scholarly journals Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Henryk Fukś ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Initial Density ◽  
Response Curves ◽  
Asynchronous Cellular Automata ◽  
Final Density ◽  
Short Cylinder ◽  
Bernoulli Measure ◽  
Single Transition ◽  
International Audience ◽  
Transition Rules

International audience We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.

scholarly journals Asynchronous Cellular Automata and Brownian Motion

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Philippe Chassaing ◽  
Lucas Gerin
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Cellular Automata ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Time Step ◽  
Interacting Particle ◽  
Asynchronous Cellular Automata ◽  
International Audience ◽  
A Cell

International audience This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.

Response Curves for Cellular Automata in One and Two Dimensions

2010 ◽  
Vol 1 (3) ◽  
pp. 85-99 ◽  
Author(s):  
Henryk Fuks ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Response Curve ◽  
Initial Density ◽  
Two Dimensions ◽  
Initial Configuration ◽  
Two Dimensional ◽  
Response Curves ◽  
Special Case

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.

scholarly journals Conservation Laws and Invariant Measures in Surjective Cellular Automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Jarkko Kari ◽  
Siamak Taati
Keyword(s):  
Cellular Automata ◽  
Conservation Laws ◽  
Arbitrary Dimension ◽  
Close Link ◽  
One Dimensional ◽  
Full Support ◽  
Markov Measure ◽  
Bernoulli Measure ◽  
International Audience ◽  
Markov Measures

International audience We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

Response Curves for Cellular Automata in One and Two Dimensions

2012 ◽  
pp. 291-305
Author(s):  
Henryk Fuks ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Response Curve ◽  
Initial Density ◽  
Two Dimensions ◽  
Initial Configuration ◽  
Two Dimensional ◽  
Response Curves ◽  
Special Case

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.

On Uniqueness of the Wulff Shape for Cellular Automata

1991 ◽  
Vol 14 (1) ◽  
pp. 75-89
Author(s):  
Paweł Wlaź
Keyword(s):  
Cellular Automata ◽  
Large Class ◽  
Growth Function ◽  
Wulff Shape ◽  
Transition Rules

In this paper, ordered transition rules are investigated. Such rules describe an increment of mono-crystals and for every rule one can calculate so called Wulff Shape. It is shown that for some large class of these rules, there exists at most one growth function which generates a given Wulff Shape.

Reversible Computation in Asynchronous Cellular Automata

2002 ◽  
pp. 220-229 ◽  
Author(s):  
Jia Lee ◽  
Ferdinand Peper ◽  
Susumu Adachi ◽  
Kenichi Morita ◽  
Shinro Mashiko
Keyword(s):  
Cellular Automata ◽  
Reversible Computation ◽  
Asynchronous Cellular Automata
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