Additive Cellular Automata

Voorhees, Burton

doi:10.1007/978-0-387-30440-3_4

Cited by 4 publications

(3 citation statements)

References 106 publications

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“…Taking single-bit or single-value perturbations to attractor states as the simplest case, the jump-graph [14][17,#20.7] represents the probabilities of jumping between basins. Apart from the philosophical/mathematical implications [6], insights into the stability and adaptability of dynamics is relevant in attractor models of memory and learning [8,10,15] and of cell differentiation in genetic networks [2,12].…”

Section: The Network-graphmentioning

confidence: 99%

The Cellular Automaton Pulsing Model, Experiments with DDLab

Wuensche¹,

Coxon²

2019

Acta Phys. Pol. B Proc. Suppl.

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The cellular automaton (CA) pulsing model [1] described the surprising phenomenon of spontaneous, sustained and robust rhythmic oscillations, pulsing dynamics, when random wiring is applied to a 2D "glider" rule running in a 3-value totalistic CA. Case studies, pulsing measures, possible mechanisms, and implications for oscillatory networks in biology were presented. In this paper we summarise the results, extend the entropydensity and density-return map plots to include a linked history, look at totalistic glider rules with neighborhoods of 3, 4 and 5, as well as 6 and 7 studied previously, introduce methods to automatically recognise the wavelength, and extend results for randomly asynchronous updating. We show how the model is implemented in DDLab to validate results, output data, and allow experiments and research by others.

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Section: The Network-graphmentioning

confidence: 99%

The Cellular Automaton Pulsing Model, Experiments with DDLab

Wuensche¹,

Coxon²

2019

Acta Phys. Pol. B Proc. Suppl.

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“…. p − 1} called the alphabet, is iteratively updated according to a specified local rule [3][4][5]. Locally, each site i ∈ Z in the lattice is characterized by the variable x i t ∈ A p at time t. An additive cellular automaton (ACA) of radius (l + r)/2 is a CA whose update rule is a linear function modulo p of the dynamical states x i+l t , x i+l−1 t , .…”

Section: Introductionmentioning

confidence: 99%

Replication of spatial patterns with reversible and additive cellular automata

Garcı́a-Morales¹

2022

J. Phys. A: Math. Theor.

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In this article, the replication of arbitrary patterns by reversible and additive cellular automata is reported. The orbit of an 1D cellular automaton operating on $p$ symbols that is both additive and reversible is explicitly given in terms of coefficients that appear in the theory of Gegenbauer polynomials. It is shown that if $p$ is an odd prime, the pattern formed after $(p-1)/2$ time steps from any arbitrary initial condition (spatially confined to a region of side less than $p$) replicates after $p+(p-1)/2$ time steps in a way that resembles budding in biological systems.

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“…The notion of parity has appeared quite often in the CA literature, even if implicitly, as it bears relevance to the related notion of additivity of CA rules [3,15]. However, the parity problem per se has not been extensively studied, particularly in comparison with the well-known benchmark CA task of density classification, where the aim is to determine the most frequent bit in the initial configuration of an odd-sized lattice, also by reaching an homogeneous configuration.…”

Section: Introductionmentioning

confidence: 99%

On the Parity Problem in One-Dimensional Cellular Automata

Betel

Oliveira

Flocchini

2012

Electron. Proc. Theor. Comput. Sci.

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We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem

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Additive Cellular Automata

Cited by 4 publications

References 106 publications

The Cellular Automaton Pulsing Model, Experiments with DDLab

The Cellular Automaton Pulsing Model, Experiments with DDLab

Replication of spatial patterns with reversible and additive cellular automata

On the Parity Problem in One-Dimensional Cellular Automata

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