Traced communication complexity

Electronic Proceedings in Theoretical Computer Science, 2009

The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular automata. But the difference here is that we look for communication protocols arising in the dynamics itself. Our work is guided by the following idea : if we are able to give a protocol describing a cellular automaton, then we can understand its behavior.

Information and Computation/information and Control, 2008

Cellular Automata can be considered discrete dynamical sys- tems and at the same time a model of parallel computation. In this paper we investigate the connections between dynamical and computa- tional properties of Cellular Automata. We propose a classification of Cellular Automata according to the language complexities which rise from the basins of attraction of subshift attractors and investigate the

2013

We study the asymptotic behaviour of symbolic computing systems, notably one-dimensional cellular automata (CA), in order to ascertain whether and at what rate the number of complex versus simple rules dominate the rule space for increasing neighbourhood range and number of symbols (or colours), and how different behaviour is distributed in the spaces of different cellular automata formalisms. Using two different measures, Shannon's block entropy and Kolmogorov complexity, the latter approximated by two different methods (lossless compressibility and block decomposition), we arrive at the same trend of larger complex behavioural fractions. We also advance a notion of asymptotic and limit behaviour for individual rules, both over initial conditions and runtimes, and we provide a formalisation of Wolfram's classification as a limit function in terms of Kolmogorov complexity.

Lecture Notes in Computer Science, 2001

We study the ability of discrete dynamical systems to transform/generate randomness in cellular spaces. Thus, we endow the space of bi-infinite sequences by a metric inspired by information distance (defined in the context of Kolmogorov complexity or algorithmic information theory). We prove structural properties of this space (non-separability, completeness, perfectness and infinite topological dimension), which turn out to be useful to understand the transformation of information performed by dynamical systems evolving on it. Finally, we focus on cellular automata and prove a dichotomy theorem: continuous cellular automata are either equivalent to the identity or to a constant one. This means that they cannot produce any amount of randomness.

Journal of cellular automata

This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodic aspect, the formulae of measure-theoretic entropy, topological entropy and topological pressure are given in closed forms and Parry measure is demonstrated to be an equilibrium measure for some potential function. For topological aspect, an example is examined to show that the exhibition of snap-back repellers for a cellular automaton infers Li-Yorke chaos. In addition, bipermutive cellular automata are optimized for the exhibition of snap-back repellers in permutive cellular automata whenever two-sided shift space is considered.

We study the orbits of reversible one-dimensional cellular automata. It is shown that the Turing degree structure of the orbits of these automata is the same as for general cellular automata. In particular there are reversible cellular automata whose orbits have arbitrary recursively enumerable degree.

Theoretical Computer Science, 2013

This paper investigates a variant of cellular automata, namely ν-CA. Indeed, ν-CA are cellular automata which can have dierent local rules at each site of their lattice. The assignment of local rules to sites of the lattice completely characterizes ν-CA. In this paper, sets of assignments sharing some interesting properties are associated with languages of bi-innite words. The complexity classes of these languages are investigated providing an initial rough classication of ν-CA.

Lecture Notes in Computer Science, 2012

ν-CA are cellular automata which can have different local rules at each site of their lattice. Indeed, the spatial distribution of local rules completely characterizes ν-CA. In this paper, sets of distributions sharing some interesting properties are associated with languages of biinfinite words. The complexity classes of these languages are investigated providing an initial rough classification of ν-CA.

Theoretical Computer Science, 2001

We present a new approach to cellular automata (CA for short) classi cation based on algorithmic complexity. We construct a parameter which is based only on the transition table of CA and measures the \randomness" of evolutions; is better, in a certain sense, than any other parameter de ned on rule tables. We investigate the relations between the classical approach based on topology and ours based on algorithmic randomness. We also compare our parameter with Langton's one: we prove that ours is theoretically better and also agrees with some practical evidences reported in literature. Finally we propose a protocol to approximate and to make experiments on CA dynamical behavior.

Theoretical Computer Science, 2015

Cellular Automata (CA) are a well-established bio-inspired model of computation that has been successfully applied in several domains. In the recent years the importance of modelling real systems more accurately has sparkled a new interest in the study of asynchronous CA (ACA). When using an ACA for modelling real systems, it is important to determine the fidelity of the model, in particular with regards to the existence (or absence) of certain dynamical behaviors. This paper is concerned with two big classes of problems: reachability and preimage existence. For each class, both an existential and a universal version are considered. The following results are proved. Reachability is PSPACE-complete, its resource bounded version is NP-complete (existential form) or coNP-complete (universal form). The preimage problem is dimension sensitive in the sense that it is NL-complete (both existential and universal form) for one-dimensional ACA while it is NP-complete (existential version) or Π P 2-complete (universal version) for higher dimension.