A guided tour of asynchronous cellular automata

Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat

Asynchronous cellular automata

This text has been proposed for the Encyclopedia of Complexity and Systems Science edited by Springer Nature and should appear in 2018.International audienceThis text is intended as an introduction to the topic of asynchronous cellular automata. We start from the simple example of the Game of Life and examine what happens to this model when it is made asynchronous (Sec. 1). We then formulate our definitions and objectives to give a mathematical description of our topic (Sec. 2). Our journey starts with the examination of the shift rule with fully asynchronous updating and from this simple example, we will progressively explore more and more rules and gain insights on the behaviour of the simplest rules (Sec. 3). As we will meet some obstacles in having a full analytical description of the asynchronous behaviour of these rules, we will turn our attention to the descriptions offered by statistical physics, and more specifically to the phase transition phenomena that occur in a wide range of rules (Sec. 4). To finish this journey, we will discuss the various problems linked to the question of asynchrony (Sec. 5) and present some openings for the readers who wish to go further (Sec. 6)

M-Asynchronous cellular automata: From fairness to quasi-fairness

4siA new model for the study of asynchronous cellular automata dynamical behavior is introduced with the main purpose of unifying several existing paradigms. The main idea is to measure the set of updating sequences to quantify the dependency of the properties under investigation from them. We propose to use the class of quasi-fair measures, namely measures that satisfy some fairness conditions on the updating sequences. Basic set properties like injectivity and surjectivity are adapted to the new setting and studied. In particular, we prove that they are dimensions sensitive properties (i.e., they are decidable in dimension 1 and undecidable in higher dimensions). A first exploration of dynamical properties is also started, some results about equicontinuity and expansivity behaviors are provided.nonenoneDennunzio Alberto; Formenti Enrico; Manzoni Luca; Mauri GiancarloDennunzio, Alberto; Formenti, Enrico; Manzoni, Luca; Mauri, Giancarl

Solution of Some Conjectures about Topological Properties of Linear Cellular Automata

AbstractWe study two dynamical properties of linear D-dimensional cellular automata over Zm namely, denseness of periodic points and topological mixing. For what concerns denseness of periodic points, we complete the work initiated in (Theoret. Comput. Sci. 174 (1997) 157, Theoret. Comput. Sci. 233 (1–2) (2000) 147, 14th Annual Symp. on Theoretical Aspects of Computer Science (STACS ’97), LNCS n. 1200, Springer, Berlin, 1997, pp. 427–438) by proving that a linear cellular automata has dense periodic points over the entire space of configurations if and only if it is surjective (as conjectured in (Cattaneo et al., 2000)). For non-surjective linear CA we give a complete characterization of the subspace where periodic points are dense. For what concerns topological mixing, we prove that this property is equivalent to transitivity and then easily checkable. Finally, we classify linear cellular automata according to the definition of chaos given by Devaney in (An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, Reading, MA, USA, 1989)

Additive cellular automata over finite abelian groups: Topological and measure theoretic properties

We study the dynamical behavior of D-dimensional (D ≥ 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [36, 43, 44, 40, 12, 11]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [12, 43] for linear CA over ℤm i.e. additive CA in which the alphabet is the cyclic group ℤm and the local rules are linear combinations with coefficients in ℤm. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over ℤnm, i.e., with the local rule defined by n × n matrices with elements in ℤm which, in turn, strictly contains the class of linear CA over ℤm. In order to further emphasize that finite abelian groups are more expressive than ℤm we prove that, contrary to what happens in ℤm, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case

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

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