Solving the density classification problem with a large diffusion and small amplification cellular automaton

Briceño, Raimundo; Espanés, Pablo Moisset de; Osses, Axel; Rapaport, Ivan

doi:10.1016/j.physd.2013.07.002

Cited by 16 publications

(27 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When relying on computer simulations only, the parameters connected with the phenomena observed, like bifurcations or phase transitions, are found only approximately. However, some studies complement an experimental study by insightful analytical research that confirms and further refined the results (see, for example, [1,4,16]).…”

mentioning

confidence: 87%

A Complete Description of the Dynamics of Legal Outer-totalistic Affine Continuous Cellular Automata

Wolnik¹,

Dembowski²,

Augustynowicz³

et al. 2021

Preprint

View full text Add to dashboard Cite

We present an investigation into the evolution and dynamics of the simplest generalization of binary cellular automata: Affine Continuous Cellular Automata (ACCAs), with [0,1] as state set and local rules that are affine in each variable. We focus on legal outer-totalistic ACCAs, an interesting class of dynamical systems that show some properties that do not occur in the binary case. A unique combination of computer simulations (sometimes quite advanced) and a panoply of analytical methods allow to lay bare the dynamics of each and every one of these cellular automata.

show abstract

mentioning

confidence: 87%

A Complete Description of the Dynamics of Legal Outer-totalistic Affine Continuous Cellular Automata

Wolnik¹,

Dembowski²,

Augustynowicz³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Despite the fact that Chau's solution uses some very interesting ideas, we will not discuss it here, as we wish to focus on nearest-neighbour rules only. Along the same vein, since we restrict our attention to ternary rules only, we will not be concerned with solutions of density classification problems which require very large number states, such as, for example, the work of Briceño et al [12].…”

Section: Classification Problemsmentioning

confidence: 99%

Explorations of Ternary Cellular Automata and Ternary Density Classification Problems

Fukś¹,

Procyk²

2019

Acta Phys. Pol. B Proc. Suppl.

View full text Add to dashboard Cite

While binary nearest-neighbour cellar automata (CA) have been studied in detail and from many different angles, the same cannot be said about ternary (three-state) CA rules. We present some results of our explorations of a small subset of the vast space of ternary rules, namely rules possessing additive invariants. We first enumerate rules with four different additive invariants, and then we investigate if any of them could be used to construct a two-rule solution of generalized density classification problem (DCP). We show that neither simple nor absolute classification is possible with a pair of ternary rules where the first rule is all-conserving and the second one is reducible to two states. Similar negative result holds for another version of DCP we propose: symmetric interval-wise DCP. Finally we show an example of a pair of rules which solve non-symmetric interval-wise DCP for initial configurations containing at least one zero.

show abstract

“…CAs and PCAs with infinite alphabets appear in the literature under different forms. In [5], CAs with alphabet E = [0, 1] were used to solve the classification problem with arbitrary precision: the classification problem consists of finding a CA such that, on any initial configuration of 1s and 0s on the line Z, the CA configuration converges to the line colored 1 if the initial fraction r of 1s is greater than 1 2 and to the line colored 0 if r < 1 2 . CAs with alphabet E = R are applicable to modeling the heat equation [18].…”

Section: Pcas With General Alphabetsmentioning

confidence: 99%

Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

Casse

2016

Adv. Appl. Probab.

View full text Add to dashboard Cite

This paper is devoted to probabilistic cellular automata (PCA) on N, Z or Z/nZ, depending of two neighbors, with a general alphabet E (finite or infinite, discrete or not).We study the following question: under which conditions does a PCA possess a Markov chain as invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCA) when the alphabet E is finite. Here we obtain conditions on the transition kernel of PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation.One of the difficulties to pass from a finite alphabet to a general alphabet comes from some problems of measurability, and a large part of this work is devoted to clarify these issues.

show abstract

Solving the density classification problem with a large diffusion and small amplification cellular automaton

Cited by 16 publications

References 13 publications

A Complete Description of the Dynamics of Legal Outer-totalistic Affine Continuous Cellular Automata

A Complete Description of the Dynamics of Legal Outer-totalistic Affine Continuous Cellular Automata

Explorations of Ternary Cellular Automata and Ternary Density Classification Problems

Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution

Contact Info

Product

Resources

About