Searcharxiv⌕ Search

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 631 records · Page 35Linked to original sources
arXiv2026

Quantum mechanics for classical transport equations

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.

quant-ph↗
arXiv2026

Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful

We prove that every probabilistic cellular automaton with strictly positive transition probabilities that admits a stationary Bernoulli measure is exponentially ergodic. Moreover, the mixing time of any finite region in such a system is logarithmic in the diameter of the region. A similar result holds in continuous time for positive-rate, finite-range interacting particle systems. The proofs use entropy, and rely on a representation of the system as a perturbation of another system with noise. The ergodic behaviour results from a competition between the accumulation of randomness due to noise and the diffusion of randomness due to local information exchange. We show that, in two and higher dimensions, the positive-rate probabilistic cellular automata that admit stationary Bernoulli measures are algorithmically indistinguishable from those that do not.

math.PR↗
arXiv2026

Universal interface fluctuations in absorbing-state phase transitions

Despite similarities between models exhibiting absorbing phase transitions (APTs) and those showing Kardar-Parisi-Zhang (KPZ) growth, the relationship between these universal fluctuations has remained elusive. We numerically study (1+1)-dimensional interfaces of (2+1)-dimensional models showing APTs of directed percolation (DP) and compact directed percolation (CDP) classes with an active boundary, finding a universal crossover from short-time APT-governed fluctuations to long-time KPZ fluctuations. Upon rescaling time and length by the APT correlation time and length, the cumulants of the interface height distributions collapse onto a single scaling function. The fluctuation properties of the discrete Domany-Kinzel model and the continuum stochastic Fisher-Kolmogorov-Petrovsky-Piskunov (sFKPP) equation coincide, indicating that the KPZ growth parameters are determined solely by fundamental properties of the APT. For the CDP sFKPP equation, a dimensionless parameter tunes both the critical interface distribution and the KPZ parameters, with the interface properties of the biased voter model recovered in a limiting case. These results uncover a universal crossover in which KPZ fluctuations emerge from APT fluctuations at long times, linking paradigmatic universality classes of nonequilibrium scale-invariant phenomena.

cond-mat.stat-mech↗
arXiv2026

The information-theoretic complexity of differentiable functions

A measure for the complexity of a differentiable function f(x) on an interval is introduced. It is based on approximations of the function by piecewise constant functions. The measure takes into account the quality of the approximation and the number of intervals in the approximating function. This measure, called the V-complexity of f(x), is shown to formalize some intuitions about the simplicity or complexity of f(x). The V-complexity is then compared to another measure of complexity, namely how compressible an approximation of f(x) is. It is hypothesized that V-complexity is equivalent to the compression measure, in the case of the Run Length Encoding and the Lempel Ziv 77 algorithms. V-complexity can be used as an ingredient in the definition of the Effective Complexity (EC) of a Complex System. When the perceived regularities of such a system are described by a differentiable function on an interval, the EC can be defined as the V-complexity of that function. EC is applied to the model of diffusion of cream in a cup of coffee. The perceived regularity of this model is given by the diffusion equation. The V-complexity of the solution of the equation starts at zero, quickly increases to a maximum and then decreases back to zero as the liquid reaches its equilibrium state. It is shown that this is also the result when a cellular automaton approach and the concept of Apparent Complexity is used.

cs.IT↗
arXiv2026

Stationary Measures and Mean Flux Depending on Multiple Conserved Quantities in a Stochastic Cellular Automaton

We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two specific local patterns. This analysis further allows us to theoretically derive the dependence of the mean flux on the conserved quantities. In particular, we recover the mean flux formula in the deterministic case by taking the zero-noise limit of the system.

math-ph↗
arXiv2026

Agnosiophobia in a virtual agent: behavioral and dynamical architecture in Lenia

All embodied agents are fundamentally patterns in physiological or other excitable media, blurring the distinction between objects and processes. Emergent patterns with complex behaviors, such as Gliders in the Game of Life and virtual patterns in Lenia, are powerful model systems in which to understand the properties and origins of behavioral traits in novel agents. To evaluate the behavior of patterns in Lenia, we introduce regions into their environment from which no sensory information is available - in effect, making creatures blind to parts of their surroundings. Complementing the conventional concept of infotaxis, we find that creatures tend to avoid these regions, a behavior we term agnosiophobia. To explain this behavior, we map each test creature's sensitivity to targeted occlusions and interpret the results in the language of dynamical systems. We observe Lenia creatures taking advantage of their freedom to change heading in order to achieve what appears to be a more fundamental goal: the preservation of their morphology. This work illustrates the beginning of an important roadmap to understand how emergent agents' behavioral propensities interact with the informational, not only tangible, topography of their world.

nlin.CG↗
arXiv2026

Three-dimensional Fundamental Diagrams of Five-neighbor Particle Cellular Automata

We analyze five-neighbor particle cellular automata whose conventional two-dimensional fundamental diagrams are multivalued, but whose mean flow is uniquely determined by introducing a second density. We first consider binary rules for which the second density is conserved, and then examine rules for which the second density is not conserved but converges asymptotically. These examples give three-dimensional fundamental diagrams in which the mean flow is determined by the particle density and the second density. We then investigate whether this single-valued structure is preserved under real-valued max-plus extensions. There are some rules where two different max-plus extensions are introduced, and numerical simulations show that both extensions preserve the same single-valued three-dimensional fundamental diagram. These observations imply that, in constructing real-valued max-plus extensions, it is important to choose the flux function and the second density consistently.

physics.soc-ph↗
arXiv2026

Dynamics in a Low-Rank Separable Field Cellular Automaton

Cellular automata (CA) are widely used to study how simple rules generate life-like behavior, typically through local interactions. Here, we introduce a Separable-Field Cellular Automaton (SFCA), in which conventional local-neighborhood interactions are replaced by updates governed by a normalized global coupling field. Each cell is updated according to q = n/n_max, with survival and birth controlled by threshold intervals S and B. Systematic scans over interval widths and positions revealed four outcome classes: extinction, fixed points, cycles, and long transients. The phase diagram was organized by the relative geometry of S and B: fixed points were enriched when B was contained within S, whereas long transients concentrated near the boundary between partial overlap and no overlap. A fine scan across this transition revealed a narrow long-transient ridge separating two distinct cycle-dominated regimes: a dense, high-change-rate branch approximating global period-2 alternation and a sparse, low-change-rate, stripe-like branch. Damage-spreading analyses were consistent with delayed selection between these cyclic attractor families and provided no evidence for sustained exponential trajectory separation. Finite-size analyses showed that the ridge persists across all tested lattice sizes, although its position shifts systematically with system size. These results show that structured long-lived dynamics can arise under separable global coupling with a field matrix of rank at most one, suggesting that nontrivial collective organization can persist under strongly compressed interaction architectures.

nlin.CG↗
arXiv2026

Universal Information-Theoretic Structure of the Quasi-Stationary Domany--Kinzel Automaton

We characterize the quasi-stationary distribution (QSD) of the bond directed-percolation line of the Domany--Kinzel automaton using a matrix-product-state representation of the probability distribution, obtained by projecting out the absorbing state and iterating the transfer matrix. Unlike moment- or sampling-based methods, this yields the full conditional distribution and direct access to information-theoretic diagnostics. The spatial structure of the QSD changes sharply across the transition: the active phase is bulk-like with finite density, whereas in the inactive phase the surviving activity collapses into a single flock -- the smallest interval containing all active sites -- occupying a vanishing fraction of the chain. Throughout the inactive phase the bipartite mutual information of the QSD equals the entropy of a single binary choice -- whether the flock lies to the left or right of the cut -- so the surviving clusters together encode just one bit of positional information, corresponding to a single effective cluster.

cond-mat.stat-mech↗
arXiv2026

Exact distribution of the output of a deep-layered machine

Deep learning builds complex global functions through the repeated composition of local functions. But the resulting distribution over global functions is unknown. We derive this distribution exactly for finite width and depth when the local functions are chosen at random. Surprisingly, depth first creates a sharply structured distribution, then erases this structure as probability drains into two absorbing states. This competition produces a crossover at a depth exponential in the number of inputs, beyond which additional layers mainly drive collapse. The transient structure may help explain why highly overparameterized networks can still generalize.

cond-mat.dis-nn↗
arXiv2026

Exact Lyapunov spectra of affine cellular automata and the parity rule on networks

The Lyapunov exponent quantifies the sensitivity of a dynamical system to perturbations, and the full Lyapunov spectrum extends this to every orthogonal direction in tangent space. For cellular automata the spectrum is almost always approximated numerically, and the approximation is delicate. We show that the affine rules, those whose update is a XOR of a subset of the inputs together with a constant, admit an exact Lyapunov spectrum. An affine rule has a configuration-independent Boolean Jacobian, so the spectrum reduces to the logarithms of the singular values of a single constant matrix, with no simulation and no limit involved. Two cases carry a closed form. For an affine cellular automaton on a periodic lattice the Jacobian is a multilevel circulant matrix, and the spectrum is the discrete Fourier transform of the rule's gradient stencil, valid in any spatial dimension. For the parity rule on an arbitrary graph the Jacobian is the adjacency matrix itself, so the Lyapunov spectrum is the logarithm of the absolute adjacency spectrum, and the maximal exponent is the logarithm of the spectral radius. The long-time amplitude of a single-site perturbation then scales with the eigenvector centrality of the seeded node. Reading the periodic lattice as the Cayley graph of an abelian group unifies the two cases. Because they are exact, the affine spectra also serve as benchmarks: they reveal numerical artefacts in previously reported spectra and turn the informal correspondence between spectral radius and dynamical sensitivity into an exact identity.

nlin.CG↗
arXiv2026

Directing Open-Ended Evolution in Artificial Life via Multi-Scale Path Divergence

Open-ended evolution (OEE) in artificial life is typically driven by uninterpretable, black-box neural-network complexity metrics, leaving life-like systems disconnected from physical theories of complexity. We introduce MSPD (Multi-Scale Path Divergence, denoted $D_P$), a renormalization-group-inspired scalar that quantifies how the heterogeneity of a system's local transition laws is organized across temporal and spatial scales. MSPD is defined at the population level as a functional of the realised trajectory and is computed as a windowed finite-resolution estimator, with consistency between the two stated as a proposition. The metric is an explicit formula and plays a dual role: as a gradient-free fitness function and as a post-hoc analytical lens on any simulation that exposes local transition laws. On a Flow-Lenia substrate we establish three claims. (C1) Under fixed-context replay of the exact pathwise objective, MSPD-optimized parameters score higher than matched random parameters. (C2) Along optimized trajectories, states whose local transition laws are more heterogeneous yield larger future divergence than matched, less-heterogeneous states under exact-state stochastic continuations, so the metric tracks intrinsic dynamics rather than injected noise. (C5) Higher MSPD corresponds to stronger scale-dependent frustration --- larger differences between the dynamics expressed at different spatial extents --- linking MSPD to the frustration criterion of biological complexity in the sense of Vanchurin et al. All three transfer to Life-like cellular automata and Particle Life++, indicating that MSPD is not specific to a single substrate. A single explicit formula thus both \emph{directs} open-ended evolution and provides a principled bridge to the physics of complexity that black-box drivers do not.

cs.NE↗
arXiv2026

West Nile virus outbreak in Italy modelled with the quantum Game of Life

In the last years, an anomalously high spreading of West Nile virus (WNV) has been observed in Italy, with particularly high peaks of infections in southern Lazio, Campania and Veneto regions. The main disease vector for WNV is represented by Culex pipiens mosquitoes, which spread human infections through their bites. Here, we investigate WNV fever epidemic diffusion during summer season 2025 in Italy through a computational approach based on a quantum version of the Game of Life (GOL) cellular automaton model. Specifically, human dynamics evolves according to the GOL rules, while stochastic dynamics of disease vectors, i.e., mosquitoes, as well as their interaction with humans, simultaneously occur. We show that this model fits the curves of cumulative infected individuals with high accuracy, either at local and average-regional level, with only optimization of mosquito birth and removal rates parameters. Furthermore, leveraging model flexibility, we show that changes in model parameters values elucidate system response to environmental variations. For instance, we quantify, e.g., the impact of mosquito spreading containment measures or sudden mosquito increasing abundance due to climatic and ecological changes. Overall, we provide a general, quantitative description of WNV infection spreading in Italy which could represent a supportive tool to test different environmental scenarios and could be useful to devise strategies for decision makers to monitor disease vector dynamics and to control consequent virus diffusion.

q-bio.PE↗
arXiv2026

It's Much Easier for Neural Networks to learn Game of Life Dynamics with the Right Activation Function: Polynomial Kolmogorov-Arnold Networks

Previous work has found a gap between the scale of neural networks that reliably learn Conway's Game of Life, and minimal networks capable of representing the classic cellular automaton with hard-coded parameter values. Viewing neural network learning as a search process suggests a dependence on networks large enough to contain sub-networks with lucky initializations (sometimes known as 'winning tickets') that actually learn the task. In this work, we reorient our perspective from discovering Life rules as a search problem back to a learning problem, and reason that with fitting inductive biases, the problem should be much more amenable to minimal networks. We find that network variants with several alternative activation functions meaningfully outperform the default choice of Rectified Linear Units, and in particular, that a 2nd degree polynomial activation function consistently learns Life dynamics with or without the benefit of learning neural weights. Our results provide an informative demonstration of the benefits of matching learning to the task at hand and challenge the easy default choice of scale for all problems. In particular, we advocate for the use of cellular automata as simple test domains for developing strategies that can benefit machine learning for science, physics-based deep learning, and interpretable machine learning.

cs.LG↗
arXiv2026

Revising price coordination in the classical and neoclassical economics based on elementary cellular automata

The coordination of prices in economics is one of the most complex phenomena. In particular, the classical and neoclassical approaches related to the economic theory provide some insights into such a complex coordination based on different formulations. However, these formulations have not been successful for explaining simple mechanisms to understand and predict a set of prices that theoretically clears all markets. Consequently, elementary cellular automata can contribute to clarify such a coordination problem by using simple computational rules to describe the theoretical bases of the classical and neoclassical economics. Therefore, we propose to use this type of cellular automata for explaining different escenarios of price coordination in which simple rules of price interactions generate stable and unstable patterns of coordination. We used an explorative data analysis based on the Shannon entropy for computing the uncertainty related to such generated patterns of coordination, and a Monte Carlo simulation approximation based on a Spearman correlation for evaluating the statistical significance of such price coordination. Findings suggested that the classical economics provides a consistent approach for understanding the coordination of prices because it emphasizes human interactions based on logical choices related to an objective data. On the other hand, the neoclassical approach does not propose any type of mechanism for describing the price coordination. The neoclassical individual is just a spectator and receiver of the unpredictable and supposed event of price coordination. As a result, by modeling the economic theory based on computational concepts, we reveal facts and believes behind the classical and neoclassical economics.

nlin.CG↗
arXiv2026

Analysis of the asymptotic density of endogamous diploid cellular automata

We investigate the asymptotic behaviour of diploid Elementary Cellular Automata (ECA), that is, the stochastic mixtures between two ECAs. In this model, each cell independently applies one rule with probability $λ$ and the other rule with probability $ 1 - λ$. Focusing on the endogamous diploids where the two ECAs are related by the reflection or conjugation symmetry, we analyse how the density varies as function of $λ$, the ``degree of mixing'' of these two rules. We propose a classification into six distinct classes depending on the profile of the density vs. $λ$ curve. We take various examples for each class and we analyse to which extent the local structure approximation succeeds to predict the asymptotic density. Our results show that for rules in which the asymptotic density depends linearly on $λ$, the local structure approximation reproduces the exact dependence of the density on $λ$. For rules with differentiable but nonlinear dependence, the approximation either becomes exact at a finite order or converges rapidly to the exact solution as the order increases. For rules with first-order phase transitions, finite-order approximations progressively approach a sharp transition profile with increasing order. In contrast, for rules exhibiting a second-order phase transition, even high-order approximations fail to capture the qualitative features of the transition. (no bifurcation is observed). Finally, we give examples of two diploid rules for which we succeed to compute the density at each time step.

nlin.CG↗
arXiv2026

The Binary Crisis Clock: Controlled by Sparse Ternary Interventions

We investigate modular Laplacian automata on triangular lattices with evolution governed by binary and ternary moduli. Extending previous studies on square lattices, we examine how lattice geometry influences long-term growth, density, fragmentation, and the emergence of self-similar structures. We further investigate whether sparse ternary interventions can stabilize predominantly binary dynamics. The experiments reveal that mask geometry is the primary determinant of large-scale morphology. Full hexagonal masks generate recurrent density crises and fragmentation, whereas triangular masks support persistent growth and reveal a threshold phenomenon governed by growth-capable nuclei. Although seed symmetry influences transient behaviour, the asymptotic morphology is inherited mainly from the mask. To control binary fragmentation, we investigate sparse developmental ternary perturbations in which a small number of carefully timed occurrences of modulus 3 are inserted into an otherwise binary sequence. A Monte Carlo optimization demonstrates that as few as three interventions are sufficient to redirect the subsequent binary evolution toward substantially denser carpet-like configurations. The effectiveness of this strategy depends primarily on the timing of the interventions rather than on their number. Analysis of the post-intervention dynamics shows that ternary shaping does not replace binary evolution. Instead, it produces denser self-similar structures, substantially reduces crisis depth, and resets the phase of the binary crisis clock. The results suggest that geometry determines the family of admissible morphologies, whereas sparse developmental perturbations select favourable long-term trajectories within that family.

nlin.CG↗