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At least 451 records · Page 25Linked to original sources

arXiv2022

Step Size is a Consequential Parameter in Continuous Cellular Automata

Step size in continuous cellular automata (CA) plays an important role in the stability and behavior of self-organizing patterns. Continous CA dynamics are defined by formula very similar to numerical estimation of physics-based ordinary differential equations, specifically Euler's method, for which a large step size is often inaccurate and unstable. Rather than asymptotically approaching more accurate estimates of CA dynamics with decreasing step size, continuous CA may support different self-organizing patterns at different ranges of step size. We discuss several examples of mobile patterns that become unstable at step sizes that are too small as well as too large. Additionally, an individual mobile pattern may exhibit qualitatively different behavior across a range of step sizes. We demonstrate examples of the effects of step size in pattern stability and qualitative behavior in continuous CA implemented in the Lenia framework and its variant, Glaberish.

nlin.CG↗

arXiv2022

Two Ways of Understanding Social Dynamics: Analyzing the Predictability of Emergence of Objects in Reddit r/place Dependent on Locality in Space and Time

Lately, studying social dynamics in interacting agents has been boosted by the power of computer models, which bring the richness of qualitative work, while offering the precision, transparency, extensiveness, and replicability of statistical and mathematical approaches. A particular set of phenomena for the study of social dynamics is Web collaborative platforms. A dataset of interest is r/place, a collaborative social experiment held in 2017 on Reddit, which consisted of a shared online canvas of 1000 pixels by 1000 pixels co-edited by over a million recorded users over 72 hours. In this paper, we designed and compared two methods to analyze the dynamics of this experiment. Our first method consisted in approximating the set of 2D cellular-automata-like rules used to generate the canvas images and how these rules change over time. The second method consisted in a convolutional neural network (CNN) that learned an approximation to the generative rules in order to generate the complex outcomes of the canvas. Our results indicate varying context-size dependencies for the predictability of different objects in r/place in time and space. They also indicate a surprising peak in difficulty to statistically infer behavioral rules towards the middle of the social experiment, while user interactions did not drop until before the end. The combination of our two approaches, one rule-based and the other statistical CNN-based, shows the ability to highlight diverse aspects of analyzing social dynamics.

physics.soc-ph↗

arXiv2022

Global Cellular Automata GCA -- A Massively Parallel Computing Model

The Global Cellular Automata (GCA) Model is a generalization of the Cellular Automata (CA) Model. The GCA model consists of a collection of cells which change their states depending on the states of their neighbors, like in the classical CA model. In generalization of the CA model, the neighbors are no longer fixed and local, they are variable and global. In the basic GCA model, a cell is structured into a data part and a pointer part. The pointer part consists of several pointers that hold addresses to global neighbors. The data rule defines the new data state, and the pointer rule define the new pointer states. The cell's state is synchronously or asynchronously updated using the new data and new pointer states. Thereby the global neighbors can be changed from generation to generation. Similar to the CA model, only the own cell's state is modified. Thereby write conflicts cannot occur, all cells can work in parallel which makes it a massively parallel model. The GCA model is related to the CROW (concurrent read owners write) model, a specific PRAM (parallel random access machine) model. Therefore many of the well-studied PRAM algorithms can be transformed into GCA algorithms. Moreover, the GCA model allows to describe a large number of data parallel applications in a suitable way. The GCA model can easily be implemented in software, efficiently interpreted on standard parallel architectures, and synthesized / configured into special hardware target architectures. This article reviews the model, applications, and hardware architectures.

cs.FL↗

arXiv2022

Affinity Classification Problem by Stochastic Cellular Automata

This work introduces a new problem, named as, affinity classification problem which is a generalization of the density classification problem. To solve this problem, we introduce temporally stochastic cellular automata where two rules are stochastically applied in each step on all cells of the automata. Our model is defined on 2-dimensional grid having affection capability. We show that this model can be used in several applications like modeling self-healing systems.

cs.FL↗

arXiv2023

Emergent structure and dynamics of tropical forest-grassland landscapes

Previous work indicates that tropical forest can exist as an alternative stable state to savanna. Therefore, perturbation by climate change or human impact may lead to crossing of a tipping point beyond which there is rapid forest dieback that is not easily reversed. A hypothesised mechanism for such bistability is a feedback between fire and vegetation, where fire spreads as a contagion process on grass patches. Theoretical models have largely implemented this mechanism implicitly, by assuming a threshold dependence of fire spread on flammable vegetation. Here, we show how the nonlinear dynamics and bistability emerge spontaneously, without assuming equations or thresholds for fire spread. We find that the forest geometry causes the nonlinearity that induces bistability. We demonstrate this in three steps. First, we model forest and fire as interacting contagion processes on grass patches, %using a cellular automaton showing that spatial structure emerges due to two counteracting effects on the forest perimeter: forest expansion by dispersal, and forest erosion by fires originating in adjacent grassland. Then, we derive a landscape-scale balance equation in which these two effects link forest geometry and dynamics: forest expands proportionally to its perimeter, while it shrinks proportionally to its perimeter weighted by adjacent grassland area. Finally, we show that these perimeter quantities introduce nonlinearity in our balance equation and lead to bistability. Relying on the link between structure and dynamics, we propose a forest resilience indicator that could be used for targeted conservation or restoration.

q-bio.PE↗

arXiv2022

Research on work zone vehicle queuing behavior based on cellular automata

A model is proposed to estimate the work zone queue length, and the cellular automata based on empirical data is used for model validation. This estimation model can be applied to work zone organization and management to improve work zone capacity and security. Relationship between the average queue length and the warning zone length can be found, and the appropriate warning zone length can be determined according to design flow. Moreover, the appropriate work zone lane-changing strategies under different design flows are found through the estimation model.

nlin.CG↗

arXiv2022

Intention to explore the role of discretization in the emergence of self-organization in certain approximations of continuous cellular automata and other complex dynamic systems

John H. Conway's Game of Life, as well as cellular automata in the larger family of Life-like CA, are discrete: the cells have a binary state space and the birth and survival transition rules are 9-bits apiece. Inspired by Life, several projects have developed continuously-valued cellular automata frameworks in 2 dimensions. These CA systems are necessarily imperfect approximations of ideal continuous systems when they are implemented in a digital computer, and this inevitably leads to discretization errors. As we know since at least the time of Poincairé's work on the three-body problem, arbitrarily small errors in a complex dynamic system can lead to substantial behavioral deviation over time. This outline of intent is based on observations in cellular automata that, in certain cases, errors are not only well-tolerated, but are essential for self-organization. This manuscript describes a set of experiments to investigate the importance of discretization in approximating continuously valued dynamic systems, how and when discretization is required for self-organization to occur, and the relevance the results might have with respect to emergent agency and analogies for consciousness.

nlin.CG↗

arXiv2022

On linear non-uniform cellular automata: duality and dynamics

For linear non-uniform cellular automata (NUCA) over an arbitrary universe, we introduce and investigate their dual linear NUCA. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Moreover, we show that linear NUCA satisfy the important shadowing property. Applications on the dual surjunctivity are also obtained.

math.DS↗

arXiv2022

Some pointwise and decidable properties of non-uniform cellular automata

For non-uniform cellular automata (NUCA) with finite memory over an arbitrary universe with multiple local transition rules, we show that pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity properties are respectively equivalent to nilpotency, periodicity, and eventual periodicity. Moreover, we prove that every linear NUCA which satisfies pointwise a polynomial equation (which may depend on the configuration) must be an eventually periodic linear NUCA. Generalizing results for higher dimensional group and linear CA, we also establish the decidability results of the above dynamical properties as well as the injectivity for arbitrary NUCA with finite memory which are local perturbations of higher dimensional linear and group CA. Some generalizations to the case of sparse global perturbations of higher dimensional linear and group CA are also obtained.

math.DS↗

arXiv2024

Emergent spacetime from purely random structures

We examine the fundamental question whether a random discrete structure with the minimal number of restrictions can converge to continuous metric space. We study the geometrical properties such as the dimensionality and the curvature emerging out of the connectivity properties of uniform random graphs. In addition we introduce a simple evolution mechanism for the graph by removing one edge per a fundamental quantum of time from an initially complete graph. We show an exponential growth of the radius of the graph, that ends up in a random structure with emergent average spatial dimension $D=3$ and zero curvature $K=0$, resembling a flat 3D manifold, that could describe the observed space in our universe and some of its geometrical properties. In addition, we introduce a generalized action for graphs based on physical quantities on different subgraph structures that helps to recover the well known properties of spacetime as described in general relativity, like time dilation due to gravity. Also, we show how various quantum mechanical concepts such as generalized uncertainty principles based on the statistical fluctuations can emerge from random discrete models. Moreover, our approach leads to a unification of space and matter-energy, for which we propose a mass-energy-space equivalence that leads to a way to transform between empty space and matter-energy via the cosmological constant.

cond-mat.dis-nn↗

arXiv2022

Cultivating the Garden of Eden

Garden of Eden (GOE) states in cellular automata are grid configurations which have no precursors, that is, they can only occur as initial conditions. Finding individual configurations that minimize or maximize some criterion of interest (e.g., grid size, density, etc.) has been a popular sport in recreational mathematics, but systematic studies of the set of GOEs for a cellular automaton have been rare. This paper presents the current results of an ongoing computational study of GOE configurations in Conway's Game of Life (GoL) cellular automaton. Specifically, we describe the current status of a map of the layout of GOEs and nonGOEs in density/size space, characterize how the density-dependent structure of the number of precursors varies with increasing grid size as we approach the point where GOEs begin to occur, provide a catalog of all known GOE configurations up to a grid size of 11x11, and initiate a study of the structure of the network of constraints that characterize GOE vs. nonGOE configurations.

nlin.CG↗

arXiv2022

Cellular Automata: Temporal Stochasticity and Computability

In this dissertation, we study temporally stochasticity in cellular automata and the behavior of such cellular automata. The work also explores the computational ability of such cellular automaton that illustrates the computability of solving the affinity classification problem. In addition to that, a cellular automaton, defined over Cayley tree, is shown as the classical searching problem solver. The proposed temporally stochastic cellular automata deals with two elementary cellular automata rules, say $f$ and $g$. The $f$ is the default rule, however, $g$ is temporally applied to the overall system with some probability $τ$ which acts as a noise in the system. After exploring the dynamics of temporally stochastic cellular automata (TSCAs), we study the dynamical behavior of these temporally stochastic cellular automata (TSCAs) to identify the TSCAs that converge to a fixed point from any seed. We apply each of the convergent TSCAs to some standard datasets and observe the effectiveness of each TSCA as a pattern classifier. It is observed that the proposed TSCA-based classifier shows competitive performance in comparison with existing classifier algorithms. We use temporally stochastic cellular automata to solve a new problem in the field of cellular automata, named as, affinity classification problem which is a generalization of the density classification problem . We show that this model can be used in several applications, like modeling self-healing systems. Finally, we introduce a new model of computing unit developed around cellular automata to reduce the workload of the Central Processing Unit (CPU) of a machine to compute. Each cell of the computing unit acts as a tiny processing element with attached memory. Such a CA is implemented on the Cayley Tree to realize efficient solutions for diverse computational problems.

nlin.CG↗

arXiv2022

Exploring the Dynamics of Fungal Cellular Automata

Cells in a fungal hyphae are separated by internal walls (septa). The septa have tiny pores that allow cytoplasm flowing between cells. Cells can close their septa blocking the flow if they are injured, preventing fluid loss from the rest of filament. This action is achieved by special organelles called Woronin bodies. Using the controllable pores as an inspiration we advance one and two-dimensional cellular automata into Elementary fungal cellular automata (EFCA) and Majority fungal automata (MFA) by adding a concept of Woronin bodies to the cell state transition rules. EFCA is a cellular automaton where the communications between neighboring cells can be blocked by the activation of the Woronin bodies (Wb), allowing or blocking the flow of information (represented by a cytoplasm and chemical elements it carries) between them. We explore a novel version of the fungal automata where the evolution of the system is only affected by the activation of the Wb. We explore two case studies: the Elementary Fungal Cellular Automata (EFCA), which is a direct application of this variant for elementary cellular automata rules, and the Majority Fungal Automata (MFA), which correspond to an application of the Wb to two dimensional automaton with majority rule with Von Neumann neighborhood. By studying the EFCA model, we analyze how the 256 elementary cellular automata rules are affected by the activation of Wb in different modes, increasing the complexity on applied rule in some cases. Also we explore how a consensus over MFA is affected when the continuous flow of information is interrupted due to the activation of Woronin bodies.

nlin.CG↗

arXiv2022

Photonic Elementary Cellular Automata for Simulation of Complex Phenomena

Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer presents new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing schemes using light.

cs.ET↗

arXiv2024

Multiband linear cellular automata and endomorphisms of algebraic vector groups

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.

math.DS↗

arXiv2023

Classification of Cellular Automata based on Statistical Mechanics

Cellular automata are a set of computational models in discrete space that have a discrete time evolution defined by neighbourhood rules. They are used to simulate many complex systems in physics and science in general. In this work, statistical mechanics and thermodynamics are used to analyse a large set of outer totalistic two-dimensional cellular automata. Thermodynamic variables and potentials are derived and computed according to three different approaches to determine if a cellular automaton rule is representing a system akin to the ideal gas, in or out of the thermodynamical equilibrium. It is suggested that this classification is sufficiently robust and predictive of interesting properties for particular set of rules.

nlin.CG↗

arXiv2022

Building Squares with Optimal State Complexity in Restricted Active Self-Assembly

Tile Automata is a recently defined model of self-assembly that borrows many concepts from cellular automata to create active self-assembling systems where changes may be occurring within an assembly without requiring attachment. This model has been shown to be powerful, but many fundamental questions have yet to be explored. Here, we study the state complexity of assembling $n \times n$ squares in seeded Tile Automata systems where growth starts from a seed and tiles may attach one at a time, similar to the abstract Tile Assembly Model. We provide optimal bounds for three classes of seeded Tile Automata systems (all without detachment), which vary in the amount of complexity allowed in the transition rules. We show that, in general, seeded Tile Automata systems require $Θ{(\log^{\frac{1}{4}} n)}$ states. For single-transition systems, where only one state may change in a transition rule, we show a bound of $Θ{(\log^{\frac{1}{3}} n)}$, and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of $Θ( (\frac{\log n}{\log \log n})^\frac{1}{2} )$.

cs.FL↗

arXiv2022

Metastable and Unstable Dynamics in multi-phase lattice Boltzmann

We quantitatively characterize the metastability in a multi-phase lattice Boltzmann model. The structure factor of density fluctuations is theoretically obtained and numerically verified to a high precision, for all simulated wave-vectors and reduced temperatures. The static structure factor is found to consistently diverge as the temperature approaches the critical-point or the density approaches the spinodal line at a sub-critical temperature. Theoretically predicted critical exponents are observed in both cases. Finally, the phase separation in the unstable branch follows the same pattern, i.e. the generation of interfaces with different topology, as observed in molecular dynamics simulations. All results can be independently reproduced through the ``idea.deploy" framework https://github.com/lullimat/idea.deploy

cond-mat.stat-mech↗