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Kůrka, Petr; Maass, Alejandro

doi:10.1023/a:1018706923831

Cited by 28 publications

(8 citation statements)

References 19 publications

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“…Before we proceed, let us define P (k) to be the column vector of all probabilities of blocks of length k arranged in lexical order. For example, for A = {0, 1}, the first three vectors P (k) are P (1) = [P (0), P (1)] T , P (2) = [P (00), P (01), P (10), P (11)] T , P (3) = [P (000), P (001), P (010), P (011), P (100), P (101), P (110),…”

Section: Minimal Entropy Extensionmentioning

confidence: 99%

“…, P (n) . We want to construct block probabilities P (n+1) which minimize the entropy h(P (n+1) ) and are consistent with block probabilities P (1) , P (2) , . .…”

Section: Minimal Entropy Extensionmentioning

confidence: 99%

“…The appropriate mathematical description of a distribution of configurations is a probability measure on X. Cellular automata (CA) are often considered as maps in the space of such probability measures [1]- [4].…”

Section: Introductionmentioning

confidence: 99%

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Minimal entropy approximation for cellular automata

Fukś¹

2014

J. Stat. Mech.

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We present a method for construction of approximate orbits of measures under the action of cellular automata which is complementary to the local structure theory. The local structure theory is based on the idea of Bayesian extension, that is, construction of a probability measure consistent with given block probabilities and maximizing entropy. If instead of maximizing entropy one minimizes it, one can develop another method for construction of approximate orbits, at the heart of which is the iteration of finitelydimensional maps, called minimal entropy maps. We present numerical evidence that minimal entropy approximation sometimes spectacularly outperforms the local structure theory in characterizing properties of cellular automata. Density response curve for elementary CA rule 26 is used to illustrate this claim.

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Section: Minimal Entropy Extensionmentioning

confidence: 99%

“…, P (n) . We want to construct block probabilities P (n+1) which minimize the entropy h(P (n+1) ) and are consistent with block probabilities P (1) , P (2) , . .…”

Section: Minimal Entropy Extensionmentioning

confidence: 99%

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Minimal entropy approximation for cellular automata

Fukś¹

2014

J. Stat. Mech.

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“…The µ-limit set Λ F (µ) = ν∈Γ F (µ) supp(ν) is of particular interest for self-organization, as discussed in [6]. Indeed, consider words that appear arbitrarily far in space-time diagrams, i.e., such that F n µ([u]) → 0 (µ-persistent words).…”

Section: Examplesmentioning

confidence: 99%

“…and symetrically if we exchange + and −. The proof relies on the fact that the behaviour of gliders automata can be characterized by some random walk process; his general idea was introduced by Kůrka & Maass in [6] and was already used in [5]. In our case, a particle appearing in a position corresponds to a minima between two concurrent random walks.…”

Section: Introductionmentioning

confidence: 99%

Entry times in automata with simple defect dynamics

Menibus

Sablik

2012

Electron. Proc. Theor. Comput. Sci.

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In this paper, we consider a simple cellular automaton with two particles of different speeds that annihilate on contact. Following a previous work by K urka et al., we study the asymptotic distribution, starting from a random configuration, of the waiting time before a particle crosses the central column after time n. Drawing a parallel between the behaviour of this automata on a random initial configuration and a certain random walk, we approximate this walk using a Brownian motion, and we obtain explicit results for a wide class of initial measures and other automata with similar dynamics

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On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures

Boyer

Poupet

Theyssier

2006

Lecture Notes in Computer Science

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Abstract. We study the notion of limit sets of cellular automata associated with probability measures (µ-limit sets). This notion was introduced by P. Kůrka and A. Maass in [1]. It is a refinement of the classical notion of ω-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words which probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterization of the persistent language for non sensitive cellular automata associated with Bernouilli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their µ-limit set) is neither recursively enumerable nor co-recursively enumerable.

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Cited by 28 publications

References 19 publications

Minimal entropy approximation for cellular automata

Minimal entropy approximation for cellular automata

Entry times in automata with simple defect dynamics

On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures

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