Computational Aspects of Asynchronous CA

This work studies some aspects of the computational power of fully asynchronous cellular automata (ACA). We deal with some notions of simulation between ACA and Turing Machines. In particular, we characterize the updating sequences specifying which are "universal", i.e., allowing a (specific family of) ACA to simulate any TM on any input. We also consider the computational cost of such simulations

Foreword: cellular automata and applications

International audienceThis special issue contains four papers presented during theworkshop, ‘‘18th International Workshop on CellularAutomata and Discrete Complex Systems’’ (Automata2012), held in La Marana, Corsica island (France) in theperiod September 19–21th, 2012.The aim of this workshop is to establish and maintain apermanent, international, multidisciplinary forum for thecollaboration of researchers in the field of Cellular Automata(CA) and Discrete Complex Systems (DCS), providea platform for presenting and discussing new ideas andresults, and support the development of theory and applicationsof CA and DCS.Typical, but not exclusive, topics of the workshop are:dynamics aspects, algorithmic, computational and complexityissues, emergent properties, formal language processing,models of parallelism and distributed systems,phenomenological descriptions, scientific modeling andpractical applications.After an additional review process, four papers wereselected and included in this special issue. They are nowpresented in an extended and improved form with respectto the already refereed workshop version that appeared inthe proceedings of Automata 2012.The paper ‘‘Computation of Functions on n Bits byAsynchronous Clocking Cellular Automata’’ by MichaelVielhaber aims at proving that different functions on binaryvectors can be computed by changing the updating schemefrom a fully synchronous to an asynchronous one on somefixed CA local rule.In their paper ‘‘Solving the Parity Problem in One–Dimensional Cellular Automata’’, Heather Betel, PedroP. B. de Oliveira, and Paola Flocchini deal with the parityproblem in one–dimensional cellular automata (CA): a CAlocal rule solves the parity problem if, starting from anyinitial configuration, the CA converges to the 0–configuration(resp., the 1–configuration) if and only if the initialconfiguration contains an even number of 1s (resp., an oddnumber of 1s). In particular, authors focus on the neighborhoodsize of CA rules solving the problem.Murillo G. Carneiro and Gina M. B. Oliveira present inthe paper ‘‘Synchronous Cellular Automata-Based Schedulerinitialized by Heuristic and modeled by a Pseudolinearneighborhood’’ two approaches based on CA to thetask scheduling problem in multiprocessor systems.The implementation of cellular automata on processorarrays is considered by Jean-Vivien Millo and Robertde Simone in the paper ‘‘Explicit routing schemes forimplementation of cellular automata on processor arrays’’.They deal with the trade-offs between the generality of theCA neighborhood and the limited expressive power providedby physical platforms. This is an extremely hot topicwhich will help in turning CA towards real extendedapplications.We would like to warmly thank the authors for theirwork and effort which made this special issue possible.Special thanks go to all referees for their valuable contributionsboth during the selection and the final reviewprocess. Finally, we also want to thank Professor GrzegorzRozenberg for offering us the opportunity to publish thisspecial issue in Natural Computing

An Easy to Check Characterization of Positive Expansivity for Additive Cellular Automata over a Finite Abelian Group

Additive cellular automata over a finite abelian group are a wide class of cellular automata (CA) that are able to exhibit most of the complex behaviors of general CA and they are often exploited for designing applications in different practical contexts. We provide an easy to check algebraic characterization of positive expansivity for Additive Cellular Automata over a finite abelian group. We stress that positive expansivity is an important property that defines a condition of strong chaos for CA and, for this reason, an easy to check characterization of positive expansivity turns out to be crucial for designing proper applications based on Additive CA and where a condition of strong chaos is required. First of all, in the paper an easy to check algebraic characterization of positive expansivity is provided for the non trivial subclass of Linear Cellular Automata over the alphabet (Z/mZ)n . Then, we show how it can be exploited to decide positive expansivity for the whole class of Additive Cellular Automata over a finite abelian group

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

An Easily Checkable Algebraic Characterization of Positive Expansivity for Additive Cellular Automata over a Finite Abelian Group

We provide an easily checkable algebraic characterization of positive expansivity for Additive Cellular Automata over a finite abelian group. First of all, an easily checkable characterization of positive expansivity is provided for the non trivial subclass of Linear Cellular Automata over the alphabet $(\Z/m\Z)^n$. Then, we show how it can be exploited to decide positive expansivity for the whole class of Additive Cellular Automata over a finite abelian group.Comment: 12 page

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 [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. 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 [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. 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 Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, 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

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

Foreword: asynchronous cellular automata and nature-inspired computation

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