A Note on the Classification of the Most Simple Asynchronous Cellular Automata

Fatès, Nazim

doi:10.1007/978-3-642-40867-0_3

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…As we have seen, analytical results have been more difficult to obtain than numerical ones, but the situation is progressively changing as more techniques from the probability theory are being developed for the specific case of cellular automata. We find it rather amazing that it is still an open question to determine the convergence time of some simple binary rules [38,39].…”

Section: Closing Wordsmentioning

confidence: 98%

A Guided Tour of Asynchronous Cellular Automata

Fatès

2013

Cellular Automata and Discrete Complex Systems

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Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We present a state of the art that covers the various approaches that deal with asynchronism in cellular automata and closely related models.Foreword: This article is the preprintof an article that is to appear in the Journal of cellular automata. It is an extended verion of the invited paper that appeared

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Section: Closing Wordsmentioning

confidence: 98%

A Guided Tour of Asynchronous Cellular Automata

Fatès

2013

Cellular Automata and Discrete Complex Systems

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“…Before tackling the analysis, let us simply observe the space-time diagram of the various rules with a rapid convergence in Figure 5. To list all the behaviours of this kind of the ECAs, one may examine by eye all the spacetime diagrams of the 88 ECAs and select those which appear to converge rapidly to a fixed point [Fat13]. In Table 5, we give a list of such candidate rules, based on whether the system has a tendency to converge to the homogeneous fixed point 0 (RCH) or to a non-homogeneous one (RCN).…”

Section: Identification Of the Rules Of This Classmentioning

confidence: 99%

A tutorial on elementary cellular automata with fully asynchronous updating

Fatès

2020

Nat Comput

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We present a picture of the convergence properties of the 256 Elementary Cellular Automata under the fully asynchronous updating, that is, when only one cell is updated at each time step. We regroup here the results which have been presented in different articles and expose a full analysis of the behaviour of finite systems with periodic boundary conditions. Our classification relies on the scaling properties of the average convergence time to a fixed point. We observe that different scaling laws can be found, which fall in one of the following classes: logarithmic, linear, quadratic, exponential and non-converging. The techniques for quantifying this behaviour rely mainly on Markov chain theory and martingales. Most behaviours can be studied analytically but there are still many rules for which obtaining a formal characterisation of their convergence properties is still an open problem.

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“…This could be done analysing the communication classes of the state space of these rules. While the classifications based on the convergence time to a fixed point remain mainly open [6], achieving this result would represent an important step in the understanding of the dynamics of asynchronous CA.…”

Section: Discussionmentioning

confidence: 99%