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Stochastic Cellular Automata Solutions to the Density Classification Problem

When Randomness Helps Computing

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Abstract

In the density classification problem, a binary cellular automaton (CA) should decide whether an initial configuration contains more 0s or more 1s. The answer is given when all cells of the CA agree on a given state. This problem is known for having no exact solution in the case of binary deterministic one-dimensional CA.

We investigate how randomness in CA may help us solve the problem. We analyse the behaviour of stochastic CA rules that perform the density classification task. We show that describing stochastic rules as a “blend” of deterministic rules allows us to derive quantitative results on the classification time and the classification time of previously studied rules.

We introduce a new rule whose effect is to spread defects and to wash them out. This stochastic rule solves the problem with an arbitrary precision, that is, its quality of classification can be made arbitrarily high, though at the price of an increase of the convergence time. We experimentally demonstrate that this rule exhibits good scaling properties and that it attains qualities of classification never reached so far.

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Notes

  1. Both terms ‘stochastic’ and ‘probabilistic’ CA are found in literature. We prefer to employ the former as etymologically the Greek word ‘stochos’ implies the idea of goal, aim, target or expectation.

  2. Note that defining rigorously the sequence of random variables x t obtained from F would require to introduce advanced tools from the probability theory.

  3. We give the “classical” rule code into parenthesis; it is obtained by converting the sequence of 8 bits of the transition table ( 0 0 0 to 1 1 1) to the corresponding decimal number.

  4. Rigorously, one needs to use the “stopped” process \(Y_{t} = X^{2}_{t\wedge T} - v \cdot(t \wedge T) \).

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Acknowledgements

The author expresses his sincere gratitude to the two anonymous referees and to his collaborators for interesting remarks and suggestions which resulted from a careful reading of the manuscript.

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  1. INRIA Nancy—Grand Est, LORIA, Nancy Université, Campus scientifique, BP 239, 54 506, Vandœuvre-lès-Nancy, France

    Nazim Fatès

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  1. Nazim Fatès
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Correspondence to Nazim Fatès.

Additional information

Extended version of “Stochastic Cellular Automata Solve the Density Classification Problem with an Arbitrary Precision”, Proceedings of STACS 2011, Dortmund, Germany.

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Fatès, N. Stochastic Cellular Automata Solutions to the Density Classification Problem. Theory Comput Syst 53, 223–242 (2013). https://doi.org/10.1007/s00224-012-9386-3

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