Exponential convergence to equilibrium in cellular automata asymptotically emulating identity

Fukś, Henryk; Soto, José Manuel Gómez

doi:10.48550/arxiv.1306.1189

Cited by 2 publications

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“…Indeed, for many other elementary CA rules for which formulae for P n (0) are either known or conjectured, the exponential convergence to P ∞ (0) can be observed most frequently. This includes 15 elementary rules which are known as asymptotic emulators of identity (Rogers and Want, 1994;Fukś and Soto, 2014). Formulae for P n (1) for the initial measure µ 1/2 for these rules are shown below.…”

Section: Convergence Of Block Probabilitiesmentioning

confidence: 99%

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Orbits of Bernoulli Measures in Cellular Automata

Fukś¹

2018

Cellular Automata

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Glossary Configuration space Set of all bisequences of symbols from the alphabet A of N symbols, A = {0, 1, . . . , N − 1}, denoted by A Z . Elements of A Z are called configurations and denoted by bold lowercase letters: x, y, etc. Block or word A finite sequence of symbols of the alphabet A . Set of all blocks of length n is denoted by A n , while the sent of all possible blocks of all lengths by A . Blocks are denoted by bold lowercase letters a, b, c, etc. Individual symbols of the block b are denoted by indexed italic form of the same letter, b = b 1 , b 2 , . . . , b n . To make formulae more compact, commas are sometimes dropped (if no confusion arises), and we simply write b = b 1 b 2 . . . b n . Cylinder set For a block b of length n, the cylinder set generated by b and anchored at i is the subset of configurations such that symbols at positions from i to i + n − 1 are fixed and equal to symbols in the block b, while the remaining symbols are arbitrary. Denoted by [b] i = {x ∈ A Z : x [i,i+n) = b}. Cellular automaton In this article, cellular automaton is understood as a map F in the space of shift-invariant probability measures over the configuration space A Z . To define F, two ingredients are needed, a positive integer r called radius and a function w : A × A 2r+1 → [0, 1], whose values are called transition probabilities. The image of a measure µ under the action of F is then defined by probabilities of cylinder sets, (F µ)([a] i ) = ∑ b∈A |a|+2r w(a|b)µ([b] i−r

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Section: Convergence Of Block Probabilitiesmentioning

confidence: 99%

“…Formulae for P n (1) for the initial measure µ 1/2 for these rules are shown below. Starred rules are those for which a formal proof has been published in the literature (see Fukś and Soto, 2014, and references therein).…”

Section: Convergence Of Block Probabilitiesmentioning

confidence: 99%

Orbits of Bernoulli Measures in Cellular Automata

Fukś¹

2018

Cellular Automata

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“…One can easily show (in a manner similar as done in [14]) that if one starts with a bi-infinite string of symbols drawn from a Bernoulli distribution where probabilities of 0, 1 and 2 are, respectively, p, q and r, then the expected proportion of sites in state k after n iterations of rule f is given by Ψ f −n (k) (p, q, r). This quantity will be called density of symbols k after n iterations of f .…”

Section: Basic Definitionmentioning

confidence: 99%

“…Cellular automata are infinitely-dimensional dynamical systems, yet a behaviour similar to hyperbolicity in finite-dimensional systems has been observed in many of them. In particular, in some binary cellular automata in one dimension, known as asymptotic emulators of identity, if the initial configuration is drawn from a Bernoulli distribution, the expected proportion of ones (or zeros) usually tends to its stationary value exponentially fast [14]. This type of behaviour is quite common in many other dynamical systems.…”

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An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

Fukś¹,

Midgley-Volpato²

2016

Acta Phys. Pol. B Proc. Suppl.

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In a recent paper [1] we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here we rederive the same formulae in a rigorous way, without resorting to any semiempirical methods. This is done by analyzing the behaviour of continuous clusters of symbols and by considering their interactions.(1)

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Exponential convergence to equilibrium in cellular automata asymptotically emulating identity

Cited by 2 publications

References 0 publications

Orbits of Bernoulli Measures in Cellular Automata

Orbits of Bernoulli Measures in Cellular Automata

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

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