is a one-to-one correspondence of quantum mechanical and CA conservation laws. We discuss the important issue of linearity, recalling that nonlinearities imply nonlocal effects in the continuous quantum mechanical description of intrinsically local discrete CA - requiring locality entails linearity. The admissible CA observables and the existence of solutions of the l-dependent dispersion relation for stationary states are mentioned, besides the construction of multipartite CA obeying the Superposition Principle. We point out problems when trying to match the deterministic CA here to those envisioned in ‘t Hooft’s CA Interpretation of Quantum Mechanics.

From deterministic cellular automata to coupled map lattices

A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit of a continuous parameter κ. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when κ is finite and nonvanishing. In the limit all RDCAs are shown to exhibit a global homogeneous fixed-point that attracts all initial conditions. A new bifurcation is discovered for RDCAs and its location is exactly determined from the linear stability analysis of the global quiescent state. In this bifurcation, fuzziness gradually begins to intrude in a purely deterministic CA-like dynamics. The mathematical method presented allows to get insight in some highly nontrivial behavior found after the bifurcation.

Integrability of a deterministic cellular automaton driven by stochastic boundaries

We propose an interacting many-body space–time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (rule 54 of Bobenko et al 1993 Commun. Math. Phys. 158 127) on a finite one-dimensional lattice , and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying the existence of a unique nonequilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. The feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.

Semipredictable dynamical systems

A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with p possible dynamical states can be decomposed at each instant of time in a superposition of N layers involving p0, p1, …, pN−1 dynamical states each, where the pk∈N,k∈[0,N−1] are divisors of p. If the divisors coincide with the prime factors of p this decomposition is unique. Conversely, we also prove that N CA working on symbols p0, p1, …, pN−1 can be composed to create a graded CA rule with N different layers. We then show that, even when the full spatiotemporal evolution can be unpredictable, certain traits (layers) can exactly be predicted. We present explicit examples of such systems involving compositions of Wolfram’s 256 elementary CA and a more complex CA rule acting on a neighborhood of two sites and 12 symbols and whose rule table corresponds to the smallest Moufang loop M12(S3, 2).

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here, we re-derive the same formulae in a rigorous way, without resorting to any semi-empirical methods. This is done by analysing the behaviour of continuous clusters of symbols and by considering their interactions.