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arXiv2024

Efficient Simulation of Non-uniform Cellular Automata with a Convolutional Neural Network

Cellular automata (CAs) and convolutional neural networks (CNNs) are closely related due to the local nature of information processing. The connection between these topics is beneficial to both related fields, for conceptual as well as practical reasons. Our contribution solidifies this connection in the case of non-uniform CAs (nuCAs), simulating a global update in the architecture of the Python package TensorFlow. Additionally, we demonstrate how the highly optimised out-of-the-box multiprocessing in TensorFlow offers interesting computational benefits, especially when simulating large numbers of nuCAs with many cells.

nlin.CG↗

arXiv2025

Effects of a Vanishing Noise on Elementary Cellular Automata Phase-Space Structure

We investigate elementary cellular automata (ECA) from the point of view of (discrete) dynamical systems. By studying small lattice sizes, we obtain the complete phase space of all minimal ECA, and, starting from a maximal entropy distribution (all configurations equiprobable), we show how the dynamics affects this distribution. We then investigate how a vanishing noise alters this phase space, connecting attractors and modifying the asymptotic probability distribution. What is interesting is that this modification not always goes in the sense of decreasing the entropy.

nlin.CG↗

arXiv2025

Solving the initial value problem for cellular automata by pattern decomposition

For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after $n$ iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.

nlin.CG↗

arXiv2023

A study on the composition of elementary cellular automata

Elementary cellular automata (ECA) are one-dimensional discrete models of computation with a small memory set that have gained significant interest since the pioneer work of Stephen Wolfram, who studied them as time-discrete dynamical systems. Each of the 256 ECA is labeled as rule $X$, where $X$ is an integer between $0$ and $255$. An important property, that is usually overlooked in computational studies, is that the composition of any two one-dimensional cellular automata is again a one-dimensional cellular automaton. In this chapter, we begin a systematic study of the composition of ECA. Intuitively speaking, we shall consider that rule $X$ has low complexity if the compositions $X \circ Y$ and $Y \circ X$ have small minimal memory sets, for many rules $Y$. Hence, we propose a new classification of ECA based on the compositions among them. We also describe all semigroups of ECA (i.e., composition-closed sets of ECA) and analyze their basic structure from the perspective of semigroup theory. In particular, we determine that the largest semigroups of ECA have $9$ elements, and have a subsemigroup of order $8$ that is $\mathcal{R}$-trivial, property which has been recently used to define random walks and Markov chains over semigroups.

nlin.CG↗

arXiv2024

Convolutional Neural Networks for Automated Cellular Automaton Classification

The emergent dynamics in spacetime diagrams of cellular automata (CAs) is often organised by means of a number of behavioural classes. Whilst classification of elementary CAs is feasible and well-studied, non-elementary CAs are generally too diverse and numerous to exhaustively classify manually. In this chapter we treat the spacetime diagram as a digital image, and implement simple computer vision techniques to perform an automated classification of elementary cellular automata into the five Li-Packard classes. In particular, we present a supervised learning task to a convolutional neural network, in such a way that it may be generalised to non-elementary CAs. If we want to do so, we must divert the algorithm's focus away from the underlying 'microscopic' local updates. We first show that previously developed deep learning approaches have in fact been trained to identify the local update rule, rather than directly focus on the mesoscopic patterns that are associated with the particular behavioural classes. By means of a well-argued neural network design, as well as a number of data augmentation techniques, we then present a convolutional neural network that performs nearly perfectly at identifying the behavioural class, without necessarily first identifying the underlying microscopic dynamics.

nlin.CG↗

arXiv2026

Ternary cellular automata induced by semigroups of order 3 are solvable

The minimal number of inputs in the local function of a non-trivial cellular automaton is two. Such a function can be viewed as as a kind of binary operation. If this operation is associative, it forms, together with the set of states, a semigroup. There are 18 semigroups of order 3 up to equivalence, and they define 18 cellular automata rules with three states. We investigate these rules with respect to solvability and show that all of them are solvable, meaning that the state of a given cell after $n$ iterations can be expressed by an explicit formula. We derive the relevant formulae for all 18 rules using some additional properties possessed by particular semigroups of order 3, such as commutativity and idempotence.

nlin.CG↗

arXiv2014

Modeling of pedestrians

Different families of models first developed for fluid mechanics have been extended to road, pedestrian, or intracellular transport. These models allow to describe the systems at different scales and to account for different aspects of dynamics. In this paper, we focus on pedestrians and illustrate the various families of models by giving an example of each type. We discuss the specificities of crowds compared to other transport systems.

physics.soc-ph↗

arXiv2016

5-State Rotation-Symmetric Number-Conserving Cellular Automata are not Strongly Universal

We study two-dimensional rotation-symmetric number-conserving cellular automata working on the von Neumann neighborhood (RNCA). It is known that such automata with 4 states or less are trivial, so we investigate the possible rules with 5 states. We give a full characterization of these automata and show that they cannot be strongly Turing universal. However, we give example of constructions that allow to embed some boolean circuit elements in a 5-states RNCA.

cs.FL↗

arXiv2015

Deterministic Abelian Sandpile and square-triangle tilings

The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome,...), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and involve different mechanisms. As will be shown elsewhere, all these structures fall under one roof, and are in fact different projections of a unique mechanism pertinent to a family of deterministic surfaces in a 4-dimensional lattice. This short note gives a description of this surface, and of the combinatorics associated to its construction.

cond-mat.stat-mech↗

arXiv2019

Multivaluedness Aspects in Self-organization, Complexity and Computations Investigations by Strong Anticipation

Since the introduction of strong anticipation by D.~Dubois the numerous investigations of concrete systems have been proposed. In proposed paper the new examples of discrete dynamical systems with anticipation are considered. The mathematical formulation of problems, possible analytical formulas for solutions and numerical examples of presumable solutions are proposed. One of the most interesting properties in such systems is presumable multivaluedness of the solutions. It can be considered from the point of view of dynamical chaos and complex behavior. We represent examples of periodic and complex solutions, attractor's properties and presumable applications in self-organization. The main peculiarity is the strong anticipation property. General new possibilities are the presumable multivaluedness of the dynamics of automata. Possible interpretations of such behavior of cellular automata are discussed. Further prospects for development of automata theory and hyper computation are proposed.

nlin.AO↗

arXiv2015

An example of computation of the density of ones in probabilistic cellular automata by direct recursion

We present a method for computing probability of occurence of 1s in a configuration obtained by iteration of a probabilistic cellular automata (PCA), starting from a random initial configuration. If the PCA is sufficiently simple, one can construct a set of words (or blocks of symbols) which is complete, meaning that probabilities of occurence of words from this set can be expressed as linear combinations of probabilities of occurence of these words at the previous time step. One can then setup and solve a recursion for block probabilities. We demonstrate an example of such PCA, which can be viewed as a simple model of diffusion of information or spread of rumors. Expressions for the density of ones are obtained for this rule using the proposed method.

nlin.CG↗

arXiv2020

Evaluating the Quality of Local Structure Approximation Using Elementary Rule 14

Cellular automata (CA) can be viewed as maps in the space of probability measures. Such maps are normally infinitely-dimensional, and in order to facilitate investigations of their properties, especially in the context of applications, finite-dimensional approximations have been proposed. The most commonly used one is known as the local structure theory, developed by H. Gutowitz et al. in 1987. In spite of the popularity of this approximation in CA research, examples of rigorous evaluations of its accuracy are lacking. In an attempt to fill this gap, we construct a local structure approximation for rule 14, and study its dynamics in a rigorous fashion, without relying on numerical experiments. We then compare the outcome with known exact results.

nlin.CG↗

arXiv2018

A gauge-invariant reversible cellular automaton

Gauge-invariance is a fundamental concept in physics---known to provide the mathematical justification for all four fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata. More precisely, we describe a step-by-step gauging procedure to enforce local symmetries upon a given Cellular Automaton. We apply it to a simple Reversible Cellular Automaton for concreteness. From a Computer Science perspective, discretized gauge theories may be applied to numerical analysis, quantum simulation, fault-tolerant (quantum) computation. From a mathematical perspective, discreteness provides a simple yet rigorous route straight to the core concepts.

cs.FL↗

arXiv2018

Regional Control of Probabilistic Cellular Automata

Probabilistic Cellular Automata are extended stochastic systems, widely used for modelling phenomena in many disciplines. The possibility of controlling their behaviour is therefore an important topic. We shall present here an approach to the problem of controlling such systems by acting only on the boundary of a target region.

nlin.CG↗

arXiv2012

Coevolving Cellular Automata with Memory for Chemical Computing: Boolean Logic Gates in the B-Z Reaction

We propose that the behaviour of non-linear media can be controlled automatically through coevolutionary systems. By extension, forms of unconventional computing, i.e., massively parallel non-linear computers, can be realised by such an approach. In this study a light-sensitive sub-excitable Belousov-Zhabotinsky reaction is controlled using various heterogeneous cellular automata. A checkerboard image comprising of varying light intensity cells is projected onto the surface of a catalyst-loaded gel resulting in rich spatio-temporal chemical wave behaviour. The coevolved cellular automata are shown to be able to control chemical activity through dynamic control of the light intensity. The approach is demonstrated through the creation of a number of simple Boolean logic gates.

cs.ET↗

arXiv2009

Computing Naturally in the Billiard Ball Model

Fredkin's Billiard Ball Model (BBM) is considered one of the fundamental models of collision-based computing, and it is essentially based on elastic collisions of mobile billiard balls. Moreover, fixed mirrors or reflectors are brought into the model to deflect balls to complete the computation. However, the use of fixed mirrors is "physically unrealistic" and makes the BBM not perfectly momentum conserving from a physical point of view, and it imposes an external architecture onto the computing substrate which is not consistent with the concept of "architectureless" in collision-based computing. In our initial attempt to reduce mirrors in the BBM, we present a class of gates: the m-counting gate, and show that certain circuits can be realized with few mirrors using this gate. We envisage that our findings can be useful in future research of collision-based computing in novel chemical and optical computing substrates.

nlin.CG↗

arXiv2011

Fuzzy cellular model for on-line traffic simulation

This paper introduces a fuzzy cellular model of road traffic that was intended for on-line applications in traffic control. The presented model uses fuzzy sets theory to deal with uncertainty of both input data and simulation results. Vehicles are modelled individually, thus various classes of them can be taken into consideration. In the proposed approach, all parameters of vehicles are described by means of fuzzy numbers. The model was implemented in a simulation of vehicles queue discharge process. Changes of the queue length were analysed in this experiment and compared to the results of NaSch cellular automata model.

cs.ET↗

arXiv2010

On Factor Universality in Symbolic Spaces

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes.

cs.DM↗