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At least 595 records · Page 33Linked to original sources

arXiv2025

Self-replication and Computational Universality

Self-replication is central to all life, and yet how it dynamically emerges in physical, non-equilibrium systems remains poorly understood. Von Neumann's pioneering work in the 1940s and subsequent developments suggest a natural hypothesis: that any physical system capable of Turing-universal computation can support self-replicating objects. In this work, we challenge this hypothesis by clarifying what computational universality means for physical systems and constructing a cellular automaton that is Turing-universal but cannot sustain non-trivial self-replication. By analogy with biology, such dynamics manifest transcription and translation but cannot instantiate replication. More broadly, our work emphasizes that the computational complexity of translating between physical dynamics and symbolic computation is inseparable from any claim of universality (exemplified by our analysis of Rule 110) and builds mathematical foundations for identifying self-replicating behavior. Our approach enables the formulation of necessary dynamical and computational conditions for a physical system to constitute a living organism.

nlin.CG↗

arXiv2025

Scaling Properties of Avalanche Activity in the Two-Dimensional Abelian Sandpile Model

We study the scaling properties of avalanche activity in the two-dimensional Abelian sandpile model. Instead of the conventional avalanche size distribution, we analyze the site activity distribution, which measures how often a site participates in avalanches when grains are added across the lattice. Using numerical simulations for system sizes up to $L = 160$, averaged over $10^4$ configurations, we determine the probability distribution $P(A, L)$ of site activities. The results show that $P(A, L)$ follows a finite-size scaling form \[ P(A, L) \sim L^{-2} F\Big(\frac{A}{L^2}\Big). \] For small values $A \ll L^2$ the scaling function behaves as \[ F(u) \sim u^{-1/2}, \quad \text{corresponding to} \quad P(A) \sim \frac{1}{L}, \] while for large activities $A \sim O(L^2)$ the distribution decays as \[ F(u) \sim \exp\big(-c_3 u - c_4 u^2\big). \] The crossover between these two regimes occurs at \[ A^* \sim 0.1 \, L^2, \] marking the threshold between typical and highly excitable sites. This characterization of local avalanche activity provides complementary information to the usual avalanche size statistics, highlighting how local regions serve as frequent conduits for critical dynamics. These results may help connect sandpile models to real-world self-organized critical systems where only partial local activity can be observed.

cond-mat.stat-mech↗

arXiv2025

Leveraging Cellular Automata for Real-Time Wildfire Spread Modeling in California

Wildfires are becoming increasingly frequent and devastating, and therefore the technology to combat them must adapt accordingly. Modern predictive models have failed to balance predictive accuracy and operational viability, resulting in consistently delayed or misinformed fire suppression and public safety efforts. The present study addresses this gap by developing and validating a predictive model based on cellular automata (CA) that incorporates key environmental variables, including vegetation density (NDVI), wind speed and direction, and topographic slope derived from open-access datasets. The presented CA framework offers a lightweight alternative to data-heavy approaches that fail in emergency contexts. Evaluation of the model using a confusion matrix against burn scars from the 2025 Pacific Palisades Fire yielded a recall of 0.860, a precision of 0.605, and an overall F1 score of 0.711 after 50 parameter optimization trials, with each simulation taking an average of 1.22 seconds. CA-based models can bridge the gap between accuracy and applicability, successfully guiding public safety and fire suppression efforts.

physics.soc-ph↗

arXiv2025

Lights Out Puzzle in p Colors: Evolution of Quiet Patterns

The Lights Out Puzzle represents a cellular automaton based on a grid of squares where clicking a square changes its state and the states of surrounding squares. A "quiet pattern" is a way to click such that in the end, no change is effected. We introduce a way to "evolve" quiet patterns in smaller grids into ones in $p$ times larger grids when the number of possible states of a square is a prime $p$. Using elliptic curves, we also find that an inverse "de-evolution" exists for most $p$. We also describe the only ways to click a grid of squares such that only 5 (the minimum) number of squares have a nonzero state.

nlin.CG↗

arXiv2025

Lu and Hamilton model for solar flares over a rewiring complex network

We present a modified Lu \& Hamilton-type model where the neighborhood relations are replaced by topological connections, which can be dynamically altered. The model represents each grid node as a flux tube, as in the classic model, but with connections evolving to capture the complex effects of magnetic reconnection. Through this framework, we analyze how the dissipated energy distribution changes, particularly focusing on the power-law exponent $α_E$, which decreases with respect to the original model due to rewiring effects. When the system is dominated by rewiring, it presents an exponential distribution exponent $β_E$, showing a faster decay of dissipated energy than in the original model. This leads to microflare-dominated dynamics at short timescales, causing the system to lose the scale-free behavior observed in both the original model (Lu \& Hamilton 1991) and in configurations where energy release is primarily driven by forcing rather than rewiring. Our results reveal a clear transition from power-law to exponential regimes as the rewiring probability increases, fundamentally altering the energy distribution characteristics of the system. In contrast, when considering topological neighbors instead of local ones, the model's dynamics become intrinsically nonlocal. This leads to scaling exponents comparable to those reported in other nonlocal dynamical systems.

astro-ph.SR↗

arXiv2026

Frobenius Revivals in Laplacian Cellular Automata: Replication, Entropy, and Information Recovery

We investigate Frobenius revivals in prime-modulus Laplacian cellular automata. The Frobenius identity yields exact superpositions of translated, coefficient-weighted seed copies at prime-power times; we distinguish this algebraic replication from their geometric separation. We investigate seed recovery from separated replicas and evaluate the correction of independent symbol errors introduced during transmission of encoded configurations. For the diagonal neighborhood, we determine the exact periods of moving corner operators on square windows and establish sufficient conditions for recovering any binary seed under a binary--ternary--binary schedule. The results connect prime-power algebraic structure with spatial pattern formation and precisely delimited forms of information recovery.

nlin.CG↗

arXiv2025

Renormalisation of Fermionic Cellular Automata

We present an exact renormalisation scheme for fermionic cellular automata on hypercubic lattices. By grouping neighbouring cells into tiles and selecting subspaces within them, multiple evolution steps on the original system correspond to a single step of an effective automaton acting on the subspaces. We derive a necessary and sufficient condition for renormalisability and fully characterise the renormalisation flow for two-cell tiles and two time steps of nearest-neighbour fermionic automata on a chain of spinless modes, identifying all fixed points.

quant-ph↗

arXiv2026

Emulating the logistic map with totalistic cellular automata

We investigate the conditions under which the mean-field formulation of a probabilistic, totalistic cellular automaton approximates the logistic equation. We show that this goal can be only fulfilled for an infinite-range neighborhood. We numerically study the corresponding one-dimensional implementation, showing that the mean-field description is obviously approached by shuffling the configuration at each time step, but also by rewiring a fraction of links, either at each time step, or using the same random sampling once and for all, in the spirit of the "small-world" mechanism. We show that it is possible to obtain a good approximation of the logistic behavior already with a fraction of rewired links different from one. We also show that there is a bifurcation cascade of the density as a function of the fraction of the rewired links, and that this scenario also holds for a deterministic, totalistic CA with the same basic symmetries of the probabilistic one.

nlin.CG↗

arXiv2026

Local Reversibility and Divergent Markov Length in 1+1-D Directed Percolation

Recent progress in open many-body quantum systems has highlighted the importance of the Markov length, the characteristic scale over which conditional correlations decay. It has been proposed that non-equilibrium phases of matter can be defined as equivalence classes of states connected by short-time evolution while maintaining a finite Markov length, a notion called local reversibility. A natural question is whether well-known classical models of non-equilibrium criticality fit within this framework. Here we investigate the Domany-Kinzel model -- which exhibits an active phase and an absorbing phase separated by a 1+1-D directed-percolation transition -- from this information-theoretic perspective. Using tensor network simulations, we provide evidence for local reversibility within the active phase. Notably, the Markov length diverges upon approaching the critical point, unlike classical equilibrium transitions where Markov length is zero due to their Gibbs character. Correspondingly, the conditional mutual information exhibits scaling consistent with directed percolation universality. Further, we analytically study the case of 1+1-D compact directed percolation, where the Markov length diverges throughout the phase diagram due to spontaneous breaking of domain-wall parity symmetry from strong to weak. Nevertheless, the conditional mutual information continues to faithfully detect the corresponding phase transition.

cond-mat.stat-mech↗

arXiv2025

Irreducible Rules and Equivalence Classes of One-dimensional Cellular Automata

One-dimensional cellular automata are discrete dynamical systems that operate on an infinite lattice of sites and are characterized by the locality and uniformity of their update rule. Permutations of the state set and isometric transformations of the lattice induce symmetry transformations on the set of local rules and the set of global maps of cellular automata, resulting in a partitioning of the set of cellular automata into equivalence classes. The concept of an irreducible local rule that depends on all its coordinates is used to analyse the equivalence classes and results on the number of equivalence classes of irreducible binary local rules and binary global maps are presented. Finally, another symmetry operator based on the scaling of neighbourhoods is introduced and the change in the number of equivalence classes is analysed.

nlin.CG↗

arXiv2025

Random state comonads encode cellular automata evaluation

Cellular automata (CA) are quintessential ALife and ubiquitous in many studies of collective behaviour and emergence, from morphogenesis to social dynamics and even brain modelling. Recently, there has been an increased interest in formalising CA, theoretically through category theory and practically in terms of a functional programming paradigm. Unfortunately, these remain either in the realm of simple implementations lacking important practical features, or too abstract and conceptually inaccessible to be useful to the ALife community at large. In this paper, we present a brief and accessible introduction to a category-theoretical model of CA computation through a practical implementation in Haskell. We instantiate arrays as comonads with state and random generators, allowing stochastic behaviour not currently supported in other known implementations. We also emphasise the importance of functional implementations for complex systems: thanks to the Curry-Howard-Lambek isomorphism, functional programs facilitate a mapping between simulation, system rules or semantics, and categorical descriptions, which may advance our understanding and development of generalised theories of emergent behaviour. Using this implementation, we show case studies of four famous CA models: first Wolfram's CA in 1D, then Conway's game of life, Greenberg-Hasings excitable cells, and the stochastic Forest Fire model in 2D, and present directions for an extension to N dimensions. Finally, we suggest that the comonadic model can encode arbitrary topologies and propose future directions for a comonadic network.

nlin.CG↗

arXiv2026

Quasicrystalline Gibbs states in 4-dimensional lattice-gas models with finite-range interactions

We construct a four-dimensional lattice-gas model with finite-range interactions that has non-periodic, "quasicrystalline" Gibbs states at low temperatures. Such Gibbs states are probability measures which are small perturbations of non-periodic ground-state configurations corresponding to stacked tilings of the plane with Ammann's aperiodic tiles. Our construction is based on the correspondence between probabilistic cellular automata and Gibbs measures on their space-time trajectories, and a classical result on noise-resilient computing with cellular automata. The cellular automaton is constructed on the basis of Ammann's tiles, which are deterministic in one direction, and has non-periodic space-time trajectories corresponding to each valid tiling. Repetitions along two extra dimensions, together with an error-correction mechanism based on Toom's model, ensure stability of the trajectories in the presence of noise.

math-ph↗

arXiv2026

Visualizing the Structure of Lenia Parameter Space

Continuous cellular automata are rocketing in popularity, yet developing a theoretical understanding of their behaviour remains a challenge. In the case of Lenia, a few fundamental open problems include determining what exactly constitutes a soliton, what is the overall structure of the parameter space, and where do the solitons occur in it. In this abstract, we present a new method to automatically classify Lenia systems into four qualitatively different dynamical classes. This allows us to detect moving solitons, and to provide an interactive visualization of Lenia's parameter space structure on our website https://lenia-explorer.vercel.app/. The results shed new light on the above-mentioned questions and lead to several observations: the existence of new soliton families for parameters where they were not believed to exist, or the universality of the phase space structure across various kernels.

nlin.CG↗

arXiv2026

Revealing Latent Self-Similarity in Cellular Automata via Recursive Gradient Profiling

Cellular automata (CA), originally developed as computational models of natural processes, have become a central subject in the study of complex systems and generative visual forms. Among them, the Ulam-Warburton Cellular Automaton (UWCA) exhibits recursive growth and fractal-like characteristics in its spatial evolution. However, exact self-similar fractal structures are typically observable only at specific generations and remain visually obscured in conventional binary renderings. This study introduces a Recursive Gradient Profile Function (RGPF) that assigns grayscale values to newly activated cells according to their generation index, enabling latent self-similar structures to emerge cumulatively in spatial visualizations. Through this gradient-based mapping, recursive geometric patterns become perceptible across scales, revealing fractal properties that are not apparent in standard representations. We further extend this approach to UWCA variants with alternative neighborhood configurations, demonstrating that these rules also produce distinct yet consistently fractal visual patterns when visualized using recursive gradient profile. Beyond computational analysis, the resulting generative forms resonate with optical and cultural phenomena such as infinity mirrors, video feedback, and mise en abyme in European art history, as well as fractal motifs found in religious architecture. These visual correspondences suggest a broader connection between complexity science, computational visualization, and cultural art and design.

nlin.CG↗

arXiv2026

LOGOS-CA: A Cellular Automaton Using Natural Language as State and Rule

Large Language Models (LLMs), trained solely on massive text data, have achieved high performance on the Winograd Schema Challenge (WSC), a benchmark proposed to measure commonsense knowledge and reasoning abilities about the real world. This suggests that the language produced by humanity describes a significant portion of the world with considerable nuance. In this study, we attempt to harness the high expressive power of language within cellular automata. Specifically, we express cell states and rules in natural language and delegate their updates to an LLM. Through this approach, cellular automata can transcend the constraints of merely numerical states and fixed rules, providing us with a richer platform for simulation. Here, we propose LOGOS-CA (Language Oriented Grid Of Statements - Cellular Automaton) as a natural framework to achieve this and examine its capabilities. We confirmed that LOGOS-CA successfully performs simple forest fire simulations and also serves as an intriguing subject for investigation from an Artificial Life (ALife) perspective. In this paper, we report the results of these experiments and discuss directions for future research using LOGOS-CA.

nlin.CG↗

arXiv2026

Thermalization in classical systems with discrete phase space

We study the emergence of statistical mechanics in isolated classical systems with local interactions and discrete phase spaces. We establish that thermalization in such systems does not require global ergodicity; instead, it arises from effective local ergodicity, where dynamics in a subsystem may appear pseudorandom. To corroborate that, we analyze the spectrum of the unitary evolution operator and propose an ansatz to describe statistical properties of local observables expanded in the eigenfunction basis - the classical counterpart of the Eigenstate Thermalization Hypothesis. Our framework provides a unified perspective on thermalization in classical and quantum systems with discrete spectra.

cond-mat.stat-mech↗

arXiv2026

A categorical framework for cellular automata

This paper proposes a generalized framework for cellular automata using the language of category theory, extending the classical definition beyond set-theoretic constraints. For an arbitrary category $\mathscr{C}$ with products, we define $\mathscr{C}$-cellular automata as morphisms $τ: A^G \to B^G$ in $\mathscr{C}$, where the alphabets $A$ and $B$ are objects in $\mathscr{C}$ and the universe is a group $G$. We show that $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ closed under finite products, and that they satisfy a categorical version of the Curtis-Hedlund-Lyndon theorem. For two arbitrary group universes $G$ and $H$, we extend our theory to define generalized $\mathscr{C}$-cellular automata as morphisms $τ: A^G \to B^H$ constructed via a group homomorphism $ϕ: H \to G$. Finally, we prove that generalized $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ with a finite weak product involving the free product of the underlying group universes. This framework unifies existing concepts and provides purely categorical proofs of foundational results in the theory of cellular automata.

cs.FL↗

arXiv2026

Integrable cellular automata on finite fields of order $2^n$

This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for $F_{2^n}$, supported by analytical proofs for $n=2$ and $n=3$. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.

nlin.SI↗