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arXiv2024

Quantum Systems from Random Probabilistic Automata

Probabilistic cellular automata with deterministic updating are quantum systems. We employ the quantum formalism for an investigation of random probabilistic cellular automata, which start with a probability distribution over initial configurations. The properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells. As an example we consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points, they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of our stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a random potential.

quant-ph↗

arXiv2024

Traffic Modeling and Forecast based on Stochastic Cell-Automata and Distributed Fiber-Optic Sensing -- A Numerical Experiment

This paper demonstrates accurate traffic modeling and forecast using stochastic cell-automata (CA) and distributed fiber-optic sensing (DFOS). Traffic congestion is a dominant issue in highways. To reduce congestion, real-time traffic control by short-term forecast is necessary. For achieving this, data assimilation using a stochastic CA model and DFOS is promising. Data assimilation with a CA enables us to model real-time traffic flow with simple processes even when rare or sudden events occur, which is challenging for usual machine learning-based methods. DFOS overcomes issues of conventional point sensors that have dead zones of observation. By estimating optimal model parameters that reproduce observed traffic flow in the simulation, future traffic flow is forecasted from the simulation. We propose an optimal model parameter estimation method using mean velocity as an extracted feature and the particle filter. In addition, an estimation methodology for the microscopic traffic situation is developed to set the initial condition of simulation for forecast in accordance with observation. The proposed methods are verified by simulation-based traffic flow. The simulation adopts the stochastic Nishinari-Fukui-Schadschneider model. The optimal model parameters are successfully derived from posterior probability distributions (PPDs) estimated from DFOS data. In contrast, those estimated from point sensors fail. The PPDs of model parameters also indicate that each parameter has different sensitivities to traffic flow. A traffic forecast up to 60 minutes later is carried out. Using optimal model parameters estimated from DFOS, the forecast error of mean velocity is approximately $\pm$10 km/h (percentage error is 18%). The error attains half of it when conventional point sensors are used. We conclude that DFOS is a powerful technique for traffic modeling and short-term forecast.

nlin.CG↗

arXiv2024

On complexity of colloid cellular automata

The colloid cellular automata do not imitate the physical structure of colloids but are governed by logical functions derived from the colloids. We analyse the space-time complexity of Boolean circuits derived from the electrical responses of colloids: ZnO (zinc oxide, an inorganic compound also known as calamine or zinc white, which naturally occurs as the mineral zincite), proteinoids (microspheres and crystals of thermal abiotic proteins), and combinations thereof to electrical stimulation. To extract Boolean circuits from colloids, we send all possible configurations of two-, four-, and eight-bit binary strings, encoded as electrical potential values, to the colloids, record their responses, and thereby infer the Boolean functions they implement. We map the discovered functions onto the cell-state transition rules of cellular automata (arrays of binary state machines that update their states synchronously according to the same rule) -- the colloid cellular automata. We then analyse the phenomenology of the space-time configurations of the automata and evaluate their complexity using measures such as compressibility, Shannon entropy, Simpson diversity, and expressivity. A hierarchy of phenomenological and measurable space-time complexity is constructed.

nlin.CG↗

arXiv2024

Operator entanglement growth quantifies complexity of cellular automata

Cellular automata (CA) exemplify systems where simple local interaction rules can lead to intricate and complex emergent phenomena at large scales. The various types of dynamical behavior of CA are usually categorized empirically into Wolfram's complexity classes. Here, we propose a quantitative measure, rooted in quantum information theory, to categorize the complexity of classical deterministic cellular automata. Specifically, we construct a Matrix Product Operator (MPO) of the transition matrix on the space of all possible CA configurations. We find that the growth of entropy of the singular value spectrum of the MPO reveals the complexity of the CA and can be used to characterize its dynamical behavior. This measure defines the concept of operator entanglement entropy for CA, demonstrating that quantum information measures can be meaningfully applied to classical deterministic systems.

nlin.CG↗

arXiv2024

The Invertibility of Cellular Automata with Menory: Correcting Errors and New Conclusions

Cellular automata with memory (CAM) are widely used in fields such as image processing, pattern recognition, simulation, and cryptography. The invertibility of CAM is generally considered to be chaotic. Paper [Invertible behavior in elementary cellular automata with memory, Juan C. Seck-Tuoh-Mora et al., Information Sciences, 2012] presented necessary and sufficient conditions for the invertibility of elementary CAM, but it contains a critical error: it classifies identity CAM as non-invertible, whereas identity CAM is undoubtedly invertible. By integrating Amoroso's algorithm and cycle graphs, we provide the correct necessary and sufficient conditions for the invertibility of one-dimensional CAM. Additionally, we link CAM to a specific type of cellular automaton that is isomorphic to CAM, behaves identically, and has easily determinable invertibility. This makes it a promising alternative tool for CAM applications.

nlin.CG↗

arXiv2025

Interactive Graph Visualization in DDLab

Interactive visualization of the basin of attraction field, the ``ibaf-graph'', is a new feature in DDLab with the same interactive functions as the ``network-graph'' and ``jump-graph''. These functions allow any node and its connected fragment to be dragged/dropped with the pointer as a graphic animation with elastic links. The fragment itself depends on the node's link setting by inputs, outputs, or either, and a distance in link-steps. Further options include graph geometry, rescaling, node display, link editing, and isolating the fragment. This article describes the three graph types, network/ibaf/jump, their selection, enhanced functions, and applications to cellular automata, discrete dynamical networks and random maps.

nlin.CG↗

arXiv2024

Emergent Dynamics in Heterogeneous Life-Like Cellular Automata

The Game of Life (GoL), one well known 2D cellular automaton, does not typically ensure interesting long-term phenotypic dynamics. Therefore, while being Turing complete, GoL cannot be said to be open-ended. In this work, we extend GoL with the opportunity for local mutations, thus enabling a heterogeneous life-like cellular automaton guided by an evolutionary inner loop. Additionally, we introduce the concept of cell ageing to ensure that cell aliveness (activated by inheritance with variation, and controlled by ageing) and actual cell computation (governed by life-like rules on local neighborhoods) are kept conceptually separated. We conduct an experimental campaign to identify suitable parameters that produce long-term phenotypic dynamics and favor genotypic innovations.

nlin.CG↗

arXiv2024

Singular knee identification to support emergence recognition in physical swarm and cellular automata trajectories

After decades of attention, emergence continues to lack a centralized mathematical definition that leads to a rigorous emergence test applicable to physical flocks and swarms, particularly those containing both deterministic elements (eg, interactions) and stochastic perturbations like measurement noise. This study develops a heuristic test based on singular value curve analysis of data matrices containing deterministic and Gaussian noise signals. The minimum detection criteria are identified, and statistical and matrix space analysis developed to determine upper and lower bounds. This study applies the analysis to representative examples by using recorded trajectories of mixed deterministic and stochastic trajectories for multi-agent, cellular automata, and biological video. Examples include Cucker Smale and Vicsek flocking, Gaussian noise and its integration, recorded observations of bird flocking, and 1D cellular automata. Ensemble simulations including measurement noise are performed to compute statistical variation and discussed relative to random matrix theory noise bounds. The results indicate singular knee analysis of recorded trajectories can detect gradated levels on a continuum of structure and noise. Across the eight singular value decay metrics considered, the angle subtended at the singular value knee emerges with the most potential for supporting cross-embodiment emergence detection, the size of noise bounds is used as an indication of required sample size, and the presence of a large fraction of singular values inside noise bounds as an indication of noise.

cs.MA↗

arXiv2024

Likelihood-based inference, identifiability and prediction using count data from lattice-based random walk models

In vitro cell biology experiments are routinely used to characterize cell migration properties under various experimental conditions. These experiments can be interpreted using lattice-based random walk models to provide insight into underlying biological mechanisms, and continuum limit partial differential equation (PDE) descriptions of the stochastic models can be used to efficiently explore model properties instead of relying on repeated stochastic simulations. Working with efficient PDE models is of high interest for parameter estimation algorithms that typically require a large number of forward model simulations. Quantitative data from cell biology experiments usually involves non-negative cell counts in different regions of the experimental images, and it is not obvious how to relate finite, noisy count data to the solutions of continuous PDE models that correspond to noise-free density profiles. In this work we illustrate how to develop and implement likelihood-based methods for parameter estimation, parameter identifiability and model prediction for lattice-based models describing collective migration with an arbitrary number of interacting subpopulations. We implement a standard additive Gaussian measurement error model as well as a new physically-motivated multinomial measurement error model that relates noisy count data with the solution of continuous PDE models. Both measurement error models lead to similar outcomes for parameter estimation and parameter identifiability, whereas the standard additive Gaussian measurement error model leads to non-physical prediction outcomes. In contrast, the new multinomial measurement error model involves a lower computational overhead for parameter estimation and identifiability analysis, as well as leading to physically meaningful model predictions.

physics.app-ph↗

arXiv2024

Interaction spaces: towards a universal mathematical theory of complex systems

We present the first steps of interaction spaces theory, a universal mathematical theory of complex systems which is able to embed cellular automata, agent based models, master equation based models, stochastic or deterministic, continuous or discrete dynamical systems, networked dynamical models, artificial neural networks and genetic algorithms in a single notion. Therefore, interaction spaces represent a common mathematical language that can be used to describe several complex systems modeling frameworks. This is the first step to start a mathematical theory of complex systems. Every notion is introduced both using an intuitive description by listing lots of examples, and using a modern mathematical language.

math-ph↗

arXiv2024

A mathematical definition of complex adaptive system as interaction space

We define a mathematical notion of complex adaptive system by following the original intuition of G.K. Zipf about the principle of least effort, an intuitive idea which is nowadays informally widespread in complex systems modeling. We call generalized evolution principle this mathematical notion of interaction spaces theory. Formalizing and generalizing Mandelbrot's ideas, we also prove that a large class of these systems satisfy a power law. We finally illustrate the notion of complex adaptive system with theorems describing a Von Thünen-like model. The latter can be easily generalized to other complex systems and describes the appearance of emergent patterns. Every notion is introduced both using an intuitive description with lots of examples, and using a modern mathematical language.

math-ph↗

arXiv2024

Thermodynamic Formalism for a family of cellular automata and duality with the shift

We will consider a family of cellular automata $Φ: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $σ$-invariant and $Φ$-invariant, where $σ$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,σ}$ and $\mathcal{L}_{B,Φ}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $σ$ and $Φ$. Via the use of a version of the involution kernel $W$ for a $(σ,Φ)$-mixed skew product $\hatΦ: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $ρ_A$ for $(\mathcal{L}_{A,σ})^*$ and eigenfunctions $ψ_B$ for $\mathcal{L}_{B,Φ}$. In another direction, considering the non-mixed extension $\hatΦ_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $Φ$, given a Lispchitz potential $\hat{A} : \{1,2,...,r\}^\mathbb{Z}\to \mathbb{R}$, we can identify a Lipschitz potential $A:\{1,2,...,r\}^\mathbb{N} \to \mathbb{R} $, in such away that relates the variational problem of $\hatΦ_n$-Topological Pressure for $\hat{A}$ with the $Φ$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $Φ$ (or $\hatΦ)$ can eventually be conjugated with another shift of finite type is irrelevant in our context.

math.DS↗

arXiv2024

Cellular Automata as a Network Topology

Cellular automata represent physical systems where both space and time are discrete, and the associated physical quantities assume a limited set of values. While previous research has applied cellular automata in modeling chemical, biological, and physical systems, its potential for modeling topological systems, specifically network topologies, remains underexplored. This paper investigates the use of cellular automata to model decentralized network topologies, which could enhance load balancing, fault tolerance, scalability, and the propagation and dissemination of information in distributed systems.

nlin.CG↗

arXiv2024

Classification of Cellular Automata based on the Hamming distance

Elementary cellular automata are the simplest form of cellular automata, studied extensively by Wolfram in the 1980s. He discovered complex behavior in some of these automata and developed a classification for all cellular automata based on their phenomenology. In this paper, we present an algorithm to classify them more effectively by measuring difference patterns using the Hamming distance. Our classification aligns with Wolfram's and further categorizes them into additional subclasses.

nlin.CG↗

arXiv2024

Unveiling the Dynamics of Employee Behavior Through Wolframs Cellular Automata

Understanding employee behavior in a workplace is critical for enhancing overall organizational performance. Despite numerous efforts to improve work environments, many organizations still need help with challenges primarily rooted in unaddressed issues or poorly understood behavioral patterns. In this paper, we have focused on recognizing this pattern and the dynamics of complex systems in organizational behavior and studying how factors influence the systems overall behavior using Wolframs Cellular Automata theory. Over the cycle, we observe how leadership influences team dynamics, influences the organization, and drives employee behavior to foster a positive environment. Keywords: Cellular Automata, Employee Behavior, team dynamics, Leadership Influence.

nlin.CG↗

arXiv2024

Intrinsic Universality in Seeded Active Tile Self-Assembly

The Tile Automata (TA) model describes self-assembly systems in which monomers can build structures and transition with an adjacent monomer to change their states. This paper shows that seeded TA is a non-committal intrinsically universal model of self-assembly. We present a single universal Tile Automata system containing approximately 4600 states that can simulate (a) the output assemblies created by any other Tile Automata system G, (b) the dynamics involved in building G's assemblies, and (c) G's internal state transitions. It does so in a non-committal way: it preserves the full non-deterministic dynamics of a tile's potential attachment or transition by selecting its state in a single step, considering all possible outcomes until the moment of selection. The system uses supertiles, each encoding the complete system being simulated. The universal system builds supertiles from its seed, each representing a single tile in G, transferring the information to simulate G to each new tile. Supertiles may also asynchronously transition states according to the rules of G. This result directly transfers to a restricted version of asynchronous Cellular Automata: pairwise Cellular Automata.

nlin.CG↗

arXiv2024

A Sensitivity Analysis of Cellular Automata and Heterogeneous Topology Networks: Partially-Local Cellular Automata and Homogeneous Homogeneous Random Boolean Networks

Elementary Cellular Automata (ECA) are a well-studied computational universe that is, despite its simple configurations, capable of impressive computational variety. Harvesting this computation in a useful way has historically shown itself to be difficult, but if combined with reservoir computing (RC), this becomes much more feasible. Furthermore, RC and ECA enable energy-efficient AI, making the combination a promising concept for Edge AI. In this work, we contrast ECA to substrates of Partially-Local CA (PLCA) and Homogeneous Homogeneous Random Boolean Networks (HHRBN). They are, in comparison, the topological heterogeneous counterparts of ECA. This represents a step from ECA towards more biological-plausible substrates. We analyse these substrates by testing on an RC benchmark (5-bit memory), using Temporal Derrida plots to estimate the sensitivity and assess the defect collapse rate. We find that, counterintuitively, disordered topology does not necessarily mean disordered computation. There are countering computational "forces" of topology imperfections leading to a higher collapse rate (order) and yet, if accounted for, an increased sensitivity to the initial condition. These observations together suggest a shrinking critical range.

nlin.CG↗

arXiv2024

The probabilistic world II : Quantum mechanics from classical statistics

This work discusses simple examples how quantum systems are obtained as subsystems of classical statistical systems. For a single qubit with arbitrary Hamiltonian and for the quantum particle in a harmonic potential we provide explicitly all steps how these quantum systems follow from an overall ''classical" probability distribution for events at all times. This overall probability distribution is the analogue of Feynman's functional integral for quantum mechanics or for the functional integral defining a quantum field theory. In our case the action and associated weight factor are real, however, defining a classical probabilistic system. Nevertheless, a unitary time-evolution of wave functions can be realized for suitable systems, in particular probabilistic automata. Based on these insights we discuss novel aspects for correlated computing not requiring the extreme isolation of quantum computers. A simple neuromorphic computer based on neurons in an active or quiet state within a probabilistic environment can learn the unitary transformations of an entangled two-qubit system. Our explicit constructions constitute a proof that no-go theorems for the embedding of quantum mechanics in classical statistics are circumvented. We show in detail how subsystems of classical statistical systems can explain various ``quantum mysteries". Conceptually our approach is a straightforward derivation starting from an overall probability distribution without invoking non-locality, acausality, contextuality, many worlds or other additional concepts. All quantum laws follow directly from the standard properties of classical probabilities.

quant-ph↗