Characterisation of Limit Measures of Higher-Dimensional Cellular Automata
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Date
2017-11
Profesor/a GuÃa
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Idioma
en
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Volume Title
Publisher
Springer New York LLC
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CC BY 4.0 DEED
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https://creativecommons.org/licenses/by/4.0/deed.es
Abstract
We consider the typical asymptotic behaviour of cellular automata of higher dimension (≥2). That is, we take an initial configuration at random according to a Bernoulli (i.i.d) probability measure, iterate some cellular automaton, and consider the (set of) limit probability measure(s) as t → ∞. In this paper, we prove that limit measures that can be reached by higher-dimensional cellular automata are completely characterised by computability conditions, as in the one-dimensional case. This implies that cellular automata have the same variety and complexity of typical asymptotic behaviours as Turing machines, and that any nontrivial property in this regard is undecidable (Rice-type theorem). These results extend to connected sets of limit measures and Cesà ro mean convergence. The main tool is the implementation of arbitrary computation in the time evolution of a cellular automata in such a way that it emerges and self-organises from a random configuration. © 2017, Springer Science+Business Media New York.
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Indexación: Scopus
Keywords
Cellular automata, Computability, Limit measure, Multidimensional, Symbolic dynamics
Citation
Theory of Computing Systems Volume 61, Issue 4, Pages 1178 - 12131 November 2017
DOI
10.1007/s00224-017-9753-1