SEARCH · Searcharxiv

Results for “nlin.CG”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 559 records · Page 31Linked to original sources

arXiv2025

Cellular Automata on Probability Measures

Classical Cellular Automata (CCAs) are a powerful computational framework widely used to model complex systems driven by local interactions. Their simplicity lies in the use of a finite set of states and a uniform local rule, yet this simplicity leads to rich and diverse dynamical behaviors. CCAs have found applications in numerous scientific fields, including quantum computing, biology, social sciences, and cryptography. However, traditional CCAs assume complete certainty in the state of all cells, which limits their ability to model systems with inherent uncertainty. This paper introduces a novel generalization of CCAs, termed Cellular Automata on Measures (CAMs), which extends the classical framework to incorporate probabilistic uncertainty. In this setting, the state of each cell is described by a probability measure, and the local rule operates on configurations of such measures. This generalization encompasses the traditional Bernoulli measure framework of CCAs and enables the study of more complex systems, including those with spatially varying probabilities. We provide a rigorous mathematical foundation for CAMs, demonstrate their applicability through concrete examples, and explore their potential to model the dynamics of random graphs. Additionally, we establish connections between CAMs and symbolic dynamics, presenting new avenues for research in random graph theory. This study lays the groundwork for future exploration of CAMs, offering a flexible and robust framework for modeling uncertainty in cellular automata and opening new directions for both theoretical analysis and practical applications.

nlin.CG↗

arXiv2025

Connections between the minimal neighborhood and the activity value of cellular automata

For a group $G$ and a finite set $A$, a cellular automaton is a transformation of the configuration space $A^G$ defined via a finite neighborhood and a local map. Although neighborhoods are not unique, every CA admits a unique minimal neighborhood, which consists on all the essential cells in $G$ that affect the behavior of the local map. An active transition of a cellular automaton is a pattern that produces a change on the current state of a cell when the local map is applied. In this paper, we study the links between the minimal neighborhood and the number of active transitions, known as the activity value, of cellular automata. Our main results state that the activity value usually imposes several restrictions on the size of the minimal neighborhood of local maps.

nlin.CG↗

arXiv2025

Conway's game Life perturbed

A stochastic modification of Conway's cellular automaton "Life" is introduced here. Any cell could be perturbed spontaneously to the opposite (dead or alive) state at any iteration with a very low probability. This probability is assumed to be so low that perturbations affect most sensibly large patterns only in single cells after they settle into stable, oscillating or moving configurations. This defines a Markov process on the set of stabilised patterns, with unboundedly growing or overly large patterns represented by a general unspecified state of "being huge". This stochastic model should approximate emergence of complexity and live processes yet more interestingly than the original Conway's game. This paper illustrates the proposed Markovian dynamics on the infinite "Life" grid with a limited set of most frequent patterns. Concrete results are presented for this new game on small square toruses, of size up to 10x10 cells.

nlin.CG↗

arXiv2025

Categorical products of cellular automata

We study two categories of cellular automata. First, for any group $G$, we consider the category $\mathcal{CA}(G)$ whose objects are configuration spaces of the form $A^G$, where $A$ is a set, and whose morphisms are cellular automata of the form $τ: A_1^G \to A_2^G$. We prove that the categorical product of two configuration spaces $A_1^G$ and $A_2^G$ in $\mathcal{CA}(G)$ is the configuration space $(A_1 \times A_2)^G$. Then, we consider the category of generalized cellular automata $\mathcal{GCA}$, whose objects are configuration spaces of the form $A^G$, where $A$ is a set and $G$ is a group, and whose morphisms are $ϕ$-cellular automata of the form $\mathcal{T} : A_1^{G_1} \to A_2^{G_2}$, where $ϕ: G_2 \to G_1$ is a group homomorphism. We prove that a categorical weak product of two configuration spaces $A_1^{G_1}$ and $A_2^{G_2}$ in $\mathcal{GCA}$ is the configuration space $(A_1 \times A_2)^{G_1 \ast G_2}$, where $G_1 \ast G_2$ is the free product of $G_1$ and $G_2$. The previous results allow us to naturally define the product of two cellular automata in $\mathcal{CA}(G)$ and the weak product of two generalized cellular automata in $\mathcal{GCA}$.

nlin.CG↗

arXiv2026

On Gottschalk's surjunctivity conjecture for non-uniform cellular automata

Gottschalk's surjunctivity conjecture for a group $G$ states that it is impossible for cellular automata (CA) over the universe $G$ with finite alphabet to produce strict embeddings of the full shift into itself. A group universe $G$ satisfying Gottschalk's surjunctivity conjecture is called a surjunctive group. The surjunctivity theorem of Gromov and Weiss shows that every sofic group is surjunctive. In this paper, we study the surjunctivity of local perturbations of CA and more generally of non-uniform cellular automata (NUCA) with finite memory and uniformly bounded singularity over surjunctive group universes. In particular, we show that such a NUCA must be invertible whenever it is reversible. We also obtain similar results which extend to the class of NUCA a certain dual-surjunctivity theorem of Capobianco, Kari, and Taati for CA.

math.DS↗

arXiv2025

Multiscale Energy Spreading in Hard-Particle Chains

We consider a one-dimensional array of particles interacting via an infinite well potential. We explore the properties of energy spreading from an initial state where only a group of particles has non-zero velocities while others are resting. We characterize anomalous diffusion of the active domain via moments and entropies of the energy distribution. Only in the special cases of a single-well potential (hard-particle gas) and of the distance between the particles being half of the potential width does the diffusion have a single scale; otherwise, a multiscale anomalous diffusion is observed.

cond-mat.stat-mech↗

arXiv2025

Regional Controllability of Cellular Automata as a SAT Problem

Controllability, one of the fundamental concepts in control theory, consists in guiding a system from an initial state to a desired one within a limited (and possibly minimum) time interval. When the objective is limited to a specific sub-region of the system's domain, the concept is referred to as regional controllability. We examine this notion in the context of Boolean one-dimensional cellular automata of finite length. Depending on the local evolution rule, we investigate whether it is possible to control the evolution of the system by imposing particular values on the boundary conditions. This approach is related to key dynamical properties of CA, specifically chain transitivity and chain mixing. We show that the control problem can be formulated as a Boolean satisfiability (SAT) problem and can thus be addressed using SAT solvers. We also show how finding shortest paths in the configuration graph allows to determine controllability properties. From our observations we can state that only peripherally-linear rules are fully controllable, while for other rules, the reachability ratio, that is, the fraction of controllable pairs of initial and final configurations, is vanishing when the system size grows.

nlin.CG↗

arXiv2025

Speculative Evolution Through 3D Cellular Automata

This project explores speculative evolution through a 3D implementation of Conway's Game of Life, using procedural simulation to generate unfamiliar extraterrestrial organic forms. By applying a volumetric optimized workflow, the raw cellular structures are smoothed into unified, bone-like geometries that resemble hypothetical non-terrestrial morphologies. The resulting forms, strange yet organic, are 3D printed as fossil-like artifacts, presenting a tangible representation of generative structures. This process situates the work at the intersection of artificial life, evolutionary modeling, and digital fabrication, illustrating how simple rules can simulate complex biological emergence and challenge conventional notions of organic form.

nlin.CG↗

arXiv2025

Reflexive Composition of Elementary State Machines, with an Application to the Reversal of Cellular Automata Rule 90

We explore the dynamics of a one-dimensional lattice of state machines on two states and two symbols sequentially updated via a process of "reflexive composition." The space of 256 machines exhibits a variety of behavior, including substitution, reversible "billiard ball" dynamics, and fractal nesting. We show that one machine generates the Sierpinski Triangle and, for a subset of boundary conditions, is isomorphic to cellular automata Rule 90 in Wolfram's naming scheme. More surprisingly, two other machines follow trajectories that map to Rule 90 in reverse. Whereas previous techniques have been developed to uncover preimages of Rule 90, this is the first study to produce such inverse dynamics naturally from the formalism itself. We argue that the system's symmetric treatment of state and message underlies its expressive power.

cs.DM↗

arXiv2025

Impact of (a)Synchronism on ECA: Towards a New Classification

In this paper, we study the effect of (a)synchronism on the dynamics of elementary cellular automata. Within the framework of our study, we choose five distinct update schemes, selected from the family of periodic update modes: parallel, sequential, block-sequential, block-parallel, and local clocks. Our main measure of complexity is the maximum period of the limit cycles in the dynamics of each rule. In this context, we present a classification of the ECA rule landscape. We classified most elementary rules into three distinct regimes: constant, linear, and superpolynomial. Surprisingly, while some rules exhibit more complex behavior under a broader class of update schemes, others show similar behavior across all the considered update schemes. Although we are able to derive upper and lower bounds for the maximum period of the limit cycles in most cases, the analysis of some rules remains open. To complement the study of the 88 elementary rules, we introduce a numerical simulation framework based on two main measurements: the energy and density of the configurations. In this context, we observe that some rules exhibit significant variability depending on the update scheme, while others remain stable, confirming what was observed as a result of the classification obtained in the theoretical analysis.

cs.DM↗

arXiv2025

Loop Patterns Formed by Cellular Automata

A Cellular Automata (CA) rule is presented that can generate "loop patterns" in a 2D grid under fixed boundary conditions. A loop is a cyclically closed path represented by one-cells enclosed by zero-cells. A loop pattern can contain several loops that are not allowed to touch each other. The problem is solved by designing an appropriate set of tiles that can overlap and which are used in the CA rule. Templates are derived from the tiles which are used for local pattern matching. In order to drive the evolution to the desired patterns, noise is injected if the templates do not match or other constraints are not fulfilled. The general CA rule can be specialized by enabling certain conditions, and the characteristics of five rule variants are explained. Simulations illustrate that the CA rule can securely evolve stable loop patterns. The preliminary theoretical analysis of the obtained loop patterns raises many interesting research problems for the future -- several of them have been briefly discussed.

nlin.CG↗

arXiv2025

Higher-order Tuning of Interface Physics in Multiphase Lattice Boltzmann

Tuning the interface properties of multiphase models is of paramount importance to the final goal of achieving a one-to-one matching with nucleation and cavitation experiments. The surface tension, at the leading order, and the Tolman length, at higher order, play a crucial role in the estimation of the free-energy barrier determining the experimentally observed nucleation rates. The lattice Boltzmann method allows for a computationally efficient modelling approach of multiphase flows, however, tuning results are concerned with the surface tension and neglect the Tolman length. We present a novel perspective that leverages all the degrees of freedom hidden in the forcing stencil of the Shan-Chen multiphase model. By means of the lattice pressure tensor we determine and tune the coefficients of higher-order derivative terms related to surface tension and Tolman length at constant interface width and density ratio. We test the method by means of both hydrostatic and dynamic simulations and demonstrate the dependence of homogeneous nucleation rates on the value of the Tolman length. This work provides a new tool that can be integrated with previously existing strategies thus marking a step forwards to a high-fidelity modelling of phase-changing fluid dynamics.

cond-mat.stat-mech↗

arXiv2025

Fast offline decoding with local message-passing automata

We present a local offline decoder for topological codes that operates according to a parallelized message-passing framework. The decoder works by passing messages between anyons, with the contents of received messages used to move nearby anyons towards one another. We prove the existence of a threshold, and show that in a system of linear size $L$, decoding terminates with an $O((\log L)^η)$ average-case runtime, where $η$ is a small constant. For the toric code subject to i.i.d Pauli noise, our decoder has $η=1$ and a threshold at a noise strength of $p_c\approx 7.3\%$.

quant-ph↗

arXiv2025

Elementary Cellular Automata as Non-Cryptographic Hash Functions

A subset of 10 of the 256 elementary cellular automata (ECA) are implemented as a hash function using an error minimization lossy compression algorithm operating on wrapped 4x4 neighborhood cells. All 256 rules are processed and 10 rules in two subsets of 8 are found to have properties that include both error minimization and maximization, unique solutions, a lossy inverse, efficient retroactive hashing, and an application to edge detection. The algorithm parallels the nested powers-of-two structure of the Fast Fourier Transform and Fast Walsh-Hadamard Transform, is implemented in Java, and is built to hash any 2 byte RGB code bitmap.

nlin.CG↗

arXiv2025

Saltatory targeting strategy in rock-paper-scissors models

We explore how strategic leaps alter the classic rock-paper-scissors dynamics in spatially structured populations. In our model, individuals can expend energy reserves to jump toward regions with a high density of individuals of the species they dominate in the spatial game. This enables them to eliminate the target organisms and gain new territory, promoting species proliferation. Through stochastic, lattice-based simulations, we show that even when the energy allocated to jumping, as opposed to random walking, is low, there is a significant shift in the cyclic dominance balance. This arises from the increased likelihood of the leaping species successfully acquiring territory. Due to the cyclical nature of the game, the dominant species becomes the one that is superior to the jumping species. We investigate how spatial patterns are affected and calculate the changes in characteristic length scales. Additionally, we quantify how saltatory targeting reshapes spatial correlations and drives shifts in population dominance. Finally, we estimate the coexistence probability and find evidence that this behavioural strategy may promote biodiversity among low-mobility organisms but jeopardise long-term coexistence in the case of high-mobility dispersal. These results underscore the profound impact of novel foraging tactics on community structure and provide concrete parameters for ecologists seeking to incorporate behavioural innovation into ecosystem models.

q-bio.PE↗

arXiv2025

Timed Prediction Problem for Sandpile Models

We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding a grain at cell p from a given configuration. The prediction problem has been shown to be P-complete in several settings, including for subsets of the Moore neighborhood, but its complexity for the von Neumann neighborhood remains open. In a previous work, we provided a complete characterization of crossover gates (a key to the implementation of non-planar monotone circuits) for these small neighborhoods, leading to P-completeness proofs with only 4 and 5 neighbors among the eight adjancent cells. In this paper, we introduce the timed setting, where the goal is to determine whether cell q becomes unstable exactly at time t. We distinguish several cases: some neighborhoods support complete timed toolkits (including timed crossover gates) and exhibit P-completeness; others admit timed crossovers but suffer from synchronization issues; planar neighborhoods provably do not admit any timed crossover; and finally, for some remaining neighborhoods, we conjecture that no timed crossover is possible.

cs.CC↗

arXiv2026

Susceptibility for extremely low external fluctuations and critical behaviour of Greenberg-Hastings neuronal model

We consider the scaling behaviour of the fluctuation susceptibility associated with the average activation in the Greenberg-Hastings neural network model and its relation to microscopic spontaneous activation. We found that, as the spontaneous activation probability tends to zero, a clear finite size scaling behaviour in the susceptibility emerges, characterized by critical exponents which follow already known scaling laws. This shows that the spontaneous activation probability plays the role of an external field conjugated to the order parameter of the dynamical activation transition. The roles of different kinds of activation mechanisms around the different dynamical phase transitions exhibited by the model are characterized numerically and using a mean field approximation.

cond-mat.dis-nn↗

arXiv2025

A perturbed cellular automaton with two phase transitions for the ergodicity

The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using Gács's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.

nlin.CG↗