ON THE STRUCTURE OF REAL-VALUED ONE-DIMENSIONAL CELLULAR AUTOMATA

2011 ◽  
Vol 21 (05) ◽  
pp. 1265-1279 ◽  
Author(s):  
XU XU ◽  
STEPHEN P. BANKS ◽  
MAHDI MAHFOUF
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Cellular Systems ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
New Type ◽  
Complex Patterns

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

scholarly journals Complex Dynamic Behaviors in Cellular Automata Rule 14

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Qi Han ◽  
Xiaofeng Liao ◽  
Chuandong Li
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Fixed Points ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulliστ-shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 14 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 14 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 14. Finally, we use diagrams to explain the attractor of rule 14, where characteristic function is used to describe that all points fall into Bernoulli-shift map after two iterations under rule 14.

scholarly journals Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zujie Bie ◽  
Qi Han ◽  
Chao Liu ◽  
Junjian Huang ◽  
Lepeng Song ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Behavior ◽  
Class Ii ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulliστ-shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalenceε52of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.

scholarly journals Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo ◽  
Mithun Chakraborty
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Boolean Functions ◽  
State Transition ◽  
The State ◽  
Transition Diagram ◽  
One Dimensional ◽  
State Transition Diagram ◽  
Linear Rule

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

scholarly journals CELLULAR AUTOMATON SUPERCOLLIDERS

2011 ◽  
Vol 22 (04) ◽  
pp. 419-439 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
CHRISTOPHER R. STEPHENS ◽  
ALEJANDRO F. HOEFLICH
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Elastic Collision ◽  
Collective Excitations ◽  
Compact Groups ◽  
Dimensional Torus ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Spatially Extended ◽  
Binary Strings

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular automaton analogous of localizations or quasi-local collective excitations traveling in a spatially extended nonlinear medium. They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyze what types of interaction occur between gliders traveling on a cellular automaton "cyclotron" and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in nonlinear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems.

scholarly journals Self-Organizing Construction Method of Offshore Structures by Cellular Automata Model

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Takeshi Ishida
Keyword(s):  
Cellular Automata ◽  
Living Organism ◽  
Offshore Structures ◽  
Structural Units ◽  
Cellular Automata Model ◽  
One Dimensional ◽  
Transition Rules ◽  
Complex Patterns ◽  
The One ◽  
Self Organizing

We propose a new algorithm to build self-organizing and self-repairing marine structures on the ocean floor, where humans and remotely operated robots cannot operate. The proposed algorithm is based on the one-dimensional cellular automata model and uses simple transition rules to produce various complex patterns. This cellular automata model can produce various complex patterns like sea shells with simple transition rules. The model can simulate the marine structure construction process with distributed cooperation control instead of central control. Like living organism is constructed with module called cell, we assume that the self-organized structure consists of unified modules (structural units). The units pile up at the bottom of the sea and a structure with the appropriate shape eventually emerges. Using the attribute of emerging patterns in the one-dimensional cellular automata model, we construct specific structures based on the local interaction of transition rules without using complex algorithms. Furthermore, the model requires smaller communication data among the units because it only relies on communication between adjacent structural units. With the proposed algorithm, in the future, it will be possible to use self-assembling structural modules without complex built-in computers.

COARSE-GRAINING INVARIANT ORBITS OF ONE-DIMENSIONAL ℤp-LINEAR CELLULAR AUTOMATA

1996 ◽  
Vol 06 (12a) ◽  
pp. 2237-2297 ◽  
Author(s):  
ANDRÉ M. BARBÉ
Keyword(s):  
Cellular Automata ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Coarse Graining ◽  
One Dimensional ◽  
State Time ◽  
All Solutions ◽  
Invariance Problem ◽  
Self Similar ◽  
Existence Of Periodic Solutions

This paper studies the properties of state-time evolution patterns of one-dimensional linear cellular automata over the field ℤp (p prime) which are invariant under certain coarse-graining operations. A procedure is developed for finding all solutions to this invariance problem. The resulting patterns display a complexity which may range from periodic over self-similar, quasi-periodic, quasi-randomlike towards randomlike. Conditions for the existence of periodic solutions are derived.

GLOBALLY SYNCHRONIZED OSCILLATIONS IN AN ONE-DIMENSIONAL CELLULAR AUTOMATON

2004 ◽  
Vol 15 (03) ◽  
pp. 409-425
Author(s):  
FRANCISCO JIMÉNEZ-MORALES ◽  
MARCO TOMASSINI
Keyword(s):  
Genetic Algorithm ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Random Noise ◽  
Regular Pattern ◽  
Final State ◽  
One Dimensional ◽  
Lattice Size ◽  
Spatial Periodicity ◽  
Temporal Periodicity

Using a genetic algorithm a population of one-dimensional binary cellular automata is evolved to perform a computational task for which the best evolved rules cause the concentration to display a period-three oscillation. One run is studied in which the final state reached by the best evolved rule consists of a regular pattern or domain Λ, plus some propagating particles. It is shown that globally synchronized period-three oscillations can be obtained if the lattice size L is a multiple of the spatial periodicity S(Λ) of the domain. When L=m.S(Λ)-1 there is a cyclic particle reaction that keeps the system in two different phases and the concentration has a temporal periodicity that depends on the lattice size. The effects of random noise on the evolved cellular automata has also been investigated.

scholarly journals Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata

2018 ◽  
Vol 4 (1) ◽  
Author(s):  
Nicolai Lang ◽  
Hans Peter Büchler
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Error Correction ◽  
Quantum Error Correction ◽  
One Dimensional ◽  
Decoding Algorithms ◽  
Topological Quantum ◽  
The One ◽  
Quantum Error

Active quantum error correction on topological codes is one of the most promising routes to long-term qubit storage. In view of future applications, the scalability of the used decoding algorithms in physical implementations is crucial. In this work, we focus on the one-dimensional Majorana chain and construct a strictly local decoder based on a self-dual cellular automaton. We study numerically and analytically its performance and exploit these results to contrive a scalable decoder with exponentially growing decoherence times in the presence of noise. Our results pave the way for scalable and modular designs of actively corrected one-dimensional topological quantum memories.

scholarly journals Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Janko Gravner ◽  
Xiaochen Liu
Keyword(s):  
Cellular Automata ◽  
Periodic Solutions ◽  
Main Tool ◽  
Random Variable ◽  
Poisson Approximation ◽  
Spatial Period ◽  
Finite Range ◽  
Poisson Random Variable ◽  
One Dimensional ◽  
Stein Method

We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.

Formal Logic of Cellular Automata

2021 ◽  
Vol 30 (2) ◽  
pp. 187-203
Author(s):  
Sukanta Das ◽  
◽  
Mihir K. Chakraborty ◽  
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Formal Logic ◽  
One Dimensional ◽  
Elementary Cellular Automata ◽  
Binary Strings

This paper develops a formal logic, named L CA , targeting modeling of one-dimensional binary cellular automata. We first develop the syntax of L CA , then give semantics to L CA in the domain of all binary strings. Then the elementary cellular automata and four-neighborhood binary cellular automata are shown as models of the logic. These instances point out that there are other models of L CA . Finally it is proved that any one-dimensional binary cellular automaton is a model of the proposed logic.

Sign in / Sign up

Export Citation Format

Share Document