scholarly journals An Information-Based Classification of Elementary Cellular Automata

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Enrico Borriello ◽  
Sara Imari Walker
Keyword(s):  
Cellular Automata ◽  
Classification Scheme ◽  
Initial Conditions ◽  
Coarse Graining ◽  
Transfer Entropy ◽  
Initial State ◽  
Elementary Cellular Automata ◽  
State Transfer ◽  
Sensitivity To Initial Conditions

We propose a novel, information-based classification of elementary cellular automata. The classification scheme proposed circumvents the problems associated with isolating whether complexity is in fact intrinsic to a dynamical rule, or if it arises merely as a product of a complex initial state. Transfer entropy variations processed by cellular automata split the 256 elementary rules into three information classes, based on sensitivity to initial conditions. These classes form a hierarchy such that coarse-graining transitions observed among elementary rules predominately occur within each information-based class or, much more rarely, down the hierarchy.

Automatic Texture Based Classification of the Dynamics of One-Dimensional Binary Cellular Automata

2019 ◽  
Vol 8 (4) ◽  
pp. 41-61
Author(s):  
Marcelo Arbori Nogueira ◽  
Pedro Paulo Balbi de Oliveira
Keyword(s):  
Cellular Automata ◽  
Dynamic Behavior ◽  
Temporal Evolution ◽  
Computational Cost ◽  
Great Variability ◽  
Texture Descriptor ◽  
One Dimensional ◽  
Binary Images ◽  
Elementary Cellular Automata

Cellular automata present great variability in their temporal evolutions due to the number of rules and initial configurations. The possibility of automatically classifying its dynamic behavior would be of great value when studying properties of its dynamics. By counting on elementary cellular automata, and considering its temporal evolution as binary images, the authors created a texture descriptor of the images - based on the neighborhood configurations of the cells in temporal evolutions - so that it could be associated to each dynamic behavior class, following the scheme of Wolfram's classic classification. It was then possible to predict the class of rules of a temporal evolution of an elementary rule in a more effective way than others in the literature in terms of precision and computational cost. By applying the classifier to the larger neighborhood space containing 4 cells, accuracy decreased to just over 70%. However, the classifier is still able to provide some information about the dynamics of an unknown larger space with reduced computational cost.

scholarly journals Periodicity and perfect state transfer in quantum walks on variants of cycles

2014 ◽  
Vol 14 (5&6) ◽  
pp. 417-438
Author(s):  
Katharine E. Barr ◽  
Tim J. Proctor ◽  
Daniel Allen ◽  
Viv M. Kendon
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Quantum Walks ◽  
High Fidelity ◽  
Perfect State ◽  
Initial State ◽  
State Transfer ◽  
Time Quantum ◽  
Perfect State Transfer ◽  
Very High

We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles $C_4$, $C_6$ and $C_8$ for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of $C_4$ could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.

Cellular Automata

2017 ◽  
Vol 29 (1) ◽  
pp. 42-50 ◽  
Author(s):  
Rupali Bhardwaj ◽  
Anil Upadhyay
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Empirical Study ◽  
Time Evolution ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dimensional Lattice ◽  
Elementary Cellular Automata

Cellular automata (CA) are discrete dynamical systems consist of a regular finite grid of cell; each cell encapsulating an equal portion of the state, and arranged spatially in a regular fashion to form an n-dimensional lattice. A cellular automata is like computers, data represented by initial configurations which is processed by time evolution to produce output. This paper is an empirical study of elementary cellular automata which includes concepts of rule equivalence, evolution of cellular automata and classification of cellular automata. In addition, explanation of behaviour of cellular automata is revealed through example.

Topological Conjugacy Classification of Elementary Cellular Automata with Majority Memory

2017 ◽  
Vol 27 (14) ◽  
pp. 1750217 ◽  
Author(s):  
Haiyun Xu ◽  
Fangyue Chen ◽  
Weifeng Jin
Keyword(s):  
Cellular Automata ◽  
Vector Space ◽  
Symbolic Dynamics ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Topological Conjugacy ◽  
Symbolic Sequence ◽  
And Topology ◽  
Elementary Cellular Automata

The topological conjugacy classification of elementary cellular automata with majority memory (ECAMs) is studied under the framework of symbolic dynamics. In the light of the conventional symbolic sequence space, the compact symbolic vector space is identified with a feasible metric and topology. A slight change is introduced to present that all global maps of ECAMs are continuous functions, thereafter generating the compact dynamical systems. By exploiting two fundamental homeomorphisms in symbolic vector space, all ECAMs are furthermore grouped into 88 equivalence classes in the sense that different mappings in the same global equivalence are mutually topologically conjugate.

scholarly journals Entropy-Based Classification of Elementary Cellular Automata under Asynchronous Updating: An Experimental Study

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 209
Author(s):  
Qin Lei ◽  
Jia Lee ◽  
Xin Huang ◽  
Shuji Kawasaki
Keyword(s):  
Cellular Automata ◽  
Complex Systems ◽  
Complete Classification ◽  
Asymptotic Density ◽  
Elementary Cellular Automata ◽  
Local Patterns ◽  
One Step ◽  
Pattern Distribution ◽  
High Uncertainty

Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems unable to distinguish the robustnesses of all AECAs. In this paper, we put forward a method that goes one step further via adopting a metric entropy (Martin, Complex Systems, 2000), with the aim of measuring the asymptotic mean entropy of local pattern distribution in the cell space of any AECA. Numerical experiments demonstrate that such an entropy-based measure can actually facilitate a complete classification of the robustnesses of all AECA models, even when all local patterns are restricted to length 1. To gain more insights into the complexity concerning the forward evolution of all AECAs, we consider another entropy defined in the form of Kolmogorov–Sinai entropy and conduct preliminary experiments on classifying their uncertainties measured in terms of the proposed entropy. The results reveal that AECAs with low uncertainty tend to converge remarkably faster than models with high uncertainty.

scholarly journals Power law behavior of Elementary Cellular Automata

2021 ◽  
Author(s):  
Jorge Laval
Keyword(s):  
Cellular Automata ◽  
Power Law ◽  
Initial Conditions ◽  
Percolation Clusters ◽  
Elementary Cellular Automata ◽  
Random Initial Conditions

This paper shows that the percolation clusters from elementary cellular automata {30, 45, 60, 86, 99, 105, 129, 150, 153, 169, 182, 183, 184, 195 and 225 exhibit strong power law behavior, either under random initial conditions, a single occupied cell, or both. Most of the tail exponents are less than unity, implying diverging means and variances of cluster sizes. The analysis presented here is admittedly coarse in an effort to expedite its dissemination.

Classifying elementary cellular automata using compressibility, diversity and sensitivity measures

2014 ◽  
Vol 25 (03) ◽  
pp. 1350098 ◽  
Author(s):  
Shigeru Ninagawa ◽  
Andrew Adamatzky
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Initial Conditions ◽  
Morphological Diversity ◽  
Compression Algorithm ◽  
One Dimensional ◽  
Elementary Cellular Automata ◽  
Eca Rules ◽  
Substantial Correlation ◽  
Elementary Cellular Automaton

An elementary cellular automaton (ECA) is a one-dimensional, synchronous, binary automaton, where each cell update depends on its own state and states of its two closest neighbors. We attempt to uncover correlations between the following measures of ECA behavior: compressibility, sensitivity and diversity. The compressibility of ECA configurations is calculated using the Lempel–Ziv (LZ) compression algorithm LZ78. The sensitivity of ECA rules to initial conditions and perturbations is evaluated using Derrida coefficients. The generative morphological diversity shows how many different neighborhood states are produced from a single nonquiescent cell. We found no significant correlation between sensitivity and compressibility. There is a substantial correlation between generative diversity and compressibility. Using sensitivity, compressibility and diversity, we uncover and characterize novel groupings of rules.

Elementary cellular automata and self-referential paradoxes

2020 ◽  
Vol 30 (3) ◽  
pp. 745-763
Author(s):  
MING HSIUNG
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Evolution Process ◽  
New Classification ◽  
Revision Process ◽  
Elementary Cellular Automata ◽  
Elementary Cellular Automaton

Abstract We associate an elementary cellular automaton with a set of self-referential sentences, whose revision process is exactly the evolution process of that automaton. A simple but useful result of this connection is that a set of self-referential sentences is paradoxical, iff (the evolution process for) the cellular automaton in question has no fixed points. We sort out several distinct kinds of paradoxes by the existence and features of the fixed points of their corresponding automata. They are finite homogeneous paradoxes and infinite homogeneous paradoxes. In some weaker sense, we will also introduce no-no-sort paradoxes and virtual paradoxes. The introduction of these paradoxes, in turn, leads to a new classification of the cellular automata.

CHAOTIC RESONANCE — METHODS AND APPLICATIONS FOR ROBUST CLASSIFICATION OF NOISY AND VARIABLE PATTERNS

2001 ◽  
Vol 11 (06) ◽  
pp. 1607-1629 ◽  
Author(s):  
ROBERT KOZMA ◽  
WALTER J. FREEMAN
Keyword(s):  
Chaotic Attractor ◽  
Additive Noise ◽  
Initial Conditions ◽  
Deterministic Chaos ◽  
Inhibitory Neuron ◽  
High Dimensional ◽  
Data Sets ◽  
Noise Sources ◽  
Sensitivity To Initial Conditions

A fundamental tenet of the theory of deterministic chaos holds that infinitesimal variation in the initial conditions of a network that is operating in the basin of a low-dimensional chaotic attractor causes the various trajectories to diverge from each other quickly. This "sensitivity to initial conditions" might seem to hold promise for signal detection, owing to an implied capacity for distinguishing small differences in patterns. However, this sensitivity is incompatible with pattern classification, because it amplifies irrelevant differences in incomplete patterns belonging to the same class, and it renders the network easily corrupted by noise. Here a theory of stochastic chaos is developed, in which aperiodic outputs with 1/f2 spectra are formed by the interaction of globally connected nodes that are individually governed by point attractors under perturbation by continuous white noise. The interaction leads to a high-dimensional global chaotic attractor that governs the entire array of nodes. An example is our spatially distributed KIII network that is derived from studies of the olfactory system, and that is stabilized by additive noise modeled on biological noise sources. Systematic parameterization of the interaction strengths corresponding to synaptic gains among nodes representing excitatory and inhibitory neuron populations enables the formation of a robust high-dimensional global chaotic attractor. Reinforcement learning from examples of patterns to be classified using habituation and association creates lower dimensional local basins, which form a global attractor landscape with one basin for each class. Thereafter, presentation of incomplete examples of a test pattern leads to confinement of the KIII network in the basin corresponding to that pattern, which constitutes many-to-one generalization. The capture after learning is expressed by a stereotypical spatial pattern of amplitude modulation of a chaotic carrier wave. Sensitivity to initial conditions is no longer an issue. Scaling of the additive noise as a parameter optimizes the classification of data sets in a manner that is comparable to stochastic resonance. The local basins constitute dynamical memories that solve difficult problems in classifying data sets that are not linearly separable. New local basins can be added quickly from very few examples without loss of existing basins. The attractor landscape enables the KIII set to provide an interface between noisy, unconstrained environments and conventional pattern classifiers. Examples given here of its robust performance include fault detection in small machine parts and the classification of spatiotemporal EEG patterns from rabbits trained to discriminate visual stimuli.

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