scholarly journals Investigating Transformational Complexity: Counting Functions a Region Induces on Another in Elementary Cellular Automata

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Martin Biehl ◽  
Olaf Witkowski
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Complex Systems ◽  
Cellular Automaton ◽  
Artificial Life ◽  
Discrete Dynamical Systems ◽  
Living Systems ◽  
Finite Region ◽  
Counting Functions ◽  
Elementary Cellular Automaton

Over the years, the field of artificial life has attempted to capture significant properties of life in artificial systems. By measuring quantities within such complex systems, the hope is to capture the reasons for the explosion of complexity in living systems. A major effort has been in discrete dynamical systems such as cellular automata, where very few rules lead to high levels of complexity. In this paper, for every elementary cellular automaton, we count the number of ways a finite region can transform an enclosed finite region. We discuss the relation of this count to existing notions of controllability, physical universality, and constructor theory. Numerically, we find that particular sizes of surrounding regions have preferred sizes of enclosed regions on which they can induce more transformations. We also find three particularly powerful rules (90, 105, 150) from this perspective.

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

scholarly journals On the Network Analysis of the State Space of Discrete Dynamical Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750062 ◽  
Author(s):  
Cheng Xu ◽  
Chengqing Li ◽  
Jinhu Lü ◽  
Shi Shu
Keyword(s):  
Dynamical Systems ◽  
Network Analysis ◽  
Cellular Automata ◽  
State Space ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dynamical Complexity ◽  
Two Parameters ◽  
Theoretical Analyses

This paper discusses the letter entitled “Network analysis of the state space of discrete dynamical systems” by A. Shreim et al. [Phys. Rev. Lett. 98, 198701 (2007)]. We found that some theoretical analyses are wrong and the proposed indicators based on two parameters of the state-mapping network cannot discriminate the dynamical complexity of the discrete dynamical systems composed of a 1D cellular automata.

Definition and Application of a Five-Parameter Characterization of One-Dimensional Cellular Automata Rule Space

2001 ◽  
Vol 7 (3) ◽  
pp. 277-301 ◽  
Author(s):  
Gina M. B. Oliveira ◽  
Pedro P. B. de Oliveira ◽  
Nizam Omar
Keyword(s):  
Phase Transition ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Dynamic Behavior ◽  
Discrete Dynamical Systems ◽  
Transition Diagram ◽  
One Dimensional ◽  
Rule Space ◽  
Spatially Extended

Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

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.

Reversible elementary cellular automaton with rule number 150 and periodic boundary conditions over 𝔽p

2015 ◽  
Vol 26 (11) ◽  
pp. 1550120 ◽  
Author(s):  
A. Martín del Rey ◽  
G. Rodríguez Sánchez
Keyword(s):  
Cellular Automata ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Discrete Dynamical Systems ◽  
Circulant Matrices ◽  
Cellular Space ◽  
Finite State ◽  
Rule Number ◽  
Elementary Cellular Automaton

The study of the reversibility of elementary cellular automata with rule number 150 over the finite state set 𝔽p and endowed with periodic boundary conditions is done. The dynamic of such discrete dynamical systems is characterized by means of characteristic circulant matrices, and their analysis allows us to state that the reversibility depends on the number of cells of the cellular space and to explicitly compute the corresponding inverse cellular automata.

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.

scholarly journals SPECTRAL PROPERTIES OF REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

2003 ◽  
Vol 14 (03) ◽  
pp. 379-395 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
SERGIO V. CHAPA VERGARA ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
HAROLD V. McINTOSH
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Spectral Properties ◽  
Neighborhood Size ◽  
Local Behavior ◽  
One Dimensional ◽  
Single Positive ◽  
Reversible Cellular Automata ◽  
Inverse Rule

Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation of every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the idempotent behavior of these matrices. The significance of this property lies in the implementation of a matrix technique for detecting whether a one-dimensional cellular automaton is reversible or not. In particular, we present a procedure using the eigenvectors of these matrices to find the inverse rule of a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided.

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