scholarly journals Surjective cellular automata far from the Garden of Eden

2013 ◽  
Vol Vol. 15 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Silvio Capobianco ◽  
Pierre Guillon ◽  
Jarkko Kari
Keyword(s):  
Cellular Automata ◽  
Automata Theory ◽  
Garden Of Eden ◽  
Amenable Groups ◽  
Von Neumann ◽  
International Audience

Automata, Logic and Semantics International audience One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.

scholarly journals On the analysis of ''simple'' 2D stochastic cellular automata

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Damien Regnault ◽  
Nicolas Schabanel ◽  
Eric Thierry
Keyword(s):  
Cellular Automata ◽  
Time Step ◽  
Energy Functions ◽  
Stochastic Cellular Automata ◽  
Von Neumann ◽  
Rich Variety ◽  
Moore Neighborhood ◽  
International Audience ◽  
Definition Of ◽  
New Phenomena

International audience Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata

Fundamental of Cellular Automata Theory

2017 ◽  
pp. 26-51
Keyword(s):  
Cellular Automata ◽  
Automata Theory ◽  
Linear Feedback ◽  
Random Number Generators ◽  
Von Neumann ◽  
One Dimensional ◽  
Asynchronous Cellular Automata ◽  
Local Functions ◽  
Definition Of

In this chapter, the author reviews the main historical aspects of the development of cellular automata. The basic structures of cellular automata are described. The classification of cellular automata is considered. A definition of a one-dimensional cellular automaton is given and the basic rules for one-dimensional cellular automata are described that allow the implementation of pseudo-random number generators. One-dimensional cellular automata with shift registers with linear feedback are compared. Synchronous two-dimensional cellular automata are considered, as well as their behavior for various using local functions. An analysis of the functioning of synchronous cellular automata for the neighborhoods of von Neumann and Moore is carried out. A lot of attention is paid to asynchronous cellular automata. The necessary definitions and rules for the behavior of asynchronous cellular automata are given.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

scholarly journals Three-state von Neumann cellular automata and pattern generation

2015 ◽  
Vol 39 (7) ◽  
pp. 2003-2024 ◽  
Author(s):  
Ugur Sahin ◽  
Selman Uguz ◽  
Hasan Akın ◽  
Irfan Siap
Keyword(s):  
Cellular Automata ◽  
Pattern Generation ◽  
Von Neumann

scholarly journals Simulation of traffic flows on road segments using cellular automata theory and quasigasdynamic approach

2019 ◽  
Vol 46 ◽  
pp. 72-90 ◽  
Author(s):  
Natalia Gennadievna Churbanova ◽  
◽  
Antonina Alexandrovna Chechina ◽  
Marina Alexandrovna Trapeznikova ◽  
Pavel Alexeevich Sokolov ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Automata Theory ◽  
Traffic Flows ◽  
Road Segments

Three-dimensional multi-species mathematical model of the aerobic granulation process based on cellular automata theory

2018 ◽  
Vol 77 (12) ◽  
pp. 2761-2771
Author(s):  
Guoqiang Zheng ◽  
Kuizu Su ◽  
Shuai Zhang ◽  
Yulan Wang ◽  
Weihong Wang
Keyword(s):  
Mathematical Model ◽  
Spatial Distribution ◽  
Cellular Automata ◽  
Three Dimensional ◽  
Granular Sludge ◽  
Automata Theory ◽  
Aerobic Granular Sludge ◽  
Granulation Process ◽  
Autotrophic Bacteria ◽  
Degrading Bacteria

Abstract Aerobic granular sludge is a kind of microbial polymer formed by self-immobilization under aerobic conditions. It has been widely studied because of its promising application in wastewater treatment. However, the granulation process of aerobic sludge is still a key factor affecting its practical application. In this paper, a three-dimensional (3D) multi-species mathematical model of aerobic granular sludge was constructed using the cellular automata (CA) theory. The growth process of aerobic granular sludge and its spatial distribution of microorganisms were studied under different conditions. The simulation results show that the aerobic granules were smaller under high shear stress and that the autotrophic bacterial content of the granular sludge interior was higher. However, the higher the dissolved oxygen concentration, the larger the size of granular sludge and the higher the content of autotrophic bacteria in the interior of the granular sludge. In addition, inhibition of toxic substances made the aerobic granule size increase more slowly, and the spatial distribution of the autotrophic bacteria and the toxic-substance-degrading bacteria were mainly located in the outer layer, with the heterotrophic bacteria mainly existing in the interior of the granular sludge.

A Language Shift Simulation Based on Cellular Automata

2011 ◽  
pp. 136-151 ◽  
Author(s):  
Francesc S. Beltran ◽  
Salvador Herrando ◽  
Violant Estreder ◽  
Doris Ferreres ◽  
Marc-Antoni Adell ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Language Shift ◽  
Automata Theory ◽  
Theory Approach ◽  
Southern Europe ◽  
Social Phenomenon ◽  
Simulation Based ◽  
The World ◽  
Traditional Language

Language extinction is a widespread social phenomenon affecting several million people throughout the world today. By the end of this century, more than 5100 of the approximately 6000 languages currently spoken around the world will have disappeared. This is mainly because of language shifts, i.e., because a community of speakers stops using their traditional language and speaks a new one in all communication settings. In this study, the authors present the properties of a cellular automaton that incorporates some assumptions from the Gaelic-Arvanitika model of language shifts and the findings on the dynamics of social impacts in the field of social psychology. To assess the cellular automaton, the authors incorporate empirical data from Valencia (a region in Southern Europe), where Catalan speakers are tending to shift towards using Spanish. Running the automaton under different scenarios, the survival or extinction of Catalan in Valencia depends on individuals’ engagement with their language. The authors discuss how a cellular automata theory approach proves to be a useful tool for understanding the language shift.

Basic Definitions

2010 ◽  
pp. 17-50
Author(s):  
Eleonora Bilotta ◽  
Pietro Pantano
Keyword(s):  
Cellular Automata ◽  
Dynamic Systems ◽  
Chaotic Behavior ◽  
Crucial Importance ◽  
Von Neumann ◽  
3 Dimensional ◽  
Straight Line ◽  
Discrete Dynamic Systems ◽  
Complex Phenomena ◽  
Emergent Behaviors

Cellular Automata (CAs) are discrete dynamic systems that exhibit chaotic behavior and self-organization and lend themselves to description in rigorous mathematical terms. The main aim of this chapter is to introduce CAs from a formal perspective. Ever since the work of von Neumann (1966), von Neumann & Burks (1970), Toffoli & Norman (1987), Wolfram (1983; 1984), and Langton (1984; 1986; 1990), specialists have recognized CAs as a model of crucial importance for complexity studies. Like other models used in the investigation of complex phenomena (Chua & Yang, 1988; Chua, 1998), a CA consists of a number of elementary components, whose interactions determine its dynamics. In this and the following chapters, we will sometimes refer to these elementary components as cells, sometimes as sites. The cells of a CA can be positioned along a straight line or on a 2 or 3-dimensional grid, creating 1-D, 2-D and 3-D CAs. Automata consisting of cells whose only possible states are 0 or 1, are Boolean Automata; automata whose cells can assume more than 2 states are multi-state CAs. In both cases, the CA contains “elementary particles” whose dynamics are governed by simple rules. These rules determine sometimes unpredictable, emergent behaviors, ranging from the simple, through the complex to the chaotic.

Making (Bio)Logical Connections

2018 ◽  
Author(s):  
Subrata Dasgupta
Keyword(s):  
Cellular Automata ◽  
Universal Theory ◽  
Transition Rule ◽  
Yale University ◽  
Von Neumann ◽  
Finite Set ◽  
A Cell ◽  
The 1960S ◽  
The University ◽  
The Brain

At first blush, computing and biology seem an odd couple, yet they formed a liaison of sorts from the very first years of the electronic digital computer. Following a seminal paper published in 1943 by neurophysiologist Warren McCulloch and mathematical logician Warren Pitts on a mathematical model of neuronal activity, John von Neumann of the Institute of Advanced Study, Princeton, presented at a symposium in 1948 a paper that compared the behaviors of computer circuits and neuronal circuits in the brain. The resulting publication was the fountainhead of what came to be called cellular automata in the 1960s. Von Neumann’s insight was the parallel between the abstraction of biological neurons (nerve cells) as natural binary (on–off) switches and the abstraction of physical computer circuit elements (at the time, relays and vacuum tubes) as artificial binary switches. His ambition was to unify the two and construct a formal universal theory. One remarkable aspect of von Neumann’s program was inspired by the biology: His universal automata must be able to self-reproduce. So his neuron-like automata must be both computational and constructive. In 1955, invited by Yale University to deliver the Silliman Lectures for 1956, von Neumann chose as his topic the relationship between the computer and the brain. He died before being able to deliver the lectures, but the unfinished manuscript was published by Yale University Press under the title The Computer and the Brain (1958). Von Neumann’s definitive writings on self-reproducing cellular automata, edited by his one-time collaborator Arthur Burks of the University of Michigan, was eventually published in 1966 as the book Theory of Self-Reproducing Automata. A possible structure of a von Neumann–style cellular automaton is depicted in Figure 7.1. It comprises a (finite or infinite) configuration of cells in which a cell can be in one of a finite set of states. The state of a cell at any time t is determined by its own state and those of its immediate neighbors in the preceding point of time t – 1, according to a state transition rule.

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