We prove a characterisation of \mu-limit sets of two-dimensional cellular automata, extending existing results in the one-dimensional case. This sets describe the typical asymptotic behaviour of the cellular automaton, getting rid of exceptional cases, when starting from the uniform measure.

Abelian cellular automata (CA) are CA which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images weak *-converge towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i.e. randomization for a wide class of initial measures (under some mixing hypotheses). First, we prove that an abelian CA randomizes in Cesaro mean if and only if it has no soliton, i.e. a finite configuration whose time evolution remains bounded in space. This characterization generalizes previously known sufficient conditions for abelian CA with scalar or commuting coefficients. Second, we exhibit examples of strong randomizers, i.e. abelian CA randomizing in simple convergence; this is the first proof of this behaviour to our knowledge. We show however that no CA with commuting coefficients can be strongly randomizing. Finally, we show that some abelian CA achieve partial randomization without being randomizing: the distribution of short finite words tends to the uniform distribution up to some threshold, but this convergence fails for larger words. Again this phenomenon cannot happen for abelian CA with commuting coefficients.

Construction of mu-Limit Sets of Two-dimensional Cellular Automata

Characterizing asymptotic randomization in abelian cellular automata

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 Cesaro 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.

https://hal.archives-ouvertes.fr/hal-01719833/document

Characterisation of Limit Measures of Higher-Dimensional Cellular Automata

We consider the typical asymptotic behaviour of cellular automata of higher dimension (greater than 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 time tends to infinity. 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 Cesaro 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.

Characterisation of limit measures of higher-dimensional cellular automata

https://drops.dagstuhl.de/opus/volltexte/2015/4919/pdf/19.pdf/

Construction of mu-Limit Sets of Two-dimensional Cellular Automata

Citations

Characterizing asymptotic randomization in abelian cellular automata

Characterisation of Limit Measures of Higher-Dimensional Cellular Automata

Characterisation of limit measures of higher-dimensional cellular automata

Related Papers (5)

On the limit sets of cellular automata

Rice's theorem for the limit sets of cellular automata

Computation theory of cellular automata

Universality and complexity in cellular automata

Two-dimensional cellular automata