From Linear to Additive Cellular Automata

This paper proves the decidability of several important properties of additive cellular automata over finite abelian groups. First of all, we prove that equicontinuity and sensitivity to initial conditions are decidable for a nontrivial subclass of additive cellular automata, namely, the linear cellular automata over \u207f, where is the ring \u2124/m\u2124. The proof of this last result has required to prove a general result on the powers of matrices over a commutative ring which is of interest in its own. Then, we extend the decidability result concerning sensitivity and equicontinuity to the whole class of additive cellular automata over a finite abelian group and for such a class we also prove the decidability of topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to it. Finally, a decidable characterization of injectivity and surjectivity for additive cellular automata over a finite abelian group is provided in terms of injectivity and surjectivity of an associated linear cellular automata over \u207f

On one-sided, D-chaotic CA without fixed points, having continuum of periodic points with period 2 and topological entropy log(p) for any prime p

A method is known by which any integer $\, n\geq2\,$ in a metric Cantor space of right-infinite words $\,\tilde{A}_{n}^{\,\mathbb N}\,$ gives a construction of a non-injective cellular automaton $\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,$ which is chaotic in Devaney sense, has a radius $\, r=1,\,$ continuum of fixed points and topological entropy $\, log(n).\,$ As a generalization of this method we present for any integer $\, n\geq2,\,$ a construction of a cellular automaton $\,(A_{n}^{\,\mathbb{N}},\, F_{n}),\,$ which has the listed properties of $\,(\tilde{A}_{n}^{\,\mathbb N},\,\tilde{F}_{n}),\,$ but has no fixed points and has continuum of periodic points with the period 2. The construction is based on properties of cellular automaton introduced here $\,(B^{\,\mathbb N},\, F)\,$ with radius $1$ defined for any prime number $\, p.\,$ We prove that $\,(B^{\,\mathbb N},\, F)\,$ is non-injective, chaotic in Devaney sense, has no fixed points, has continuum of periodic points with the period $2$ and topological entropy \(\, log(p).\,\

Directional Dynamics along Arbitrary Curves in Cellular Automata

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers

Non-Uniform Cellular Automata: classes, dynamics, and decidability

The dynamical behavior of non-uniform cellular automata is compared with the one of classical cellular automata. Several differences and similarities are pointed out by a series of examples. Decidability of basic properties like surjectivity and injectivity is also established. The final part studies a strong form of equicontinuity property specially suited for non-uniform cellular automata.Comment: Paper submitted to an international journal on June 9, 2011. This is an extended and improved version of the conference paper: G. Cattaneo, A. Dennunzio, E. Formenti, and J. Provillard. "Non-uniform cellular automata". In Proceedings of LATA 2009, volume 5457 of Lecture Notes in Computer Science, pages 302-313. Springe

On the directional dynamics of additive cellular automata

Abstract We continue the study of cellular automata (CA) directional dynamics, i.e. the behavior of the joint action of CA and shift maps. This notion has been investigated for general CA in the case of expansive dynamics by Boyle and by Sablik for sensitivity and equicontinuity. In this paper we give a detailed classification for the class of additive CA providing non-trivial examples for some classes of Sablik's classification. Moreover, we extend the directional dynamics studies by considering also factor languages and attractors

On the directional dynamics of additive cellular automata

We continue the study of cellular automata (CA) directional dynamics, i.e. , the behavior of the joint action of CA and shift maps. This notion has been investigated for general CA in the case of expansive dynamics by Boyle and Lind; and by Sablik for sensitivity and equicontinuity. In this paper we give a detailed classification for the class of additive CA providing non-trivial examples for some classes of Sablik\u2019s classification. Moreover, we extend the directional dynamics studies by considering also factor languages and attractors