scholarly journals Automaticity and Invariant Measures of Linear Cellular Automata

2019 ◽  
Vol 72 (6) ◽  
pp. 1691-1726
Author(s):  
Eric Rowland ◽  
Reem Yassawi
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Invariant Measures ◽  
Initial Conditions ◽  
Invariant Set ◽  
Initial Condition ◽  
Invariant Subset ◽  
Left Shift ◽  
Shift Map ◽  
Spacetime Diagram

AbstractWe show that spacetime diagrams of linear cellular automata $\unicode[STIX]{x1D6F7}:\,\mathbb{F}_{p}^{\mathbb{Z}}\rightarrow \mathbb{F}_{p}^{\mathbb{Z}}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions that are eventually constant. Each automatic spacetime diagram defines a $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant subset of $\mathbb{F}_{p}^{\mathbb{Z}}$, where $\unicode[STIX]{x1D70E}$ is the left shift map, and if the initial condition is not eventually periodic, then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measures on $\mathbb{F}_{3}^{\mathbb{Z}}$. Finally, given a linear cellular automaton $\unicode[STIX]{x1D6F7}$, we construct a nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measure on $\mathbb{F}_{p}^{\mathbb{Z}}$ for all but finitely many $p$.

scholarly journals Measure rigidity for algebraic bipermutative cellular automata

2007 ◽  
Vol 27 (6) ◽  
pp. 1965-1990 ◽  
Author(s):  
MATHIEU SABLIK
Keyword(s):  
Cellular Automata ◽  
Probability Measure ◽  
Cellular Automaton ◽  
Compact Group ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Invariant Subgroup ◽  
Shift Map ◽  
Bernoulli Measure ◽  
Measure Rigidity

AbstractLet $({\mathcal {A}^{\mathbb {Z}}} ,F)$ be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the $ {\mathbb {N}} \times {\mathbb {Z}} $-action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group $ {\mathcal {A}^{\mathbb {Z}}} $. This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on $ {\mathcal {A}^{\mathbb {N}}} $.

Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

Emergent Structures

2010 ◽  
pp. 83-113
Author(s):  
Eleonora Bilotta ◽  
Pietro Pantano
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Initial Conditions ◽  
Natural Sciences ◽  
Forward Problem ◽  
Second Step ◽  
Large Collection ◽  
Parameter Values

There are two classes of problem in the study of Cellular Automata. The forward. problem is the problem of determining the properties of the system. Solutions often consist of finding quantities that are computable on a rules table and characterizing the behavior of the rule upon repeated iterations, starting from different initial conditions. Solutions to the backwards problem begin with the properties of a system and find a rule or a set of rules which have these properties. This is especially useful in the application of Cellular Automata to the natural sciences, when researchers deal with a large collection of phenomena (Gutowitz, 1989). Another approach is to identify the basic structures of a Cellular Automaton (Adamatzky, 1995). Once these are known it becomes possible to develop specific models for particular systems and to detect general principles applicable to a wide variety of systems (Wolfram, 1984; Lam, 1998). According to Adamatzky, the identification of a system consists of two related steps, namely specification and estimation. In specification we choose a useful and efficient description of the system: perhaps an equation and a set of parameters. The second step involves the estimation of parameter values for the equation: exploiting measures of similarity.

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 Distributed Control of a Manufacturing System with One-Dimensional Cellular Automata

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Irving Barragan-Vite ◽  
Juan C. Seck-Tuoh-Mora ◽  
Norberto Hernandez-Romero ◽  
Joselito Medina-Marin ◽  
Eva S. Hernandez-Gress
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Distributed Control ◽  
Manufacturing Process ◽  
Manufacturing Systems ◽  
Manufacturing System ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Global Dynamics ◽  
One Dimensional

We present a distributed control modeling approach for an automated manufacturing system based on the dynamics of one-dimensional cellular automata. This is inspired by the fact that both cellular automata and manufacturing systems are discrete dynamical systems where local interactions given among their elements (resources) can lead to complex dynamics, despite the simple rules governing such interactions. The cellular automaton model developed in this study focuses on two states of the resources of a manufacturing system, namely, busy or idle. However, the interaction among the resources such as whether they are shared at different stages of the manufacturing process determines the global dynamics of the system. A procedure is shown to obtain the local evolution rule of the automaton based on the relationships among the resources and the material flow through the manufacturing process. The resulting distributed control of the manufacturing system appears to be heterarchical, and the evolution of the cellular automaton exhibits a Class II behavior for some given disordered initial conditions.

Cellular Automata Metrics

2010 ◽  
pp. 114-149
Author(s):  
Eleonora Bilotta ◽  
Pietro Pantano
Keyword(s):  
Complex System ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Initial Conditions ◽  
Biological Systems ◽  
Complexity Science ◽  
Complex Behavior ◽  
Multiple Components ◽  
Complex Phenomena

There have been many attempts to understand complexity and to represent it in terms of computable quantities. To date, however, these attempts have had little success. Although we find complexity in a broad range of scientific domains, precise definitions escape our grasp (Bak, 1996; Morin, 2001; Prigogine & Stengers, 1984). One of the key models in complexity science is the Cellular Automaton (CA), a class of system in which small changes in the initial conditions or in local rules can provoke unpredictable behavior (Wolfram, 1984; Wolfram, 2002; Langton, 1986; 1990). The key issue, here as in other kinds of complex system, is to discover the rules governing the emergence of complex phenomena. If such rules were known we could use them to model and predict the behavior of complex physical and biological systems. Taking it for granted that complex behavior is the result of interactions among multiple components of a larger system; we can ask a number of fundamental questions.

Is it a Right Assumption That B and Ge are Distributed Randomly After Growing a Strained HBT-Structure?

2002 ◽  
Vol 716 ◽  
Author(s):  
Victor I. Kol'dyaev
Keyword(s):  
Surface Segregation ◽  
Initial Conditions ◽  
Initial Condition ◽  
Basic Assumptions ◽  
Transient Diffusion ◽  
Segregation Process ◽  
Main Observation

AbstractIt is accepted that surface Ge atoms are considered to be responsible for the surface B segregation process. A set of original experiments is carried out. A main observation from the B and Ge profiles grown at different conditions shows that at certain conditions B is taking initiative and determine the Ge surface segregation process. basic assumptions are suggested to self-consistently explain these original experimental features and what is observed in the literature. These results have a strong implication for modeling the B diffusion in Si1-xGex where the initial conditions should be formulated accounting for the correlation in B and Ge distribution. A new assumption for the initial condition to be “all B atoms are captured by Ge” is regarded as a right one implicating that there is no any transient diffusion representing the B capturing kinetics.

Simulating Self-Regeneration and Self-Replication Processes Using Movable Cellular Automata with a Mutual Equilibrium Neighborhood

2020 ◽  
Vol 29 (4) ◽  
pp. 741-757
Author(s):  
Kateryna Hazdiuk ◽  
◽  
Volodymyr Zhikharevich ◽  
Serhiy Ostapov ◽  
◽  
...  
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Three Dimensional ◽  
Computer Models ◽  
The Self ◽  
Model Construction ◽  
Natural Phenomena ◽  
Different Types ◽  
Movable Cellular Automata ◽  
Self Replication

This paper deals with the issue of model construction of the self-regeneration and self-replication processes using movable cellular automata (MCAs). The rules of cellular automaton (CA) interactions are found according to the concept of equilibrium neighborhood. The method is implemented by establishing these rules between different types of cellular automata (CAs). Several models for two- and three-dimensional cases are described, which depict both stable and unstable structures. As a result, computer models imitating such natural phenomena as self-replication and self-regeneration are obtained and graphically presented.

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.

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