scholarly journals Maximal Temporal Period of a Periodic Solution Generated by a One-Dimensional Cellular Automaton

2021 ◽  
Vol 30 (3) ◽  
pp. 239-272
Author(s):  
Janko Gravner ◽  
◽  
Xiaochen Liu ◽  
Keyword(s):  
Periodic Solution ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Upper Bound ◽  
Multiplicative Group ◽  
Spatial Period ◽  
Open Problems ◽  
One Dimensional ◽  
Spatial Periodicity ◽  
Temporal And Spatial

One-dimensional cellular automata evolutions with both temporal and spatial periodicity are studied. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period σ and number of states n. When σ = 2, 3, 4 or 6, and the rules are restricted to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. Non-additive rules are also constructed with temporal period on the same order as the trivial upper bound n σ . Experimental results, open problems and possible extensions of the results are also discussed.

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 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.

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 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 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.

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.

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.

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.

A Study of Chaos in Cellular Automata

2018 ◽  
Vol 28 (03) ◽  
pp. 1830008 ◽  
Author(s):  
Supreeti Kamilya ◽  
Sukanta Das
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Equivalence Class ◽  
Binary Relation ◽  
Equivalence Relation ◽  
De Bruijn Graph ◽  
One Dimensional ◽  
De Bruijn ◽  
Better Than

This paper presents a study of chaos in one-dimensional cellular automata (CAs). The communication of information from one part of the system to another has been taken into consideration in this study. This communication is formalized as a binary relation over the set of cells. It is shown that this relation is an equivalence relation and all the cells form a single equivalence class when the cellular automaton (CA) is chaotic. However, the communication between two cells is sometimes blocked in some CAs by a subconfiguration which appears in between the cells during evolution. This blocking of communication by a subconfiguration has been analyzed in this paper with the help of de Bruijn graph. We identify two types of blocking — full and partial. Finally a parameter has been developed for the CAs. We show that the proposed parameter performs better than the existing parameters.

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